Optimal. Leaf size=90 \[ \frac{b x (b c-a d)}{2 a^3 \left (a+b x^2\right )}+\frac{2 b c-a d}{a^3 x}+\frac{\sqrt{b} (5 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{c}{3 a^2 x^3} \]
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Rubi [A] time = 0.106381, antiderivative size = 90, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {456, 1261, 205} \[ \frac{b x (b c-a d)}{2 a^3 \left (a+b x^2\right )}+\frac{2 b c-a d}{a^3 x}+\frac{\sqrt{b} (5 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{c}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Rule 456
Rule 1261
Rule 205
Rubi steps
\begin{align*} \int \frac{c+d x^2}{x^4 \left (a+b x^2\right )^2} \, dx &=\frac{b (b c-a d) x}{2 a^3 \left (a+b x^2\right )}-\frac{1}{2} b \int \frac{-\frac{2 c}{a b}+\frac{2 (b c-a d) x^2}{a^2 b}-\frac{(b c-a d) x^4}{a^3}}{x^4 \left (a+b x^2\right )} \, dx\\ &=\frac{b (b c-a d) x}{2 a^3 \left (a+b x^2\right )}-\frac{1}{2} b \int \left (-\frac{2 c}{a^2 b x^4}-\frac{2 (-2 b c+a d)}{a^3 b x^2}+\frac{-5 b c+3 a d}{a^3 \left (a+b x^2\right )}\right ) \, dx\\ &=-\frac{c}{3 a^2 x^3}+\frac{2 b c-a d}{a^3 x}+\frac{b (b c-a d) x}{2 a^3 \left (a+b x^2\right )}+\frac{(b (5 b c-3 a d)) \int \frac{1}{a+b x^2} \, dx}{2 a^3}\\ &=-\frac{c}{3 a^2 x^3}+\frac{2 b c-a d}{a^3 x}+\frac{b (b c-a d) x}{2 a^3 \left (a+b x^2\right )}+\frac{\sqrt{b} (5 b c-3 a d) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.070273, size = 90, normalized size = 1. \[ -\frac{b x (a d-b c)}{2 a^3 \left (a+b x^2\right )}+\frac{2 b c-a d}{a^3 x}-\frac{\sqrt{b} (3 a d-5 b c) \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{2 a^{7/2}}-\frac{c}{3 a^2 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.011, size = 110, normalized size = 1.2 \begin{align*} -{\frac{c}{3\,{a}^{2}{x}^{3}}}-{\frac{d}{{a}^{2}x}}+2\,{\frac{bc}{{a}^{3}x}}-{\frac{bdx}{2\,{a}^{2} \left ( b{x}^{2}+a \right ) }}+{\frac{{b}^{2}cx}{2\,{a}^{3} \left ( b{x}^{2}+a \right ) }}-{\frac{3\,bd}{2\,{a}^{2}}\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}}+{\frac{5\,{b}^{2}c}{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 = 1.52035, size = 529, normalized size = 5.88 \begin{align*} \left [\frac{6 \,{\left (5 \, b^{2} c - 3 \, a b d\right )} x^{4} - 4 \, a^{2} c + 4 \,{\left (5 \, a b c - 3 \, a^{2} d\right )} x^{2} - 3 \,{\left ({\left (5 \, b^{2} c - 3 \, a b d\right )} x^{5} +{\left (5 \, a b c - 3 \, a^{2} d\right )} x^{3}\right )} \sqrt{-\frac{b}{a}} \log \left (\frac{b x^{2} - 2 \, a x \sqrt{-\frac{b}{a}} - a}{b x^{2} + a}\right )}{12 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}, \frac{3 \,{\left (5 \, b^{2} c - 3 \, a b d\right )} x^{4} - 2 \, a^{2} c + 2 \,{\left (5 \, a b c - 3 \, a^{2} d\right )} x^{2} + 3 \,{\left ({\left (5 \, b^{2} c - 3 \, a b d\right )} x^{5} +{\left (5 \, a b c - 3 \, a^{2} d\right )} x^{3}\right )} \sqrt{\frac{b}{a}} \arctan \left (x \sqrt{\frac{b}{a}}\right )}{6 \,{\left (a^{3} b x^{5} + a^{4} x^{3}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.795143, size = 184, normalized size = 2.04 \begin{align*} \frac{\sqrt{- \frac{b}{a^{7}}} \left (3 a d - 5 b c\right ) \log{\left (- \frac{a^{4} \sqrt{- \frac{b}{a^{7}}} \left (3 a d - 5 b c\right )}{3 a b d - 5 b^{2} c} + x \right )}}{4} - \frac{\sqrt{- \frac{b}{a^{7}}} \left (3 a d - 5 b c\right ) \log{\left (\frac{a^{4} \sqrt{- \frac{b}{a^{7}}} \left (3 a d - 5 b c\right )}{3 a b d - 5 b^{2} c} + x \right )}}{4} - \frac{2 a^{2} c + x^{4} \left (9 a b d - 15 b^{2} c\right ) + x^{2} \left (6 a^{2} d - 10 a b c\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 = 1.16169, size = 116, normalized size = 1.29 \begin{align*} \frac{{\left (5 \, b^{2} c - 3 \, a b d\right )} \arctan \left (\frac{b x}{\sqrt{a b}}\right )}{2 \, \sqrt{a b} a^{3}} + \frac{b^{2} c x - a b d x}{2 \,{\left (b x^{2} + a\right )} a^{3}} + \frac{6 \, b c x^{2} - 3 \, a d x^{2} - a c}{3 \, a^{3} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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