Optimal. Leaf size=66 \[ -\frac {1}{4 a^2 x^4}+\frac {b}{a^3 x^2}+\frac {b^2}{2 a^3 \left (a+b x^2\right )}+\frac {3 b^2 \log (x)}{a^4}-\frac {3 b^2 \log \left (a+b x^2\right )}{2 a^4} \]
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Rubi [A]
time = 0.03, antiderivative size = 66, 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 {3 b^2 \log \left (a+b x^2\right )}{2 a^4}+\frac {3 b^2 \log (x)}{a^4}+\frac {b^2}{2 a^3 \left (a+b x^2\right )}+\frac {b}{a^3 x^2}-\frac {1}{4 a^2 x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 46
Rule 272
Rubi steps
\begin {align*} \int \frac {1}{x^5 \left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {1}{x^3 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {1}{a^2 x^3}-\frac {2 b}{a^3 x^2}+\frac {3 b^2}{a^4 x}-\frac {b^3}{a^3 (a+b x)^2}-\frac {3 b^3}{a^4 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{4 a^2 x^4}+\frac {b}{a^3 x^2}+\frac {b^2}{2 a^3 \left (a+b x^2\right )}+\frac {3 b^2 \log (x)}{a^4}-\frac {3 b^2 \log \left (a+b x^2\right )}{2 a^4}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 57, normalized size = 0.86 \begin {gather*} \frac {a \left (-\frac {a}{x^4}+\frac {4 b}{x^2}+\frac {2 b^2}{a+b x^2}\right )+12 b^2 \log (x)-6 b^2 \log \left (a+b x^2\right )}{4 a^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 | \(-\frac {b^{3} \left (-\frac {a}{b \left (b \,x^{2}+a \right )}+\frac {3 \ln \left (b \,x^{2}+a \right )}{b}\right )}{2 a^{4}}-\frac {1}{4 a^{2} x^{4}}+\frac {b}{a^{3} x^{2}}+\frac {3 b^{2} \ln \left (x \right )}{a^{4}}\) | \(65\) |
norman | \(\frac {-\frac {1}{4 a}+\frac {3 b \,x^{2}}{4 a^{2}}-\frac {3 b^{3} x^{6}}{2 a^{4}}}{x^{4} \left (b \,x^{2}+a \right )}+\frac {3 b^{2} \ln \left (x \right )}{a^{4}}-\frac {3 b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{4}}\) | \(67\) |
risch | \(\frac {\frac {3 b^{2} x^{4}}{2 a^{3}}+\frac {3 b \,x^{2}}{4 a^{2}}-\frac {1}{4 a}}{x^{4} \left (b \,x^{2}+a \right )}+\frac {3 b^{2} \ln \left (x \right )}{a^{4}}-\frac {3 b^{2} \ln \left (b \,x^{2}+a \right )}{2 a^{4}}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 70, normalized size = 1.06 \begin {gather*} \frac {6 \, b^{2} x^{4} + 3 \, a b x^{2} - a^{2}}{4 \, {\left (a^{3} b x^{6} + a^{4} x^{4}\right )}} - \frac {3 \, b^{2} \log \left (b x^{2} + a\right )}{2 \, a^{4}} + \frac {3 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.38, size = 90, normalized size = 1.36 \begin {gather*} \frac {6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - a^{3} - 6 \, {\left (b^{3} x^{6} + a b^{2} x^{4}\right )} \log \left (b x^{2} + a\right ) + 12 \, {\left (b^{3} x^{6} + a b^{2} x^{4}\right )} \log \left (x\right )}{4 \, {\left (a^{4} b x^{6} + a^{5} x^{4}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.19, size = 68, normalized size = 1.03 \begin {gather*} \frac {- a^{2} + 3 a b x^{2} + 6 b^{2} x^{4}}{4 a^{4} x^{4} + 4 a^{3} b x^{6}} + \frac {3 b^{2} \log {\left (x \right )}}{a^{4}} - \frac {3 b^{2} \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 = 1.17, size = 86, normalized size = 1.30 \begin {gather*} \frac {3 \, b^{2} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {3 \, b^{2} \log \left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac {3 \, b^{3} x^{2} + 4 \, a b^{2}}{2 \, {\left (b x^{2} + a\right )} a^{4}} - \frac {9 \, b^{2} x^{4} - 4 \, a b x^{2} + a^{2}}{4 \, a^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.80, size = 67, normalized size = 1.02 \begin {gather*} \frac {\frac {3\,b\,x^2}{4\,a^2}-\frac {1}{4\,a}+\frac {3\,b^2\,x^4}{2\,a^3}}{b\,x^6+a\,x^4}-\frac {3\,b^2\,\ln \left (b\,x^2+a\right )}{2\,a^4}+\frac {3\,b^2\,\ln \left (x\right )}{a^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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