Optimal. Leaf size=122 \[ -\frac {b (2 A b-a B) \log \left (a+b x^3\right )}{a^5}+\frac {3 b \log (x) (2 A b-a B)}{a^5}+\frac {b (3 A b-2 a B)}{3 a^4 \left (a+b x^3\right )}+\frac {3 A b-a B}{3 a^4 x^3}+\frac {b (A b-a B)}{6 a^3 \left (a+b x^3\right )^2}-\frac {A}{6 a^3 x^6} \]
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Rubi [A] time = 0.13, antiderivative size = 122, 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 (3 A b-2 a B)}{3 a^4 \left (a+b x^3\right )}+\frac {b (A b-a B)}{6 a^3 \left (a+b x^3\right )^2}+\frac {3 A b-a B}{3 a^4 x^3}-\frac {b (2 A b-a B) \log \left (a+b x^3\right )}{a^5}+\frac {3 b \log (x) (2 A b-a B)}{a^5}-\frac {A}{6 a^3 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 )^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{x^3 (a+b x)^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {A}{a^3 x^3}+\frac {-3 A b+a B}{a^4 x^2}-\frac {3 b (-2 A b+a B)}{a^5 x}+\frac {b^2 (-A b+a B)}{a^3 (a+b x)^3}+\frac {b^2 (-3 A b+2 a B)}{a^4 (a+b x)^2}+\frac {3 b^2 (-2 A b+a B)}{a^5 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {A}{6 a^3 x^6}+\frac {3 A b-a B}{3 a^4 x^3}+\frac {b (A b-a B)}{6 a^3 \left (a+b x^3\right )^2}+\frac {b (3 A b-2 a B)}{3 a^4 \left (a+b x^3\right )}+\frac {3 b (2 A b-a B) \log (x)}{a^5}-\frac {b (2 A b-a B) \log \left (a+b x^3\right )}{a^5}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 108, normalized size = 0.89 \begin {gather*} \frac {\frac {a^2 b (A b-a B)}{\left (a+b x^3\right )^2}-\frac {a^2 A}{x^6}+\frac {2 a b (3 A b-2 a B)}{a+b x^3}-\frac {2 a (a B-3 A b)}{x^3}+6 b (a B-2 A b) \log \left (a+b x^3\right )+18 b \log (x) (2 A b-a B)}{6 a^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 {A+B x^3}{x^7 \left (a+b x^3\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [B] time = 0.64, size = 229, normalized size = 1.88 \begin {gather*} -\frac {6 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{9} + 9 \, {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{6} + A a^{4} + 2 \, {\left (B a^{4} - 2 \, A a^{3} b\right )} x^{3} - 6 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{12} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{9} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{6}\right )} \log \left (b x^{3} + a\right ) + 18 \, {\left ({\left (B a b^{3} - 2 \, A b^{4}\right )} x^{12} + 2 \, {\left (B a^{2} b^{2} - 2 \, A a b^{3}\right )} x^{9} + {\left (B a^{3} b - 2 \, A a^{2} b^{2}\right )} x^{6}\right )} \log \relax (x)}{6 \, {\left (a^{5} b^{2} x^{12} + 2 \, a^{6} b x^{9} + a^{7} x^{6}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 131, normalized size = 1.07 \begin {gather*} -\frac {3 \, {\left (B a b - 2 \, A b^{2}\right )} \log \left ({\left | x \right |}\right )}{a^{5}} + \frac {{\left (B a b^{2} - 2 \, A b^{3}\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{a^{5} b} - \frac {6 \, B a b^{2} x^{9} - 12 \, A b^{3} x^{9} + 9 \, B a^{2} b x^{6} - 18 \, A a b^{2} x^{6} + 2 \, B a^{3} x^{3} - 4 \, A a^{2} b x^{3} + A a^{3}}{6 \, {\left (b x^{6} + a x^{3}\right )}^{2} a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 147, normalized size = 1.20 \begin {gather*} \frac {A \,b^{2}}{6 \left (b \,x^{3}+a \right )^{2} a^{3}}-\frac {B b}{6 \left (b \,x^{3}+a \right )^{2} a^{2}}+\frac {A \,b^{2}}{\left (b \,x^{3}+a \right ) a^{4}}+\frac {6 A \,b^{2} \ln \relax (x )}{a^{5}}-\frac {2 A \,b^{2} \ln \left (b \,x^{3}+a \right )}{a^{5}}-\frac {2 B b}{3 \left (b \,x^{3}+a \right ) a^{3}}-\frac {3 B b \ln \relax (x )}{a^{4}}+\frac {B b \ln \left (b \,x^{3}+a \right )}{a^{4}}+\frac {A b}{a^{4} x^{3}}-\frac {B}{3 a^{3} x^{3}}-\frac {A}{6 a^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.52, size = 136, normalized size = 1.11 \begin {gather*} -\frac {6 \, {\left (B a b^{2} - 2 \, A b^{3}\right )} x^{9} + 9 \, {\left (B a^{2} b - 2 \, A a b^{2}\right )} x^{6} + A a^{3} + 2 \, {\left (B a^{3} - 2 \, A a^{2} b\right )} x^{3}}{6 \, {\left (a^{4} b^{2} x^{12} + 2 \, a^{5} b x^{9} + a^{6} x^{6}\right )}} + \frac {{\left (B a b - 2 \, A b^{2}\right )} \log \left (b x^{3} + a\right )}{a^{5}} - \frac {{\left (B a b - 2 \, A b^{2}\right )} \log \left (x^{3}\right )}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 130, normalized size = 1.07 \begin {gather*} \frac {\frac {x^3\,\left (2\,A\,b-B\,a\right )}{3\,a^2}-\frac {A}{6\,a}+\frac {b^2\,x^9\,\left (2\,A\,b-B\,a\right )}{a^4}+\frac {3\,b\,x^6\,\left (2\,A\,b-B\,a\right )}{2\,a^3}}{a^2\,x^6+2\,a\,b\,x^9+b^2\,x^{12}}-\frac {\ln \left (b\,x^3+a\right )\,\left (2\,A\,b^2-B\,a\,b\right )}{a^5}+\frac {\ln \relax (x)\,\left (6\,A\,b^2-3\,B\,a\,b\right )}{a^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 3.55, size = 133, normalized size = 1.09 \begin {gather*} \frac {- A a^{3} + x^{9} \left (12 A b^{3} - 6 B a b^{2}\right ) + x^{6} \left (18 A a b^{2} - 9 B a^{2} b\right ) + x^{3} \left (4 A a^{2} b - 2 B a^{3}\right )}{6 a^{6} x^{6} + 12 a^{5} b x^{9} + 6 a^{4} b^{2} x^{12}} - \frac {3 b \left (- 2 A b + B a\right ) \log {\relax (x )}}{a^{5}} + \frac {b \left (- 2 A b + B a\right ) \log {\left (\frac {a}{b} + x^{3} \right )}}{a^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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