Optimal. Leaf size=107 \[ \frac {a^3 (A b-a B)}{6 b^5 \left (a+b x^3\right )^2}-\frac {a^2 (3 A b-4 a B)}{3 b^5 \left (a+b x^3\right )}-\frac {a (A b-2 a B) \log \left (a+b x^3\right )}{b^5}+\frac {x^3 (A b-3 a B)}{3 b^4}+\frac {B x^6}{6 b^3} \]
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Rubi [A] time = 0.14, antiderivative size = 107, 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 (3 A b-4 a B)}{3 b^5 \left (a+b x^3\right )}+\frac {a^3 (A b-a B)}{6 b^5 \left (a+b x^3\right )^2}+\frac {x^3 (A b-3 a B)}{3 b^4}-\frac {a (A b-2 a B) \log \left (a+b x^3\right )}{b^5}+\frac {B x^6}{6 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {x^{11} \left (A+B x^3\right )}{\left (a+b x^3\right )^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {x^3 (A+B x)}{(a+b x)^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {A b-3 a B}{b^4}+\frac {B x}{b^3}+\frac {a^3 (-A b+a B)}{b^4 (a+b x)^3}-\frac {a^2 (-3 A b+4 a B)}{b^4 (a+b x)^2}+\frac {3 a (-A b+2 a B)}{b^4 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {(A b-3 a B) x^3}{3 b^4}+\frac {B x^6}{6 b^3}+\frac {a^3 (A b-a B)}{6 b^5 \left (a+b x^3\right )^2}-\frac {a^2 (3 A b-4 a B)}{3 b^5 \left (a+b x^3\right )}-\frac {a (A b-2 a B) \log \left (a+b x^3\right )}{b^5}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 94, normalized size = 0.88 \begin {gather*} \frac {\frac {a^3 (A b-a B)}{\left (a+b x^3\right )^2}+\frac {2 a^2 (4 a B-3 A b)}{a+b x^3}+2 b x^3 (A b-3 a B)+6 a (2 a B-A b) \log \left (a+b x^3\right )+b^2 B x^6}{6 b^5} \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^{11} \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 = 0.64, size = 179, normalized size = 1.67 \begin {gather*} \frac {B b^{4} x^{12} - 2 \, {\left (2 \, B a b^{3} - A b^{4}\right )} x^{9} - {\left (11 \, B a^{2} b^{2} - 4 \, A a b^{3}\right )} x^{6} + 7 \, B a^{4} - 5 \, A a^{3} b + 2 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{3} + 6 \, {\left ({\left (2 \, B a^{2} b^{2} - A a b^{3}\right )} x^{6} + 2 \, B a^{4} - A a^{3} b + 2 \, {\left (2 \, B a^{3} b - A a^{2} b^{2}\right )} x^{3}\right )} \log \left (b x^{3} + a\right )}{6 \, {\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 131, normalized size = 1.22 \begin {gather*} \frac {{\left (2 \, B a^{2} - A a b\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{b^{5}} + \frac {B b^{3} x^{6} - 6 \, B a b^{2} x^{3} + 2 \, A b^{3} x^{3}}{6 \, b^{6}} - \frac {18 \, B a^{2} b^{2} x^{6} - 9 \, A a b^{3} x^{6} + 28 \, B a^{3} b x^{3} - 12 \, A a^{2} b^{2} x^{3} + 11 \, B a^{4} - 4 \, A a^{3} b}{6 \, {\left (b x^{3} + a\right )}^{2} b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 134, normalized size = 1.25 \begin {gather*} \frac {B \,x^{6}}{6 b^{3}}+\frac {A \,x^{3}}{3 b^{3}}-\frac {B a \,x^{3}}{b^{4}}+\frac {A \,a^{3}}{6 \left (b \,x^{3}+a \right )^{2} b^{4}}-\frac {B \,a^{4}}{6 \left (b \,x^{3}+a \right )^{2} b^{5}}-\frac {A \,a^{2}}{\left (b \,x^{3}+a \right ) b^{4}}-\frac {A a \ln \left (b \,x^{3}+a \right )}{b^{4}}+\frac {4 B \,a^{3}}{3 \left (b \,x^{3}+a \right ) b^{5}}+\frac {2 B \,a^{2} \ln \left (b \,x^{3}+a \right )}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.54, size = 115, normalized size = 1.07 \begin {gather*} \frac {7 \, B a^{4} - 5 \, A a^{3} b + 2 \, {\left (4 \, B a^{3} b - 3 \, A a^{2} b^{2}\right )} x^{3}}{6 \, {\left (b^{7} x^{6} + 2 \, a b^{6} x^{3} + a^{2} b^{5}\right )}} + \frac {B b x^{6} - 2 \, {\left (3 \, B a - A b\right )} x^{3}}{6 \, b^{4}} + \frac {{\left (2 \, B a^{2} - A a b\right )} \log \left (b x^{3} + a\right )}{b^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 117, normalized size = 1.09 \begin {gather*} \frac {\frac {7\,B\,a^4-5\,A\,a^3\,b}{6\,b}+x^3\,\left (\frac {4\,B\,a^3}{3}-A\,a^2\,b\right )}{a^2\,b^4+2\,a\,b^5\,x^3+b^6\,x^6}+x^3\,\left (\frac {A}{3\,b^3}-\frac {B\,a}{b^4}\right )+\frac {\ln \left (b\,x^3+a\right )\,\left (2\,B\,a^2-A\,a\,b\right )}{b^5}+\frac {B\,x^6}{6\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.94, size = 112, normalized size = 1.05 \begin {gather*} \frac {B x^{6}}{6 b^{3}} + \frac {a \left (- A b + 2 B a\right ) \log {\left (a + b x^{3} \right )}}{b^{5}} + x^{3} \left (\frac {A}{3 b^{3}} - \frac {B a}{b^{4}}\right ) + \frac {- 5 A a^{3} b + 7 B a^{4} + x^{3} \left (- 6 A a^{2} b^{2} + 8 B a^{3} b\right )}{6 a^{2} b^{5} + 12 a b^{6} x^{3} + 6 b^{7} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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