Optimal. Leaf size=78 \[ \frac{d \log \left (a+c x^2\right )}{2 c^2}-\frac{3 \sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 c^{5/2}}-\frac{x^2 (d+e x)}{2 c \left (a+c x^2\right )}+\frac{3 e x}{2 c^2} \]
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Rubi [A] time = 0.0448865, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278, Rules used = {819, 774, 635, 205, 260} \[ \frac{d \log \left (a+c x^2\right )}{2 c^2}-\frac{3 \sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 c^{5/2}}-\frac{x^2 (d+e x)}{2 c \left (a+c x^2\right )}+\frac{3 e x}{2 c^2} \]
Antiderivative was successfully verified.
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Rule 819
Rule 774
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{x^3 (d+e x)}{\left (a+c x^2\right )^2} \, dx &=-\frac{x^2 (d+e x)}{2 c \left (a+c x^2\right )}+\frac{\int \frac{x (2 a d+3 a e x)}{a+c x^2} \, dx}{2 a c}\\ &=\frac{3 e x}{2 c^2}-\frac{x^2 (d+e x)}{2 c \left (a+c x^2\right )}+\frac{\int \frac{-3 a^2 e+2 a c d x}{a+c x^2} \, dx}{2 a c^2}\\ &=\frac{3 e x}{2 c^2}-\frac{x^2 (d+e x)}{2 c \left (a+c x^2\right )}+\frac{d \int \frac{x}{a+c x^2} \, dx}{c}-\frac{(3 a e) \int \frac{1}{a+c x^2} \, dx}{2 c^2}\\ &=\frac{3 e x}{2 c^2}-\frac{x^2 (d+e x)}{2 c \left (a+c x^2\right )}-\frac{3 \sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 c^{5/2}}+\frac{d \log \left (a+c x^2\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.0509549, size = 75, normalized size = 0.96 \[ \frac{a d+a e x}{2 c^2 \left (a+c x^2\right )}+\frac{d \log \left (a+c x^2\right )}{2 c^2}-\frac{3 \sqrt{a} e \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 c^{5/2}}+\frac{e x}{c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 76, normalized size = 1. \begin{align*}{\frac{ex}{{c}^{2}}}+{\frac{aex}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{ad}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{d\ln \left ( c{x}^{2}+a \right ) }{2\,{c}^{2}}}-{\frac{3\,ae}{2\,{c}^{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 = 1.55213, size = 417, normalized size = 5.35 \begin{align*} \left [\frac{4 \, c e x^{3} + 6 \, a e x + 3 \,{\left (c e x^{2} + a e\right )} \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{a}{c}} - a}{c x^{2} + a}\right ) + 2 \, a d + 2 \,{\left (c d x^{2} + a d\right )} \log \left (c x^{2} + a\right )}{4 \,{\left (c^{3} x^{2} + a c^{2}\right )}}, \frac{2 \, c e x^{3} + 3 \, a e x - 3 \,{\left (c e x^{2} + a e\right )} \sqrt{\frac{a}{c}} \arctan \left (\frac{c x \sqrt{\frac{a}{c}}}{a}\right ) + a d +{\left (c d x^{2} + a d\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (c^{3} x^{2} + a c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.89868, size = 162, normalized size = 2.08 \begin{align*} \left (\frac{d}{2 c^{2}} - \frac{3 e \sqrt{- a c^{5}}}{4 c^{5}}\right ) \log{\left (x + \frac{- 4 c^{2} \left (\frac{d}{2 c^{2}} - \frac{3 e \sqrt{- a c^{5}}}{4 c^{5}}\right ) + 2 d}{3 e} \right )} + \left (\frac{d}{2 c^{2}} + \frac{3 e \sqrt{- a c^{5}}}{4 c^{5}}\right ) \log{\left (x + \frac{- 4 c^{2} \left (\frac{d}{2 c^{2}} + \frac{3 e \sqrt{- a c^{5}}}{4 c^{5}}\right ) + 2 d}{3 e} \right )} + \frac{a d + a e x}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{e x}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1099, size = 90, normalized size = 1.15 \begin{align*} -\frac{3 \, a \arctan \left (\frac{c x}{\sqrt{a c}}\right ) e}{2 \, \sqrt{a c} c^{2}} + \frac{x e}{c^{2}} + \frac{d \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{a x e + a d}{2 \,{\left (c x^{2} + a\right )} c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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