Optimal. Leaf size=88 \[ -\frac {a^2 (A b-a B)}{6 b^4 \left (a+b x^3\right )^2}+\frac {a (2 A b-3 a B)}{3 b^4 \left (a+b x^3\right )}+\frac {(A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac {B x^3}{3 b^3} \]
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Rubi [A] time = 0.09, antiderivative size = 88, 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)}{6 b^4 \left (a+b x^3\right )^2}+\frac {a (2 A b-3 a B)}{3 b^4 \left (a+b x^3\right )}+\frac {(A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac {B x^3}{3 b^3} \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 )^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^2 (A+B x)}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {B}{b^3}-\frac {a^2 (-A b+a B)}{b^3 (a+b x)^3}+\frac {a (-2 A b+3 a B)}{b^3 (a+b x)^2}+\frac {A b-3 a B}{b^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {B x^3}{3 b^3}-\frac {a^2 (A b-a B)}{6 b^4 \left (a+b x^3\right )^2}+\frac {a (2 A b-3 a B)}{3 b^4 \left (a+b x^3\right )}+\frac {(A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 92, normalized size = 1.05 \begin {gather*} \frac {2 a A b-3 a^2 B}{3 b^4 \left (a+b x^3\right )}+\frac {a^3 B-a^2 A b}{6 b^4 \left (a+b x^3\right )^2}+\frac {(A b-3 a B) \log \left (a+b x^3\right )}{3 b^4}+\frac {B x^3}{3 b^3} \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 )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.63, size = 142, normalized size = 1.61 \begin {gather*} \frac {2 \, B b^{3} x^{9} + 4 \, B a b^{2} x^{6} - 5 \, B a^{3} + 3 \, A a^{2} b - 4 \, {\left (B a^{2} b - A a b^{2}\right )} x^{3} - 2 \, {\left ({\left (3 \, B a b^{2} - A b^{3}\right )} x^{6} + 3 \, B a^{3} - A a^{2} b + 2 \, {\left (3 \, B a^{2} b - A a b^{2}\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} 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 = 93, normalized size = 1.06 \begin {gather*} \frac {B x^{3}}{3 \, b^{3}} - \frac {{\left (3 \, B a - A b\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, b^{4}} + \frac {9 \, B a b^{2} x^{6} - 3 \, A b^{3} x^{6} + 12 \, B a^{2} b x^{3} - 2 \, A a b^{2} x^{3} + 4 \, B a^{3}}{6 \, {\left (b x^{3} + a\right )}^{2} b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 110, normalized size = 1.25 \begin {gather*} \frac {B \,x^{3}}{3 b^{3}}-\frac {A \,a^{2}}{6 \left (b \,x^{3}+a \right )^{2} b^{3}}+\frac {B \,a^{3}}{6 \left (b \,x^{3}+a \right )^{2} b^{4}}+\frac {2 A a}{3 \left (b \,x^{3}+a \right ) b^{3}}+\frac {A \ln \left (b \,x^{3}+a \right )}{3 b^{3}}-\frac {B \,a^{2}}{\left (b \,x^{3}+a \right ) b^{4}}-\frac {B a \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 = 0.52, size = 94, normalized size = 1.07 \begin {gather*} \frac {B x^{3}}{3 \, b^{3}} - \frac {5 \, B a^{3} - 3 \, A a^{2} b + 2 \, {\left (3 \, B a^{2} b - 2 \, A a b^{2}\right )} x^{3}}{6 \, {\left (b^{6} x^{6} + 2 \, a b^{5} x^{3} + a^{2} b^{4}\right )}} - \frac {{\left (3 \, B 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 = 2.40, size = 94, normalized size = 1.07 \begin {gather*} \frac {B\,x^3}{3\,b^3}-\frac {x^3\,\left (B\,a^2-\frac {2\,A\,a\,b}{3}\right )+\frac {5\,B\,a^3-3\,A\,a^2\,b}{6\,b}}{a^2\,b^3+2\,a\,b^4\,x^3+b^5\,x^6}+\frac {\ln \left (b\,x^3+a\right )\,\left (A\,b-3\,B\,a\right )}{3\,b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.94, size = 94, normalized size = 1.07 \begin {gather*} \frac {B x^{3}}{3 b^{3}} + \frac {3 A a^{2} b - 5 B a^{3} + x^{3} \left (4 A a b^{2} - 6 B a^{2} b\right )}{6 a^{2} b^{4} + 12 a b^{5} x^{3} + 6 b^{6} x^{6}} - \frac {\left (- A b + 3 B a\right ) \log {\left (a + b x^{3} \right )}}{3 b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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