Optimal. Leaf size=91 \[ -\frac {\log \left (a-b x^2\right )}{2 a^5}+\frac {\log (x)}{a^5}+\frac {1}{2 a^4 \left (a-b x^2\right )}+\frac {1}{4 a^3 \left (a-b x^2\right )^2}+\frac {1}{6 a^2 \left (a-b x^2\right )^3}+\frac {1}{8 a \left (a-b x^2\right )^4} \]
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Rubi [A] time = 0.06, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {266, 44} \[ \frac {1}{2 a^4 \left (a-b x^2\right )}+\frac {1}{4 a^3 \left (a-b x^2\right )^2}+\frac {1}{6 a^2 \left (a-b x^2\right )^3}-\frac {\log \left (a-b x^2\right )}{2 a^5}+\frac {\log (x)}{a^5}+\frac {1}{8 a \left (a-b x^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 44
Rule 266
Rubi steps
\begin {align*} \int \frac {1}{x \left (a-b x^2\right )^5} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {1}{x (a-b x)^5} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {1}{a^5 x}+\frac {b}{a (a-b x)^5}+\frac {b}{a^2 (a-b x)^4}+\frac {b}{a^3 (a-b x)^3}+\frac {b}{a^4 (a-b x)^2}+\frac {b}{a^5 (a-b x)}\right ) \, dx,x,x^2\right )\\ &=\frac {1}{8 a \left (a-b x^2\right )^4}+\frac {1}{6 a^2 \left (a-b x^2\right )^3}+\frac {1}{4 a^3 \left (a-b x^2\right )^2}+\frac {1}{2 a^4 \left (a-b x^2\right )}+\frac {\log (x)}{a^5}-\frac {\log \left (a-b x^2\right )}{2 a^5}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 67, normalized size = 0.74 \[ \frac {\frac {a \left (25 a^3-52 a^2 b x^2+42 a b^2 x^4-12 b^3 x^6\right )}{\left (a-b x^2\right )^4}-12 \log \left (a-b x^2\right )+24 \log (x)}{24 a^5} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 180, normalized size = 1.98 \[ -\frac {12 \, a b^{3} x^{6} - 42 \, a^{2} b^{2} x^{4} + 52 \, a^{3} b x^{2} - 25 \, a^{4} + 12 \, {\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \log \left (b x^{2} - a\right ) - 24 \, {\left (b^{4} x^{8} - 4 \, a b^{3} x^{6} + 6 \, a^{2} b^{2} x^{4} - 4 \, a^{3} b x^{2} + a^{4}\right )} \log \relax (x)}{24 \, {\left (a^{5} b^{4} x^{8} - 4 \, a^{6} b^{3} x^{6} + 6 \, a^{7} b^{2} x^{4} - 4 \, a^{8} b x^{2} + a^{9}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.61, size = 85, normalized size = 0.93 \[ \frac {\log \left (x^{2}\right )}{2 \, a^{5}} - \frac {\log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{5}} + \frac {25 \, b^{4} x^{8} - 112 \, a b^{3} x^{6} + 192 \, a^{2} b^{2} x^{4} - 152 \, a^{3} b x^{2} + 50 \, a^{4}}{24 \, {\left (b x^{2} - a\right )}^{4} a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 87, normalized size = 0.96 \[ \frac {1}{8 \left (b \,x^{2}-a \right )^{4} a}-\frac {1}{6 \left (b \,x^{2}-a \right )^{3} a^{2}}+\frac {1}{4 \left (b \,x^{2}-a \right )^{2} a^{3}}-\frac {1}{2 \left (b \,x^{2}-a \right ) a^{4}}+\frac {\ln \relax (x )}{a^{5}}-\frac {\ln \left (b \,x^{2}-a \right )}{2 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.39, size = 106, normalized size = 1.16 \[ -\frac {12 \, b^{3} x^{6} - 42 \, a b^{2} x^{4} + 52 \, a^{2} b x^{2} - 25 \, a^{3}}{24 \, {\left (a^{4} b^{4} x^{8} - 4 \, a^{5} b^{3} x^{6} + 6 \, a^{6} b^{2} x^{4} - 4 \, a^{7} b x^{2} + a^{8}\right )}} - \frac {\log \left (b x^{2} - a\right )}{2 \, a^{5}} + \frac {\log \left (x^{2}\right )}{2 \, a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.24, size = 101, normalized size = 1.11 \[ \frac {\ln \relax (x)}{a^5}+\frac {\frac {25}{24\,a}-\frac {13\,b\,x^2}{6\,a^2}+\frac {7\,b^2\,x^4}{4\,a^3}-\frac {b^3\,x^6}{2\,a^4}}{a^4-4\,a^3\,b\,x^2+6\,a^2\,b^2\,x^4-4\,a\,b^3\,x^6+b^4\,x^8}-\frac {\ln \left (a-b\,x^2\right )}{2\,a^5} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.64, size = 104, normalized size = 1.14 \[ - \frac {- 25 a^{3} + 52 a^{2} b x^{2} - 42 a b^{2} x^{4} + 12 b^{3} x^{6}}{24 a^{8} - 96 a^{7} b x^{2} + 144 a^{6} b^{2} x^{4} - 96 a^{5} b^{3} x^{6} + 24 a^{4} b^{4} x^{8}} + \frac {\log {\relax (x )}}{a^{5}} - \frac {\log {\left (- \frac {a}{b} + x^{2} \right )}}{2 a^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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