Optimal. Leaf size=140 \[ \frac{d x^5 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}+\frac{a^{3/2} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{11/2}}+\frac{d^2 x^7 (3 b c-a d)}{7 b^2}+\frac{x^3 (b c-a d)^3}{3 b^4}-\frac{a x (b c-a d)^3}{b^5}+\frac{d^3 x^9}{9 b} \]
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Rubi [A] time = 0.0971003, antiderivative size = 140, 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} \[ \frac{d x^5 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}+\frac{a^{3/2} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{11/2}}+\frac{d^2 x^7 (3 b c-a d)}{7 b^2}+\frac{x^3 (b c-a d)^3}{3 b^4}-\frac{a x (b c-a d)^3}{b^5}+\frac{d^3 x^9}{9 b} \]
Antiderivative was successfully verified.
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Rule 461
Rule 205
Rubi steps
\begin{align*} \int \frac{x^4 \left (c+d x^2\right )^3}{a+b x^2} \, dx &=\int \left (-\frac{a (b c-a d)^3}{b^5}+\frac{(b c-a d)^3 x^2}{b^4}+\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^4}{b^3}+\frac{d^2 (3 b c-a d) x^6}{b^2}+\frac{d^3 x^8}{b}+\frac{a^2 b^3 c^3-3 a^3 b^2 c^2 d+3 a^4 b c d^2-a^5 d^3}{b^5 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{a (b c-a d)^3 x}{b^5}+\frac{(b c-a d)^3 x^3}{3 b^4}+\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac{d^2 (3 b c-a d) x^7}{7 b^2}+\frac{d^3 x^9}{9 b}+\frac{\left (a^2 (b c-a d)^3\right ) \int \frac{1}{a+b x^2} \, dx}{b^5}\\ &=-\frac{a (b c-a d)^3 x}{b^5}+\frac{(b c-a d)^3 x^3}{3 b^4}+\frac{d \left (3 b^2 c^2-3 a b c d+a^2 d^2\right ) x^5}{5 b^3}+\frac{d^2 (3 b c-a d) x^7}{7 b^2}+\frac{d^3 x^9}{9 b}+\frac{a^{3/2} (b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{11/2}}\\ \end{align*}
Mathematica [A] time = 0.0481145, size = 140, normalized size = 1. \[ \frac{d x^5 \left (a^2 d^2-3 a b c d+3 b^2 c^2\right )}{5 b^3}-\frac{a^{3/2} (a d-b c)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{b^{11/2}}+\frac{d^2 x^7 (3 b c-a d)}{7 b^2}+\frac{x^3 (b c-a d)^3}{3 b^4}+\frac{a x (a d-b c)^3}{b^5}+\frac{d^3 x^9}{9 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.003, size = 276, normalized size = 2. \begin{align*}{\frac{{d}^{3}{x}^{9}}{9\,b}}-{\frac{{x}^{7}a{d}^{3}}{7\,{b}^{2}}}+{\frac{3\,{x}^{7}c{d}^{2}}{7\,b}}+{\frac{{x}^{5}{a}^{2}{d}^{3}}{5\,{b}^{3}}}-{\frac{3\,{x}^{5}ac{d}^{2}}{5\,{b}^{2}}}+{\frac{3\,{x}^{5}{c}^{2}d}{5\,b}}-{\frac{{x}^{3}{a}^{3}{d}^{3}}{3\,{b}^{4}}}+{\frac{{x}^{3}{a}^{2}c{d}^{2}}{{b}^{3}}}-{\frac{{x}^{3}a{c}^{2}d}{{b}^{2}}}+{\frac{{x}^{3}{c}^{3}}{3\,b}}+{\frac{{a}^{4}{d}^{3}x}{{b}^{5}}}-3\,{\frac{{a}^{3}c{d}^{2}x}{{b}^{4}}}+3\,{\frac{{a}^{2}{c}^{2}dx}{{b}^{3}}}-{\frac{a{c}^{3}x}{{b}^{2}}}-{\frac{{a}^{5}{d}^{3}}{{b}^{5}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+3\,{\frac{{a}^{4}c{d}^{2}}{{b}^{4}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-3\,{\frac{{a}^{3}{c}^{2}d}{{b}^{3}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{{a}^{2}{c}^{3}}{{b}^{2}}\arctan \left ({bx{\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 = 1.49598, size = 964, normalized size = 6.89 \begin{align*} \left [\frac{70 \, b^{4} d^{3} x^{9} + 90 \,{\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{7} + 126 \,{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{5} + 210 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3} - 315 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt{-\frac{a}{b}} \log \left (\frac{b x^{2} - 2 \, b x \sqrt{-\frac{a}{b}} - a}{b x^{2} + a}\right ) - 630 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{630 \, b^{5}}, \frac{35 \, b^{4} d^{3} x^{9} + 45 \,{\left (3 \, b^{4} c d^{2} - a b^{3} d^{3}\right )} x^{7} + 63 \,{\left (3 \, b^{4} c^{2} d - 3 \, a b^{3} c d^{2} + a^{2} b^{2} d^{3}\right )} x^{5} + 105 \,{\left (b^{4} c^{3} - 3 \, a b^{3} c^{2} d + 3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{3} + 315 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \sqrt{\frac{a}{b}} \arctan \left (\frac{b x \sqrt{\frac{a}{b}}}{a}\right ) - 315 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} x}{315 \, b^{5}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.814542, size = 338, normalized size = 2.41 \begin{align*} \frac{\sqrt{- \frac{a^{3}}{b^{11}}} \left (a d - b c\right )^{3} \log{\left (- \frac{b^{5} \sqrt{- \frac{a^{3}}{b^{11}}} \left (a d - b c\right )^{3}}{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}} + x \right )}}{2} - \frac{\sqrt{- \frac{a^{3}}{b^{11}}} \left (a d - b c\right )^{3} \log{\left (\frac{b^{5} \sqrt{- \frac{a^{3}}{b^{11}}} \left (a d - b c\right )^{3}}{a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}} + x \right )}}{2} + \frac{d^{3} x^{9}}{9 b} - \frac{x^{7} \left (a d^{3} - 3 b c d^{2}\right )}{7 b^{2}} + \frac{x^{5} \left (a^{2} d^{3} - 3 a b c d^{2} + 3 b^{2} c^{2} d\right )}{5 b^{3}} - \frac{x^{3} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}\right )}{3 b^{4}} + \frac{x \left (a^{4} d^{3} - 3 a^{3} b c d^{2} + 3 a^{2} b^{2} c^{2} d - a b^{3} c^{3}\right )}{b^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.20489, size = 325, normalized size = 2.32 \begin{align*} \frac{{\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} b^{5}} + \frac{35 \, b^{8} d^{3} x^{9} + 135 \, b^{8} c d^{2} x^{7} - 45 \, a b^{7} d^{3} x^{7} + 189 \, b^{8} c^{2} d x^{5} - 189 \, a b^{7} c d^{2} x^{5} + 63 \, a^{2} b^{6} d^{3} x^{5} + 105 \, b^{8} c^{3} x^{3} - 315 \, a b^{7} c^{2} d x^{3} + 315 \, a^{2} b^{6} c d^{2} x^{3} - 105 \, a^{3} b^{5} d^{3} x^{3} - 315 \, a b^{7} c^{3} x + 945 \, a^{2} b^{6} c^{2} d x - 945 \, a^{3} b^{5} c d^{2} x + 315 \, a^{4} b^{4} d^{3} x}{315 \, b^{9}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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