Optimal. Leaf size=52 \[ \frac {2 b \log \left (a+b x^3\right )}{3 a^3}-\frac {2 b \log (x)}{a^3}-\frac {b}{3 a^2 \left (a+b x^3\right )}-\frac {1}{3 a^2 x^3} \]
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Rubi [A] time = 0.04, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 44} \[ -\frac {b}{3 a^2 \left (a+b x^3\right )}+\frac {2 b \log \left (a+b x^3\right )}{3 a^3}-\frac {2 b \log (x)}{a^3}-\frac {1}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{x^2 (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \operatorname {Subst}\left (\int \left (\frac {1}{a^2 x^2}-\frac {2 b}{a^3 x}+\frac {b^2}{a^2 (a+b x)^2}+\frac {2 b^2}{a^3 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {1}{3 a^2 x^3}-\frac {b}{3 a^2 \left (a+b x^3\right )}-\frac {2 b \log (x)}{a^3}+\frac {2 b \log \left (a+b x^3\right )}{3 a^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 41, normalized size = 0.79 \[ -\frac {a \left (\frac {b}{a+b x^3}+\frac {1}{x^3}\right )-2 b \log \left (a+b x^3\right )+6 b \log (x)}{3 a^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 73, normalized size = 1.40 \[ -\frac {2 \, a b x^{3} + a^{2} - 2 \, {\left (b^{2} x^{6} + a b x^{3}\right )} \log \left (b x^{3} + a\right ) + 6 \, {\left (b^{2} x^{6} + a b x^{3}\right )} \log \relax (x)}{3 \, {\left (a^{3} b x^{6} + a^{4} x^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 51, normalized size = 0.98 \[ \frac {2 \, b \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{3}} - \frac {2 \, b \log \left ({\left | x \right |}\right )}{a^{3}} - \frac {2 \, b x^{3} + a}{3 \, {\left (b x^{6} + a x^{3}\right )} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 47, normalized size = 0.90 \[ -\frac {b}{3 \left (b \,x^{3}+a \right ) a^{2}}-\frac {2 b \ln \relax (x )}{a^{3}}+\frac {2 b \ln \left (b \,x^{3}+a \right )}{3 a^{3}}-\frac {1}{3 a^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.33, size = 53, normalized size = 1.02 \[ -\frac {2 \, b x^{3} + a}{3 \, {\left (a^{2} b x^{6} + a^{3} x^{3}\right )}} + \frac {2 \, b \log \left (b x^{3} + a\right )}{3 \, a^{3}} - \frac {2 \, b \log \left (x^{3}\right )}{3 \, a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 53, normalized size = 1.02 \[ \frac {2\,b\,\ln \left (b\,x^3+a\right )}{3\,a^3}-\frac {\frac {1}{3\,a}+\frac {2\,b\,x^3}{3\,a^2}}{b\,x^6+a\,x^3}-\frac {2\,b\,\ln \relax (x)}{a^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.56, size = 54, normalized size = 1.04 \[ \frac {- a - 2 b x^{3}}{3 a^{3} x^{3} + 3 a^{2} b x^{6}} - \frac {2 b \log {\relax (x )}}{a^{3}} + \frac {2 b \log {\left (\frac {a}{b} + x^{3} \right )}}{3 a^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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