Optimal. Leaf size=150 \[ \frac {-a^2 d^2-2 b^2 c^2}{3 b^2 d \sqrt {c+d x^3} (b c-a d)^2}-\frac {a^2}{3 b^2 \left (a+b x^3\right ) \sqrt {c+d x^3} (b c-a d)}+\frac {a (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{5/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 149, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 89, 78, 63, 208} \begin {gather*} -\frac {a^2 d^2+2 b^2 c^2}{3 b^2 d \sqrt {c+d x^3} (b c-a d)^2}-\frac {a^2}{3 b^2 \left (a+b x^3\right ) \sqrt {c+d x^3} (b c-a d)}+\frac {a (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 63
Rule 78
Rule 89
Rule 208
Rule 446
Rubi steps
\begin {align*} \int \frac {x^8}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^{3/2}} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2}{(a+b x)^2 (c+d x)^{3/2}} \, dx,x,x^3\right )\\ &=-\frac {a^2}{3 b^2 (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}+\frac {\operatorname {Subst}\left (\int \frac {-\frac {1}{2} a (2 b c+a d)+b (b c-a d) x}{(a+b x) (c+d x)^{3/2}} \, dx,x,x^3\right )}{3 b^2 (b c-a d)}\\ &=-\frac {2 b^2 c^2+a^2 d^2}{3 b^2 d (b c-a d)^2 \sqrt {c+d x^3}}-\frac {a^2}{3 b^2 (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {(a (4 b c-a d)) \operatorname {Subst}\left (\int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx,x,x^3\right )}{6 b (b c-a d)^2}\\ &=-\frac {2 b^2 c^2+a^2 d^2}{3 b^2 d (b c-a d)^2 \sqrt {c+d x^3}}-\frac {a^2}{3 b^2 (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}-\frac {(a (4 b c-a d)) \operatorname {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x^3}\right )}{3 b d (b c-a d)^2}\\ &=-\frac {2 b^2 c^2+a^2 d^2}{3 b^2 d (b c-a d)^2 \sqrt {c+d x^3}}-\frac {a^2}{3 b^2 (b c-a d) \left (a+b x^3\right ) \sqrt {c+d x^3}}+\frac {a (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{3 b^{3/2} (b c-a d)^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 134, normalized size = 0.89 \begin {gather*} \frac {\frac {a (4 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3}}{\sqrt {b c-a d}}\right )}{(b c-a d)^{5/2}}-\frac {\sqrt {b} \left (a^2 d \left (c+d x^3\right )+2 a b c^2+2 b^2 c^2 x^3\right )}{d \left (a+b x^3\right ) \sqrt {c+d x^3} (b c-a d)^2}}{3 b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.25, size = 151, normalized size = 1.01 \begin {gather*} \frac {\left (4 a b c-a^2 d\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^3} \sqrt {a d-b c}}{b c-a d}\right )}{3 b^{3/2} (a d-b c)^{5/2}}+\frac {-a^2 c d-a^2 d^2 x^3-2 a b c^2-2 b^2 c^2 x^3}{3 b d \left (a+b x^3\right ) \sqrt {c+d x^3} (b c-a d)^2} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.92, size = 746, normalized size = 4.97 \begin {gather*} \left [-\frac {{\left ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x^{3} + 2 \, b c - a d - 2 \, \sqrt {d x^{3} + c} \sqrt {b^{2} c - a b d}}{b x^{3} + a}\right ) + 2 \, {\left (2 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d - a^{3} b c d^{2} + {\left (2 \, b^{4} c^{3} - 2 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{6 \, {\left (a b^{5} c^{4} d - 3 \, a^{2} b^{4} c^{3} d^{2} + 3 \, a^{3} b^{3} c^{2} d^{3} - a^{4} b^{2} c d^{4} + {\left (b^{6} c^{3} d^{2} - 3 \, a b^{5} c^{2} d^{3} + 3 \, a^{2} b^{4} c d^{4} - a^{3} b^{3} d^{5}\right )} x^{6} + {\left (b^{6} c^{4} d - 2 \, a b^{5} c^{3} d^{2} + 2 \, a^{3} b^{3} c d^{4} - a^{4} b^{2} d^{5}\right )} x^{3}\right )}}, -\frac {{\left ({\left (4 \, a b^{2} c d^{2} - a^{2} b d^{3}\right )} x^{6} + 4 \, a^{2} b c^{2} d - a^{3} c d^{2} + {\left (4 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{3}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {d x^{3} + c} \sqrt {-b^{2} c + a b d}}{b d x^{3} + b c}\right ) + {\left (2 \, a b^{3} c^{3} - a^{2} b^{2} c^{2} d - a^{3} b c d^{2} + {\left (2 \, b^{4} c^{3} - 2 \, a b^{3} c^{2} d + a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3}\right )} \sqrt {d x^{3} + c}}{3 \, {\left (a b^{5} c^{4} d - 3 \, a^{2} b^{4} c^{3} d^{2} + 3 \, a^{3} b^{3} c^{2} d^{3} - a^{4} b^{2} c d^{4} + {\left (b^{6} c^{3} d^{2} - 3 \, a b^{5} c^{2} d^{3} + 3 \, a^{2} b^{4} c d^{4} - a^{3} b^{3} d^{5}\right )} x^{6} + {\left (b^{6} c^{4} d - 2 \, a b^{5} c^{3} d^{2} + 2 \, a^{3} b^{3} c d^{4} - a^{4} b^{2} d^{5}\right )} x^{3}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 195, normalized size = 1.30 \begin {gather*} -\frac {{\left (4 \, a b c - a^{2} d\right )} \arctan \left (\frac {\sqrt {d x^{3} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{3 \, {\left (b^{3} c^{2} - 2 \, a b^{2} c d + a^{2} b d^{2}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, {\left (d x^{3} + c\right )} b^{2} c^{2} - 2 \, b^{2} c^{3} + 2 \, a b c^{2} d + {\left (d x^{3} + c\right )} a^{2} d^{2}}{3 \, {\left (b^{3} c^{2} d - 2 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} {\left ({\left (d x^{3} + c\right )}^{\frac {3}{2}} b - \sqrt {d x^{3} + c} b c + \sqrt {d x^{3} + c} a d\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.36, size = 978, normalized size = 6.52
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.90, size = 367, normalized size = 2.45 \begin {gather*} \frac {\sqrt {d\,x^3+c}\,\left (x^3\,\left (\frac {\left (\frac {3\,b\,d\,\left (a\,d+b\,c\right )-b\,d\,\left (a\,d+2\,b\,c\right )}{3\,\left (a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}-\frac {b\,d\,\left (a\,d+b\,c\right )}{a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d}\right )\,\left (a\,d+b\,c\right )}{b\,d}+\frac {a\,b\,c\,d}{a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d}\right )+\frac {a\,c\,\left (\frac {3\,b\,d\,\left (a\,d+b\,c\right )-b\,d\,\left (a\,d+2\,b\,c\right )}{3\,\left (a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d\right )}-\frac {b\,d\,\left (a\,d+b\,c\right )}{a^2\,b\,d^3-2\,a\,b^2\,c\,d^2+b^3\,c^2\,d}\right )}{b\,d}\right )}{b\,d\,x^6+\left (a\,d+b\,c\right )\,x^3+a\,c}+\frac {a\,\ln \left (\frac {2\,b\,c-a\,d+b\,d\,x^3+\sqrt {b}\,\sqrt {d\,x^3+c}\,\sqrt {a\,d-b\,c}\,2{}\mathrm {i}}{b\,x^3+a}\right )\,\left (a\,d-4\,b\,c\right )\,1{}\mathrm {i}}{6\,b^{3/2}\,{\left (a\,d-b\,c\right )}^{5/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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