Optimal. Leaf size=104 \[ \frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{9/2}}-\frac{b x^5 (b c-2 a d)}{5 d^2}+\frac{x^3 (b c-a d)^2}{3 d^3}-\frac{c x (b c-a d)^2}{d^4}+\frac{b^2 x^7}{7 d} \]
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Rubi [A] time = 0.0737845, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {461, 205} \[ \frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{9/2}}-\frac{b x^5 (b c-2 a d)}{5 d^2}+\frac{x^3 (b c-a d)^2}{3 d^3}-\frac{c x (b c-a d)^2}{d^4}+\frac{b^2 x^7}{7 d} \]
Antiderivative was successfully verified.
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Rule 461
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4 \left (a+b x^2\right )^2}{c+d x^2} \, dx &=\int \left (-\frac{c (b c-a d)^2}{d^4}+\frac{(b c-a d)^2 x^2}{d^3}-\frac{b (b c-2 a d) x^4}{d^2}+\frac{b^2 x^6}{d}+\frac{b^2 c^4-2 a b c^3 d+a^2 c^2 d^2}{d^4 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac{c (b c-a d)^2 x}{d^4}+\frac{(b c-a d)^2 x^3}{3 d^3}-\frac{b (b c-2 a d) x^5}{5 d^2}+\frac{b^2 x^7}{7 d}+\frac{\left (c^2 (b c-a d)^2\right ) \int \frac{1}{c+d x^2} \, dx}{d^4}\\ &=-\frac{c (b c-a d)^2 x}{d^4}+\frac{(b c-a d)^2 x^3}{3 d^3}-\frac{b (b c-2 a d) x^5}{5 d^2}+\frac{b^2 x^7}{7 d}+\frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{9/2}}\\ \end{align*}
Mathematica [A] time = 0.0930724, size = 104, normalized size = 1. \[ \frac{c^{3/2} (b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{d^{9/2}}-\frac{b x^5 (b c-2 a d)}{5 d^2}+\frac{x^3 (a d-b c)^2}{3 d^3}-\frac{c x (b c-a d)^2}{d^4}+\frac{b^2 x^7}{7 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 176, normalized size = 1.7 \begin{align*}{\frac{{b}^{2}{x}^{7}}{7\,d}}+{\frac{2\,{x}^{5}ab}{5\,d}}-{\frac{{x}^{5}{b}^{2}c}{5\,{d}^{2}}}+{\frac{{x}^{3}{a}^{2}}{3\,d}}-{\frac{2\,{x}^{3}abc}{3\,{d}^{2}}}+{\frac{{x}^{3}{b}^{2}{c}^{2}}{3\,{d}^{3}}}-{\frac{{a}^{2}cx}{{d}^{2}}}+2\,{\frac{ab{c}^{2}x}{{d}^{3}}}-{\frac{{b}^{2}{c}^{3}x}{{d}^{4}}}+{\frac{{a}^{2}{c}^{2}}{{d}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-2\,{\frac{ab{c}^{3}}{{d}^{3}\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+{\frac{{b}^{2}{c}^{4}}{{d}^{4}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}} \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 = 1.30505, size = 649, normalized size = 6.24 \begin{align*} \left [\frac{30 \, b^{2} d^{3} x^{7} - 42 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{5} + 70 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + 105 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt{-\frac{c}{d}} \log \left (\frac{d x^{2} + 2 \, d x \sqrt{-\frac{c}{d}} - c}{d x^{2} + c}\right ) - 210 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x}{210 \, d^{4}}, \frac{15 \, b^{2} d^{3} x^{7} - 21 \,{\left (b^{2} c d^{2} - 2 \, a b d^{3}\right )} x^{5} + 35 \,{\left (b^{2} c^{2} d - 2 \, a b c d^{2} + a^{2} d^{3}\right )} x^{3} + 105 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} \sqrt{\frac{c}{d}} \arctan \left (\frac{d x \sqrt{\frac{c}{d}}}{c}\right ) - 105 \,{\left (b^{2} c^{3} - 2 \, a b c^{2} d + a^{2} c d^{2}\right )} x}{105 \, d^{4}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.65842, size = 240, normalized size = 2.31 \begin{align*} \frac{b^{2} x^{7}}{7 d} - \frac{\sqrt{- \frac{c^{3}}{d^{9}}} \left (a d - b c\right )^{2} \log{\left (- \frac{d^{4} \sqrt{- \frac{c^{3}}{d^{9}}} \left (a d - b c\right )^{2}}{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}} + x \right )}}{2} + \frac{\sqrt{- \frac{c^{3}}{d^{9}}} \left (a d - b c\right )^{2} \log{\left (\frac{d^{4} \sqrt{- \frac{c^{3}}{d^{9}}} \left (a d - b c\right )^{2}}{a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}} + x \right )}}{2} + \frac{x^{5} \left (2 a b d - b^{2} c\right )}{5 d^{2}} + \frac{x^{3} \left (a^{2} d^{2} - 2 a b c d + b^{2} c^{2}\right )}{3 d^{3}} - \frac{x \left (a^{2} c d^{2} - 2 a b c^{2} d + b^{2} c^{3}\right )}{d^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20064, size = 207, normalized size = 1.99 \begin{align*} \frac{{\left (b^{2} c^{4} - 2 \, a b c^{3} d + a^{2} c^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} d^{4}} + \frac{15 \, b^{2} d^{6} x^{7} - 21 \, b^{2} c d^{5} x^{5} + 42 \, a b d^{6} x^{5} + 35 \, b^{2} c^{2} d^{4} x^{3} - 70 \, a b c d^{5} x^{3} + 35 \, a^{2} d^{6} x^{3} - 105 \, b^{2} c^{3} d^{3} x + 210 \, a b c^{2} d^{4} x - 105 \, a^{2} c d^{5} x}{105 \, d^{7}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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