Optimal. Leaf size=52 \[ -\frac {1}{3 a^2 x^3}-\frac {b}{3 a^2 \left (a+b x^3\right )}-\frac {2 b \log (x)}{a^3}+\frac {2 b \log \left (a+b x^3\right )}{3 a^3} \]
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Rubi [A]
time = 0.03, 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 = {272, 46}
\begin {gather*} \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} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^4 \left (a+b x^3\right )^2} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)^2} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {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 = 0.03, size = 41, normalized size = 0.79 \begin {gather*} -\frac {a \left (\frac {1}{x^3}+\frac {b}{a+b x^3}\right )+6 b \log (x)-2 b \log \left (a+b x^3\right )}{3 a^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 55, normalized size = 1.06
method | result | size |
norman | \(\frac {-\frac {1}{3 a}+\frac {2 b^{2} x^{6}}{3 a^{3}}}{x^{3} \left (b \,x^{3}+a \right )}-\frac {2 b \ln \left (x \right )}{a^{3}}+\frac {2 b \ln \left (b \,x^{3}+a \right )}{3 a^{3}}\) | \(54\) |
default | \(\frac {b^{2} \left (-\frac {a}{b \left (b \,x^{3}+a \right )}+\frac {2 \ln \left (b \,x^{3}+a \right )}{b}\right )}{3 a^{3}}-\frac {1}{3 a^{2} x^{3}}-\frac {2 b \ln \left (x \right )}{a^{3}}\) | \(55\) |
risch | \(\frac {-\frac {2 b \,x^{3}}{3 a^{2}}-\frac {1}{3 a}}{x^{3} \left (b \,x^{3}+a \right )}-\frac {2 b \ln \left (x \right )}{a^{3}}+\frac {2 b \ln \left (-b \,x^{3}-a \right )}{3 a^{3}}\) | \(55\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 53, normalized size = 1.02 \begin {gather*} -\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}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 73, normalized size = 1.40 \begin {gather*} -\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 \left (x\right )}{3 \, {\left (a^{3} b x^{6} + a^{4} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.22, size = 54, normalized size = 1.04 \begin {gather*} \frac {- a - 2 b x^{3}}{3 a^{3} x^{3} + 3 a^{2} b x^{6}} - \frac {2 b \log {\left (x \right )}}{a^{3}} + \frac {2 b \log {\left (\frac {a}{b} + x^{3} \right )}}{3 a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.35, size = 51, normalized size = 0.98 \begin {gather*} \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}} \end {gather*}
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 \begin {gather*} \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 \left (x\right )}{a^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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