Optimal. Leaf size=127 \[ \frac{x \left (3 a^2 d^2-6 a b c d+5 b^2 c^2\right )}{6 a^3 \left (a+b x^2\right )}+\frac{c (5 b c-6 a d)}{3 a^3 x}+\frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} \sqrt{b}}-\frac{c^2}{3 a x^3 \left (a+b x^2\right )} \]
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Rubi [A] time = 0.142764, antiderivative size = 125, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {462, 456, 453, 205} \[ \frac{x \left (\frac{b c (5 b c-6 a d)}{a^2}+3 d^2\right )}{6 a \left (a+b x^2\right )}+\frac{c (5 b c-6 a d)}{3 a^3 x}+\frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} \sqrt{b}}-\frac{c^2}{3 a x^3 \left (a+b x^2\right )} \]
Antiderivative was successfully verified.
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Rule 462
Rule 456
Rule 453
Rule 205
Rubi steps
\begin{align*} \int \frac{\left (c+d x^2\right )^2}{x^4 \left (a+b x^2\right )^2} \, dx &=-\frac{c^2}{3 a x^3 \left (a+b x^2\right )}+\frac{\int \frac{-c (5 b c-6 a d)+3 a d^2 x^2}{x^2 \left (a+b x^2\right )^2} \, dx}{3 a}\\ &=-\frac{c^2}{3 a x^3 \left (a+b x^2\right )}+\frac{\left (3 d^2+\frac{b c (5 b c-6 a d)}{a^2}\right ) x}{6 a \left (a+b x^2\right )}-\frac{\int \frac{\frac{2 c (5 b c-6 a d)}{a}-\left (\frac{5 b^2 c^2}{a^2}-\frac{6 b c d}{a}+3 d^2\right ) x^2}{x^2 \left (a+b x^2\right )} \, dx}{6 a}\\ &=\frac{c (5 b c-6 a d)}{3 a^3 x}-\frac{c^2}{3 a x^3 \left (a+b x^2\right )}+\frac{\left (3 d^2+\frac{b c (5 b c-6 a d)}{a^2}\right ) x}{6 a \left (a+b x^2\right )}+\frac{((b c-a d) (5 b c-a d)) \int \frac{1}{a+b x^2} \, dx}{2 a^3}\\ &=\frac{c (5 b c-6 a d)}{3 a^3 x}-\frac{c^2}{3 a x^3 \left (a+b x^2\right )}+\frac{\left (3 d^2+\frac{b c (5 b c-6 a d)}{a^2}\right ) x}{6 a \left (a+b x^2\right )}+\frac{(b c-a d) (5 b c-a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} \sqrt{b}}\\ \end{align*}
Mathematica [A] time = 0.0695077, size = 107, normalized size = 0.84 \[ \frac{\left (a^2 d^2-6 a b c d+5 b^2 c^2\right ) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2} \sqrt{b}}+\frac{x (a d-b c)^2}{2 a^3 \left (a+b x^2\right )}-\frac{2 c (a d-b c)}{a^3 x}-\frac{c^2}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 161, normalized size = 1.3 \begin{align*} -{\frac{{c}^{2}}{3\,{a}^{2}{x}^{3}}}-2\,{\frac{cd}{{a}^{2}x}}+2\,{\frac{b{c}^{2}}{{a}^{3}x}}+{\frac{x{d}^{2}}{2\,a \left ( b{x}^{2}+a \right ) }}-{\frac{bcxd}{{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}{c}^{2}x}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{{d}^{2}}{2\,a}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}-3\,{\frac{bcd}{{a}^{2}\sqrt{ab}}\arctan \left ({\frac{bx}{\sqrt{ab}}} \right ) }+{\frac{5\,{b}^{2}{c}^{2}}{2\,{a}^{3}}\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 = 1.56387, size = 737, normalized size = 5.8 \begin{align*} \left [-\frac{4 \, a^{3} b c^{2} - 6 \,{\left (5 \, a b^{3} c^{2} - 6 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} - 4 \,{\left (5 \, a^{2} b^{2} c^{2} - 6 \, a^{3} b c d\right )} x^{2} + 3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{5} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3}\right )} \sqrt{-a b} \log \left (\frac{b x^{2} - 2 \, \sqrt{-a b} x - a}{b x^{2} + a}\right )}{12 \,{\left (a^{4} b^{2} x^{5} + a^{5} b x^{3}\right )}}, -\frac{2 \, a^{3} b c^{2} - 3 \,{\left (5 \, a b^{3} c^{2} - 6 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} x^{4} - 2 \,{\left (5 \, a^{2} b^{2} c^{2} - 6 \, a^{3} b c d\right )} x^{2} - 3 \,{\left ({\left (5 \, b^{3} c^{2} - 6 \, a b^{2} c d + a^{2} b d^{2}\right )} x^{5} +{\left (5 \, a b^{2} c^{2} - 6 \, a^{2} b c d + a^{3} d^{2}\right )} x^{3}\right )} \sqrt{a b} \arctan \left (\frac{\sqrt{a b} x}{a}\right )}{6 \,{\left (a^{4} b^{2} x^{5} + a^{5} b x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.18685, size = 248, normalized size = 1.95 \begin{align*} - \frac{\sqrt{- \frac{1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right ) \log{\left (- \frac{a^{4} \sqrt{- \frac{1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right )}{a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}} + x \right )}}{4} + \frac{\sqrt{- \frac{1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right ) \log{\left (\frac{a^{4} \sqrt{- \frac{1}{a^{7} b}} \left (a d - 5 b c\right ) \left (a d - b c\right )}{a^{2} d^{2} - 6 a b c d + 5 b^{2} c^{2}} + x \right )}}{4} + \frac{- 2 a^{2} c^{2} + x^{4} \left (3 a^{2} d^{2} - 18 a b c d + 15 b^{2} c^{2}\right ) + x^{2} \left (- 12 a^{2} c d + 10 a b c^{2}\right )}{6 a^{4} x^{3} + 6 a^{3} b x^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.16318, size = 151, normalized size = 1.19 \begin{align*} \frac{{\left (5 \, b^{2} c^{2} - 6 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{3}} + \frac{b^{2} c^{2} x - 2 \, a b c d x + a^{2} d^{2} x}{2 \,{\left (b x^{2} + a\right )} a^{3}} + \frac{6 \, b c^{2} x^{2} - 6 \, a c d x^{2} - a c^{2}}{3 \, a^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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