Optimal. Leaf size=66 \[ -\frac{a^2}{3 c x^3}-\frac{a (2 b c-a d)}{c^2 x}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} \sqrt{d}} \]
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Rubi [A] time = 0.0547288, antiderivative size = 66, 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} \[ -\frac{a^2}{3 c x^3}-\frac{a (2 b c-a d)}{c^2 x}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} \sqrt{d}} \]
Antiderivative was successfully verified.
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Rule 461
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (a+b x^2\right )^2}{x^4 \left (c+d x^2\right )} \, dx &=\int \left (\frac{a^2}{c x^4}-\frac{a (-2 b c+a d)}{c^2 x^2}+\frac{(b c-a d)^2}{c^2 \left (c+d x^2\right )}\right ) \, dx\\ &=-\frac{a^2}{3 c x^3}-\frac{a (2 b c-a d)}{c^2 x}+\frac{(b c-a d)^2 \int \frac{1}{c+d x^2} \, dx}{c^2}\\ &=-\frac{a^2}{3 c x^3}-\frac{a (2 b c-a d)}{c^2 x}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} \sqrt{d}}\\ \end{align*}
Mathematica [A] time = 0.0520988, size = 64, normalized size = 0.97 \[ -\frac{a^2}{3 c x^3}+\frac{a (a d-2 b c)}{c^2 x}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c}}\right )}{c^{5/2} \sqrt{d}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 98, normalized size = 1.5 \begin{align*}{\frac{{a}^{2}{d}^{2}}{{c}^{2}}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-2\,{\frac{abd}{c\sqrt{cd}}\arctan \left ({\frac{dx}{\sqrt{cd}}} \right ) }+{{b}^{2}\arctan \left ({dx{\frac{1}{\sqrt{cd}}}} \right ){\frac{1}{\sqrt{cd}}}}-{\frac{{a}^{2}}{3\,c{x}^{3}}}+{\frac{{a}^{2}d}{{c}^{2}x}}-2\,{\frac{ab}{cx}} \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 = 1.31396, size = 408, normalized size = 6.18 \begin{align*} \left [-\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-c d} x^{3} \log \left (\frac{d x^{2} - 2 \, \sqrt{-c d} x - c}{d x^{2} + c}\right ) + 2 \, a^{2} c^{2} d + 6 \,{\left (2 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{2}}{6 \, c^{3} d x^{3}}, \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{c d} x^{3} \arctan \left (\frac{\sqrt{c d} x}{c}\right ) - a^{2} c^{2} d - 3 \,{\left (2 \, a b c^{2} d - a^{2} c d^{2}\right )} x^{2}}{3 \, c^{3} d x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.782587, size = 172, normalized size = 2.61 \begin{align*} - \frac{\sqrt{- \frac{1}{c^{5} d}} \left (a d - b c\right )^{2} \log{\left (- \frac{c^{3} \sqrt{- \frac{1}{c^{5} d}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{\sqrt{- \frac{1}{c^{5} d}} \left (a d - b c\right )^{2} \log{\left (\frac{c^{3} \sqrt{- \frac{1}{c^{5} d}} \left (a d - b c\right )^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )}}{2} + \frac{- a^{2} c + x^{2} \left (3 a^{2} d - 6 a b c\right )}{3 c^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.15619, size = 96, normalized size = 1.45 \begin{align*} \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{d x}{\sqrt{c d}}\right )}{\sqrt{c d} c^{2}} - \frac{6 \, a b c x^{2} - 3 \, a^{2} d x^{2} + a^{2} c}{3 \, c^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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