Optimal. Leaf size=74 \[ -\frac{5 x^3}{8 a^2 \left (a x^2+b\right )}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{7/2}}+\frac{15 x}{8 a^3}-\frac{x^5}{4 a \left (a x^2+b\right )^2} \]
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Rubi [A] time = 0.0253844, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.444, Rules used = {193, 288, 321, 205} \[ -\frac{5 x^3}{8 a^2 \left (a x^2+b\right )}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{7/2}}+\frac{15 x}{8 a^3}-\frac{x^5}{4 a \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 193
Rule 288
Rule 321
Rule 205
Rubi steps
\begin{align*} \int \frac{1}{\left (a+\frac{b}{x^2}\right )^3} \, dx &=\int \frac{x^6}{\left (b+a x^2\right )^3} \, dx\\ &=-\frac{x^5}{4 a \left (b+a x^2\right )^2}+\frac{5 \int \frac{x^4}{\left (b+a x^2\right )^2} \, dx}{4 a}\\ &=-\frac{x^5}{4 a \left (b+a x^2\right )^2}-\frac{5 x^3}{8 a^2 \left (b+a x^2\right )}+\frac{15 \int \frac{x^2}{b+a x^2} \, dx}{8 a^2}\\ &=\frac{15 x}{8 a^3}-\frac{x^5}{4 a \left (b+a x^2\right )^2}-\frac{5 x^3}{8 a^2 \left (b+a x^2\right )}-\frac{(15 b) \int \frac{1}{b+a x^2} \, dx}{8 a^3}\\ &=\frac{15 x}{8 a^3}-\frac{x^5}{4 a \left (b+a x^2\right )^2}-\frac{5 x^3}{8 a^2 \left (b+a x^2\right )}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.0436836, size = 66, normalized size = 0.89 \[ \frac{8 a^2 x^5+25 a b x^3+15 b^2 x}{8 a^3 \left (a x^2+b\right )^2}-\frac{15 \sqrt{b} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{7/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.008, size = 63, normalized size = 0.9 \begin{align*}{\frac{x}{{a}^{3}}}+{\frac{9\,b{x}^{3}}{8\,{a}^{2} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{7\,{b}^{2}x}{8\,{a}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{15\,b}{8\,{a}^{3}}\arctan \left ({ax{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \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 = 1.51831, size = 425, normalized size = 5.74 \begin{align*} \left [\frac{16 \, a^{2} x^{5} + 50 \, a b x^{3} + 30 \, b^{2} x + 15 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right )}{16 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}}, \frac{8 \, a^{2} x^{5} + 25 \, a b x^{3} + 15 \, b^{2} x - 15 \,{\left (a^{2} x^{4} + 2 \, a b x^{2} + b^{2}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a x \sqrt{\frac{b}{a}}}{b}\right )}{8 \,{\left (a^{5} x^{4} + 2 \, a^{4} b x^{2} + a^{3} b^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.722368, size = 107, normalized size = 1.45 \begin{align*} \frac{15 \sqrt{- \frac{b}{a^{7}}} \log{\left (- a^{3} \sqrt{- \frac{b}{a^{7}}} + x \right )}}{16} - \frac{15 \sqrt{- \frac{b}{a^{7}}} \log{\left (a^{3} \sqrt{- \frac{b}{a^{7}}} + x \right )}}{16} + \frac{9 a b x^{3} + 7 b^{2} x}{8 a^{5} x^{4} + 16 a^{4} b x^{2} + 8 a^{3} b^{2}} + \frac{x}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.19519, size = 73, normalized size = 0.99 \begin{align*} -\frac{15 \, b \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{3}} + \frac{x}{a^{3}} + \frac{9 \, a b x^{3} + 7 \, b^{2} x}{8 \,{\left (a x^{2} + b\right )}^{2} a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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