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

Optimal. Leaf size=207 \[ \frac {\sqrt {a^2 x^3+b^2} \left (a^2 x^3+2 b^2+2 c x^3\right )}{6 b^2 x^6}-\frac {\sqrt {b-a} \left (3 a^2 b-a c\right ) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {b-a}}\right )}{3 b^{5/2}}-\frac {\sqrt {a+b} \left (3 a^2 b+a c\right ) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )}{3 b^{5/2}}+\frac {\left (-a^4+12 a^2 b^2+2 a^2 c\right ) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{6 b^3} \]

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Rubi [A]  time = 0.91, antiderivative size = 359, normalized size of antiderivative = 1.73, number of steps used = 23, number of rules used = 9, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.173, Rules used = {6725, 266, 47, 51, 63, 208, 50, 444, 205} \begin {gather*} \frac {c \sqrt {a^2 x^3+b^2}}{3 b^2 x^3}-\frac {2 a^2 \sqrt {a^2 x^3+b^2}}{b^2}+\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 )}{b}+\frac {\sqrt {a^2 x^3+b^2}}{3 x^6}-\frac {a \sqrt {a-b} (3 a b-c) \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-b}}\right )}{3 b^{5/2}}-\frac {a \sqrt {a+b} (3 a b+c) \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )}{3 b^{5/2}}+\frac {a (3 a b-c) \sqrt {a^2 x^3+b^2}}{3 b^3}+\frac {a (3 a b+c) \sqrt {a^2 x^3+b^2}}{3 b^3}+\frac {a^2 c \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )}{3 b^3}-\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]

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

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 205

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

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 444

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[m
- n + 1, 0]

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

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Mathematica [C]  time = 1.09, size = 321, normalized size = 1.55 \begin {gather*} \frac {\frac {3 b^4 c \left (a^2 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 x^3+b^2\right )}{x^3 \sqrt {a^2 x^3+b^2}}+18 a^2 b^4 \left (b \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{b}\right )-\sqrt {a^2 x^3+b^2}\right )-3 a b^3 (3 a b-c) \left (\sqrt {b} \sqrt {a-b} \tan ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a-b}}\right )-\sqrt {a^2 x^3+b^2}\right )-3 a b^3 (3 a b+c) \left (\sqrt {b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {a^2 x^3+b^2}}{\sqrt {b} \sqrt {a+b}}\right )-\sqrt {a^2 x^3+b^2}\right )+4 a^4 \left (a^2 x^3+b^2\right )^{3/2} \, _2F_1\left (\frac {3}{2},3;\frac {5}{2};\frac {a^2 x^3}{b^2}+1\right )}{9 b^6} \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*a*b^3*(3*a*b - c)*(-Sqrt[b^2 + a^2*x^3] + Sqrt[a - b]*Sqrt[b]*ArcTan[Sqrt[b^2 + a^2*x^3]/(Sqrt[a - b]*Sqrt
[b])]) + 18*a^2*b^4*(-Sqrt[b^2 + a^2*x^3] + b*ArcTanh[Sqrt[b^2 + a^2*x^3]/b]) - 3*a*b^3*(3*a*b + c)*(-Sqrt[b^2
+ a^2*x^3] + Sqrt[b]*Sqrt[a + b]*ArcTanh[Sqrt[b^2 + a^2*x^3]/(Sqrt[b]*Sqrt[a + b])]) + (3*b^4*c*(b^2 + a^2*x^
3 + a^2*x^3*Sqrt[1 + (a^2*x^3)/b^2]*ArcTanh[Sqrt[1 + (a^2*x^3)/b^2]]))/(x^3*Sqrt[b^2 + a^2*x^3]) + 4*a^4*(b^2
+ a^2*x^3)^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, 1 + (a^2*x^3)/b^2])/(9*b^6)

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IntegrateAlgebraic [A]  time = 0.63, size = 207, 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-b} \left (3 a^2 b-a c\right ) \tan ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {a-b} \sqrt {b}}\right )}{3 b^{5/2}}+\frac {\left (-a^4+12 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}-\frac {\sqrt {a+b} \left (3 a^2 b+a c\right ) \tanh ^{-1}\left (\frac {\sqrt {b^2+a^2 x^3}}{\sqrt {b} \sqrt {a+b}}\right )}{3 b^{5/2}} \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 - b]*(3*a^2*b - a*c)*ArcTan[Sqrt[b^2 +
a^2*x^3]/(Sqrt[a - b]*Sqrt[b])])/(3*b^(5/2)) + ((-a^4 + 12*a^2*b^2 + 2*a^2*c)*ArcTanh[Sqrt[b^2 + a^2*x^3]/b])
/(6*b^3) - (Sqrt[a + b]*(3*a^2*b + a*c)*ArcTanh[Sqrt[b^2 + a^2*x^3]/(Sqrt[b]*Sqrt[a + b])])/(3*b^(5/2))

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fricas [A]  time = 1.41, size = 1051, normalized size = 5.08 \begin {gather*} \left [-\frac {2 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \sqrt {-\frac {a - b}{b}} \log \left (\frac {a^{2} x^{3} - a b + 2 \, b^{2} + 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {-\frac {a - b}{b}}}{a x^{3} + b}\right ) - 2 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \sqrt {\frac {a + b}{b}} \log \left (\frac {a^{2} x^{3} + a b + 2 \, b^{2} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {\frac {a + b}{b}}}{a x^{3} - b}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) - 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}, \frac {4 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + 2 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \sqrt {\frac {a + b}{b}} \log \left (\frac {a^{2} x^{3} + a b + 2 \, b^{2} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {\frac {a + b}{b}}}{a x^{3} - b}\right ) - {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}, \frac {4 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) - 2 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \sqrt {-\frac {a - b}{b}} \log \left (\frac {a^{2} x^{3} - a b + 2 \, b^{2} + 2 \, \sqrt {a^{2} x^{3} + b^{2}} b \sqrt {-\frac {a - b}{b}}}{a x^{3} + b}\right ) - {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}, \frac {4 \, {\left (3 \, a^{2} b^{2} + a b c\right )} x^{6} \sqrt {-\frac {a + b}{b}} \arctan \left (\frac {b \sqrt {-\frac {a + b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) + 4 \, {\left (3 \, a^{2} b^{2} - a b c\right )} x^{6} \sqrt {\frac {a - b}{b}} \arctan \left (\frac {b \sqrt {\frac {a - b}{b}}}{\sqrt {a^{2} x^{3} + b^{2}}}\right ) - {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + {\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} x^{6} \log \left (-b + \sqrt {a^{2} x^{3} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{3} + b^{2}} {\left ({\left (a^{2} b + 2 \, b c\right )} x^{3} + 2 \, b^{3}\right )}}{12 \, b^{3} x^{6}}\right ] \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="fricas")

[Out]

[-1/12*(2*(3*a^2*b^2 - a*b*c)*x^6*sqrt(-(a - b)/b)*log((a^2*x^3 - a*b + 2*b^2 + 2*sqrt(a^2*x^3 + b^2)*b*sqrt(-
(a - b)/b))/(a*x^3 + b)) - 2*(3*a^2*b^2 + a*b*c)*x^6*sqrt((a + b)/b)*log((a^2*x^3 + a*b + 2*b^2 - 2*sqrt(a^2*x
^3 + b^2)*b*sqrt((a + b)/b))/(a*x^3 - b)) + (a^4 - 12*a^2*b^2 - 2*a^2*c)*x^6*log(b + sqrt(a^2*x^3 + b^2)) - (a
^4 - 12*a^2*b^2 - 2*a^2*c)*x^6*log(-b + sqrt(a^2*x^3 + b^2)) - 2*sqrt(a^2*x^3 + b^2)*((a^2*b + 2*b*c)*x^3 + 2*
b^3))/(b^3*x^6), 1/12*(4*(3*a^2*b^2 - a*b*c)*x^6*sqrt((a - b)/b)*arctan(b*sqrt((a - b)/b)/sqrt(a^2*x^3 + b^2))
+ 2*(3*a^2*b^2 + a*b*c)*x^6*sqrt((a + b)/b)*log((a^2*x^3 + a*b + 2*b^2 - 2*sqrt(a^2*x^3 + b^2)*b*sqrt((a + b)
/b))/(a*x^3 - b)) - (a^4 - 12*a^2*b^2 - 2*a^2*c)*x^6*log(b + sqrt(a^2*x^3 + b^2)) + (a^4 - 12*a^2*b^2 - 2*a^2*
c)*x^6*log(-b + sqrt(a^2*x^3 + b^2)) + 2*sqrt(a^2*x^3 + b^2)*((a^2*b + 2*b*c)*x^3 + 2*b^3))/(b^3*x^6), 1/12*(4
*(3*a^2*b^2 + a*b*c)*x^6*sqrt(-(a + b)/b)*arctan(b*sqrt(-(a + b)/b)/sqrt(a^2*x^3 + b^2)) - 2*(3*a^2*b^2 - a*b*
c)*x^6*sqrt(-(a - b)/b)*log((a^2*x^3 - a*b + 2*b^2 + 2*sqrt(a^2*x^3 + b^2)*b*sqrt(-(a - b)/b))/(a*x^3 + b)) -
(a^4 - 12*a^2*b^2 - 2*a^2*c)*x^6*log(b + sqrt(a^2*x^3 + b^2)) + (a^4 - 12*a^2*b^2 - 2*a^2*c)*x^6*log(-b + sqrt
(a^2*x^3 + b^2)) + 2*sqrt(a^2*x^3 + b^2)*((a^2*b + 2*b*c)*x^3 + 2*b^3))/(b^3*x^6), 1/12*(4*(3*a^2*b^2 + a*b*c)
*x^6*sqrt(-(a + b)/b)*arctan(b*sqrt(-(a + b)/b)/sqrt(a^2*x^3 + b^2)) + 4*(3*a^2*b^2 - a*b*c)*x^6*sqrt((a - b)/
b)*arctan(b*sqrt((a - b)/b)/sqrt(a^2*x^3 + b^2)) - (a^4 - 12*a^2*b^2 - 2*a^2*c)*x^6*log(b + sqrt(a^2*x^3 + b^2
)) + (a^4 - 12*a^2*b^2 - 2*a^2*c)*x^6*log(-b + sqrt(a^2*x^3 + b^2)) + 2*sqrt(a^2*x^3 + b^2)*((a^2*b + 2*b*c)*x
^3 + 2*b^3))/(b^3*x^6)]

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giac [A]  time = 0.19, size = 308, normalized size = 1.49 \begin {gather*} -\frac {{\left (3 \, a^{3} b - 3 \, a^{2} b^{2} - a^{2} c + a b c\right )} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {a b - b^{2}}}\right )}{3 \, \sqrt {a b - b^{2}} b^{2}} + \frac {{\left (3 \, a^{3} b + 3 \, a^{2} b^{2} + a^{2} c + a b c\right )} \arctan \left (\frac {\sqrt {a^{2} x^{3} + b^{2}}}{\sqrt {-a b - b^{2}}}\right )}{3 \, \sqrt {-a b - b^{2}} b^{2}} - \frac {{\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} \log \left ({\left | b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right )}{12 \, b^{3}} + \frac {{\left (a^{4} - 12 \, a^{2} b^{2} - 2 \, a^{2} c\right )} \log \left ({\left | -b + \sqrt {a^{2} x^{3} + b^{2}} \right |}\right )}{12 \, b^{3}} + \frac {\sqrt {a^{2} x^{3} + b^{2}} a^{4} b^{2} + {\left (a^{2} x^{3} + b^{2}\right )}^{\frac {3}{2}} a^{4} - 2 \, \sqrt {a^{2} x^{3} + b^{2}} a^{2} b^{2} c + 2 \, {\left (a^{2} x^{3} + b^{2}\right )}^{\frac {3}{2}} a^{2} c}{6 \, a^{4} b^{2} x^{6}} \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]

-1/3*(3*a^3*b - 3*a^2*b^2 - a^2*c + a*b*c)*arctan(sqrt(a^2*x^3 + b^2)/sqrt(a*b - b^2))/(sqrt(a*b - b^2)*b^2) +
1/3*(3*a^3*b + 3*a^2*b^2 + a^2*c + a*b*c)*arctan(sqrt(a^2*x^3 + b^2)/sqrt(-a*b - b^2))/(sqrt(-a*b - b^2)*b^2)
- 1/12*(a^4 - 12*a^2*b^2 - 2*a^2*c)*log(abs(b + sqrt(a^2*x^3 + b^2)))/b^3 + 1/12*(a^4 - 12*a^2*b^2 - 2*a^2*c)
*log(abs(-b + sqrt(a^2*x^3 + b^2)))/b^3 + 1/6*(sqrt(a^2*x^3 + b^2)*a^4*b^2 + (a^2*x^3 + b^2)^(3/2)*a^4 - 2*sqr
t(a^2*x^3 + b^2)*a^2*b^2*c + 2*(a^2*x^3 + b^2)^(3/2)*a^2*c)/(a^4*b^2*x^6)

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maple [C]  time = 0.71, size = 984, normalized size = 4.75

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 {i \left (6 a^{2} b +6 a \,b^{2}+2 a c +2 b c \right ) \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\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 (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, a -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a -\left (-a \,b^{2}\right )^{\frac {2}{3}}\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 ^{2} a -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, b^{2}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 b^{2}}{2 b \left (a +b \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 \left (a +b \right ) \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{2} b}-\frac {2 \left (a^{2}-12 b^{2}-2 c \right ) \arctanh \left (\frac {\sqrt {a^{2} x^{3}+b^{2}}}{\sqrt {b^{2}}}\right )}{3 \sqrt {b^{2}}}-\frac {i \left (-6 a^{2} b +6 a \,b^{2}+2 a c -2 b c \right ) \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\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 (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, a -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a -\left (-a \,b^{2}\right )^{\frac {2}{3}}\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 ^{2} a -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, b^{2}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 b^{2}}{2 b \left (a -b \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 \left (a -b \right ) \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{2} b}\right )}{4 b^{2}}\)	\(984\)
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 (3 a b +c \right ) \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}}{3 a}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}-b \right )}{\sum }\frac {\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 (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, a -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a -\left (-a \,b^{2}\right )^{\frac {2}{3}}\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 ^{2} a -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, b^{2}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 b^{2}}{2 b \left (a +b \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 \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{2}}\right )}{2 b^{3}}+\frac {a^{2} \left (3 a b -c \right ) \left (\frac {2 \sqrt {a^{2} x^{3}+b^{2}}}{3 a}+\frac {i \sqrt {2}\, \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\RootOf \left (a \,\textit {\_Z}^{3}+b \right )}{\sum }\frac {\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 (i \left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha \sqrt {3}\, a -i \sqrt {3}\, \left (-a \,b^{2}\right )^{\frac {2}{3}}+2 \underline {\hspace {1.25 ex}}\alpha ^{2} a^{2}-\left (-a \,b^{2}\right )^{\frac {1}{3}} \underline {\hspace {1.25 ex}}\alpha a -\left (-a \,b^{2}\right )^{\frac {2}{3}}\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 ^{2} a -i \left (-a \,b^{2}\right )^{\frac {2}{3}} \sqrt {3}\, \underline {\hspace {1.25 ex}}\alpha +i \sqrt {3}\, b^{2}-3 \left (-a \,b^{2}\right )^{\frac {2}{3}} \underline {\hspace {1.25 ex}}\alpha -3 b^{2}}{2 b \left (a -b \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 \sqrt {a^{2} x^{3}+b^{2}}}\right )}{3 a^{2}}\right )}{2 b^{3}}-\frac {3 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}}}\)	\(1088\)
elliptic	\(\text {Expression too large to display}\)	\(3271\)

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*(1/3*I*(6*a^2*b+6*a*b^2+2*a*c+2*b*c)/a^2/b
*2^(1/2)*sum(1/(a+b)*(-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)*(I*(-a*b^2)^(1/3)*_alpha*3^(
1/2)*a-I*3^(1/2)*(-a*b^2)^(2/3)+2*_alpha^2*a^2-(-a*b^2)^(1/3)*_alpha*a-(-a*b^2)^(2/3))*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^2*a-I*(-a*b^2)^(2/3)*3^(1/2)*_alpha+I*3^(1/2)*b^2-3*(-a*b^2)^(2/3)*_alpha-3*b^2)/b/(a+b),
(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^3*
a-b))-2/3*(a^2-12*b^2-2*c)*arctanh((a^2*x^3+b^2)^(1/2)/(b^2)^(1/2))/(b^2)^(1/2)-1/3*I*(-6*a^2*b+6*a*b^2+2*a*c-
2*b*c)/a^2/b*2^(1/2)*sum(1/(a-b)*(-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)*(I*(-a*b^2)^(1/3
)*_alpha*3^(1/2)*a-I*3^(1/2)*(-a*b^2)^(2/3)+2*_alpha^2*a^2-(-a*b^2)^(1/3)*_alpha*a-(-a*b^2)^(2/3))*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^2*a-I*(-a*b^2)^(2/3)*3^(1/2)*_alpha+I*3^(1/2)*b^2-3*(-a*b^2)^(2/3)*_alpha-3*b^
2)/b/(a-b),(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=R
ootOf(_Z^3*a+b)))

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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 = 9.50, size = 248, normalized size = 1.20 \begin {gather*} \frac {\sqrt {a^2\,x^3+b^2}}{3\,x^6}+\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+12\,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 {a\,\ln \left (\frac {a\,b+2\,b^2+a^2\,x^3-2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {a+b}}{b-a\,x^3}\right )\,\sqrt {a+b}\,\left (c+3\,a\,b\right )}{6\,b^{5/2}}+\frac {a\,\ln \left (\frac {2\,b^2-a\,b+a^2\,x^3+2\,\sqrt {b}\,\sqrt {a^2\,x^3+b^2}\,\sqrt {b-a}}{a\,x^3+b}\right )\,\sqrt {b-a}\,\left (c-3\,a\,b\right )}{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]

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