Optimal. Leaf size=97 \[ -\frac {b (3 A b-2 a B) \log \left (a+b x^3\right )}{3 a^4}+\frac {b \log (x) (3 A b-2 a B)}{a^4}+\frac {b (A b-a B)}{3 a^3 \left (a+b x^3\right )}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {A}{6 a^2 x^6} \]
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Rubi [A] time = 0.10, antiderivative size = 97, 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 {b (A b-a B)}{3 a^3 \left (a+b x^3\right )}+\frac {2 A b-a B}{3 a^3 x^3}-\frac {b (3 A b-2 a B) \log \left (a+b x^3\right )}{3 a^4}+\frac {b \log (x) (3 A b-2 a B)}{a^4}-\frac {A}{6 a^2 x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x^7 \left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{x^3 (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {A}{a^2 x^3}+\frac {-2 A b+a B}{a^3 x^2}-\frac {b (-3 A b+2 a B)}{a^4 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)^2}+\frac {b^2 (-3 A b+2 a B)}{a^4 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {A}{6 a^2 x^6}+\frac {2 A b-a B}{3 a^3 x^3}+\frac {b (A b-a B)}{3 a^3 \left (a+b x^3\right )}+\frac {b (3 A b-2 a B) \log (x)}{a^4}-\frac {b (3 A b-2 a B) \log \left (a+b x^3\right )}{3 a^4}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 85, normalized size = 0.88 \begin {gather*} -\frac {\frac {a^2 A}{x^6}+\frac {2 a b (a B-A b)}{a+b x^3}+\frac {2 a (a B-2 A b)}{x^3}+2 b (3 A b-2 a B) \log \left (a+b x^3\right )-6 b \log (x) (3 A b-2 a B)}{6 a^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 {A+B x^3}{x^7 \left (a+b x^3\right )^2} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.46, size = 154, normalized size = 1.59 \begin {gather*} -\frac {2 \, {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{6} + A a^{3} + {\left (2 \, B a^{3} - 3 \, A a^{2} b\right )} x^{3} - 2 \, {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{9} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{6}\right )} \log \left (b x^{3} + a\right ) + 6 \, {\left ({\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} x^{9} + {\left (2 \, B a^{2} b - 3 \, A a b^{2}\right )} x^{6}\right )} \log \relax (x)}{6 \, {\left (a^{4} b x^{9} + a^{5} x^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 149, normalized size = 1.54 \begin {gather*} -\frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (2 \, B a b^{2} - 3 \, A b^{3}\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{4} b} - \frac {2 \, B a b^{2} x^{3} - 3 \, A b^{3} x^{3} + 3 \, B a^{2} b - 4 \, A a b^{2}}{3 \, {\left (b x^{3} + a\right )} a^{4}} + \frac {6 \, B a b x^{6} - 9 \, A b^{2} x^{6} - 2 \, B a^{2} x^{3} + 4 \, A a b x^{3} - A a^{2}}{6 \, a^{4} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 116, normalized size = 1.20 \begin {gather*} \frac {A \,b^{2}}{3 \left (b \,x^{3}+a \right ) a^{3}}+\frac {3 A \,b^{2} \ln \relax (x )}{a^{4}}-\frac {A \,b^{2} \ln \left (b \,x^{3}+a \right )}{a^{4}}-\frac {B b}{3 \left (b \,x^{3}+a \right ) a^{2}}-\frac {2 B b \ln \relax (x )}{a^{3}}+\frac {2 B b \ln \left (b \,x^{3}+a \right )}{3 a^{3}}+\frac {2 A b}{3 a^{3} x^{3}}-\frac {B}{3 a^{2} x^{3}}-\frac {A}{6 a^{2} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 106, normalized size = 1.09 \begin {gather*} -\frac {2 \, {\left (2 \, B a b - 3 \, A b^{2}\right )} x^{6} + {\left (2 \, B a^{2} - 3 \, A a b\right )} x^{3} + A a^{2}}{6 \, {\left (a^{3} b x^{9} + a^{4} x^{6}\right )}} + \frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (b x^{3} + a\right )}{3 \, a^{4}} - \frac {{\left (2 \, B a b - 3 \, A b^{2}\right )} \log \left (x^{3}\right )}{3 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.14, size = 100, normalized size = 1.03 \begin {gather*} \frac {\frac {x^3\,\left (3\,A\,b-2\,B\,a\right )}{6\,a^2}-\frac {A}{6\,a}+\frac {b\,x^6\,\left (3\,A\,b-2\,B\,a\right )}{3\,a^3}}{b\,x^9+a\,x^6}-\frac {\ln \left (b\,x^3+a\right )\,\left (3\,A\,b^2-2\,B\,a\,b\right )}{3\,a^4}+\frac {\ln \relax (x)\,\left (3\,A\,b^2-2\,B\,a\,b\right )}{a^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.40, size = 100, normalized size = 1.03 \begin {gather*} \frac {- A a^{2} + x^{6} \left (6 A b^{2} - 4 B a b\right ) + x^{3} \left (3 A a b - 2 B a^{2}\right )}{6 a^{4} x^{6} + 6 a^{3} b x^{9}} - \frac {b \left (- 3 A b + 2 B a\right ) \log {\relax (x )}}{a^{4}} + \frac {b \left (- 3 A b + 2 B a\right ) \log {\left (\frac {a}{b} + x^{3} \right )}}{3 a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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