Optimal. Leaf size=77 \[ -\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{5/2}}+\frac{d^2 x (3 b c-a d)}{b^2}-\frac{c^3}{a x}+\frac{d^3 x^3}{3 b} \]
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Rubi [A] time = 0.064137, antiderivative size = 77, 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} \[ -\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{5/2}}+\frac{d^2 x (3 b c-a d)}{b^2}-\frac{c^3}{a x}+\frac{d^3 x^3}{3 b} \]
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^2 \left (a+b x^2\right )} \, dx &=\int \left (\frac{d^2 (3 b c-a d)}{b^2}+\frac{c^3}{a x^2}+\frac{d^3 x^2}{b}+\frac{(-b c+a d)^3}{a b^2 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{c^3}{a x}+\frac{d^2 (3 b c-a d) x}{b^2}+\frac{d^3 x^3}{3 b}-\frac{(b c-a d)^3 \int \frac{1}{a+b x^2} \, dx}{a b^2}\\ &=-\frac{c^3}{a x}+\frac{d^2 (3 b c-a d) x}{b^2}+\frac{d^3 x^3}{3 b}-\frac{(b c-a d)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.0322889, size = 76, normalized size = 0.99 \[ \frac{(a d-b c)^3 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} b^{5/2}}+\frac{d^2 x (3 b c-a d)}{b^2}-\frac{c^3}{a x}+\frac{d^3 x^3}{3 b} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.005, size = 135, normalized size = 1.8 \begin{align*}{\frac{{d}^{3}{x}^{3}}{3\,b}}-{\frac{{d}^{3}ax}{{b}^{2}}}+3\,{\frac{{d}^{2}xc}{b}}-{\frac{{c}^{3}}{ax}}+{\frac{{a}^{2}{d}^{3}}{{b}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-3\,{\frac{ac{d}^{2}}{b\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+3\,{\frac{{c}^{2}d}{\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }-{\frac{b{c}^{3}}{a}\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 = 1.51257, size = 520, normalized size = 6.75 \begin{align*} \left [\frac{2 \, a^{2} b^{2} d^{3} x^{4} - 6 \, a b^{3} 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 \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 6 \,{\left (3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2}}{6 \, a^{2} b^{3} x}, \frac{a^{2} b^{2} d^{3} x^{4} - 3 \, a b^{3} 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 \arctan \left (\frac{\sqrt{a b} x}{a}\right ) + 3 \,{\left (3 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} x^{2}}{3 \, a^{2} b^{3} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.931007, size = 221, normalized size = 2.87 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{3} b^{5}}} \left (a d - b c\right )^{3} \log{\left (- \frac{a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} \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^{3} b^{5}}} \left (a d - b c\right )^{3} \log{\left (\frac{a^{2} b^{2} \sqrt{- \frac{1}{a^{3} b^{5}}} \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^{3}}{3 b} - \frac{x \left (a d^{3} - 3 b c d^{2}\right )}{b^{2}} - \frac{c^{3}}{a x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13286, size = 140, normalized size = 1.82 \begin{align*} -\frac{c^{3}}{a x} - \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 b^{2}} + \frac{b^{2} d^{3} x^{3} + 9 \, b^{2} c d^{2} x - 3 \, a b d^{3} x}{3 \, b^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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