Optimal. Leaf size=96 \[ \frac{c d \log \left (a+c x^2\right )}{a^3}-\frac{2 c d \log (x)}{a^3}-\frac{3 \sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{d}{a^2 x^2}-\frac{3 e}{2 a^2 x}+\frac{d+e x}{2 a x^2 \left (a+c x^2\right )} \]
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Rubi [A] time = 0.0836267, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {823, 801, 635, 205, 260} \[ \frac{c d \log \left (a+c x^2\right )}{a^3}-\frac{2 c d \log (x)}{a^3}-\frac{3 \sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{d}{a^2 x^2}-\frac{3 e}{2 a^2 x}+\frac{d+e x}{2 a x^2 \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 823
Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{d+e x}{x^3 \left (a+c x^2\right )^2} \, dx &=\frac{d+e x}{2 a x^2 \left (a+c x^2\right )}-\frac{\int \frac{-4 a c d-3 a c e x}{x^3 \left (a+c x^2\right )} \, dx}{2 a^2 c}\\ &=\frac{d+e x}{2 a x^2 \left (a+c x^2\right )}-\frac{\int \left (-\frac{4 c d}{x^3}-\frac{3 c e}{x^2}+\frac{4 c^2 d}{a x}+\frac{c^2 (3 a e-4 c d x)}{a \left (a+c x^2\right )}\right ) \, dx}{2 a^2 c}\\ &=-\frac{d}{a^2 x^2}-\frac{3 e}{2 a^2 x}+\frac{d+e x}{2 a x^2 \left (a+c x^2\right )}-\frac{2 c d \log (x)}{a^3}-\frac{c \int \frac{3 a e-4 c d x}{a+c x^2} \, dx}{2 a^3}\\ &=-\frac{d}{a^2 x^2}-\frac{3 e}{2 a^2 x}+\frac{d+e x}{2 a x^2 \left (a+c x^2\right )}-\frac{2 c d \log (x)}{a^3}+\frac{\left (2 c^2 d\right ) \int \frac{x}{a+c x^2} \, dx}{a^3}-\frac{(3 c e) \int \frac{1}{a+c x^2} \, dx}{2 a^2}\\ &=-\frac{d}{a^2 x^2}-\frac{3 e}{2 a^2 x}+\frac{d+e x}{2 a x^2 \left (a+c x^2\right )}-\frac{3 \sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{5/2}}-\frac{2 c d \log (x)}{a^3}+\frac{c d \log \left (a+c x^2\right )}{a^3}\\ \end{align*}
Mathematica [A] time = 0.108234, size = 82, normalized size = 0.85 \[ -\frac{\frac{a c (d+e x)}{a+c x^2}-2 c d \log \left (a+c x^2\right )+3 \sqrt{a} \sqrt{c} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )+\frac{a d}{x^2}+\frac{2 a e}{x}+4 c d \log (x)}{2 a^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.013, size = 97, normalized size = 1. \begin{align*} -{\frac{d}{2\,{a}^{2}{x}^{2}}}-{\frac{e}{{a}^{2}x}}-2\,{\frac{cd\ln \left ( x \right ) }{{a}^{3}}}-{\frac{cex}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{cd}{2\,{a}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{cd\ln \left ( c{x}^{2}+a \right ) }{{a}^{3}}}-{\frac{3\,ce}{2\,{a}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 = 1.61931, size = 620, normalized size = 6.46 \begin{align*} \left [-\frac{6 \, a c e x^{3} + 4 \, a c d x^{2} + 4 \, a^{2} e x + 2 \, a^{2} d - 3 \,{\left (a c e x^{4} + a^{2} e x^{2}\right )} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} - 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) - 4 \,{\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (c x^{2} + a\right ) + 8 \,{\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (x\right )}{4 \,{\left (a^{3} c x^{4} + a^{4} x^{2}\right )}}, -\frac{3 \, a c e x^{3} + 2 \, a c d x^{2} + 2 \, a^{2} e x + a^{2} d + 3 \,{\left (a c e x^{4} + a^{2} e x^{2}\right )} \sqrt{\frac{c}{a}} \arctan \left (x \sqrt{\frac{c}{a}}\right ) - 2 \,{\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (c x^{2} + a\right ) + 4 \,{\left (c^{2} d x^{4} + a c d x^{2}\right )} \log \left (x\right )}{2 \,{\left (a^{3} c x^{4} + a^{4} x^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.12207, size = 396, normalized size = 4.12 \begin{align*} \left (\frac{c d}{a^{3}} - \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) \log{\left (x + \frac{- 64 a^{6} d \left (\frac{c d}{a^{3}} - \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right )^{2} - 12 a^{4} e^{2} \left (\frac{c d}{a^{3}} - \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) - 64 a^{3} c d^{2} \left (\frac{c d}{a^{3}} - \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) - 24 a c d e^{2} + 128 c^{2} d^{3}}{9 a c e^{3} + 144 c^{2} d^{2} e} \right )} + \left (\frac{c d}{a^{3}} + \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) \log{\left (x + \frac{- 64 a^{6} d \left (\frac{c d}{a^{3}} + \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right )^{2} - 12 a^{4} e^{2} \left (\frac{c d}{a^{3}} + \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) - 64 a^{3} c d^{2} \left (\frac{c d}{a^{3}} + \frac{3 e \sqrt{- a^{7} c}}{4 a^{6}}\right ) - 24 a c d e^{2} + 128 c^{2} d^{3}}{9 a c e^{3} + 144 c^{2} d^{2} e} \right )} - \frac{a d + 2 a e x + 2 c d x^{2} + 3 c e x^{3}}{2 a^{3} x^{2} + 2 a^{2} c x^{4}} - \frac{2 c d \log{\left (x \right )}}{a^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.18226, size = 128, normalized size = 1.33 \begin{align*} -\frac{3 \, c \arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{2 \, \sqrt{a c} a^{2}} + \frac{c d \log \left (c x^{2} + a\right )}{a^{3}} - \frac{2 \, c d \log \left ({\left | x \right |}\right )}{a^{3}} - \frac{3 \, a c x^{3} e + 2 \, a c d x^{2} + 2 \, a^{2} x e + a^{2} d}{2 \,{\left (c x^{2} + a\right )} a^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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