Optimal. Leaf size=80 \[ \frac {2 b^3 \log \left (a+b x^2\right )}{a^5}-\frac {4 b^3 \log (x)}{a^5}-\frac {b^3}{2 a^4 \left (a+b x^2\right )}-\frac {3 b^2}{2 a^4 x^2}+\frac {b}{2 a^3 x^4}-\frac {1}{6 a^2 x^6} \]
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Rubi [A] time = 0.05, antiderivative size = 80, 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 = {266, 44} \[ -\frac {b^3}{2 a^4 \left (a+b x^2\right )}-\frac {3 b^2}{2 a^4 x^2}+\frac {2 b^3 \log \left (a+b x^2\right )}{a^5}-\frac {4 b^3 \log (x)}{a^5}+\frac {b}{2 a^3 x^4}-\frac {1}{6 a^2 x^6} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x^7 \left (a+b x^2\right )^2} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x^4 (a+b x)^2} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^2 x^4}-\frac {2 b}{a^3 x^3}+\frac {3 b^2}{a^4 x^2}-\frac {4 b^3}{a^5 x}+\frac {b^4}{a^4 (a+b x)^2}+\frac {4 b^4}{a^5 (a+b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac {1}{6 a^2 x^6}+\frac {b}{2 a^3 x^4}-\frac {3 b^2}{2 a^4 x^2}-\frac {b^3}{2 a^4 \left (a+b x^2\right )}-\frac {4 b^3 \log (x)}{a^5}+\frac {2 b^3 \log \left (a+b x^2\right )}{a^5}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 68, normalized size = 0.85 \[ \frac {a \left (-\frac {a^2}{x^6}-\frac {3 b^3}{a+b x^2}+\frac {3 a b}{x^4}-\frac {9 b^2}{x^2}\right )+12 b^3 \log \left (a+b x^2\right )-24 b^3 \log (x)}{6 a^5} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.61, size = 99, normalized size = 1.24 \[ -\frac {12 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 2 \, a^{3} b x^{2} + a^{4} - 12 \, {\left (b^{4} x^{8} + a b^{3} x^{6}\right )} \log \left (b x^{2} + a\right ) + 24 \, {\left (b^{4} x^{8} + a b^{3} x^{6}\right )} \log \relax (x)}{6 \, {\left (a^{5} b x^{8} + a^{6} x^{6}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.64, size = 99, normalized size = 1.24 \[ -\frac {2 \, b^{3} \log \left (x^{2}\right )}{a^{5}} + \frac {2 \, b^{3} \log \left ({\left | b x^{2} + a \right |}\right )}{a^{5}} - \frac {4 \, b^{4} x^{2} + 5 \, a b^{3}}{2 \, {\left (b x^{2} + a\right )} a^{5}} + \frac {22 \, b^{3} x^{6} - 9 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - a^{3}}{6 \, a^{5} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 73, normalized size = 0.91 \[ -\frac {b^{3}}{2 \left (b \,x^{2}+a \right ) a^{4}}-\frac {4 b^{3} \ln \relax (x )}{a^{5}}+\frac {2 b^{3} \ln \left (b \,x^{2}+a \right )}{a^{5}}-\frac {3 b^{2}}{2 a^{4} x^{2}}+\frac {b}{2 a^{3} x^{4}}-\frac {1}{6 a^{2} x^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 79, normalized size = 0.99 \[ -\frac {12 \, b^{3} x^{6} + 6 \, a b^{2} x^{4} - 2 \, a^{2} b x^{2} + a^{3}}{6 \, {\left (a^{4} b x^{8} + a^{5} x^{6}\right )}} + \frac {2 \, b^{3} \log \left (b x^{2} + a\right )}{a^{5}} - \frac {2 \, b^{3} \log \left (x^{2}\right )}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.12, size = 78, normalized size = 0.98 \[ \frac {2\,b^3\,\ln \left (b\,x^2+a\right )}{a^5}-\frac {\frac {1}{6\,a}-\frac {b\,x^2}{3\,a^2}+\frac {b^2\,x^4}{a^3}+\frac {2\,b^3\,x^6}{a^4}}{b\,x^8+a\,x^6}-\frac {4\,b^3\,\ln \relax (x)}{a^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.47, size = 78, normalized size = 0.98 \[ \frac {- a^{3} + 2 a^{2} b x^{2} - 6 a b^{2} x^{4} - 12 b^{3} x^{6}}{6 a^{5} x^{6} + 6 a^{4} b x^{8}} - \frac {4 b^{3} \log {\relax (x )}}{a^{5}} + \frac {2 b^{3} \log {\left (\frac {a}{b} + x^{2} \right )}}{a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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