Optimal. Leaf size=118 \[ \frac{105}{128 a^4 x \left (a-b x^2\right )}+\frac{21}{64 a^3 x \left (a-b x^2\right )^2}+\frac{3}{16 a^2 x \left (a-b x^2\right )^3}+\frac{315 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{11/2}}-\frac{315}{128 a^5 x}+\frac{1}{8 a x \left (a-b x^2\right )^4} \]
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Rubi [A] time = 0.0487778, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 3, integrand size = 14, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {290, 325, 208} \[ \frac{105}{128 a^4 x \left (a-b x^2\right )}+\frac{21}{64 a^3 x \left (a-b x^2\right )^2}+\frac{3}{16 a^2 x \left (a-b x^2\right )^3}+\frac{315 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{11/2}}-\frac{315}{128 a^5 x}+\frac{1}{8 a x \left (a-b x^2\right )^4} \]
Antiderivative was successfully verified.
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Rule 290
Rule 325
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{x^2 \left (a-b x^2\right )^5} \, dx &=\frac{1}{8 a x \left (a-b x^2\right )^4}+\frac{9 \int \frac{1}{x^2 \left (a-b x^2\right )^4} \, dx}{8 a}\\ &=\frac{1}{8 a x \left (a-b x^2\right )^4}+\frac{3}{16 a^2 x \left (a-b x^2\right )^3}+\frac{21 \int \frac{1}{x^2 \left (a-b x^2\right )^3} \, dx}{16 a^2}\\ &=\frac{1}{8 a x \left (a-b x^2\right )^4}+\frac{3}{16 a^2 x \left (a-b x^2\right )^3}+\frac{21}{64 a^3 x \left (a-b x^2\right )^2}+\frac{105 \int \frac{1}{x^2 \left (a-b x^2\right )^2} \, dx}{64 a^3}\\ &=\frac{1}{8 a x \left (a-b x^2\right )^4}+\frac{3}{16 a^2 x \left (a-b x^2\right )^3}+\frac{21}{64 a^3 x \left (a-b x^2\right )^2}+\frac{105}{128 a^4 x \left (a-b x^2\right )}+\frac{315 \int \frac{1}{x^2 \left (a-b x^2\right )} \, dx}{128 a^4}\\ &=-\frac{315}{128 a^5 x}+\frac{1}{8 a x \left (a-b x^2\right )^4}+\frac{3}{16 a^2 x \left (a-b x^2\right )^3}+\frac{21}{64 a^3 x \left (a-b x^2\right )^2}+\frac{105}{128 a^4 x \left (a-b x^2\right )}+\frac{(315 b) \int \frac{1}{a-b x^2} \, dx}{128 a^5}\\ &=-\frac{315}{128 a^5 x}+\frac{1}{8 a x \left (a-b x^2\right )^4}+\frac{3}{16 a^2 x \left (a-b x^2\right )^3}+\frac{21}{64 a^3 x \left (a-b x^2\right )^2}+\frac{105}{128 a^4 x \left (a-b x^2\right )}+\frac{315 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.0530645, size = 92, normalized size = 0.78 \[ \frac{\frac{\sqrt{a} \left (-1533 a^2 b^2 x^4+837 a^3 b x^2-128 a^4+1155 a b^3 x^6-315 b^4 x^8\right )}{x \left (a-b x^2\right )^4}+315 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{128 a^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 78, normalized size = 0.7 \begin{align*} -{\frac{1}{{a}^{5}x}}-{\frac{b}{{a}^{5}} \left ({\frac{1}{ \left ( b{x}^{2}-a \right ) ^{4}} \left ({\frac{187\,{b}^{3}{x}^{7}}{128}}-{\frac{643\,a{b}^{2}{x}^{5}}{128}}+{\frac{765\,{a}^{2}b{x}^{3}}{128}}-{\frac{325\,{a}^{3}x}{128}} \right ) }-{\frac{315}{128}{\it Artanh} \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.28183, size = 721, normalized size = 6.11 \begin{align*} \left [-\frac{630 \, b^{4} x^{8} - 2310 \, a b^{3} x^{6} + 3066 \, a^{2} b^{2} x^{4} - 1674 \, a^{3} b x^{2} + 256 \, a^{4} - 315 \,{\left (b^{4} x^{9} - 4 \, a b^{3} x^{7} + 6 \, a^{2} b^{2} x^{5} - 4 \, a^{3} b x^{3} + a^{4} x\right )} \sqrt{\frac{b}{a}} \log \left (\frac{b x^{2} + 2 \, a x \sqrt{\frac{b}{a}} + a}{b x^{2} - a}\right )}{256 \,{\left (a^{5} b^{4} x^{9} - 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} - 4 \, a^{8} b x^{3} + a^{9} x\right )}}, -\frac{315 \, b^{4} x^{8} - 1155 \, a b^{3} x^{6} + 1533 \, a^{2} b^{2} x^{4} - 837 \, a^{3} b x^{2} + 128 \, a^{4} + 315 \,{\left (b^{4} x^{9} - 4 \, a b^{3} x^{7} + 6 \, a^{2} b^{2} x^{5} - 4 \, a^{3} b x^{3} + a^{4} x\right )} \sqrt{-\frac{b}{a}} \arctan \left (x \sqrt{-\frac{b}{a}}\right )}{128 \,{\left (a^{5} b^{4} x^{9} - 4 \, a^{6} b^{3} x^{7} + 6 \, a^{7} b^{2} x^{5} - 4 \, a^{8} b x^{3} + a^{9} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.84662, size = 155, normalized size = 1.31 \begin{align*} - \frac{315 \sqrt{\frac{b}{a^{11}}} \log{\left (- \frac{a^{6} \sqrt{\frac{b}{a^{11}}}}{b} + x \right )}}{256} + \frac{315 \sqrt{\frac{b}{a^{11}}} \log{\left (\frac{a^{6} \sqrt{\frac{b}{a^{11}}}}{b} + x \right )}}{256} - \frac{128 a^{4} - 837 a^{3} b x^{2} + 1533 a^{2} b^{2} x^{4} - 1155 a b^{3} x^{6} + 315 b^{4} x^{8}}{128 a^{9} x - 512 a^{8} b x^{3} + 768 a^{7} b^{2} x^{5} - 512 a^{6} b^{3} x^{7} + 128 a^{5} b^{4} x^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 2.37003, size = 112, normalized size = 0.95 \begin{align*} -\frac{315 \, b \arctan \left (\frac{b x}{\sqrt{-a b}}\right )}{128 \, \sqrt{-a b} a^{5}} - \frac{1}{a^{5} x} - \frac{187 \, b^{4} x^{7} - 643 \, a b^{3} x^{5} + 765 \, a^{2} b^{2} x^{3} - 325 \, a^{3} b x}{128 \,{\left (b x^{2} - a\right )}^{4} a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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