Optimal. Leaf size=64 \[ \frac{c (b c-2 a d)}{a^2 x}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}-\frac{c^2}{3 a x^3} \]
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Rubi [A] time = 0.0546262, antiderivative size = 64, 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} \[ \frac{c (b c-2 a d)}{a^2 x}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}-\frac{c^2}{3 a x^3} \]
Antiderivative was successfully verified.
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Rule 461
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^2}{x^4 \left (a+b x^2\right )} \, dx &=\int \left (\frac{c^2}{a x^4}+\frac{c (-b c+2 a d)}{a^2 x^2}+\frac{(-b c+a d)^2}{a^2 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{c^2}{3 a x^3}+\frac{c (b c-2 a d)}{a^2 x}+\frac{(b c-a d)^2 \int \frac{1}{a+b x^2} \, dx}{a^2}\\ &=-\frac{c^2}{3 a x^3}+\frac{c (b c-2 a d)}{a^2 x}+\frac{(b c-a d)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0618717, size = 66, normalized size = 1.03 \[ -\frac{c (2 a d-b c)}{a^2 x}+\frac{(a d-b c)^2 \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{5/2} \sqrt{b}}-\frac{c^2}{3 a x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.006, size = 98, normalized size = 1.5 \begin{align*} -{\frac{{c}^{2}}{3\,a{x}^{3}}}-2\,{\frac{cd}{ax}}+{\frac{b{c}^{2}}{{a}^{2}x}}+{{d}^{2}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-2\,{\frac{bcd}{a\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{{b}^{2}{c}^{2}}{{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 = 1.5038, size = 408, normalized size = 6.38 \begin{align*} \left [-\frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{-a b} x^{3} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right ) + 2 \, a^{2} b c^{2} - 6 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d\right )} x^{2}}{6 \, a^{3} b x^{3}}, \frac{3 \,{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \sqrt{a b} x^{3} \arctan \left (\frac{\sqrt{a b} x}{a}\right ) - a^{2} b c^{2} + 3 \,{\left (a b^{2} c^{2} - 2 \, a^{2} b c d\right )} x^{2}}{3 \, a^{3} b x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.786519, size = 172, normalized size = 2.69 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{5} b}} \left (a d - b c\right )^{2} \log{\left (- \frac{a^{3} \sqrt{- \frac{1}{a^{5} b}} \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}{a^{5} b}} \left (a d - b c\right )^{2} \log{\left (\frac{a^{3} \sqrt{- \frac{1}{a^{5} b}} \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 c^{2} + x^{2} \left (6 a c d - 3 b c^{2}\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 = 1.18474, size = 97, normalized size = 1.52 \begin{align*} \frac{{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{\sqrt{a b} a^{2}} + \frac{3 \, b c^{2} x^{2} - 6 \, a c d x^{2} - a c^{2}}{3 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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