Optimal. Leaf size=106 \[ \frac{2 b}{a^5 \left (a-b x^2\right )}+\frac{3 b}{4 a^4 \left (a-b x^2\right )^2}+\frac{b}{3 a^3 \left (a-b x^2\right )^3}+\frac{b}{8 a^2 \left (a-b x^2\right )^4}-\frac{5 b \log \left (a-b x^2\right )}{2 a^6}+\frac{5 b \log (x)}{a^6}-\frac{1}{2 a^5 x^2} \]
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Rubi [A] time = 0.0870354, antiderivative size = 106, normalized size of antiderivative = 1., 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} \[ \frac{2 b}{a^5 \left (a-b x^2\right )}+\frac{3 b}{4 a^4 \left (a-b x^2\right )^2}+\frac{b}{3 a^3 \left (a-b x^2\right )^3}+\frac{b}{8 a^2 \left (a-b x^2\right )^4}-\frac{5 b \log \left (a-b x^2\right )}{2 a^6}+\frac{5 b \log (x)}{a^6}-\frac{1}{2 a^5 x^2} \]
Antiderivative was successfully verified.
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Rule 266
Rule 44
Rubi steps
\begin{align*} \int \frac{1}{x^3 \left (a-b x^2\right )^5} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{1}{x^2 (a-b x)^5} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left (\frac{1}{a^5 x^2}+\frac{5 b}{a^6 x}+\frac{b^2}{a^2 (a-b x)^5}+\frac{2 b^2}{a^3 (a-b x)^4}+\frac{3 b^2}{a^4 (a-b x)^3}+\frac{4 b^2}{a^5 (a-b x)^2}+\frac{5 b^2}{a^6 (a-b x)}\right ) \, dx,x,x^2\right )\\ &=-\frac{1}{2 a^5 x^2}+\frac{b}{8 a^2 \left (a-b x^2\right )^4}+\frac{b}{3 a^3 \left (a-b x^2\right )^3}+\frac{3 b}{4 a^4 \left (a-b x^2\right )^2}+\frac{2 b}{a^5 \left (a-b x^2\right )}+\frac{5 b \log (x)}{a^6}-\frac{5 b \log \left (a-b x^2\right )}{2 a^6}\\ \end{align*}
Mathematica [A] time = 0.0631878, size = 83, normalized size = 0.78 \[ \frac{\frac{a \left (-260 a^2 b^2 x^4+125 a^3 b x^2-12 a^4+210 a b^3 x^6-60 b^4 x^8\right )}{x^2 \left (a-b x^2\right )^4}-60 b \log \left (a-b x^2\right )+120 b \log (x)}{24 a^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.015, size = 102, normalized size = 1. \begin{align*} -{\frac{1}{2\,{a}^{5}{x}^{2}}}+5\,{\frac{b\ln \left ( x \right ) }{{a}^{6}}}-2\,{\frac{b}{{a}^{5} \left ( b{x}^{2}-a \right ) }}-{\frac{b}{3\,{a}^{3} \left ( b{x}^{2}-a \right ) ^{3}}}+{\frac{b}{8\,{a}^{2} \left ( b{x}^{2}-a \right ) ^{4}}}+{\frac{3\,b}{4\,{a}^{4} \left ( b{x}^{2}-a \right ) ^{2}}}-{\frac{5\,b\ln \left ( b{x}^{2}-a \right ) }{2\,{a}^{6}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.12544, size = 166, normalized size = 1.57 \begin{align*} -\frac{60 \, b^{4} x^{8} - 210 \, a b^{3} x^{6} + 260 \, a^{2} b^{2} x^{4} - 125 \, a^{3} b x^{2} + 12 \, a^{4}}{24 \,{\left (a^{5} b^{4} x^{10} - 4 \, a^{6} b^{3} x^{8} + 6 \, a^{7} b^{2} x^{6} - 4 \, a^{8} b x^{4} + a^{9} x^{2}\right )}} - \frac{5 \, b \log \left (b x^{2} - a\right )}{2 \, a^{6}} + \frac{5 \, b \log \left (x^{2}\right )}{2 \, a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.27491, size = 440, normalized size = 4.15 \begin{align*} -\frac{60 \, a b^{4} x^{8} - 210 \, a^{2} b^{3} x^{6} + 260 \, a^{3} b^{2} x^{4} - 125 \, a^{4} b x^{2} + 12 \, a^{5} + 60 \,{\left (b^{5} x^{10} - 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} - 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (b x^{2} - a\right ) - 120 \,{\left (b^{5} x^{10} - 4 \, a b^{4} x^{8} + 6 \, a^{2} b^{3} x^{6} - 4 \, a^{3} b^{2} x^{4} + a^{4} b x^{2}\right )} \log \left (x\right )}{24 \,{\left (a^{6} b^{4} x^{10} - 4 \, a^{7} b^{3} x^{8} + 6 \, a^{8} b^{2} x^{6} - 4 \, a^{9} b x^{4} + a^{10} x^{2}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 2.81119, size = 126, normalized size = 1.19 \begin{align*} - \frac{12 a^{4} - 125 a^{3} b x^{2} + 260 a^{2} b^{2} x^{4} - 210 a b^{3} x^{6} + 60 b^{4} x^{8}}{24 a^{9} x^{2} - 96 a^{8} b x^{4} + 144 a^{7} b^{2} x^{6} - 96 a^{6} b^{3} x^{8} + 24 a^{5} b^{4} x^{10}} + \frac{5 b \log{\left (x \right )}}{a^{6}} - \frac{5 b \log{\left (- \frac{a}{b} + x^{2} \right )}}{2 a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.55494, size = 143, normalized size = 1.35 \begin{align*} \frac{5 \, b \log \left (x^{2}\right )}{2 \, a^{6}} - \frac{5 \, b \log \left ({\left | b x^{2} - a \right |}\right )}{2 \, a^{6}} - \frac{5 \, b x^{2} + a}{2 \, a^{6} x^{2}} + \frac{125 \, b^{5} x^{8} - 548 \, a b^{4} x^{6} + 912 \, a^{2} b^{3} x^{4} - 688 \, a^{3} b^{2} x^{2} + 202 \, a^{4} b}{24 \,{\left (b x^{2} - a\right )}^{4} a^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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