Optimal. Leaf size=67 \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 \sqrt{a} c^{3/2}}+\frac{e \log \left (a+c x^2\right )}{2 c^2}-\frac{x (d+e x)}{2 c \left (a+c x^2\right )} \]
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Rubi [A] time = 0.0307889, antiderivative size = 67, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {819, 635, 205, 260} \[ \frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 \sqrt{a} c^{3/2}}+\frac{e \log \left (a+c x^2\right )}{2 c^2}-\frac{x (d+e x)}{2 c \left (a+c x^2\right )} \]
Antiderivative was successfully verified.
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Rule 819
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{x^2 (d+e x)}{\left (a+c x^2\right )^2} \, dx &=-\frac{x (d+e x)}{2 c \left (a+c x^2\right )}+\frac{\int \frac{a d+2 a e x}{a+c x^2} \, dx}{2 a c}\\ &=-\frac{x (d+e x)}{2 c \left (a+c x^2\right )}+\frac{d \int \frac{1}{a+c x^2} \, dx}{2 c}+\frac{e \int \frac{x}{a+c x^2} \, dx}{c}\\ &=-\frac{x (d+e x)}{2 c \left (a+c x^2\right )}+\frac{d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 \sqrt{a} c^{3/2}}+\frac{e \log \left (a+c x^2\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.0385828, size = 62, normalized size = 0.93 \[ \frac{\frac{a e-c d x}{a+c x^2}+\frac{\sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a}}+e \log \left (a+c x^2\right )}{2 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.007, size = 61, normalized size = 0.9 \begin{align*}{\frac{1}{c{x}^{2}+a} \left ( -{\frac{dx}{2\,c}}+{\frac{ae}{2\,{c}^{2}}} \right ) }+{\frac{e\ln \left ( c{x}^{2}+a \right ) }{2\,{c}^{2}}}+{\frac{d}{2\,c}\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 = 1.52568, size = 406, normalized size = 6.06 \begin{align*} \left [-\frac{2 \, a c d x - 2 \, a^{2} e +{\left (c d x^{2} + a d\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) - 2 \,{\left (a c e x^{2} + a^{2} e\right )} \log \left (c x^{2} + a\right )}{4 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}, -\frac{a c d x - a^{2} e -{\left (c d x^{2} + a d\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (a c e x^{2} + a^{2} e\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (a c^{3} x^{2} + a^{2} c^{2}\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.844594, size = 162, normalized size = 2.42 \begin{align*} \left (\frac{e}{2 c^{2}} - \frac{d \sqrt{- a c^{5}}}{4 a c^{4}}\right ) \log{\left (x + \frac{4 a c^{2} \left (\frac{e}{2 c^{2}} - \frac{d \sqrt{- a c^{5}}}{4 a c^{4}}\right ) - 2 a e}{c d} \right )} + \left (\frac{e}{2 c^{2}} + \frac{d \sqrt{- a c^{5}}}{4 a c^{4}}\right ) \log{\left (x + \frac{4 a c^{2} \left (\frac{e}{2 c^{2}} + \frac{d \sqrt{- a c^{5}}}{4 a c^{4}}\right ) - 2 a e}{c d} \right )} - \frac{- a e + c d x}{2 a c^{2} + 2 c^{3} x^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.10044, size = 84, normalized size = 1.25 \begin{align*} \frac{d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} c} + \frac{e \log \left (c x^{2} + a\right )}{2 \, c^{2}} - \frac{d x - \frac{a e}{c}}{2 \,{\left (c x^{2} + a\right )} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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