Optimal. Leaf size=66 \[ -\frac {1}{4 b^2 x^4}+\frac {a}{b^3 x^2}+\frac {a^2}{2 b^3 \left (b+a x^2\right )}+\frac {3 a^2 \log (x)}{b^4}-\frac {3 a^2 \log \left (b+a x^2\right )}{2 b^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, Rules used = {269, 272, 46}
\begin {gather*} -\frac {3 a^2 \log \left (a x^2+b\right )}{2 b^4}+\frac {3 a^2 \log (x)}{b^4}+\frac {a^2}{2 b^3 \left (a x^2+b\right )}+\frac {a}{b^3 x^2}-\frac {1}{4 b^2 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 269
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{\left (a+\frac {b}{x^2}\right )^2 x^9} \, dx &=\int \frac {1}{x^5 \left (b+a x^2\right )^2} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 (b+a x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{b^2 x^3}-\frac {2 a}{b^3 x^2}+\frac {3 a^2}{b^4 x}-\frac {a^3}{b^3 (b+a x)^2}-\frac {3 a^3}{b^4 (b+a x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 b^2 x^4}+\frac {a}{b^3 x^2}+\frac {a^2}{2 b^3 \left (b+a x^2\right )}+\frac {3 a^2 \log (x)}{b^4}-\frac {3 a^2 \log \left (b+a x^2\right )}{2 b^4}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 57, normalized size = 0.86 \begin {gather*} \frac {b \left (-\frac {b}{x^4}+\frac {4 a}{x^2}+\frac {2 a^2}{b+a x^2}\right )+12 a^2 \log (x)-6 a^2 \log \left (b+a x^2\right )}{4 b^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 65, normalized size = 0.98
method | result | size |
default | \(-\frac {a^{3} \left (-\frac {b}{a \left (a \,x^{2}+b \right )}+\frac {3 \ln \left (a \,x^{2}+b \right )}{a}\right )}{2 b^{4}}-\frac {1}{4 b^{2} x^{4}}+\frac {a}{b^{3} x^{2}}+\frac {3 a^{2} \ln \left (x \right )}{b^{4}}\) | \(65\) |
risch | \(\frac {\frac {3 a^{2} x^{4}}{2 b^{3}}+\frac {3 a \,x^{2}}{4 b^{2}}-\frac {1}{4 b}}{\left (a \,x^{2}+b \right ) x^{4}}+\frac {3 a^{2} \ln \left (x \right )}{b^{4}}-\frac {3 a^{2} \ln \left (a \,x^{2}+b \right )}{2 b^{4}}\) | \(67\) |
norman | \(\frac {-\frac {x^{4}}{4 b}+\frac {3 a \,x^{6}}{4 b^{2}}-\frac {3 a^{3} x^{10}}{2 b^{4}}}{\left (a \,x^{2}+b \right ) x^{8}}+\frac {3 a^{2} \ln \left (x \right )}{b^{4}}-\frac {3 a^{2} \ln \left (a \,x^{2}+b \right )}{2 b^{4}}\) | \(70\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 70, normalized size = 1.06 \begin {gather*} \frac {6 \, a^{2} x^{4} + 3 \, a b x^{2} - b^{2}}{4 \, {\left (a b^{3} x^{6} + b^{4} x^{4}\right )}} - \frac {3 \, a^{2} \log \left (a x^{2} + b\right )}{2 \, b^{4}} + \frac {3 \, a^{2} \log \left (x^{2}\right )}{2 \, b^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 90, normalized size = 1.36 \begin {gather*} \frac {6 \, a^{2} b x^{4} + 3 \, a b^{2} x^{2} - b^{3} - 6 \, {\left (a^{3} x^{6} + a^{2} b x^{4}\right )} \log \left (a x^{2} + b\right ) + 12 \, {\left (a^{3} x^{6} + a^{2} b x^{4}\right )} \log \left (x\right )}{4 \, {\left (a b^{4} x^{6} + b^{5} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.20, size = 68, normalized size = 1.03 \begin {gather*} \frac {3 a^{2} \log {\left (x \right )}}{b^{4}} - \frac {3 a^{2} \log {\left (x^{2} + \frac {b}{a} \right )}}{2 b^{4}} + \frac {6 a^{2} x^{4} + 3 a b x^{2} - b^{2}}{4 a b^{3} x^{6} + 4 b^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.87, size = 86, normalized size = 1.30 \begin {gather*} \frac {3 \, a^{2} \log \left (x^{2}\right )}{2 \, b^{4}} - \frac {3 \, a^{2} \log \left ({\left | a x^{2} + b \right |}\right )}{2 \, b^{4}} + \frac {3 \, a^{3} x^{2} + 4 \, a^{2} b}{2 \, {\left (a x^{2} + b\right )} b^{4}} - \frac {9 \, a^{2} x^{4} - 4 \, a b x^{2} + b^{2}}{4 \, b^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.08, size = 67, normalized size = 1.02 \begin {gather*} \frac {\frac {3\,a\,x^2}{4\,b^2}-\frac {1}{4\,b}+\frac {3\,a^2\,x^4}{2\,b^3}}{a\,x^6+b\,x^4}-\frac {3\,a^2\,\ln \left (a\,x^2+b\right )}{2\,b^4}+\frac {3\,a^2\,\ln \left (x\right )}{b^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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