Optimal. Leaf size=83 \[ \frac{c^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{c d}{a^2 x}+\frac{c e \log \left (a+c x^2\right )}{2 a^2}-\frac{c e \log (x)}{a^2}-\frac{d}{3 a x^3}-\frac{e}{2 a x^2} \]
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Rubi [A] time = 0.0577717, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {801, 635, 205, 260} \[ \frac{c^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}+\frac{c d}{a^2 x}+\frac{c e \log \left (a+c x^2\right )}{2 a^2}-\frac{c e \log (x)}{a^2}-\frac{d}{3 a x^3}-\frac{e}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{d+e x}{x^4 \left (a+c x^2\right )} \, dx &=\int \left (\frac{d}{a x^4}+\frac{e}{a x^3}-\frac{c d}{a^2 x^2}-\frac{c e}{a^2 x}+\frac{c^2 (d+e x)}{a^2 \left (a+c x^2\right )}\right ) \, dx\\ &=-\frac{d}{3 a x^3}-\frac{e}{2 a x^2}+\frac{c d}{a^2 x}-\frac{c e \log (x)}{a^2}+\frac{c^2 \int \frac{d+e x}{a+c x^2} \, dx}{a^2}\\ &=-\frac{d}{3 a x^3}-\frac{e}{2 a x^2}+\frac{c d}{a^2 x}-\frac{c e \log (x)}{a^2}+\frac{\left (c^2 d\right ) \int \frac{1}{a+c x^2} \, dx}{a^2}+\frac{\left (c^2 e\right ) \int \frac{x}{a+c x^2} \, dx}{a^2}\\ &=-\frac{d}{3 a x^3}-\frac{e}{2 a x^2}+\frac{c d}{a^2 x}+\frac{c^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}-\frac{c e \log (x)}{a^2}+\frac{c e \log \left (a+c x^2\right )}{2 a^2}\\ \end{align*}
Mathematica [A] time = 0.0419439, size = 77, normalized size = 0.93 \[ \frac{c^{3/2} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{5/2}}-\frac{-3 c e x^3 \log \left (a+c x^2\right )+2 a d+3 a e x-6 c d x^2+6 c e x^3 \log (x)}{6 a^2 x^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 75, normalized size = 0.9 \begin{align*} -{\frac{d}{3\,a{x}^{3}}}-{\frac{e}{2\,a{x}^{2}}}+{\frac{cd}{{a}^{2}x}}-{\frac{ec\ln \left ( x \right ) }{{a}^{2}}}+{\frac{ec\ln \left ( c{x}^{2}+a \right ) }{2\,{a}^{2}}}+{\frac{{c}^{2}d}{{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 = 1.62692, size = 404, normalized size = 4.87 \begin{align*} \left [\frac{3 \, c d x^{3} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{2} + 2 \, a x \sqrt{-\frac{c}{a}} - a}{c x^{2} + a}\right ) + 3 \, c e x^{3} \log \left (c x^{2} + a\right ) - 6 \, c e x^{3} \log \left (x\right ) + 6 \, c d x^{2} - 3 \, a e x - 2 \, a d}{6 \, a^{2} x^{3}}, \frac{6 \, c d x^{3} \sqrt{\frac{c}{a}} \arctan \left (x \sqrt{\frac{c}{a}}\right ) + 3 \, c e x^{3} \log \left (c x^{2} + a\right ) - 6 \, c e x^{3} \log \left (x\right ) + 6 \, c d x^{2} - 3 \, a e x - 2 \, a d}{6 \, a^{2} x^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 1.88224, size = 408, normalized size = 4.92 \begin{align*} \left (\frac{c e}{2 a^{2}} - \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right ) \log{\left (x + \frac{12 a^{6} e \left (\frac{c e}{2 a^{2}} - \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right )^{2} + 6 a^{4} c e^{2} \left (\frac{c e}{2 a^{2}} - \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right ) + 2 a^{3} c^{2} d^{2} \left (\frac{c e}{2 a^{2}} - \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right ) - 6 a^{2} c^{2} e^{3} + 2 a c^{3} d^{2} e}{9 a c^{3} d e^{2} + c^{4} d^{3}} \right )} + \left (\frac{c e}{2 a^{2}} + \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right ) \log{\left (x + \frac{12 a^{6} e \left (\frac{c e}{2 a^{2}} + \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right )^{2} + 6 a^{4} c e^{2} \left (\frac{c e}{2 a^{2}} + \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right ) + 2 a^{3} c^{2} d^{2} \left (\frac{c e}{2 a^{2}} + \frac{d \sqrt{- a^{5} c^{3}}}{2 a^{5}}\right ) - 6 a^{2} c^{2} e^{3} + 2 a c^{3} d^{2} e}{9 a c^{3} d e^{2} + c^{4} d^{3}} \right )} - \frac{c e \log{\left (x \right )}}{a^{2}} + \frac{- 2 a d - 3 a e x + 6 c d x^{2}}{6 a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10915, size = 103, normalized size = 1.24 \begin{align*} \frac{c^{2} d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} a^{2}} + \frac{c e \log \left (c x^{2} + a\right )}{2 \, a^{2}} - \frac{c e \log \left ({\left | x \right |}\right )}{a^{2}} + \frac{6 \, c d x^{2} - 3 \, a x e - 2 \, a d}{6 \, a^{2} x^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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