Optimal. Leaf size=50 \[ \frac {(A b-a B) \log \left (a+b x^3\right )}{3 a^2}-\frac {\log (x) (A b-a B)}{a^2}-\frac {A}{3 a x^3} \]
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Rubi [A] time = 0.05, antiderivative size = 50, 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 b-a B) \log \left (a+b x^3\right )}{3 a^2}-\frac {\log (x) (A b-a B)}{a^2}-\frac {A}{3 a x^3} \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^4 \left (a+b x^3\right )} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {A+B x}{x^2 (a+b x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {A}{a x^2}+\frac {-A b+a B}{a^2 x}-\frac {b (-A b+a B)}{a^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {A}{3 a x^3}-\frac {(A b-a B) \log (x)}{a^2}+\frac {(A b-a B) \log \left (a+b x^3\right )}{3 a^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 49, normalized size = 0.98 \begin {gather*} \frac {(A b-a B) \log \left (a+b x^3\right )}{3 a^2}+\frac {\log (x) (a B-A b)}{a^2}-\frac {A}{3 a x^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 {A+B x^3}{x^4 \left (a+b x^3\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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fricas [A] time = 0.50, size = 47, normalized size = 0.94 \begin {gather*} -\frac {{\left (B a - A b\right )} x^{3} \log \left (b x^{3} + a\right ) - 3 \, {\left (B a - A b\right )} x^{3} \log \relax (x) + A a}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 69, normalized size = 1.38 \begin {gather*} \frac {{\left (B a - A b\right )} \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {{\left (B a b - A b^{2}\right )} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2} b} - \frac {B a x^{3} - A b x^{3} + A a}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 56, normalized size = 1.12 \begin {gather*} -\frac {A b \ln \relax (x )}{a^{2}}+\frac {A b \ln \left (b \,x^{3}+a \right )}{3 a^{2}}+\frac {B \ln \relax (x )}{a}-\frac {B \ln \left (b \,x^{3}+a \right )}{3 a}-\frac {A}{3 a \,x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 48, normalized size = 0.96 \begin {gather*} -\frac {{\left (B a - A b\right )} \log \left (b x^{3} + a\right )}{3 \, a^{2}} + \frac {{\left (B a - A b\right )} \log \left (x^{3}\right )}{3 \, a^{2}} - \frac {A}{3 \, a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.40, size = 46, normalized size = 0.92 \begin {gather*} \frac {\ln \left (b\,x^3+a\right )\,\left (A\,b-B\,a\right )}{3\,a^2}-\frac {A}{3\,a\,x^3}-\frac {\ln \relax (x)\,\left (A\,b-B\,a\right )}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.36, size = 41, normalized size = 0.82 \begin {gather*} - \frac {A}{3 a x^{3}} + \frac {\left (- A b + B a\right ) \log {\relax (x )}}{a^{2}} - \frac {\left (- A b + B a\right ) \log {\left (\frac {a}{b} + x^{3} \right )}}{3 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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