Optimal. Leaf size=68 \[ -\frac {A \log \left (a+b x^3\right )}{3 a^3}+\frac {A \log (x)}{a^3}+\frac {A}{3 a^2 \left (a+b x^3\right )}+\frac {A b-a B}{6 a b \left (a+b x^3\right )^2} \]
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Rubi [A] time = 0.06, antiderivative size = 68, 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} \[ \frac {A}{3 a^2 \left (a+b x^3\right )}-\frac {A \log \left (a+b x^3\right )}{3 a^3}+\frac {A \log (x)}{a^3}+\frac {A b-a B}{6 a b \left (a+b x^3\right )^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {A+B x^3}{x \left (a+b x^3\right )^3} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{x (a+b x)^3} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {A}{a^3 x}+\frac {-A b+a B}{a (a+b x)^3}-\frac {A b}{a^2 (a+b x)^2}-\frac {A b}{a^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=\frac {A b-a B}{6 a b \left (a+b x^3\right )^2}+\frac {A}{3 a^2 \left (a+b x^3\right )}+\frac {A \log (x)}{a^3}-\frac {A \log \left (a+b x^3\right )}{3 a^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 59, normalized size = 0.87 \[ \frac {\frac {a \left (-a^2 B+3 a A b+2 A b^2 x^3\right )}{b \left (a+b x^3\right )^2}-2 A \log \left (a+b x^3\right )+6 A \log (x)}{6 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 119, normalized size = 1.75 \[ \frac {2 \, A a b^{2} x^{3} - B a^{3} + 3 \, A a^{2} b - 2 \, {\left (A b^{3} x^{6} + 2 \, A a b^{2} x^{3} + A a^{2} b\right )} \log \left (b x^{3} + a\right ) + 6 \, {\left (A b^{3} x^{6} + 2 \, A a b^{2} x^{3} + A a^{2} b\right )} \log \relax (x)}{6 \, {\left (a^{3} b^{3} x^{6} + 2 \, a^{4} b^{2} x^{3} + a^{5} b\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 74, normalized size = 1.09 \[ -\frac {A \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} + \frac {A \log \left ({\left | x \right |}\right )}{a^{3}} + \frac {3 \, A b^{3} x^{6} + 8 \, A a b^{2} x^{3} - B a^{3} + 6 \, A a^{2} b}{6 \, {\left (b x^{3} + a\right )}^{2} a^{3} b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 68, normalized size = 1.00 \[ \frac {A}{6 \left (b \,x^{3}+a \right )^{2} a}-\frac {B}{6 \left (b \,x^{3}+a \right )^{2} b}+\frac {A}{3 \left (b \,x^{3}+a \right ) a^{2}}+\frac {A \ln \relax (x )}{a^{3}}-\frac {A \ln \left (b \,x^{3}+a \right )}{3 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 77, normalized size = 1.13 \[ \frac {2 \, A b^{2} x^{3} - B a^{2} + 3 \, A a b}{6 \, {\left (a^{2} b^{3} x^{6} + 2 \, a^{3} b^{2} x^{3} + a^{4} b\right )}} - \frac {A \log \left (b x^{3} + a\right )}{3 \, a^{3}} + \frac {A \log \left (x^{3}\right )}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.16, size = 71, normalized size = 1.04 \[ \frac {\frac {3\,A\,b-B\,a}{6\,a\,b}+\frac {A\,b\,x^3}{3\,a^2}}{a^2+2\,a\,b\,x^3+b^2\,x^6}-\frac {A\,\ln \left (b\,x^3+a\right )}{3\,a^3}+\frac {A\,\ln \relax (x)}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.48, size = 75, normalized size = 1.10 \[ \frac {A \log {\relax (x )}}{a^{3}} - \frac {A \log {\left (\frac {a}{b} + x^{3} \right )}}{3 a^{3}} + \frac {3 A a b + 2 A b^{2} x^{3} - B a^{2}}{6 a^{4} b + 12 a^{3} b^{2} x^{3} + 6 a^{2} b^{3} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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