Optimal. Leaf size=63 \[ -\frac {1}{6 a x^6}+\frac {b}{4 a^2 x^4}-\frac {b^2}{2 a^3 x^2}-\frac {b^3 \log (x)}{a^4}+\frac {b^3 \log \left (a+b x^2\right )}{2 a^4} \]
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Rubi [A]
time = 0.02, antiderivative size = 63, 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^3 \log \left (a+b x^2\right )}{2 a^4}-\frac {b^3 \log (x)}{a^4}-\frac {b^2}{2 a^3 x^2}+\frac {b}{4 a^2 x^4}-\frac {1}{6 a x^6} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (a+b x^2\right )} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^4 (a+b x)} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a x^4}-\frac {b}{a^2 x^3}+\frac {b^2}{a^3 x^2}-\frac {b^3}{a^4 x}+\frac {b^4}{a^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{6 a x^6}+\frac {b}{4 a^2 x^4}-\frac {b^2}{2 a^3 x^2}-\frac {b^3 \log (x)}{a^4}+\frac {b^3 \log \left (a+b x^2\right )}{2 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 63, normalized size = 1.00 \begin {gather*} -\frac {1}{6 a x^6}+\frac {b}{4 a^2 x^4}-\frac {b^2}{2 a^3 x^2}-\frac {b^3 \log (x)}{a^4}+\frac {b^3 \log \left (a+b x^2\right )}{2 a^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 56, normalized size = 0.89
method | result | size |
default | \(-\frac {1}{6 a \,x^{6}}+\frac {b}{4 a^{2} x^{4}}-\frac {b^{2}}{2 a^{3} x^{2}}-\frac {b^{3} \ln \left (x \right )}{a^{4}}+\frac {b^{3} \ln \left (b \,x^{2}+a \right )}{2 a^{4}}\) | \(56\) |
norman | \(\frac {-\frac {1}{6 a}+\frac {b \,x^{2}}{4 a^{2}}-\frac {b^{2} x^{4}}{2 a^{3}}}{x^{6}}-\frac {b^{3} \ln \left (x \right )}{a^{4}}+\frac {b^{3} \ln \left (b \,x^{2}+a \right )}{2 a^{4}}\) | \(58\) |
risch | \(\frac {-\frac {1}{6 a}+\frac {b \,x^{2}}{4 a^{2}}-\frac {b^{2} x^{4}}{2 a^{3}}}{x^{6}}-\frac {b^{3} \ln \left (x \right )}{a^{4}}+\frac {b^{3} \ln \left (-b \,x^{2}-a \right )}{2 a^{4}}\) | \(61\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 58, normalized size = 0.92 \begin {gather*} \frac {b^{3} \log \left (b x^{2} + a\right )}{2 \, a^{4}} - \frac {b^{3} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {6 \, b^{2} x^{4} - 3 \, a b x^{2} + 2 \, a^{2}}{12 \, a^{3} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.16, size = 58, normalized size = 0.92 \begin {gather*} \frac {6 \, b^{3} x^{6} \log \left (b x^{2} + a\right ) - 12 \, b^{3} x^{6} \log \left (x\right ) - 6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, a^{4} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.15, size = 56, normalized size = 0.89 \begin {gather*} \frac {- 2 a^{2} + 3 a b x^{2} - 6 b^{2} x^{4}}{12 a^{3} x^{6}} - \frac {b^{3} \log {\left (x \right )}}{a^{4}} + \frac {b^{3} \log {\left (\frac {a}{b} + x^{2} \right )}}{2 a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 70, normalized size = 1.11 \begin {gather*} -\frac {b^{3} \log \left (x^{2}\right )}{2 \, a^{4}} + \frac {b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac {11 \, b^{3} x^{6} - 6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, a^{4} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.64, size = 58, normalized size = 0.92 \begin {gather*} \frac {b^3\,\ln \left (b\,x^2+a\right )}{2\,a^4}-\frac {\frac {1}{6\,a}-\frac {b\,x^2}{4\,a^2}+\frac {b^2\,x^4}{2\,a^3}}{x^6}-\frac {b^3\,\ln \left (x\right )}{a^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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