Optimal. Leaf size=85 \[ \frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{9/2}}-\frac{7 x^5}{8 a^2 \left (a x^2+b\right )}-\frac{35 b x}{8 a^4}+\frac{35 x^3}{24 a^3}-\frac{x^7}{4 a \left (a x^2+b\right )^2} \]
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Rubi [A] time = 0.0358001, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {263, 288, 302, 205} \[ \frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{9/2}}-\frac{7 x^5}{8 a^2 \left (a x^2+b\right )}-\frac{35 b x}{8 a^4}+\frac{35 x^3}{24 a^3}-\frac{x^7}{4 a \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 288
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^2}{\left (a+\frac{b}{x^2}\right )^3} \, dx &=\int \frac{x^8}{\left (b+a x^2\right )^3} \, dx\\ &=-\frac{x^7}{4 a \left (b+a x^2\right )^2}+\frac{7 \int \frac{x^6}{\left (b+a x^2\right )^2} \, dx}{4 a}\\ &=-\frac{x^7}{4 a \left (b+a x^2\right )^2}-\frac{7 x^5}{8 a^2 \left (b+a x^2\right )}+\frac{35 \int \frac{x^4}{b+a x^2} \, dx}{8 a^2}\\ &=-\frac{x^7}{4 a \left (b+a x^2\right )^2}-\frac{7 x^5}{8 a^2 \left (b+a x^2\right )}+\frac{35 \int \left (-\frac{b}{a^2}+\frac{x^2}{a}+\frac{b^2}{a^2 \left (b+a x^2\right )}\right ) \, dx}{8 a^2}\\ &=-\frac{35 b x}{8 a^4}+\frac{35 x^3}{24 a^3}-\frac{x^7}{4 a \left (b+a x^2\right )^2}-\frac{7 x^5}{8 a^2 \left (b+a x^2\right )}+\frac{\left (35 b^2\right ) \int \frac{1}{b+a x^2} \, dx}{8 a^4}\\ &=-\frac{35 b x}{8 a^4}+\frac{35 x^3}{24 a^3}-\frac{x^7}{4 a \left (b+a x^2\right )^2}-\frac{7 x^5}{8 a^2 \left (b+a x^2\right )}+\frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0437503, size = 77, normalized size = 0.91 \[ \frac{35 b^{3/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{9/2}}-\frac{56 a^2 b x^5-8 a^3 x^7+175 a b^2 x^3+105 b^3 x}{24 a^4 \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.01, size = 77, normalized size = 0.9 \begin{align*}{\frac{{x}^{3}}{3\,{a}^{3}}}-3\,{\frac{bx}{{a}^{4}}}-{\frac{13\,{b}^{2}{x}^{3}}{8\,{a}^{3} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{11\,{b}^{3}x}{8\,{a}^{4} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{35\,{b}^{2}}{8\,{a}^{4}}\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 = 1.47591, size = 493, normalized size = 5.8 \begin{align*} \left [\frac{16 \, a^{3} x^{7} - 112 \, a^{2} b x^{5} - 350 \, a b^{2} x^{3} - 210 \, b^{3} x + 105 \,{\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} + 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right )}{48 \,{\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}}, \frac{8 \, a^{3} x^{7} - 56 \, a^{2} b x^{5} - 175 \, a b^{2} x^{3} - 105 \, b^{3} x + 105 \,{\left (a^{2} b x^{4} + 2 \, a b^{2} x^{2} + b^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a x \sqrt{\frac{b}{a}}}{b}\right )}{24 \,{\left (a^{6} x^{4} + 2 \, a^{5} b x^{2} + a^{4} b^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.783442, size = 131, normalized size = 1.54 \begin{align*} - \frac{35 \sqrt{- \frac{b^{3}}{a^{9}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b^{3}}{a^{9}}}}{b} + x \right )}}{16} + \frac{35 \sqrt{- \frac{b^{3}}{a^{9}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b^{3}}{a^{9}}}}{b} + x \right )}}{16} - \frac{13 a b^{2} x^{3} + 11 b^{3} x}{8 a^{6} x^{4} + 16 a^{5} b x^{2} + 8 a^{4} b^{2}} + \frac{x^{3}}{3 a^{3}} - \frac{3 b x}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.12967, size = 99, normalized size = 1.16 \begin{align*} \frac{35 \, b^{2} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{4}} - \frac{13 \, a b^{2} x^{3} + 11 \, b^{3} x}{8 \,{\left (a x^{2} + b\right )}^{2} a^{4}} + \frac{a^{6} x^{3} - 9 \, a^{5} b x}{3 \, a^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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