Optimal. Leaf size=74 \[ \frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} b^{3/2}}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{c^3}{3 a x^3}+\frac{d^3 x}{b} \]
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Rubi [A] time = 0.0657843, antiderivative size = 74, 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} \[ \frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} b^{3/2}}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{c^3}{3 a x^3}+\frac{d^3 x}{b} \]
Antiderivative was successfully verified.
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Rule 461
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^3}{x^4 \left (a+b x^2\right )} \, dx &=\int \left (\frac{d^3}{b}+\frac{c^3}{a x^4}+\frac{c^2 (-b c+3 a d)}{a^2 x^2}-\frac{(-b c+a d)^3}{a^2 b \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{c^3}{3 a x^3}+\frac{c^2 (b c-3 a d)}{a^2 x}+\frac{d^3 x}{b}+\frac{(b c-a d)^3 \int \frac{1}{a+b x^2} \, dx}{a^2 b}\\ &=-\frac{c^3}{3 a x^3}+\frac{c^2 (b c-3 a d)}{a^2 x}+\frac{d^3 x}{b}+\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} b^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0390994, size = 74, normalized size = 1. \[ \frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} b^{3/2}}+\frac{c^2 (b c-3 a d)}{a^2 x}-\frac{c^3}{3 a x^3}+\frac{d^3 x}{b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.007, size = 135, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}x}{b}}-{\frac{{c}^{3}}{3\,a{x}^{3}}}-3\,{\frac{{c}^{2}d}{ax}}+{\frac{b{c}^{3}}{{a}^{2}x}}-{\frac{a{d}^{3}}{b}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+3\,{\frac{c{d}^{2}}{\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-3\,{\frac{b{c}^{2}d}{a\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{{b}^{2}{c}^{3}}{{a}^{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 = 1.4524, size = 531, normalized size = 7.18 \begin{align*} \left [\frac{6 \, a^{3} b d^{3} x^{4} - 2 \, a^{2} b^{2} c^{3} + 3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{-a b} x^{3} \log \left (\frac{b x^{2} + 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 6 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}}{6 \, a^{3} b^{2} x^{3}}, \frac{3 \, a^{3} b d^{3} x^{4} - a^{2} b^{2} c^{3} + 3 \,{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \sqrt{a b} x^{3} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + 3 \,{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d\right )} x^{2}}{3 \, a^{3} b^{2} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.23825, size = 221, normalized size = 2.99 \begin{align*} \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right )^{3} \log{\left (- \frac{a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} - \frac{\sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right )^{3} \log{\left (\frac{a^{3} b \sqrt{- \frac{1}{a^{5} b^{3}}} \left (a d - b c\right )^{3}}{a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - b^{3} c^{3}} + x \right )}}{2} + \frac{d^{3} x}{b} - \frac{a c^{3} + x^{2} \left (9 a c^{2} d - 3 b c^{3}\right )}{3 a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15046, size = 135, normalized size = 1.82 \begin{align*} \frac{d^{3} x}{b} + \frac{{\left (b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2} b} + \frac{3 \, b c^{3} x^{2} - 9 \, a c^{2} d x^{2} - a c^{3}}{3 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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