Optimal. Leaf size=79 \[ \frac{7 b^2 x}{2 a^4}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{9/2}}-\frac{7 b x^3}{6 a^3}+\frac{7 x^5}{10 a^2}-\frac{x^7}{2 a \left (a x^2+b\right )} \]
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Rubi [A] time = 0.0327662, antiderivative size = 79, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {263, 288, 302, 205} \[ \frac{7 b^2 x}{2 a^4}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{9/2}}-\frac{7 b x^3}{6 a^3}+\frac{7 x^5}{10 a^2}-\frac{x^7}{2 a \left (a x^2+b\right )} \]
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 )^2} \, dx &=\int \frac{x^8}{\left (b+a x^2\right )^2} \, dx\\ &=-\frac{x^7}{2 a \left (b+a x^2\right )}+\frac{7 \int \frac{x^6}{b+a x^2} \, dx}{2 a}\\ &=-\frac{x^7}{2 a \left (b+a x^2\right )}+\frac{7 \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}{2 a}\\ &=\frac{7 b^2 x}{2 a^4}-\frac{7 b x^3}{6 a^3}+\frac{7 x^5}{10 a^2}-\frac{x^7}{2 a \left (b+a x^2\right )}-\frac{\left (7 b^3\right ) \int \frac{1}{b+a x^2} \, dx}{2 a^4}\\ &=\frac{7 b^2 x}{2 a^4}-\frac{7 b x^3}{6 a^3}+\frac{7 x^5}{10 a^2}-\frac{x^7}{2 a \left (b+a x^2\right )}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0455872, size = 71, normalized size = 0.9 \[ \frac{x \left (6 a^2 x^4+\frac{15 b^3}{a x^2+b}-20 a b x^2+90 b^2\right )}{30 a^4}-\frac{7 b^{5/2} \tan ^{-1}\left (\frac{\sqrt{a} x}{\sqrt{b}}\right )}{2 a^{9/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 68, normalized size = 0.9 \begin{align*}{\frac{{x}^{5}}{5\,{a}^{2}}}-{\frac{2\,b{x}^{3}}{3\,{a}^{3}}}+3\,{\frac{{b}^{2}x}{{a}^{4}}}+{\frac{{b}^{3}x}{2\,{a}^{4} \left ( a{x}^{2}+b \right ) }}-{\frac{7\,{b}^{3}}{2\,{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 = 1.48904, size = 409, normalized size = 5.18 \begin{align*} \left [\frac{12 \, a^{3} x^{7} - 28 \, a^{2} b x^{5} + 140 \, a b^{2} x^{3} + 210 \, b^{3} x + 105 \,{\left (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 )}{60 \,{\left (a^{5} x^{2} + a^{4} b\right )}}, \frac{6 \, a^{3} x^{7} - 14 \, a^{2} b x^{5} + 70 \, a b^{2} x^{3} + 105 \, b^{3} x - 105 \,{\left (a b^{2} x^{2} + b^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (\frac{a x \sqrt{\frac{b}{a}}}{b}\right )}{30 \,{\left (a^{5} x^{2} + a^{4} b\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.580015, size = 124, normalized size = 1.57 \begin{align*} \frac{b^{3} x}{2 a^{5} x^{2} + 2 a^{4} b} + \frac{7 \sqrt{- \frac{b^{5}}{a^{9}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{b^{5}}{a^{9}}}}{b^{2}} + x \right )}}{4} - \frac{7 \sqrt{- \frac{b^{5}}{a^{9}}} \log{\left (\frac{a^{4} \sqrt{- \frac{b^{5}}{a^{9}}}}{b^{2}} + x \right )}}{4} + \frac{x^{5}}{5 a^{2}} - \frac{2 b x^{3}}{3 a^{3}} + \frac{3 b^{2} x}{a^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15071, size = 99, normalized size = 1.25 \begin{align*} -\frac{7 \, b^{3} \arctan \left (\frac{a x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{4}} + \frac{b^{3} x}{2 \,{\left (a x^{2} + b\right )} a^{4}} + \frac{3 \, a^{8} x^{5} - 10 \, a^{7} b x^{3} + 45 \, a^{6} b^{2} x}{15 \, a^{10}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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