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

Optimal. Leaf size=227 \[ \frac {\left (\sqrt [4]{-1} c \sqrt {a-i b}-(-1)^{3/4} a b \sqrt {a-i b}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-i b}}\right )}{3 a \sqrt {b}}+\frac {\left (-\sqrt [4]{-1} c \sqrt {a+i b}-(-1)^{3/4} a b \sqrt {a+i b}\right ) \tanh ^{-1}\left (\frac {(1+i) \sqrt {a^2 x^3+b^2}}{\sqrt {2} \sqrt {b} \sqrt {a+i b}}\right )}{3 a \sqrt {b}}+\frac {2}{3} \sqrt {a^2 x^3+b^2}-\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right ) \]

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Rubi [B]  time = 3.16, antiderivative size = 609, normalized size of antiderivative = 2.68, number of steps used = 18, number of rules used = 13, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.260, Rules used = {6725, 266, 50, 63, 208, 6715, 825, 827, 1169, 634, 618, 206, 628} \begin {gather*} \frac {2}{3} \sqrt {a^2 x^3+b^2}-\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )-\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+b}}+\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {\sqrt {a^2+b^2}+b}}+\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}}-\frac {\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} \sqrt {b} \sqrt {a^2+b^2} \sqrt {b-\sqrt {a^2+b^2}}} \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*(b^2 + a^2*x^6)),x]

[Out]

(2*Sqrt[b^2 + a^2*x^3])/3 - (4*b*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])/3 + ((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]*Sqrt[b]*Sqrt[a^2 + b^2]*Sqrt[b - Sqrt[a^2 + b^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]*Sqrt[b]*Sqrt[a^2 + b^2]*Sqrt[b - Sqrt[a^2 + b^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*Sqr
t[2]*Sqrt[b]*Sqrt[a^2 + b^2]*Sqrt[b + Sqrt[a^2 + b^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]*Sqrt
[b]*Sqrt[a^2 + b^2]*Sqrt[b + Sqrt[a^2 + b^2]])

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

________________________________________________________________________________________

Mathematica [A]  time = 0.44, size = 267, normalized size = 1.18 \begin {gather*} \frac {-\sqrt {-a^2} c \sqrt {\sqrt {-a^2}-b} \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {\sqrt {-a^2}-b}}\right )+\sqrt {\sqrt {-a^2}+b} \left (a^2 b+\sqrt {-a^2} c\right ) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {\sqrt {-a^2}+b}}\right )+2 a^2 \sqrt {b} \sqrt {a^2 x^3+b^2}+a^2 b \sqrt {\sqrt {-a^2}-b} \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {\sqrt {-a^2}-b}}\right )-4 a^2 b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 a^2 \sqrt {b}} \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*(b^2 + a^2*x^6)),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.49, size = 223, normalized size = 0.98 \begin {gather*} \frac {2}{3} \sqrt {b^2+a^2 x^3}+\frac {\left (-(-1)^{3/4} a \sqrt {a-i b} b+\sqrt [4]{-1} \sqrt {a-i b} c\right ) \tan ^{-1}\left (\frac {\sqrt [4]{-1} \sqrt {b^2+a^2 x^3}}{\sqrt {a-i b} \sqrt {b}}\right )}{3 a \sqrt {b}}+\frac {\left (-\sqrt [4]{-1} a \sqrt {a+i b} b+(-1)^{3/4} \sqrt {a+i b} c\right ) \tan ^{-1}\left (\frac {(-1)^{3/4} \sqrt {b^2+a^2 x^3}}{\sqrt {a+i b} \sqrt {b}}\right )}{3 a \sqrt {b}}-\frac {4}{3} b \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{b}\right ) \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*(b^2 + a^2*x^6)),x]

[Out]

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

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fricas [B]  time = 11.45, size = 9711, normalized size = 42.78

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/(a^2*x^6+b^2),x, algorithm="fricas")

[Out]

1/12*(4*sqrt(2)*a^4*b^2*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2 - (a^4*b^4 - 2*a
^4*b^2*c - a^2*b^2*c^2)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^
6*b^4 + 4*a^4*b^4*c - 4*a^2*b^2*c^3 + a^2*c^4 - 2*(a^4*b^2 - 2*a^2*b^4)*c^2))*sqrt((a^4*b^4 + 4*a^2*b^4*c - 4*
b^2*c^3 + c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^2))*((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b
^4)*c^2)/(a^4*b^2))^(3/4)*arctan((sqrt(2)*sqrt(a^2*x^3 + b^2)*((a^10*b^8 + 2*a^8*b^8*c + 2*a^6*b^6*c^3 - a^6*b
^4*c^4)*sqrt((a^4*b^4 + 4*a^2*b^4*c - 4*b^2*c^3 + c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^2))*sqrt((a^6*b^4 + a^
4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)) + (a^10*b^10 - a^8*b^8*c^2 - 5*a^6*b^6*c^4 - 3
*a^4*b^4*c^6 + a^4*b^2*c^7 + (a^6*b^4 + 2*a^4*b^6)*c^5 - (a^8*b^6 - 4*a^6*b^8)*c^3 - (a^10*b^8 - 2*a^8*b^10)*c
)*sqrt((a^4*b^4 + 4*a^2*b^4*c - 4*b^2*c^3 + c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^2)))*sqrt((a^6*b^4 + a^4*b^6
 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2 - (a^4*b^4 - 2*a^4*b^2*c - a^2*b^2*c^2)*sqrt((a^6*b^4 + a^4*b^6
 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^6*b^4 + 4*a^4*b^4*c - 4*a^2*b^2*c^3 + a^2*c^4 -
 2*(a^4*b^2 - 2*a^2*b^4)*c^2))*((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2))^(
3/4) + sqrt(2)*(a^6*b^4*sqrt((a^4*b^4 + 4*a^2*b^4*c - 4*b^2*c^3 + c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^2))*sq
rt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)) + (a^6*b^6 - a^6*b^4*c + a^4*b
^4*c^2 - a^4*b^2*c^3)*sqrt((a^4*b^4 + 4*a^2*b^4*c - 4*b^2*c^3 + c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^2)))*sqr
t((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2 - (a^4*b^4 - 2*a^4*b^2*c - a^2*b^2*c^2)*sqr
t((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^6*b^4 + 4*a^4*b^4*c - 4*a^2
*b^2*c^3 + a^2*c^4 - 2*(a^4*b^2 - 2*a^2*b^4)*c^2))*sqrt((a^10*b^10 + a^8*b^12 + (a^2*b^2 + b^4)*c^8 - 4*(a^2*b
^4 + b^6)*c^7 + 4*(a^2*b^6 + b^8)*c^6 - 4*(a^4*b^6 + a^2*b^8)*c^5 - 2*(a^6*b^6 - 3*a^4*b^8 - 4*a^2*b^10)*c^4 +
 4*(a^6*b^8 + a^4*b^10)*c^3 + (a^12*b^8 + a^10*b^10 + (a^4 + a^2*b^2)*c^8 - 4*(a^4*b^2 + a^2*b^4)*c^7 + 4*(a^4
*b^4 + a^2*b^6)*c^6 - 4*(a^6*b^4 + a^4*b^6)*c^5 - 2*(a^8*b^4 - 3*a^6*b^6 - 4*a^4*b^8)*c^4 + 4*(a^8*b^6 + a^6*b
^8)*c^3 + 4*(a^8*b^8 + a^6*b^10)*c^2 + 4*(a^10*b^8 + a^8*b^10)*c)*x^3 + 4*(a^6*b^10 + a^4*b^12)*c^2 + sqrt(2)*
(a^10*b^8 + a^8*b^10 + (a^4*b^2 + a^2*b^4)*c^6 - 4*(a^4*b^4 + a^2*b^6)*c^5 - (a^6*b^4 - 3*a^4*b^6 - 4*a^2*b^8)
*c^4 - (a^8*b^6 - 3*a^6*b^8 - 4*a^4*b^10)*c^2 + 4*(a^8*b^8 + a^6*b^10)*c + (a^8*b^8 + 5*a^4*b^4*c^4 - a^4*b^2*
c^5 + 2*(a^6*b^4 - 4*a^4*b^6)*c^3 - 2*(3*a^6*b^6 - 2*a^4*b^8)*c^2 - (a^8*b^6 - 4*a^6*b^8)*c)*sqrt((a^6*b^4 + a
^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))*sqrt(a^2*x^3 + b^2)*sqrt((a^6*b^4 + a^4*b^6
+ (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2 - (a^4*b^4 - 2*a^4*b^2*c - a^2*b^2*c^2)*sqrt((a^6*b^4 + a^4*b^6
+ (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^6*b^4 + 4*a^4*b^4*c - 4*a^2*b^2*c^3 + a^2*c^4 -
2*(a^4*b^2 - 2*a^2*b^4)*c^2))*((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2))^(1
/4) + 4*(a^8*b^10 + a^6*b^12)*c + (a^10*b^8 + a^8*b^10 + (a^4*b^2 + a^2*b^4)*c^6 - 4*(a^4*b^4 + a^2*b^6)*c^5 -
 (a^6*b^4 - 3*a^4*b^6 - 4*a^2*b^8)*c^4 - (a^8*b^6 - 3*a^6*b^8 - 4*a^4*b^10)*c^2 + 4*(a^8*b^8 + a^6*b^10)*c)*sq
rt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^2 + b^2))*((a^6*b^4 + a^4*
b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2))^(3/4) + (a^12*b^10 + a^10*b^12 - (a^4*b^2 + a^2*
b^4)*c^8 + 2*(a^4*b^4 + a^2*b^6)*c^7 - 2*(a^6*b^4 + a^4*b^6)*c^6 + 6*(a^6*b^6 + a^4*b^8)*c^5 + 6*(a^8*b^8 + a^
6*b^10)*c^3 + 2*(a^10*b^8 + a^8*b^10)*c^2 + 2*(a^10*b^10 + a^8*b^12)*c)*sqrt((a^4*b^4 + 4*a^2*b^4*c - 4*b^2*c^
3 + c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^2))*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4
)*c^2)/(a^4*b^2)) + (a^12*b^12 + a^10*b^14 - (a^2*b^2 + b^4)*c^10 + 2*(a^2*b^4 + b^6)*c^9 - 3*(a^4*b^4 + a^2*b
^6)*c^8 + 8*(a^4*b^6 + a^2*b^8)*c^7 - 2*(a^6*b^6 + a^4*b^8)*c^6 + 12*(a^6*b^8 + a^4*b^10)*c^5 + 2*(a^8*b^8 + a
^6*b^10)*c^4 + 8*(a^8*b^10 + a^6*b^12)*c^3 + 3*(a^10*b^10 + a^8*b^12)*c^2 + 2*(a^10*b^12 + a^8*b^14)*c)*sqrt((
a^4*b^4 + 4*a^2*b^4*c - 4*b^2*c^3 + c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^2)))/(a^14*b^12 + a^12*b^14 + (a^2 +
 b^2)*c^12 - 4*(a^2*b^2 + b^4)*c^11 + 2*(a^4*b^2 + 3*a^2*b^4 + 2*b^6)*c^10 - 12*(a^4*b^4 + a^2*b^6)*c^9 - (a^6
*b^4 - 15*a^4*b^6 - 16*a^2*b^8)*c^8 - 8*(a^6*b^6 + a^4*b^8)*c^7 - 4*(a^8*b^6 - 5*a^6*b^8 - 6*a^4*b^10)*c^6 + 8
*(a^8*b^8 + a^6*b^10)*c^5 - (a^10*b^8 - 15*a^8*b^10 - 16*a^6*b^12)*c^4 + 12*(a^10*b^10 + a^8*b^12)*c^3 + 2*(a^
12*b^10 + 3*a^10*b^12 + 2*a^8*b^14)*c^2 + 4*(a^12*b^12 + a^10*b^14)*c)) + 4*sqrt(2)*a^4*b^2*sqrt((a^6*b^4 + a^
4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2 - (a^4*b^4 - 2*a^4*b^2*c - a^2*b^2*c^2)*sqrt((a^6*b^4 + a^
4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^6*b^4 + 4*a^4*b^4*c - 4*a^2*b^2*c^3 + a^2*
c^4 - 2*(a^4*b^2 - 2*a^2*b^4)*c^2))*sqrt((a^4*b^4 + 4*a^2*b^4*c - 4*b^2*c^3 + c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(
a^2*b^2))*((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2))^(3/4)*arctan((sqrt(2)*
sqrt(a^2*x^3 + b^2)*((a^10*b^8 + 2*a^8*b^8*c + 2*a^6*b^6*c^3 - a^6*b^4*c^4)*sqrt((a^4*b^4 + 4*a^2*b^4*c - 4*b^
2*c^3 + c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^2))*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2
*b^4)*c^2)/(a^4*b^2)) + (a^10*b^10 - a^8*b^8*c^2 - 5*a^6*b^6*c^4 - 3*a^4*b^4*c^6 + a^4*b^2*c^7 + (a^6*b^4 + 2*
a^4*b^6)*c^5 - (a^8*b^6 - 4*a^6*b^8)*c^3 - (a^10*b^8 - 2*a^8*b^10)*c)*sqrt((a^4*b^4 + 4*a^2*b^4*c - 4*b^2*c^3
+ c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^2)))*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)
*c^2 - (a^4*b^4 - 2*a^4*b^2*c - a^2*b^2*c^2)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)
*c^2)/(a^4*b^2)))/(a^6*b^4 + 4*a^4*b^4*c - 4*a^2*b^2*c^3 + a^2*c^4 - 2*(a^4*b^2 - 2*a^2*b^4)*c^2))*((a^6*b^4 +
 a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2))^(3/4) + sqrt(2)*(a^6*b^4*sqrt((a^4*b^4 + 4*
a^2*b^4*c - 4*b^2*c^3 + c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^2))*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 +
2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)) + (a^6*b^6 - a^6*b^4*c + a^4*b^4*c^2 - a^4*b^2*c^3)*sqrt((a^4*b^4 + 4*a^
2*b^4*c - 4*b^2*c^3 + c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^2)))*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2
*(a^4*b^2 + a^2*b^4)*c^2 - (a^4*b^4 - 2*a^4*b^2*c - a^2*b^2*c^2)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2
*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^6*b^4 + 4*a^4*b^4*c - 4*a^2*b^2*c^3 + a^2*c^4 - 2*(a^4*b^2 - 2*a^2*b^
4)*c^2))*sqrt((a^10*b^10 + a^8*b^12 + (a^2*b^2 + b^4)*c^8 - 4*(a^2*b^4 + b^6)*c^7 + 4*(a^2*b^6 + b^8)*c^6 - 4*
(a^4*b^6 + a^2*b^8)*c^5 - 2*(a^6*b^6 - 3*a^4*b^8 - 4*a^2*b^10)*c^4 + 4*(a^6*b^8 + a^4*b^10)*c^3 + (a^12*b^8 +
a^10*b^10 + (a^4 + a^2*b^2)*c^8 - 4*(a^4*b^2 + a^2*b^4)*c^7 + 4*(a^4*b^4 + a^2*b^6)*c^6 - 4*(a^6*b^4 + a^4*b^6
)*c^5 - 2*(a^8*b^4 - 3*a^6*b^6 - 4*a^4*b^8)*c^4 + 4*(a^8*b^6 + a^6*b^8)*c^3 + 4*(a^8*b^8 + a^6*b^10)*c^2 + 4*(
a^10*b^8 + a^8*b^10)*c)*x^3 + 4*(a^6*b^10 + a^4*b^12)*c^2 - sqrt(2)*(a^10*b^8 + a^8*b^10 + (a^4*b^2 + a^2*b^4)
*c^6 - 4*(a^4*b^4 + a^2*b^6)*c^5 - (a^6*b^4 - 3*a^4*b^6 - 4*a^2*b^8)*c^4 - (a^8*b^6 - 3*a^6*b^8 - 4*a^4*b^10)*
c^2 + 4*(a^8*b^8 + a^6*b^10)*c + (a^8*b^8 + 5*a^4*b^4*c^4 - a^4*b^2*c^5 + 2*(a^6*b^4 - 4*a^4*b^6)*c^3 - 2*(3*a
^6*b^6 - 2*a^4*b^8)*c^2 - (a^8*b^6 - 4*a^6*b^8)*c)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^
2*b^4)*c^2)/(a^4*b^2)))*sqrt(a^2*x^3 + b^2)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*
c^2 - (a^4*b^4 - 2*a^4*b^2*c - a^2*b^2*c^2)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*
c^2)/(a^4*b^2)))/(a^6*b^4 + 4*a^4*b^4*c - 4*a^2*b^2*c^3 + a^2*c^4 - 2*(a^4*b^2 - 2*a^2*b^4)*c^2))*((a^6*b^4 +
a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2))^(1/4) + 4*(a^8*b^10 + a^6*b^12)*c + (a^10*b^
8 + a^8*b^10 + (a^4*b^2 + a^2*b^4)*c^6 - 4*(a^4*b^4 + a^2*b^6)*c^5 - (a^6*b^4 - 3*a^4*b^6 - 4*a^2*b^8)*c^4 - (
a^8*b^6 - 3*a^6*b^8 - 4*a^4*b^10)*c^2 + 4*(a^8*b^8 + a^6*b^10)*c)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 +
2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^2 + b^2))*((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b
^4)*c^2)/(a^4*b^2))^(3/4) - (a^12*b^10 + a^10*b^12 - (a^4*b^2 + a^2*b^4)*c^8 + 2*(a^4*b^4 + a^2*b^6)*c^7 - 2*(
a^6*b^4 + a^4*b^6)*c^6 + 6*(a^6*b^6 + a^4*b^8)*c^5 + 6*(a^8*b^8 + a^6*b^10)*c^3 + 2*(a^10*b^8 + a^8*b^10)*c^2
+ 2*(a^10*b^10 + a^8*b^12)*c)*sqrt((a^4*b^4 + 4*a^2*b^4*c - 4*b^2*c^3 + c^4 - 2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^
2))*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)) - (a^12*b^12 + a^10*b^14
 - (a^2*b^2 + b^4)*c^10 + 2*(a^2*b^4 + b^6)*c^9 - 3*(a^4*b^4 + a^2*b^6)*c^8 + 8*(a^4*b^6 + a^2*b^8)*c^7 - 2*(a
^6*b^6 + a^4*b^8)*c^6 + 12*(a^6*b^8 + a^4*b^10)*c^5 + 2*(a^8*b^8 + a^6*b^10)*c^4 + 8*(a^8*b^10 + a^6*b^12)*c^3
 + 3*(a^10*b^10 + a^8*b^12)*c^2 + 2*(a^10*b^12 + a^8*b^14)*c)*sqrt((a^4*b^4 + 4*a^2*b^4*c - 4*b^2*c^3 + c^4 -
2*(a^2*b^2 - 2*b^4)*c^2)/(a^2*b^2)))/(a^14*b^12 + a^12*b^14 + (a^2 + b^2)*c^12 - 4*(a^2*b^2 + b^4)*c^11 + 2*(a
^4*b^2 + 3*a^2*b^4 + 2*b^6)*c^10 - 12*(a^4*b^4 + a^2*b^6)*c^9 - (a^6*b^4 - 15*a^4*b^6 - 16*a^2*b^8)*c^8 - 8*(a
^6*b^6 + a^4*b^8)*c^7 - 4*(a^8*b^6 - 5*a^6*b^8 - 6*a^4*b^10)*c^6 + 8*(a^8*b^8 + a^6*b^10)*c^5 - (a^10*b^8 - 15
*a^8*b^10 - 16*a^6*b^12)*c^4 + 12*(a^10*b^10 + a^8*b^12)*c^3 + 2*(a^12*b^10 + 3*a^10*b^12 + 2*a^8*b^14)*c^2 +
4*(a^12*b^12 + a^10*b^14)*c)) + sqrt(2)*(a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2 + (a^
4*b^4 - 2*a^4*b^2*c - a^2*b^2*c^2)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4
*b^2)))*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2 - (a^4*b^4 - 2*a^4*b^2*c - a^2*b
^2*c^2)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^6*b^4 + 4*a^4*b^
4*c - 4*a^2*b^2*c^3 + a^2*c^4 - 2*(a^4*b^2 - 2*a^2*b^4)*c^2))*((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b
^2 + a^2*b^4)*c^2)/(a^4*b^2))^(1/4)*log((a^10*b^10 + a^8*b^12 + (a^2*b^2 + b^4)*c^8 - 4*(a^2*b^4 + b^6)*c^7 +
4*(a^2*b^6 + b^8)*c^6 - 4*(a^4*b^6 + a^2*b^8)*c^5 - 2*(a^6*b^6 - 3*a^4*b^8 - 4*a^2*b^10)*c^4 + 4*(a^6*b^8 + a^
4*b^10)*c^3 + (a^12*b^8 + a^10*b^10 + (a^4 + a^2*b^2)*c^8 - 4*(a^4*b^2 + a^2*b^4)*c^7 + 4*(a^4*b^4 + a^2*b^6)*
c^6 - 4*(a^6*b^4 + a^4*b^6)*c^5 - 2*(a^8*b^4 - 3*a^6*b^6 - 4*a^4*b^8)*c^4 + 4*(a^8*b^6 + a^6*b^8)*c^3 + 4*(a^8
*b^8 + a^6*b^10)*c^2 + 4*(a^10*b^8 + a^8*b^10)*c)*x^3 + 4*(a^6*b^10 + a^4*b^12)*c^2 + sqrt(2)*(a^10*b^8 + a^8*
b^10 + (a^4*b^2 + a^2*b^4)*c^6 - 4*(a^4*b^4 + a^2*b^6)*c^5 - (a^6*b^4 - 3*a^4*b^6 - 4*a^2*b^8)*c^4 - (a^8*b^6
- 3*a^6*b^8 - 4*a^4*b^10)*c^2 + 4*(a^8*b^8 + a^6*b^10)*c + (a^8*b^8 + 5*a^4*b^4*c^4 - a^4*b^2*c^5 + 2*(a^6*b^4
 - 4*a^4*b^6)*c^3 - 2*(3*a^6*b^6 - 2*a^4*b^8)*c^2 - (a^8*b^6 - 4*a^6*b^8)*c)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 +
b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))*sqrt(a^2*x^3 + b^2)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^
4 + 2*(a^4*b^2 + a^2*b^4)*c^2 - (a^4*b^4 - 2*a^4*b^2*c - a^2*b^2*c^2)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^
4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^6*b^4 + 4*a^4*b^4*c - 4*a^2*b^2*c^3 + a^2*c^4 - 2*(a^4*b^2 - 2*a
^2*b^4)*c^2))*((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2))^(1/4) + 4*(a^8*b^1
0 + a^6*b^12)*c + (a^10*b^8 + a^8*b^10 + (a^4*b^2 + a^2*b^4)*c^6 - 4*(a^4*b^4 + a^2*b^6)*c^5 - (a^6*b^4 - 3*a^
4*b^6 - 4*a^2*b^8)*c^4 - (a^8*b^6 - 3*a^6*b^8 - 4*a^4*b^10)*c^2 + 4*(a^8*b^8 + a^6*b^10)*c)*sqrt((a^6*b^4 + a^
4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^2 + b^2)) - sqrt(2)*(a^6*b^4 + a^4*b^6 + (
a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2 + (a^4*b^4 - 2*a^4*b^2*c - a^2*b^2*c^2)*sqrt((a^6*b^4 + a^4*b^6 + (
a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2
 + a^2*b^4)*c^2 - (a^4*b^4 - 2*a^4*b^2*c - a^2*b^2*c^2)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2
 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^6*b^4 + 4*a^4*b^4*c - 4*a^2*b^2*c^3 + a^2*c^4 - 2*(a^4*b^2 - 2*a^2*b^4)*c^2))*
((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2))^(1/4)*log((a^10*b^10 + a^8*b^12
+ (a^2*b^2 + b^4)*c^8 - 4*(a^2*b^4 + b^6)*c^7 + 4*(a^2*b^6 + b^8)*c^6 - 4*(a^4*b^6 + a^2*b^8)*c^5 - 2*(a^6*b^6
 - 3*a^4*b^8 - 4*a^2*b^10)*c^4 + 4*(a^6*b^8 + a^4*b^10)*c^3 + (a^12*b^8 + a^10*b^10 + (a^4 + a^2*b^2)*c^8 - 4*
(a^4*b^2 + a^2*b^4)*c^7 + 4*(a^4*b^4 + a^2*b^6)*c^6 - 4*(a^6*b^4 + a^4*b^6)*c^5 - 2*(a^8*b^4 - 3*a^6*b^6 - 4*a
^4*b^8)*c^4 + 4*(a^8*b^6 + a^6*b^8)*c^3 + 4*(a^8*b^8 + a^6*b^10)*c^2 + 4*(a^10*b^8 + a^8*b^10)*c)*x^3 + 4*(a^6
*b^10 + a^4*b^12)*c^2 - sqrt(2)*(a^10*b^8 + a^8*b^10 + (a^4*b^2 + a^2*b^4)*c^6 - 4*(a^4*b^4 + a^2*b^6)*c^5 - (
a^6*b^4 - 3*a^4*b^6 - 4*a^2*b^8)*c^4 - (a^8*b^6 - 3*a^6*b^8 - 4*a^4*b^10)*c^2 + 4*(a^8*b^8 + a^6*b^10)*c + (a^
8*b^8 + 5*a^4*b^4*c^4 - a^4*b^2*c^5 + 2*(a^6*b^4 - 4*a^4*b^6)*c^3 - 2*(3*a^6*b^6 - 2*a^4*b^8)*c^2 - (a^8*b^6 -
 4*a^6*b^8)*c)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))*sqrt(a^2*x^3
 + b^2)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2 - (a^4*b^4 - 2*a^4*b^2*c - a^2*b
^2*c^2)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)))/(a^6*b^4 + 4*a^4*b^
4*c - 4*a^2*b^2*c^3 + a^2*c^4 - 2*(a^4*b^2 - 2*a^2*b^4)*c^2))*((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b
^2 + a^2*b^4)*c^2)/(a^4*b^2))^(1/4) + 4*(a^8*b^10 + a^6*b^12)*c + (a^10*b^8 + a^8*b^10 + (a^4*b^2 + a^2*b^4)*c
^6 - 4*(a^4*b^4 + a^2*b^6)*c^5 - (a^6*b^4 - 3*a^4*b^6 - 4*a^2*b^8)*c^4 - (a^8*b^6 - 3*a^6*b^8 - 4*a^4*b^10)*c^
2 + 4*(a^8*b^8 + a^6*b^10)*c)*sqrt((a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)/(a^4*b^2)
))/(a^2 + b^2)) - 8*(a^6*b^5 + a^4*b^7 + (a^2*b + b^3)*c^4 + 2*(a^4*b^3 + a^2*b^5)*c^2)*log(b + sqrt(a^2*x^3 +
 b^2)) + 8*(a^6*b^5 + a^4*b^7 + (a^2*b + b^3)*c^4 + 2*(a^4*b^3 + a^2*b^5)*c^2)*log(-b + sqrt(a^2*x^3 + b^2)) +
 8*(a^6*b^4 + a^4*b^6 + (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)*sqrt(a^2*x^3 + b^2))/(a^6*b^4 + a^4*b^6 +
 (a^2 + b^2)*c^4 + 2*(a^4*b^2 + a^2*b^4)*c^2)

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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/(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)]=[-25,89]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)]
=[11,12]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)]=[-33,-91]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 c
hoice was done assuming [abs(b)]=[-45,-7]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Err
or: Bad Argument ValueWarning, need to choose a branch for the root of a polynomial with parameters. This migh
t be wrong.The choice was done assuming [abs(b)]=[-14,70]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 param
eters. This might be wrong.The choice was done assuming [abs(b)]=[84,29]sym2poly/r2sym(const gen & e,const ind
ex_m & i,const vecteur & l) Error: Bad Argument ValueWarning, need to choose a branch for the root of a polyno
mial with parameters. This might be wrong.The choice was done assuming [abs(b)]=[-27,-21]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)]=[-43,4]sym2po
ly/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)
]=[-60,90]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)]=[-10,65]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argumen
t 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)]=[-37,53]sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Er
ror: Bad Argument Valuesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument V
aluesym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(
const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Valuesym2poly/r2sym(const gen & e,const
 index_m & i,const vecteur & l) Error: Bad Argument ValueEvaluation time: 6.42Done

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maple [C]  time = 0.39, size = 703, normalized size = 3.10

method result size
default \(\frac {2 \sqrt {a^{2} x^{3}+b^{2}}}{3}-\frac {4 b^{2} \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}-\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a^{2} \textit {\_Z}^{6}+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}}\) \(703\)
elliptic \(\text {Expression too large to display}\) \(12449\)

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/(a^2*x^6+b^2),x,method=_RETURNVERBOSE)

[Out]

2/3*(a^2*x^3+b^2)^(1/2)-4/3*b^2*arctanh((a^2*x^3+b^2)^(1/2)/(b^2)^(1/2))/(b^2)^(1/2)-1/6*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)*_alpha*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)*_alpha^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}\,{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/(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), x)

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mupad [B]  time = 13.68, size = 246, normalized size = 1.08 \begin {gather*} \frac {2\,\sqrt {a^2\,x^3+b^2}}{3}+\frac {2\,b\,\ln \left (\frac {\left (b+\sqrt {a^2\,x^3+b^2}\right )\,{\left (b-\sqrt {a^2\,x^3+b^2}\right )}^3}{x^6}\right )}{3}+\frac {\sqrt {\frac {1}{36}{}\mathrm {i}}\,\ln \left (\frac {2\,{\left (-1\right )}^{1/4}\,b^2+{\left (-1\right )}^{1/4}\,a^2\,x^3-2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {a+b\,1{}\mathrm {i}}-{\left (-1\right )}^{3/4}\,a\,b}{a\,x^3+b\,1{}\mathrm {i}}\right )\,\sqrt {a+b\,1{}\mathrm {i}}\,\left (c+a\,b\,1{}\mathrm {i}\right )}{a\,\sqrt {b}}+\frac {\sqrt {\frac {1}{36}{}\mathrm {i}}\,\ln \left (\frac {2\,{\left (-1\right )}^{1/4}\,b^2+{\left (-1\right )}^{1/4}\,a^2\,x^3+{\left (-1\right )}^{3/4}\,a\,b+2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {-a+b\,1{}\mathrm {i}}}{-a\,x^3+b\,1{}\mathrm {i}}\right )\,\sqrt {-a+b\,1{}\mathrm {i}}\,\left (c-a\,b\,1{}\mathrm {i}\right )}{a\,\sqrt {b}} \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*(b^2 + a^2*x^6)),x)

[Out]

(2*(b^2 + a^2*x^3)^(1/2))/3 + (2*b*log(((b + (b^2 + a^2*x^3)^(1/2))*(b - (b^2 + a^2*x^3)^(1/2))^3)/x^6))/3 + (
(1i/36)^(1/2)*log((2*(-1)^(1/4)*b^2 + (-1)^(1/4)*a^2*x^3 - 2*b^(1/2)*(b^2 + a^2*x^3)^(1/2)*(a + b*1i)^(1/2) -
(-1)^(3/4)*a*b)/(b*1i + a*x^3))*(a + b*1i)^(1/2)*(c + a*b*1i))/(a*b^(1/2)) + ((1i/36)^(1/2)*log((2*(-1)^(1/4)*
b^2 + (-1)^(1/4)*a^2*x^3 + (-1)^(3/4)*a*b + 2*b^(1/2)*(b^2 + a^2*x^3)^(1/2)*(b*1i - a)^(1/2))/(b*1i - a*x^3))*
(b*1i - a)^(1/2)*(c - a*b*1i))/(a*b^(1/2))

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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/(a**2*x**6+b**2),x)

[Out]

Timed out

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