Optimal. Leaf size=98 \[ \frac{63 b^2 x}{8 a^5}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{11/2}}-\frac{9 x^7}{8 a^2 \left (a x^2+b\right )}-\frac{21 b x^3}{8 a^4}+\frac{63 x^5}{40 a^3}-\frac{x^9}{4 a \left (a x^2+b\right )^2} \]
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Rubi [A] time = 0.0411233, antiderivative size = 98, 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} \[ \frac{63 b^2 x}{8 a^5}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{11/2}}-\frac{9 x^7}{8 a^2 \left (a x^2+b\right )}-\frac{21 b x^3}{8 a^4}+\frac{63 x^5}{40 a^3}-\frac{x^9}{4 a \left (a x^2+b\right )^2} \]
Antiderivative was successfully verified.
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Rule 263
Rule 288
Rule 302
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4}{\left (a+\frac{b}{x^2}\right )^3} \, dx &=\int \frac{x^{10}}{\left (b+a x^2\right )^3} \, dx\\ &=-\frac{x^9}{4 a \left (b+a x^2\right )^2}+\frac{9 \int \frac{x^8}{\left (b+a x^2\right )^2} \, dx}{4 a}\\ &=-\frac{x^9}{4 a \left (b+a x^2\right )^2}-\frac{9 x^7}{8 a^2 \left (b+a x^2\right )}+\frac{63 \int \frac{x^6}{b+a x^2} \, dx}{8 a^2}\\ &=-\frac{x^9}{4 a \left (b+a x^2\right )^2}-\frac{9 x^7}{8 a^2 \left (b+a x^2\right )}+\frac{63 \int \left (\frac{b^2}{a^3}-\frac{b x^2}{a^2}+\frac{x^4}{a}-\frac{b^3}{a^3 \left (b+a x^2\right )}\right ) \, dx}{8 a^2}\\ &=\frac{63 b^2 x}{8 a^5}-\frac{21 b x^3}{8 a^4}+\frac{63 x^5}{40 a^3}-\frac{x^9}{4 a \left (b+a x^2\right )^2}-\frac{9 x^7}{8 a^2 \left (b+a x^2\right )}-\frac{\left (63 b^3\right ) \int \frac{1}{b+a x^2} \, dx}{8 a^5}\\ &=\frac{63 b^2 x}{8 a^5}-\frac{21 b x^3}{8 a^4}+\frac{63 x^5}{40 a^3}-\frac{x^9}{4 a \left (b+a x^2\right )^2}-\frac{9 x^7}{8 a^2 \left (b+a x^2\right )}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.0463756, size = 88, normalized size = 0.9 \[ \frac{168 a^2 b^2 x^5-24 a^3 b x^7+8 a^4 x^9+525 a b^3 x^3+315 b^4 x}{40 a^5 \left (a x^2+b\right )^2}-\frac{63 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{8 a^{11/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 88, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{5\,{a}^{3}}}-{\frac{b{x}^{3}}{{a}^{4}}}+6\,{\frac{{b}^{2}x}{{a}^{5}}}+{\frac{17\,{b}^{3}{x}^{3}}{8\,{a}^{4} \left ( a{x}^{2}+b \right ) ^{2}}}+{\frac{15\,{b}^{4}x}{8\,{a}^{5} \left ( a{x}^{2}+b \right ) ^{2}}}-{\frac{63\,{b}^{3}}{8\,{a}^{5}}\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 = 1.48464, size = 547, normalized size = 5.58 \begin{align*} \left [\frac{16 \, a^{4} x^{9} - 48 \, a^{3} b x^{7} + 336 \, a^{2} b^{2} x^{5} + 1050 \, a b^{3} x^{3} + 630 \, b^{4} x + 315 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{a x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - b}{a x^{2} + b}\right )}{80 \,{\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}}, \frac{8 \, a^{4} x^{9} - 24 \, a^{3} b x^{7} + 168 \, a^{2} b^{2} x^{5} + 525 \, a b^{3} x^{3} + 315 \, b^{4} x - 315 \,{\left (a^{2} b^{2} x^{4} + 2 \, a b^{3} x^{2} + b^{4}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a x \sqrt{\frac{b}{a}}}{b}\right )}{40 \,{\left (a^{7} x^{4} + 2 \, a^{6} b x^{2} + a^{5} b^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.808986, size = 144, normalized size = 1.47 \begin{align*} \frac{63 \sqrt{- \frac{b^{5}}{a^{11}}} \log{\left (- \frac{a^{5} \sqrt{- \frac{b^{5}}{a^{11}}}}{b^{2}} + x \right )}}{16} - \frac{63 \sqrt{- \frac{b^{5}}{a^{11}}} \log{\left (\frac{a^{5} \sqrt{- \frac{b^{5}}{a^{11}}}}{b^{2}} + x \right )}}{16} + \frac{17 a b^{3} x^{3} + 15 b^{4} x}{8 a^{7} x^{4} + 16 a^{6} b x^{2} + 8 a^{5} b^{2}} + \frac{x^{5}}{5 a^{3}} - \frac{b x^{3}}{a^{4}} + \frac{6 b^{2} x}{a^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14722, size = 113, normalized size = 1.15 \begin{align*} -\frac{63 \, b^{3} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{8 \, \sqrt{a b} a^{5}} + \frac{17 \, a b^{3} x^{3} + 15 \, b^{4} x}{8 \,{\left (a x^{2} + b\right )}^{2} a^{5}} + \frac{a^{12} x^{5} - 5 \, a^{11} b x^{3} + 30 \, a^{10} b^{2} x}{5 \, a^{15}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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