Optimal. Leaf size=82 \[ -\frac {a^2 (A b-a B)}{3 b^4 \left (a+b x^3\right )}-\frac {a (2 A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac {x^3 (A b-2 a B)}{3 b^3}+\frac {B x^6}{6 b^2} \]
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Rubi [A] time = 0.09, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {446, 77} \begin {gather*} -\frac {a^2 (A b-a B)}{3 b^4 \left (a+b x^3\right )}+\frac {x^3 (A b-2 a B)}{3 b^3}-\frac {a (2 A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac {B x^6}{6 b^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {x^8 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2 (A+B x)}{(a+b x)^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {A b-2 a B}{b^3}+\frac {B x}{b^2}-\frac {a^2 (-A b+a B)}{b^3 (a+b x)^2}+\frac {a (-2 A b+3 a B)}{b^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {(A b-2 a B) x^3}{3 b^3}+\frac {B x^6}{6 b^2}-\frac {a^2 (A b-a B)}{3 b^4 \left (a+b x^3\right )}-\frac {a (2 A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 72, normalized size = 0.88 \begin {gather*} \frac {\frac {2 a^2 (a B-A b)}{a+b x^3}+2 b x^3 (A b-2 a B)+2 a (3 a B-2 A b) \log \left (a+b x^3\right )+b^2 B x^6}{6 b^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^8 \left (A+B x^3\right )}{\left (a+b x^3\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.57, size = 121, normalized size = 1.48 \begin {gather*} \frac {B b^{3} x^{9} - {\left (3 \, B a b^{2} - 2 \, A b^{3}\right )} x^{6} + 2 \, B a^{3} - 2 \, A a^{2} b - 2 \, {\left (2 \, B a^{2} b - A a b^{2}\right )} x^{3} + 2 \, {\left (3 \, B a^{3} - 2 \, A a^{2} b + {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \, {\left (b^{5} x^{3} + a b^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 106, normalized size = 1.29 \begin {gather*} \frac {{\left (3 \, B a^{2} - 2 \, A a b\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{4}} + \frac {B b^{2} x^{6} - 4 \, B a b x^{3} + 2 \, A b^{2} x^{3}}{6 \, b^{4}} - \frac {3 \, B a^{2} b x^{3} - 2 \, A a b^{2} x^{3} + 2 \, B a^{3} - A a^{2} b}{3 \, {\left (b x^{3} + a\right )} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 97, normalized size = 1.18 \begin {gather*} \frac {B \,x^{6}}{6 b^{2}}+\frac {A \,x^{3}}{3 b^{2}}-\frac {2 B a \,x^{3}}{3 b^{3}}-\frac {A \,a^{2}}{3 \left (b \,x^{3}+a \right ) b^{3}}-\frac {2 A a \ln \left (b \,x^{3}+a \right )}{3 b^{3}}+\frac {B \,a^{3}}{3 \left (b \,x^{3}+a \right ) b^{4}}+\frac {B \,a^{2} \ln \left (b \,x^{3}+a \right )}{b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 82, normalized size = 1.00 \begin {gather*} \frac {B a^{3} - A a^{2} b}{3 \, {\left (b^{5} x^{3} + a b^{4}\right )}} + \frac {B b x^{6} - 2 \, {\left (2 \, B a - A b\right )} x^{3}}{6 \, b^{3}} + \frac {{\left (3 \, B a^{2} - 2 \, A a b\right )} \log \left (b x^{3} + a\right )}{3 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.08, size = 86, normalized size = 1.05 \begin {gather*} x^3\,\left (\frac {A}{3\,b^2}-\frac {2\,B\,a}{3\,b^3}\right )+\frac {\ln \left (b\,x^3+a\right )\,\left (3\,B\,a^2-2\,A\,a\,b\right )}{3\,b^4}+\frac {B\,x^6}{6\,b^2}+\frac {B\,a^3-A\,a^2\,b}{3\,b\,\left (b^4\,x^3+a\,b^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.14, size = 82, normalized size = 1.00 \begin {gather*} \frac {B x^{6}}{6 b^{2}} + \frac {a \left (- 2 A b + 3 B a\right ) \log {\left (a + b x^{3} \right )}}{3 b^{4}} + x^{3} \left (\frac {A}{3 b^{2}} - \frac {2 B a}{3 b^{3}}\right ) + \frac {- A a^{2} b + B a^{3}}{3 a b^{4} + 3 b^{5} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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