Optimal. Leaf size=35 \[ -\frac {1}{5 a x^5}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^5\right )}{5 a^2} \]
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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^5\right )}{5 a^2}-\frac {b \log (x)}{a^2}-\frac {1}{5 a x^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^6 \left (a+b x^5\right )} \, dx &=\frac {1}{5} \text {Subst}\left (\int \frac {1}{x^2 (a+b x)} \, dx,x,x^5\right )\\ &=\frac {1}{5} \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^5\right )\\ &=-\frac {1}{5 a x^5}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^5\right )}{5 a^2}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 35, normalized size = 1.00 \begin {gather*} -\frac {1}{5 a x^5}-\frac {b \log (x)}{a^2}+\frac {b \log \left (a+b x^5\right )}{5 a^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 32, normalized size = 0.91
method | result | size |
default | \(-\frac {1}{5 a \,x^{5}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b \,x^{5}+a \right )}{5 a^{2}}\) | \(32\) |
norman | \(-\frac {1}{5 a \,x^{5}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (b \,x^{5}+a \right )}{5 a^{2}}\) | \(32\) |
risch | \(-\frac {1}{5 a \,x^{5}}-\frac {b \ln \left (x \right )}{a^{2}}+\frac {b \ln \left (-b \,x^{5}-a \right )}{5 a^{2}}\) | \(35\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 33, normalized size = 0.94 \begin {gather*} \frac {b \log \left (b x^{5} + a\right )}{5 \, a^{2}} - \frac {b \log \left (x^{5}\right )}{5 \, a^{2}} - \frac {1}{5 \, a x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 33, normalized size = 0.94 \begin {gather*} \frac {b x^{5} \log \left (b x^{5} + a\right ) - 5 \, b x^{5} \log \left (x\right ) - a}{5 \, a^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 31, normalized size = 0.89 \begin {gather*} - \frac {1}{5 a x^{5}} - \frac {b \log {\left (x \right )}}{a^{2}} + \frac {b \log {\left (\frac {a}{b} + x^{5} \right )}}{5 a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.12, size = 42, normalized size = 1.20 \begin {gather*} \frac {b \log \left ({\left | b x^{5} + a \right |}\right )}{5 \, a^{2}} - \frac {b \log \left ({\left | x \right |}\right )}{a^{2}} + \frac {b x^{5} - a}{5 \, a^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.07, size = 31, normalized size = 0.89 \begin {gather*} \frac {b\,\ln \left (b\,x^5+a\right )}{5\,a^2}-\frac {1}{5\,a\,x^5}-\frac {b\,\ln \left (x\right )}{a^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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