Optimal. Leaf size=35 \[ -\frac {1}{3 a x^3}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^3\right )}{3 a^2} \]
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Rubi [A]
time = 0.02, antiderivative size = 35, 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 {b \log \left (a+b x^3\right )}{3 a^2}-\frac {b \log (x)}{a^2}-\frac {1}{3 a 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 )} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {1}{a x^2}-\frac {b}{a^2 x}+\frac {b^2}{a^2 (a+b x)}\right ) \, dx,x,x^3\right )\\ &=-\frac {1}{3 a x^3}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^3\right )}{3 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.00 \begin {gather*} -\frac {1}{3 a x^3}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^3\right )}{3 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.12, size = 32, normalized size = 0.91
method | result | size |
default | \(-\frac {1}{3 a \,x^{3}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b \,x^{3}+a \right )}{3 a^{2}}\) | \(32\) |
norman | \(-\frac {1}{3 a \,x^{3}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b \,x^{3}+a \right )}{3 a^{2}}\) | \(32\) |
risch | \(-\frac {1}{3 a \,x^{3}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (-b \,x^{3}-a \right )}{3 a^{2}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 33, normalized size = 0.94 \begin {gather*} \frac {b \log \left (b x^{3} + a\right )}{3 \, a^{2}} - \frac {b \log \left (x^{3}\right )}{3 \, a^{2}} - \frac {1}{3 \, a x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 33, normalized size = 0.94 \begin {gather*} \frac {b x^{3} \log \left (b x^{3} + a\right ) - 3 \, b x^{3} \log \left (x\right ) - a}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.14, size = 31, normalized size = 0.89 \begin {gather*} - \frac {1}{3 a x^{3}} - \frac {b \log {\left (x \right )}}{a^{2}} + \frac {b \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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Giac [A]
time = 1.42, size = 42, normalized size = 1.20 \begin {gather*} \frac {b \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} - \frac {b \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {b x^{3} - a}{3 \, a^{2} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 31, normalized size = 0.89 \begin {gather*} \frac {b\,\ln \left (b\,x^3+a\right )}{3\,a^2}-\frac {1}{3\,a\,x^3}-\frac {b\,\ln \left (x\right )}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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