3.28.9 \(\int \frac {\sqrt {b^2+a^2 x^3} (2 b^2+c x^3+a^2 x^6)}{x^7 (b^2+a^2 x^6)} \, dx\)

Optimal. Leaf size=247 \[ \frac {\sqrt {a^2 x^3+b^2} \left (-a^2 x^3-2 b^2-2 c x^3\right )}{6 b^2 x^6}+\frac {\sqrt {a-i b} \left ((-1)^{3/4} a^2 b-\sqrt [4]{-1} a c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-i b}}\right )}{3 b^{5/2}}-\frac {\sqrt {a+i b} \left ((-1)^{3/4} a c-\sqrt [4]{-1} a^2 b\right ) \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+i b}}\right )}{3 b^{5/2}}+\frac {\left (a^4+4 a^2 b^2-2 a^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{6 b^3} \]

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Rubi [B]  time = 3.60, antiderivative size = 744, normalized size of antiderivative = 3.01, number of steps used = 27, number of rules used = 15, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {6725, 266, 47, 51, 63, 208, 50, 6715, 825, 827, 1169, 634, 618, 206, 628} \begin {gather*} -\frac {c \sqrt {a^2 x^3+b^2}}{3 b^2 x^3}-\frac {a^2 \sqrt {a^2 x^3+b^2}}{6 b^2 x^3}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 b}-\frac {\sqrt {a^2 x^3+b^2}}{3 x^6}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 b^3}+\frac {a^2 \left (\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \log \left (-\sqrt {2} \sqrt {b} \sqrt {\sqrt {a^2+b^2}+b} \sqrt {a^2 x^3+b^2}+b \left (\sqrt {a^2+b^2}+b\right )+a^2 x^3\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+b}}-\frac {a^2 \left (\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \log \left (\sqrt {2} \sqrt {b} \sqrt {\sqrt {a^2+b^2}+b} \sqrt {a^2 x^3+b^2}+b \left (\sqrt {a^2+b^2}+b\right )+a^2 x^3\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+b}}-\frac {a^2 \left (-\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {\sqrt {a^2+b^2}+b}-\sqrt {2} \sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\frac {a^2 \left (-\sqrt {a^2+b^2} \left (b^2-c\right )+a^2 b+b^3\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {a^2 x^3+b^2}+\sqrt {b} \sqrt {\sqrt {a^2+b^2}+b}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{6 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[b^2 + a^2*x^3]*(2*b^2 + c*x^3 + a^2*x^6))/(x^7*(b^2 + a^2*x^6)),x]

[Out]

-1/3*Sqrt[b^2 + a^2*x^3]/x^6 - (a^2*Sqrt[b^2 + a^2*x^3])/(6*b^2*x^3) - (c*Sqrt[b^2 + a^2*x^3])/(3*b^2*x^3) + (
a^4*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/(6*b^3) + (2*a^2*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/(3*b) - (a^2*c*ArcTanh[Sq
rt[b^2 + a^2*x^3]/b])/(3*b^3) - (a^2*(a^2*b + b^3 - Sqrt[a^2 + b^2]*(b^2 - c))*ArcTanh[(Sqrt[b]*Sqrt[b + Sqrt[
a^2 + b^2]] - Sqrt[2]*Sqrt[b^2 + a^2*x^3])/(Sqrt[b]*Sqrt[b - Sqrt[a^2 + b^2]])])/(3*Sqrt[2]*b^(5/2)*Sqrt[a^2 +
b^2]*Sqrt[b - Sqrt[a^2 + b^2]]) + (a^2*(a^2*b + b^3 - Sqrt[a^2 + b^2]*(b^2 - c))*ArcTanh[(Sqrt[b]*Sqrt[b + Sq
rt[a^2 + b^2]] + Sqrt[2]*Sqrt[b^2 + a^2*x^3])/(Sqrt[b]*Sqrt[b - Sqrt[a^2 + b^2]])])/(3*Sqrt[2]*b^(5/2)*Sqrt[a^
2 + b^2]*Sqrt[b - Sqrt[a^2 + b^2]]) + (a^2*(a^2*b + b^3 + Sqrt[a^2 + b^2]*(b^2 - c))*Log[b*(b + Sqrt[a^2 + b^2
]) + a^2*x^3 - Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[a^2 + b^2]]*Sqrt[b^2 + a^2*x^3]])/(6*Sqrt[2]*b^(5/2)*Sqrt[a^2 + b
^2]*Sqrt[b + Sqrt[a^2 + b^2]]) - (a^2*(a^2*b + b^3 + Sqrt[a^2 + b^2]*(b^2 - c))*Log[b*(b + Sqrt[a^2 + b^2]) +
a^2*x^3 + Sqrt[2]*Sqrt[b]*Sqrt[b + Sqrt[a^2 + b^2]]*Sqrt[b^2 + a^2*x^3]])/(6*Sqrt[2]*b^(5/2)*Sqrt[a^2 + b^2]*S
qrt[b + Sqrt[a^2 + b^2]])

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*(m + n + 2))/((b*c - a*d)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 825

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g*(d + e*x)^m)/
(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d + c*e*f)*x, x])/(a + c*x^2), x], x] /
; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 827

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2, Subst[Int[(e*f
- d*g + g*x^2)/(c*d^2 + a*e^2 - 2*c*d*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0]

Rule 1169

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a/c, 2]}, With[{r =
Rt[2*q - b/c, 2]}, Dist[1/(2*c*q*r), Int[(d*r - (d - e*q)*x)/(q - r*x + x^2), x], x] + Dist[1/(2*c*q*r), Int[(
d*r + (d - e*q)*x)/(q + r*x + x^2), x], x]]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2
- b*d*e + a*e^2, 0] && NegQ[b^2 - 4*a*c]

Rule 6715

Int[(u_)*(x_)^(m_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1), u, x], x], x, x^(m + 1)], x] /
; FreeQ[m, x] && NeQ[m, -1] && FunctionOfQ[x^(m + 1), u, x]

Rule 6725

Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a + b*x^n), x]}, Int[v, x]
/; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {\sqrt {b^2+a^2 x^3} \left (2 b^2+c x^3+a^2 x^6\right )}{x^7 \left (b^2+a^2 x^6\right )} \, dx &=\int \left (\frac {2 \sqrt {b^2+a^2 x^3}}{x^7}+\frac {c \sqrt {b^2+a^2 x^3}}{b^2 x^4}-\frac {a^2 \sqrt {b^2+a^2 x^3}}{b^2 x}+\frac {a^2 x^2 \sqrt {b^2+a^2 x^3} \left (-c+a^2 x^3\right )}{b^2 \left (b^2+a^2 x^6\right )}\right ) \, dx\\ &=2 \int \frac {\sqrt {b^2+a^2 x^3}}{x^7} \, dx-\frac {a^2 \int \frac {\sqrt {b^2+a^2 x^3}}{x} \, dx}{b^2}+\frac {a^2 \int \frac {x^2 \sqrt {b^2+a^2 x^3} \left (-c+a^2 x^3\right )}{b^2+a^2 x^6} \, dx}{b^2}+\frac {c \int \frac {\sqrt {b^2+a^2 x^3}}{x^4} \, dx}{b^2}\\ &=\frac {2}{3} \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x^3} \, dx,x,x^3\right )-\frac {a^2 \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x} \, dx,x,x^3\right )}{3 b^2}+\frac {a^2 \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x} \left (-c+a^2 x\right )}{b^2+a^2 x^2} \, dx,x,x^3\right )}{3 b^2}+\frac {c \operatorname {Subst}\left (\int \frac {\sqrt {b^2+a^2 x}}{x^2} \, dx,x,x^3\right )}{3 b^2}\\ &=-\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}-\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}+\frac {1}{6} a^2 \operatorname {Subst}\left (\int \frac {1}{x^2 \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )-\frac {1}{3} a^2 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )+\frac {\operatorname {Subst}\left (\int \frac {-a^2 b^2 \left (a^2+c\right )+a^4 \left (b^2-c\right ) x}{\sqrt {b^2+a^2 x} \left (b^2+a^2 x^2\right )} \, dx,x,x^3\right )}{3 b^2}+\frac {\left (a^2 c\right ) \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{6 b^2}\\ &=-\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}-\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}-\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}-\frac {2}{3} \operatorname {Subst}\left (\int \frac {1}{-\frac {b^2}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )+\frac {2 \operatorname {Subst}\left (\int \frac {-a^4 b^2 \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )+a^4 \left (b^2-c\right ) x^2}{a^4 b^2+a^2 b^4-2 a^2 b^2 x^2+a^2 x^4} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 b^2}-\frac {a^4 \operatorname {Subst}\left (\int \frac {1}{x \sqrt {b^2+a^2 x}} \, dx,x,x^3\right )}{12 b^2}+\frac {c \operatorname {Subst}\left (\int \frac {1}{-\frac {b^2}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 b^2}\\ &=-\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}-\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}-\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{-\frac {b^2}{a^2}+\frac {x^2}{a^2}} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 b^2}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \left (-a^4 b^2 \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )\right )-\left (-a^4 b^2 \left (b^2-c\right )-a^4 b \sqrt {a^2+b^2} \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )\right ) x}{b \sqrt {a^2+b^2}-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 \sqrt {2} a^2 b^{7/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}+\frac {\operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \left (-a^4 b^2 \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )\right )+\left (-a^4 b^2 \left (b^2-c\right )-a^4 b \sqrt {a^2+b^2} \left (b^2-c\right )-a^4 b^2 \left (a^2+c\right )\right ) x}{b \sqrt {a^2+b^2}+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{3 \sqrt {2} a^2 b^{7/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}\\ &=-\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}-\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}-\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}+\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {\left (a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b \sqrt {a^2+b^2}-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 b^2 \sqrt {a^2+b^2}}-\frac {\left (a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b \sqrt {a^2+b^2}+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 b^2 \sqrt {a^2+b^2}}+\frac {\left (a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \operatorname {Subst}\left (\int \frac {-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+2 x}{b \sqrt {a^2+b^2}-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}-\frac {\left (a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \operatorname {Subst}\left (\int \frac {\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+2 x}{b \sqrt {a^2+b^2}+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} x+x^2} \, dx,x,\sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}\\ &=-\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}-\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}-\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}+\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}+\frac {a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \log \left (b \left (b+\sqrt {a^2+b^2}\right )+a^2 x^3-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}-\frac {a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \log \left (b \left (b+\sqrt {a^2+b^2}\right )+a^2 x^3+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}+\frac {\left (a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 b \left (b-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+2 \sqrt {b^2+a^2 x^3}\right )}{3 b^2 \sqrt {a^2+b^2}}+\frac {\left (a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right )\right ) \operatorname {Subst}\left (\int \frac {1}{2 b \left (b-\sqrt {a^2+b^2}\right )-x^2} \, dx,x,\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+2 \sqrt {b^2+a^2 x^3}\right )}{3 b^2 \sqrt {a^2+b^2}}\\ &=-\frac {\sqrt {b^2+a^2 x^3}}{3 x^6}-\frac {a^2 \sqrt {b^2+a^2 x^3}}{6 b^2 x^3}-\frac {c \sqrt {b^2+a^2 x^3}}{3 b^2 x^3}+\frac {a^4 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3}+\frac {2 a^2 \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b}-\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{3 b^3}-\frac {a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}-\sqrt {2} \sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\frac {a^2 \left (a^2 b+b^3-\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {b+\sqrt {a^2+b^2}}+\sqrt {2} \sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {b-\sqrt {a^2+b^2}}}\right )}{3 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}+\frac {a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \log \left (b \left (b+\sqrt {a^2+b^2}\right )+a^2 x^3-\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}-\frac {a^2 \left (a^2 b+b^3+\sqrt {a^2+b^2} \left (b^2-c\right )\right ) \log \left (b \left (b+\sqrt {a^2+b^2}\right )+a^2 x^3+\sqrt {2} \sqrt {b} \sqrt {b+\sqrt {a^2+b^2}} \sqrt {b^2+a^2 x^3}\right )}{6 \sqrt {2} b^{5/2} \sqrt {a^2+b^2} \sqrt {b+\sqrt {a^2+b^2}}}\\ \end {align*}

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Mathematica [C]  time = 0.81, size = 442, normalized size = 1.79 \begin {gather*} \frac {-3 b^{7/2} \left (-\sqrt {-a^2} c x^3 \sqrt {\sqrt {-a^2}-b} \sqrt {a^2 x^3+b^2} \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {\sqrt {-a^2}-b}}\right )+x^3 \sqrt {\sqrt {-a^2}+b} \left (a^2 b+\sqrt {-a^2} c\right ) \sqrt {a^2 x^3+b^2} \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {\sqrt {-a^2}+b}}\right )+a^2 \sqrt {b} c x^3 \sqrt {\frac {a^2 x^3}{b^2}+1} \tanh ^{-1}\left (\sqrt {\frac {a^2 x^3}{b^2}+1}\right )+a^2 b x^3 \sqrt {\sqrt {-a^2}-b} \sqrt {a^2 x^3+b^2} \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {\sqrt {-a^2}-b}}\right )-2 a^2 b^{3/2} x^3 \sqrt {a^2 x^3+b^2} \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )+a^2 \sqrt {b} c x^3+b^{5/2} c\right )-4 a^4 x^3 \left (a^2 x^3+b^2\right )^2 \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {a^2 x^3}{b^2}+1\right )}{9 b^6 x^3 \sqrt {a^2 x^3+b^2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[b^2 + a^2*x^3]*(2*b^2 + c*x^3 + a^2*x^6))/(x^7*(b^2 + a^2*x^6)),x]

[Out]

(-3*b^(7/2)*(b^(5/2)*c + a^2*Sqrt[b]*c*x^3 + a^2*Sqrt[Sqrt[-a^2] - b]*b*x^3*Sqrt[b^2 + a^2*x^3]*ArcTan[Sqrt[b^
2 + a^2*x^3]/(Sqrt[Sqrt[-a^2] - b]*Sqrt[b])] - Sqrt[-a^2]*Sqrt[Sqrt[-a^2] - b]*c*x^3*Sqrt[b^2 + a^2*x^3]*ArcTa
n[Sqrt[b^2 + a^2*x^3]/(Sqrt[Sqrt[-a^2] - b]*Sqrt[b])] - 2*a^2*b^(3/2)*x^3*Sqrt[b^2 + a^2*x^3]*ArcTanh[Sqrt[b^2
+ a^2*x^3]/b] + Sqrt[Sqrt[-a^2] + b]*(a^2*b + Sqrt[-a^2]*c)*x^3*Sqrt[b^2 + a^2*x^3]*ArcTanh[Sqrt[b^2 + a^2*x^
3]/(Sqrt[b]*Sqrt[Sqrt[-a^2] + b])] + a^2*Sqrt[b]*c*x^3*Sqrt[1 + (a^2*x^3)/b^2]*ArcTanh[Sqrt[1 + (a^2*x^3)/b^2]
]) - 4*a^4*x^3*(b^2 + a^2*x^3)^2*Hypergeometric2F1[3/2, 3, 5/2, 1 + (a^2*x^3)/b^2])/(9*b^6*x^3*Sqrt[b^2 + a^2*
x^3])

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IntegrateAlgebraic [A]  time = 0.82, size = 247, normalized size = 1.00 \begin {gather*} \frac {\sqrt {b^2+a^2 x^3} \left (-2 b^2-a^2 x^3-2 c x^3\right )}{6 b^2 x^6}+\frac {\sqrt {a-i b} \left ((-1)^{3/4} a^2 b-\sqrt [4]{-1} a c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {b^2+a^2 x^3}}{\sqrt {a-i b} \sqrt {b}}\right )}{3 b^{5/2}}-\frac {\sqrt {a+i b} \left (-\sqrt [4]{-1} a^2 b+(-1)^{3/4} a c\right ) \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {b^2+a^2 x^3}}{\sqrt {a+i b} \sqrt {b}}\right )}{3 b^{5/2}}+\frac {\left (a^4+4 a^2 b^2-2 a^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right )}{6 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[b^2 + a^2*x^3]*(2*b^2 + c*x^3 + a^2*x^6))/(x^7*(b^2 + a^2*x^6)),x]

[Out]

(Sqrt[b^2 + a^2*x^3]*(-2*b^2 - a^2*x^3 - 2*c*x^3))/(6*b^2*x^6) + (Sqrt[a - I*b]*((-1)^(3/4)*a^2*b - (-1)^(1/4)
*a*c)*ArcTan[((-1)^(1/4)*Sqrt[b^2 + a^2*x^3])/(Sqrt[a - I*b]*Sqrt[b])])/(3*b^(5/2)) - (Sqrt[a + I*b]*(-((-1)^(
1/4)*a^2*b) + (-1)^(3/4)*a*c)*ArcTan[((-1)^(3/4)*Sqrt[b^2 + a^2*x^3])/(Sqrt[a + I*b]*Sqrt[b])])/(3*b^(5/2)) +
((a^4 + 4*a^2*b^2 - 2*a^2*c)*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/(6*b^3)

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fricas [B]  time = 9.27, size = 10278, normalized size = 41.61

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^3+b^2)^(1/2)*(a^2*x^6+c*x^3+2*b^2)/x^7/(a^2*x^6+b^2),x, algorithm="fricas")

[Out]

-1/12*(4*sqrt(2)*b^13*x^6*sqrt((a^8*b^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*b^8
- 2*a^2*b^6*c - b^6*c^2)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(
a^8*b^4 + 4*a^6*b^4*c - 4*a^4*b^2*c^3 + a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6 + (a^6 +
a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)^(3/4)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 -
2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10)*arctan((sqrt(2)*sqrt(a^2*x^3 + b^2)*((a^8*b^22 + 2*a^6*b^22*c + 2*a^4*b^20
*c^3 - a^4*b^18*c^4)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)*sqrt((a
^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10) + (a^12*b^20 - a^10*b^18*
c^2 - 5*a^8*b^16*c^4 - 3*a^6*b^14*c^6 + a^6*b^12*c^7 + (a^8*b^14 + 2*a^6*b^16)*c^5 - (a^10*b^16 - 4*a^8*b^18)*
c^3 - (a^12*b^18 - 2*a^10*b^20)*c)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6
*b^4)*c^2)/b^10))*sqrt((a^8*b^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*b^8 - 2*a^2
*b^6*c - b^6*c^2)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(a^8*b^4
+ 4*a^6*b^4*c - 4*a^4*b^2*c^3 + a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)
*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)^(3/4) + sqrt(2)*(b^18*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 +
2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*
b^4)*c^2)/b^10) + (a^4*b^16 - a^4*b^14*c + a^2*b^14*c^2 - a^2*b^12*c^3)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b
^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10))*sqrt((a^8*b^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*
b^2 + a^4*b^4)*c^2 - (a^2*b^8 - 2*a^2*b^6*c - b^6*c^2)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8
*b^2 + a^6*b^4)*c^2)/b^10))/(a^8*b^4 + 4*a^6*b^4*c - 4*a^4*b^2*c^3 + a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*s
qrt((a^18*b^10 + a^16*b^12 + (a^10*b^2 + a^8*b^4)*c^8 - 4*(a^10*b^4 + a^8*b^6)*c^7 + 4*(a^10*b^6 + a^8*b^8)*c^
6 - 4*(a^12*b^6 + a^10*b^8)*c^5 - 2*(a^14*b^6 - 3*a^12*b^8 - 4*a^10*b^10)*c^4 + 4*(a^14*b^8 + a^12*b^10)*c^3 +
(a^20*b^8 + a^18*b^10 + (a^12 + a^10*b^2)*c^8 - 4*(a^12*b^2 + a^10*b^4)*c^7 + 4*(a^12*b^4 + a^10*b^6)*c^6 - 4
*(a^14*b^4 + a^12*b^6)*c^5 - 2*(a^16*b^4 - 3*a^14*b^6 - 4*a^12*b^8)*c^4 + 4*(a^16*b^6 + a^14*b^8)*c^3 + 4*(a^1
6*b^8 + a^14*b^10)*c^2 + 4*(a^18*b^8 + a^16*b^10)*c)*x^3 + 4*(a^14*b^10 + a^12*b^12)*c^2 + sqrt(2)*(a^16*b^10
+ a^14*b^12 + (a^10*b^4 + a^8*b^6)*c^6 - 4*(a^10*b^6 + a^8*b^8)*c^5 - (a^12*b^6 - 3*a^10*b^8 - 4*a^8*b^10)*c^4
- (a^14*b^8 - 3*a^12*b^10 - 4*a^10*b^12)*c^2 + 4*(a^14*b^10 + a^12*b^12)*c + (a^10*b^14 + 5*a^6*b^10*c^4 - a^
6*b^8*c^5 + 2*(a^8*b^10 - 4*a^6*b^12)*c^3 - 2*(3*a^8*b^12 - 2*a^6*b^14)*c^2 - (a^10*b^12 - 4*a^8*b^14)*c)*sqrt
((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))*sqrt(a^2*x^3 + b^2)*sqrt((a^8*b
^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*b^8 - 2*a^2*b^6*c - b^6*c^2)*sqrt((a^10*
b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(a^8*b^4 + 4*a^6*b^4*c - 4*a^4*b^2*c^3
+ a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*
c^2)/b^10)^(1/4) + 4*(a^16*b^10 + a^14*b^12)*c + (a^14*b^12 + a^12*b^14 + (a^8*b^6 + a^6*b^8)*c^6 - 4*(a^8*b^8
+ a^6*b^10)*c^5 - (a^10*b^8 - 3*a^8*b^10 - 4*a^6*b^12)*c^4 - (a^12*b^10 - 3*a^10*b^12 - 4*a^8*b^14)*c^2 + 4*(
a^12*b^12 + a^10*b^14)*c)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(
a^2 + b^2))*((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)^(3/4) + (a^16*b^18 +
a^14*b^20 - (a^8*b^10 + a^6*b^12)*c^8 + 2*(a^8*b^12 + a^6*b^14)*c^7 - 2*(a^10*b^12 + a^8*b^14)*c^6 + 6*(a^10*
b^14 + a^8*b^16)*c^5 + 6*(a^12*b^16 + a^10*b^18)*c^3 + 2*(a^14*b^16 + a^12*b^18)*c^2 + 2*(a^14*b^18 + a^12*b^2
0)*c)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)*sqrt((a^10*b^4 + 4*a^8
*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10) + (a^20*b^16 + a^18*b^18 - (a^10*b^6 + a
^8*b^8)*c^10 + 2*(a^10*b^8 + a^8*b^10)*c^9 - 3*(a^12*b^8 + a^10*b^10)*c^8 + 8*(a^12*b^10 + a^10*b^12)*c^7 - 2*
(a^14*b^10 + a^12*b^12)*c^6 + 12*(a^14*b^12 + a^12*b^14)*c^5 + 2*(a^16*b^12 + a^14*b^14)*c^4 + 8*(a^16*b^14 +
a^14*b^16)*c^3 + 3*(a^18*b^14 + a^16*b^16)*c^2 + 2*(a^18*b^16 + a^16*b^18)*c)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4
*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10))/(a^26*b^12 + a^24*b^14 + (a^14 + a^12*b^2)*c^12 -
4*(a^14*b^2 + a^12*b^4)*c^11 + 2*(a^16*b^2 + 3*a^14*b^4 + 2*a^12*b^6)*c^10 - 12*(a^16*b^4 + a^14*b^6)*c^9 - (
a^18*b^4 - 15*a^16*b^6 - 16*a^14*b^8)*c^8 - 8*(a^18*b^6 + a^16*b^8)*c^7 - 4*(a^20*b^6 - 5*a^18*b^8 - 6*a^16*b^
10)*c^6 + 8*(a^20*b^8 + a^18*b^10)*c^5 - (a^22*b^8 - 15*a^20*b^10 - 16*a^18*b^12)*c^4 + 12*(a^22*b^10 + a^20*b
^12)*c^3 + 2*(a^24*b^10 + 3*a^22*b^12 + 2*a^20*b^14)*c^2 + 4*(a^24*b^12 + a^22*b^14)*c)) + 4*sqrt(2)*b^13*x^6*
sqrt((a^8*b^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*b^8 - 2*a^2*b^6*c - b^6*c^2)*
sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(a^8*b^4 + 4*a^6*b^4*c - 4*
a^4*b^2*c^3 + a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2
+ a^6*b^4)*c^2)/b^10)^(3/4)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c
^2)/b^10)*arctan((sqrt(2)*sqrt(a^2*x^3 + b^2)*((a^8*b^22 + 2*a^6*b^22*c + 2*a^4*b^20*c^3 - a^4*b^18*c^4)*sqrt(
(a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*
a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10) + (a^12*b^20 - a^10*b^18*c^2 - 5*a^8*b^16*c^4 - 3*a
^6*b^14*c^6 + a^6*b^12*c^7 + (a^8*b^14 + 2*a^6*b^16)*c^5 - (a^10*b^16 - 4*a^8*b^18)*c^3 - (a^12*b^18 - 2*a^10*
b^20)*c)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10))*sqrt((a^
8*b^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*b^8 - 2*a^2*b^6*c - b^6*c^2)*sqrt((a^
10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(a^8*b^4 + 4*a^6*b^4*c - 4*a^4*b^2*
c^3 + a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^
4)*c^2)/b^10)^(3/4) + sqrt(2)*(b^18*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2
)/b^10)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10) + (a^4*b^1
6 - a^4*b^14*c + a^2*b^14*c^2 - a^2*b^12*c^3)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2*(a^8*
b^2 - 2*a^6*b^4)*c^2)/b^10))*sqrt((a^8*b^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*
b^8 - 2*a^2*b^6*c - b^6*c^2)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)
)/(a^8*b^4 + 4*a^6*b^4*c - 4*a^4*b^2*c^3 + a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*sqrt((a^18*b^10 + a^16*b^12
+ (a^10*b^2 + a^8*b^4)*c^8 - 4*(a^10*b^4 + a^8*b^6)*c^7 + 4*(a^10*b^6 + a^8*b^8)*c^6 - 4*(a^12*b^6 + a^10*b^8
)*c^5 - 2*(a^14*b^6 - 3*a^12*b^8 - 4*a^10*b^10)*c^4 + 4*(a^14*b^8 + a^12*b^10)*c^3 + (a^20*b^8 + a^18*b^10 + (
a^12 + a^10*b^2)*c^8 - 4*(a^12*b^2 + a^10*b^4)*c^7 + 4*(a^12*b^4 + a^10*b^6)*c^6 - 4*(a^14*b^4 + a^12*b^6)*c^5
- 2*(a^16*b^4 - 3*a^14*b^6 - 4*a^12*b^8)*c^4 + 4*(a^16*b^6 + a^14*b^8)*c^3 + 4*(a^16*b^8 + a^14*b^10)*c^2 + 4
*(a^18*b^8 + a^16*b^10)*c)*x^3 + 4*(a^14*b^10 + a^12*b^12)*c^2 - sqrt(2)*(a^16*b^10 + a^14*b^12 + (a^10*b^4 +
a^8*b^6)*c^6 - 4*(a^10*b^6 + a^8*b^8)*c^5 - (a^12*b^6 - 3*a^10*b^8 - 4*a^8*b^10)*c^4 - (a^14*b^8 - 3*a^12*b^10
- 4*a^10*b^12)*c^2 + 4*(a^14*b^10 + a^12*b^12)*c + (a^10*b^14 + 5*a^6*b^10*c^4 - a^6*b^8*c^5 + 2*(a^8*b^10 -
4*a^6*b^12)*c^3 - 2*(3*a^8*b^12 - 2*a^6*b^14)*c^2 - (a^10*b^12 - 4*a^8*b^14)*c)*sqrt((a^10*b^4 + a^8*b^6 + (a^
6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))*sqrt(a^2*x^3 + b^2)*sqrt((a^8*b^4 + a^6*b^6 + (a^4 + a^2*
b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*b^8 - 2*a^2*b^6*c - b^6*c^2)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4
*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(a^8*b^4 + 4*a^6*b^4*c - 4*a^4*b^2*c^3 + a^4*c^4 - 2*(a^6*b^2 -
2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)^(1/4) + 4*(a^16
*b^10 + a^14*b^12)*c + (a^14*b^12 + a^12*b^14 + (a^8*b^6 + a^6*b^8)*c^6 - 4*(a^8*b^8 + a^6*b^10)*c^5 - (a^10*b
^8 - 3*a^8*b^10 - 4*a^6*b^12)*c^4 - (a^12*b^10 - 3*a^10*b^12 - 4*a^8*b^14)*c^2 + 4*(a^12*b^12 + a^10*b^14)*c)*
sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(a^2 + b^2))*((a^10*b^4 + a
^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)^(3/4) - (a^16*b^18 + a^14*b^20 - (a^8*b^10 + a
^6*b^12)*c^8 + 2*(a^8*b^12 + a^6*b^14)*c^7 - 2*(a^10*b^12 + a^8*b^14)*c^6 + 6*(a^10*b^14 + a^8*b^16)*c^5 + 6*(
a^12*b^16 + a^10*b^18)*c^3 + 2*(a^14*b^16 + a^12*b^18)*c^2 + 2*(a^14*b^18 + a^12*b^20)*c)*sqrt((a^10*b^4 + a^8
*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a
^6*c^4 - 2*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10) - (a^20*b^16 + a^18*b^18 - (a^10*b^6 + a^8*b^8)*c^10 + 2*(a^10*b^8
+ a^8*b^10)*c^9 - 3*(a^12*b^8 + a^10*b^10)*c^8 + 8*(a^12*b^10 + a^10*b^12)*c^7 - 2*(a^14*b^10 + a^12*b^12)*c^
6 + 12*(a^14*b^12 + a^12*b^14)*c^5 + 2*(a^16*b^12 + a^14*b^14)*c^4 + 8*(a^16*b^14 + a^14*b^16)*c^3 + 3*(a^18*b
^14 + a^16*b^16)*c^2 + 2*(a^18*b^16 + a^16*b^18)*c)*sqrt((a^10*b^4 + 4*a^8*b^4*c - 4*a^6*b^2*c^3 + a^6*c^4 - 2
*(a^8*b^2 - 2*a^6*b^4)*c^2)/b^10))/(a^26*b^12 + a^24*b^14 + (a^14 + a^12*b^2)*c^12 - 4*(a^14*b^2 + a^12*b^4)*c
^11 + 2*(a^16*b^2 + 3*a^14*b^4 + 2*a^12*b^6)*c^10 - 12*(a^16*b^4 + a^14*b^6)*c^9 - (a^18*b^4 - 15*a^16*b^6 - 1
6*a^14*b^8)*c^8 - 8*(a^18*b^6 + a^16*b^8)*c^7 - 4*(a^20*b^6 - 5*a^18*b^8 - 6*a^16*b^10)*c^6 + 8*(a^20*b^8 + a^
18*b^10)*c^5 - (a^22*b^8 - 15*a^20*b^10 - 16*a^18*b^12)*c^4 + 12*(a^22*b^10 + a^20*b^12)*c^3 + 2*(a^24*b^10 +
3*a^22*b^12 + 2*a^20*b^14)*c^2 + 4*(a^24*b^12 + a^22*b^14)*c)) - (a^14*b^4 + 5*a^12*b^6 + 4*a^10*b^8 - 2*(a^8
+ a^6*b^2)*c^5 + (a^10 + 5*a^8*b^2 + 4*a^6*b^4)*c^4 - 4*(a^10*b^2 + a^8*b^4)*c^3 + 2*(a^12*b^2 + 5*a^10*b^4 +
4*a^8*b^6)*c^2 - 2*(a^12*b^4 + a^10*b^6)*c)*x^6*log(b + sqrt(a^2*x^3 + b^2)) + (a^14*b^4 + 5*a^12*b^6 + 4*a^10
*b^8 - 2*(a^8 + a^6*b^2)*c^5 + (a^10 + 5*a^8*b^2 + 4*a^6*b^4)*c^4 - 4*(a^10*b^2 + a^8*b^4)*c^3 + 2*(a^12*b^2 +
5*a^10*b^4 + 4*a^8*b^6)*c^2 - 2*(a^12*b^4 + a^10*b^6)*c)*x^6*log(-b + sqrt(a^2*x^3 + b^2)) + sqrt(2)*((a^4*b^
11 - 2*a^4*b^9*c - a^2*b^9*c^2)*x^6*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2
)/b^10) + (a^10*b^7 + a^8*b^9 + (a^6*b^3 + a^4*b^5)*c^4 + 2*(a^8*b^5 + a^6*b^7)*c^2)*x^6)*sqrt((a^8*b^4 + a^6*
b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*b^8 - 2*a^2*b^6*c - b^6*c^2)*sqrt((a^10*b^4 + a^8
*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(a^8*b^4 + 4*a^6*b^4*c - 4*a^4*b^2*c^3 + a^4*c^
4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10
)^(1/4)*log((a^18*b^10 + a^16*b^12 + (a^10*b^2 + a^8*b^4)*c^8 - 4*(a^10*b^4 + a^8*b^6)*c^7 + 4*(a^10*b^6 + a^8
*b^8)*c^6 - 4*(a^12*b^6 + a^10*b^8)*c^5 - 2*(a^14*b^6 - 3*a^12*b^8 - 4*a^10*b^10)*c^4 + 4*(a^14*b^8 + a^12*b^1
0)*c^3 + (a^20*b^8 + a^18*b^10 + (a^12 + a^10*b^2)*c^8 - 4*(a^12*b^2 + a^10*b^4)*c^7 + 4*(a^12*b^4 + a^10*b^6)
*c^6 - 4*(a^14*b^4 + a^12*b^6)*c^5 - 2*(a^16*b^4 - 3*a^14*b^6 - 4*a^12*b^8)*c^4 + 4*(a^16*b^6 + a^14*b^8)*c^3
+ 4*(a^16*b^8 + a^14*b^10)*c^2 + 4*(a^18*b^8 + a^16*b^10)*c)*x^3 + 4*(a^14*b^10 + a^12*b^12)*c^2 + sqrt(2)*(a^
16*b^10 + a^14*b^12 + (a^10*b^4 + a^8*b^6)*c^6 - 4*(a^10*b^6 + a^8*b^8)*c^5 - (a^12*b^6 - 3*a^10*b^8 - 4*a^8*b
^10)*c^4 - (a^14*b^8 - 3*a^12*b^10 - 4*a^10*b^12)*c^2 + 4*(a^14*b^10 + a^12*b^12)*c + (a^10*b^14 + 5*a^6*b^10*
c^4 - a^6*b^8*c^5 + 2*(a^8*b^10 - 4*a^6*b^12)*c^3 - 2*(3*a^8*b^12 - 2*a^6*b^14)*c^2 - (a^10*b^12 - 4*a^8*b^14)
*c)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))*sqrt(a^2*x^3 + b^2)*sqr
t((a^8*b^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*b^8 - 2*a^2*b^6*c - b^6*c^2)*sqr
t((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(a^8*b^4 + 4*a^6*b^4*c - 4*a^4
*b^2*c^3 + a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a
^6*b^4)*c^2)/b^10)^(1/4) + 4*(a^16*b^10 + a^14*b^12)*c + (a^14*b^12 + a^12*b^14 + (a^8*b^6 + a^6*b^8)*c^6 - 4*
(a^8*b^8 + a^6*b^10)*c^5 - (a^10*b^8 - 3*a^8*b^10 - 4*a^6*b^12)*c^4 - (a^12*b^10 - 3*a^10*b^12 - 4*a^8*b^14)*c
^2 + 4*(a^12*b^12 + a^10*b^14)*c)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/
b^10))/(a^2 + b^2)) - sqrt(2)*((a^4*b^11 - 2*a^4*b^9*c - a^2*b^9*c^2)*x^6*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^
4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10) + (a^10*b^7 + a^8*b^9 + (a^6*b^3 + a^4*b^5)*c^4 + 2*(a^8*b^5 + a
^6*b^7)*c^2)*x^6)*sqrt((a^8*b^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 - (a^2*b^8 - 2*a^2
*b^6*c - b^6*c^2)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(a^8*b^4
+ 4*a^6*b^4*c - 4*a^4*b^2*c^3 + a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)
*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)^(1/4)*log((a^18*b^10 + a^16*b^12 + (a^10*b^2 + a^8*b^4)*c^8 - 4*(a^10*
b^4 + a^8*b^6)*c^7 + 4*(a^10*b^6 + a^8*b^8)*c^6 - 4*(a^12*b^6 + a^10*b^8)*c^5 - 2*(a^14*b^6 - 3*a^12*b^8 - 4*a
^10*b^10)*c^4 + 4*(a^14*b^8 + a^12*b^10)*c^3 + (a^20*b^8 + a^18*b^10 + (a^12 + a^10*b^2)*c^8 - 4*(a^12*b^2 + a
^10*b^4)*c^7 + 4*(a^12*b^4 + a^10*b^6)*c^6 - 4*(a^14*b^4 + a^12*b^6)*c^5 - 2*(a^16*b^4 - 3*a^14*b^6 - 4*a^12*b
^8)*c^4 + 4*(a^16*b^6 + a^14*b^8)*c^3 + 4*(a^16*b^8 + a^14*b^10)*c^2 + 4*(a^18*b^8 + a^16*b^10)*c)*x^3 + 4*(a^
14*b^10 + a^12*b^12)*c^2 - sqrt(2)*(a^16*b^10 + a^14*b^12 + (a^10*b^4 + a^8*b^6)*c^6 - 4*(a^10*b^6 + a^8*b^8)*
c^5 - (a^12*b^6 - 3*a^10*b^8 - 4*a^8*b^10)*c^4 - (a^14*b^8 - 3*a^12*b^10 - 4*a^10*b^12)*c^2 + 4*(a^14*b^10 + a
^12*b^12)*c + (a^10*b^14 + 5*a^6*b^10*c^4 - a^6*b^8*c^5 + 2*(a^8*b^10 - 4*a^6*b^12)*c^3 - 2*(3*a^8*b^12 - 2*a^
6*b^14)*c^2 - (a^10*b^12 - 4*a^8*b^14)*c)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^
4)*c^2)/b^10))*sqrt(a^2*x^3 + b^2)*sqrt((a^8*b^4 + a^6*b^6 + (a^4 + a^2*b^2)*c^4 + 2*(a^6*b^2 + a^4*b^4)*c^2 -
(a^2*b^8 - 2*a^2*b^6*c - b^6*c^2)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)
/b^10))/(a^8*b^4 + 4*a^6*b^4*c - 4*a^4*b^2*c^3 + a^4*c^4 - 2*(a^6*b^2 - 2*a^4*b^4)*c^2))*((a^10*b^4 + a^8*b^6
+ (a^6 + a^4*b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10)^(1/4) + 4*(a^16*b^10 + a^14*b^12)*c + (a^14*b^12 + a^
12*b^14 + (a^8*b^6 + a^6*b^8)*c^6 - 4*(a^8*b^8 + a^6*b^10)*c^5 - (a^10*b^8 - 3*a^8*b^10 - 4*a^6*b^12)*c^4 - (a
^12*b^10 - 3*a^10*b^12 - 4*a^8*b^14)*c^2 + 4*(a^12*b^12 + a^10*b^14)*c)*sqrt((a^10*b^4 + a^8*b^6 + (a^6 + a^4*
b^2)*c^4 + 2*(a^8*b^2 + a^6*b^4)*c^2)/b^10))/(a^2 + b^2)) + 2*(2*a^10*b^7 + 2*a^8*b^9 + 2*(a^6*b^3 + a^4*b^5)*
c^4 + (a^12*b^5 + a^10*b^7 + 2*(a^6*b + a^4*b^3)*c^5 + (a^8*b + a^6*b^3)*c^4 + 4*(a^8*b^3 + a^6*b^5)*c^3 + 2*(
a^10*b^3 + a^8*b^5)*c^2 + 2*(a^10*b^5 + a^8*b^7)*c)*x^3 + 4*(a^8*b^5 + a^6*b^7)*c^2)*sqrt(a^2*x^3 + b^2))/((a^
10*b^7 + a^8*b^9 + (a^6*b^3 + a^4*b^5)*c^4 + 2*(a^8*b^5 + a^6*b^7)*c^2)*x^6)

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^3+b^2)^(1/2)*(a^2*x^6+c*x^3+2*b^2)/x^7/(a^2*x^6+b^2),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const ge
n & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, need to choose a branch for the r
oot of a polynomial with parameters. This might be wrong.The choice was done assuming [abs(b)]=[-4,-35]sym2pol
y/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, need to choose a
branch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [abs(b)]
=[12,21]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, ne
ed to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done as
suming [abs(b)]=[24,16]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument V
alueWarning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The cho
ice was done assuming [abs(b)]=[-71,69]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error
: Bad Argument ValueWarning, need to choose a branch for the root of a polynomial with parameters. This might
be wrong.The choice was done assuming [abs(b)]=[-22,57]sym2poly/r2sym(const gen & e,const index_m & i,const ve
cteur & l) Error: Bad Argument ValueWarning, need to choose a branch for the root of a polynomial with paramet
ers. This might be wrong.The choice was done assuming [abs(b)]=[84,29]sym2poly/r2sym(const gen & e,const index
_m & i,const vecteur & l) Error: Bad Argument ValueWarning, need to choose a branch for the root of a polynomi
al with parameters. This might be wrong.The choice was done assuming [abs(b)]=[-27,-21]sym2poly/r2sym(const ge
n & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, need to choose a branch for the r
oot of a polynomial with parameters. This might be wrong.The choice was done assuming [abs(b)]=[-43,4]sym2poly
/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, need to choose a b
ranch for the root of a polynomial with parameters. This might be wrong.The choice was done assuming [abs(b)]=
[-60,90]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument ValueWarning, ne
ed to choose a branch for the root of a polynomial with parameters. This might be wrong.The choice was done as
suming [abs(b)]=[-10,65]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument
ValueWarning, need to choose a branch for the root of a polynomial with parameters. This might be wrong.The ch
oice was done assuming [abs(b)]=[-4,-25]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Erro
r: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Val
uesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(co
nst gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const i
ndex_m & i,const vecteur & l) Error: Bad Argument ValueEvaluation time: 7.12Done

________________________________________________________________________________________

maple [C]  time = 0.42, size = 746, normalized size = 3.02

method	result	size
risch	\(-\frac {\sqrt {a^{2} x^{3}+b^{2}}\, \left (a^{2} x^{3}+2 x^{3} c +2 b^{2}\right )}{6 b^{2} x^{6}}-\frac {a^{2} \left (-\frac {2 \left (a^{2}+4 b^{2}-2 c \right ) \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}-\frac {2 i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{6} a^{2}+b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} b^{2}+\underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} c +a^{2} b^{2}+b^{2} c \right ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i a \left (2 x +\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (2 a^{2} \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{5}-b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}\right )+i a^{3} \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}-i a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}-a^{3} \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4}-a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3} \left (-a \,b^{2}\right )^{\frac {2}{3}}-i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b^{2} a +i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, b^{2}+\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \,b^{2} a +\left (-a \,b^{2}\right )^{\frac {2}{3}} b^{2}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a^{3}-i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a^{2}+i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} b^{2}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} a^{2}-2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,b^{2}+i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,b^{2}-3 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} b^{2}-i \sqrt {3}\, b^{4}+3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \,b^{2}+3 b^{4}}{2 b^{2} \left (a^{2}+b^{2}\right )}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{3} \left (a^{2}+b^{2}\right ) \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{3} b^{2}}\right )}{4 b^{2}}\)	\(746\)
default	\(\frac {c \left (-\frac {\sqrt {a^{2} x^{3}+b^{2}}}{3 x^{3}}-\frac {a^{2} \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}\right )}{b^{2}}-\frac {a^{2} \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}}{3}-\frac {2 b^{2} \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}\right )}{b^{2}}-\frac {\sqrt {a^{2} x^{3}+b^{2}}}{3 x^{6}}-\frac {a^{2} \sqrt {a^{2} x^{3}+b^{2}}}{6 b^{2} x^{3}}+\frac {a^{4} \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{6 b^{2} \sqrt {b^{2}}}+\frac {a^{2} \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}}{3}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (\textit {\_Z}^{6} a^{2}+b^{2}\right )}{\sum }\frac {\left (-\underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} b^{2}+\underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} c +a^{2} b^{2}+b^{2} c \right ) \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {2}\, \sqrt {\frac {i a \left (2 x +\frac {-i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {\frac {a \left (x -\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{-3 \left (-a \,b^{2}\right )^{\frac {1}{3}}+i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \sqrt {-\frac {i a \left (2 x +\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}+\left (-a \,b^{2}\right )^{\frac {1}{3}}}{a}\right )}{2 \left (-a \,b^{2}\right )^{\frac {1}{3}}}}\, \left (2 a^{2} \left (a^{2} \underline {\hspace {1.25 ex}}\alpha ^{5}-b^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}\right )+i a^{3} \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} \sqrt {3}-i a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3} \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}-a^{3} \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4}-a^{2} \underline {\hspace {1.25 ex}}\alpha ^{3} \left (-a \,b^{2}\right )^{\frac {2}{3}}-i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, b^{2} a +i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, b^{2}+\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \,b^{2} a +\left (-a \,b^{2}\right )^{\frac {2}{3}} b^{2}\right ) \EllipticPi \left (\frac {\sqrt {3}\, \sqrt {\frac {i \left (x +\frac {\left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}-\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right ) \sqrt {3}\, a}{\left (-a \,b^{2}\right )^{\frac {1}{3}}}}}{3}, \frac {2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{5} a^{3}-i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{4} a^{2}+i \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} b^{2}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha ^{4} a^{2}-2 i \left (-a \,b^{2}\right )^{\frac {1}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha ^{2} a \,b^{2}+i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha \,b^{2}-3 \underline {\hspace {1.25 ex}}\alpha ^{3} a^{2} b^{2}-i \sqrt {3}\, b^{4}+3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha \,b^{2}+3 b^{4}}{2 b^{2} \left (a^{2}+b^{2}\right )}, \sqrt {\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{a \left (-\frac {3 \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}+\frac {i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {1}{3}}}{2 a}\right )}}\right )}{2 \underline {\hspace {1.25 ex}}\alpha ^{3} \left (a^{2}+b^{2}\right ) \sqrt {a^{2} x^{3}+b^{2}}}\right )}{6 a^{3} b^{2}}\right )}{b^{2}}\)	\(864\)
elliptic	\(\text {Expression too large to display}\)	\(12546\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a^2*x^3+b^2)^(1/2)*(a^2*x^6+c*x^3+2*b^2)/x^7/(a^2*x^6+b^2),x,method=_RETURNVERBOSE)

[Out]

-1/6*(a^2*x^3+b^2)^(1/2)*(a^2*x^3+2*c*x^3+2*b^2)/b^2/x^6-1/4/b^2*a^2*(-2/3*(a^2+4*b^2-2*c)*arctanh((a^2*x^3+b^
2)^(1/2)/(b^2)^(1/2))/(b^2)^(1/2)-2/3*I/a^3/b^2*2^(1/2)*sum((-_alpha^3*a^2*b^2+_alpha^3*a^2*c+a^2*b^2+b^2*c)/_
alpha^3/(a^2+b^2)*(-a*b^2)^(1/3)*(1/2*I*a*(2*x+1/a*(-I*3^(1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))
^(1/2)*(a*(x-1/a*(-a*b^2)^(1/3))/(-3*(-a*b^2)^(1/3)+I*3^(1/2)*(-a*b^2)^(1/3)))^(1/2)*(-1/2*I*a*(2*x+1/a*(I*3^(
1/2)*(-a*b^2)^(1/3)+(-a*b^2)^(1/3)))/(-a*b^2)^(1/3))^(1/2)/(a^2*x^3+b^2)^(1/2)*(2*a^2*(_alpha^5*a^2-_alpha^2*b
^2)+I*a^3*(-a*b^2)^(1/3)*_alpha^4*3^(1/2)-I*a^2*_alpha^3*3^(1/2)*(-a*b^2)^(2/3)-a^3*(-a*b^2)^(1/3)*_alpha^4-a^
2*_alpha^3*(-a*b^2)^(2/3)-I*(-a*b^2)^(1/3)*_alpha*3^(1/2)*b^2*a+I*(-a*b^2)^(2/3)*3^(1/2)*b^2+(-a*b^2)^(1/3)*_a
lpha*b^2*a+(-a*b^2)^(2/3)*b^2)*EllipticPi(1/3*3^(1/2)*(I*(x+1/2/a*(-a*b^2)^(1/3)-1/2*I*3^(1/2)/a*(-a*b^2)^(1/3
))*3^(1/2)*a/(-a*b^2)^(1/3))^(1/2),1/2*(2*I*(-a*b^2)^(1/3)*3^(1/2)*_alpha^5*a^3-I*(-a*b^2)^(2/3)*3^(1/2)*_alph
a^4*a^2+I*3^(1/2)*_alpha^3*a^2*b^2-3*(-a*b^2)^(2/3)*_alpha^4*a^2-2*I*(-a*b^2)^(1/3)*3^(1/2)*_alpha^2*a*b^2+I*(
-a*b^2)^(2/3)*3^(1/2)*_alpha*b^2-3*_alpha^3*a^2*b^2-I*3^(1/2)*b^4+3*(-a*b^2)^(2/3)*_alpha*b^2+3*b^4)/b^2/(a^2+
b^2),(I*3^(1/2)/a*(-a*b^2)^(1/3)/(-3/2/a*(-a*b^2)^(1/3)+1/2*I*3^(1/2)/a*(-a*b^2)^(1/3)))^(1/2)),_alpha=RootOf(
_Z^6*a^2+b^2)))

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {{\left (a^{2} x^{6} + c x^{3} + 2 \, b^{2}\right )} \sqrt {a^{2} x^{3} + b^{2}}}{{\left (a^{2} x^{6} + b^{2}\right )} x^{7}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a^2*x^3+b^2)^(1/2)*(a^2*x^6+c*x^3+2*b^2)/x^7/(a^2*x^6+b^2),x, algorithm="maxima")

[Out]

integrate((a^2*x^6 + c*x^3 + 2*b^2)*sqrt(a^2*x^3 + b^2)/((a^2*x^6 + b^2)*x^7), x)

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mupad [B]  time = 14.50, size = 274, normalized size = 1.11 \begin {gather*} \frac {a^2\,\ln \left (\frac {{\left (b+\sqrt {a^2\,x^3+b^2}\right )}^3\,\left (b-\sqrt {a^2\,x^3+b^2}\right )}{x^6}\right )\,\left (a^2+4\,b^2-2\,c\right )}{12\,b^3}-\frac {\sqrt {a^2\,x^3+b^2}\,\left (a^2+2\,c\right )}{6\,b^2\,x^3}-\frac {\sqrt {a^2\,x^3+b^2}}{3\,x^6}+\frac {a\,\ln \left (\frac {2\,b^2-a\,b\,1{}\mathrm {i}+a^2\,x^3+\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {-b+a\,1{}\mathrm {i}}\,2{}\mathrm {i}}{a\,x^3+b\,1{}\mathrm {i}}\right )\,\left (-a\,b+c\,1{}\mathrm {i}\right )\,\sqrt {-b+a\,1{}\mathrm {i}}\,1{}\mathrm {i}}{6\,b^{5/2}}+\frac {a\,\ln \left (\frac {a\,b\,1{}\mathrm {i}+2\,b^2+a^2\,x^3-2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {b+a\,1{}\mathrm {i}}}{-a\,x^3+b\,1{}\mathrm {i}}\right )\,\left (a\,b+c\,1{}\mathrm {i}\right )\,\sqrt {b+a\,1{}\mathrm {i}}}{6\,b^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((b^2 + a^2*x^3)^(1/2)*(c*x^3 + 2*b^2 + a^2*x^6))/(x^7*(b^2 + a^2*x^6)),x)

[Out]

(a^2*log(((b + (b^2 + a^2*x^3)^(1/2))^3*(b - (b^2 + a^2*x^3)^(1/2)))/x^6)*(a^2 - 2*c + 4*b^2))/(12*b^3) - ((b^
2 + a^2*x^3)^(1/2)*(2*c + a^2))/(6*b^2*x^3) - (b^2 + a^2*x^3)^(1/2)/(3*x^6) + (a*log((2*b^2 - a*b*1i + a^2*x^3
+ b^(1/2)*(b^2 + a^2*x^3)^(1/2)*(a*1i - b)^(1/2)*2i)/(b*1i + a*x^3))*(c*1i - a*b)*(a*1i - b)^(1/2)*1i)/(6*b^(
5/2)) + (a*log((a*b*1i + 2*b^2 + a^2*x^3 - 2*b^(1/2)*(b^2 + a^2*x^3)^(1/2)*(a*1i + b)^(1/2))/(b*1i - a*x^3))*(
c*1i + a*b)*(a*1i + b)^(1/2))/(6*b^(5/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a**2*x**3+b**2)**(1/2)*(a**2*x**6+c*x**3+2*b**2)/x**7/(a**2*x**6+b**2),x)

[Out]

Timed out

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