Optimal. Leaf size=61 \[ -\frac{\sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{3/2}}-\frac{a e \log \left (a+c x^2\right )}{2 c^2}+\frac{d x}{c}+\frac{e x^2}{2 c} \]
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Rubi [A] time = 0.0441019, antiderivative size = 61, 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} \[ -\frac{\sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{3/2}}-\frac{a e \log \left (a+c x^2\right )}{2 c^2}+\frac{d x}{c}+\frac{e x^2}{2 c} \]
Antiderivative was successfully verified.
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Rule 801
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{x^2 (d+e x)}{a+c x^2} \, dx &=\int \left (\frac{d}{c}+\frac{e x}{c}-\frac{a d+a e x}{c \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{d x}{c}+\frac{e x^2}{2 c}-\frac{\int \frac{a d+a e x}{a+c x^2} \, dx}{c}\\ &=\frac{d x}{c}+\frac{e x^2}{2 c}-\frac{(a d) \int \frac{1}{a+c x^2} \, dx}{c}-\frac{(a e) \int \frac{x}{a+c x^2} \, dx}{c}\\ &=\frac{d x}{c}+\frac{e x^2}{2 c}-\frac{\sqrt{a} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{c^{3/2}}-\frac{a e \log \left (a+c x^2\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.0209669, size = 56, normalized size = 0.92 \[ \frac{-2 \sqrt{a} \sqrt{c} d \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )-a e \log \left (a+c x^2\right )+c x (2 d+e x)}{2 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.004, size = 53, normalized size = 0.9 \begin{align*}{\frac{e{x}^{2}}{2\,c}}+{\frac{dx}{c}}-{\frac{ae\ln \left ( c{x}^{2}+a \right ) }{2\,{c}^{2}}}-{\frac{ad}{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 = 1.54537, size = 285, normalized size = 4.67 \begin{align*} \left [\frac{c e x^{2} + c d \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{2} - 2 \, c x \sqrt{-\frac{a}{c}} - a}{c x^{2} + a}\right ) + 2 \, c d x - a e \log \left (c x^{2} + a\right )}{2 \, c^{2}}, \frac{c e x^{2} - 2 \, c d \sqrt{\frac{a}{c}} \arctan \left (\frac{c x \sqrt{\frac{a}{c}}}{a}\right ) + 2 \, c d x - a e \log \left (c x^{2} + a\right )}{2 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.591244, size = 151, normalized size = 2.48 \begin{align*} \left (- \frac{a e}{2 c^{2}} - \frac{d \sqrt{- a c^{5}}}{2 c^{4}}\right ) \log{\left (x + \frac{- a e - 2 c^{2} \left (- \frac{a e}{2 c^{2}} - \frac{d \sqrt{- a c^{5}}}{2 c^{4}}\right )}{c d} \right )} + \left (- \frac{a e}{2 c^{2}} + \frac{d \sqrt{- a c^{5}}}{2 c^{4}}\right ) \log{\left (x + \frac{- a e - 2 c^{2} \left (- \frac{a e}{2 c^{2}} + \frac{d \sqrt{- a c^{5}}}{2 c^{4}}\right )}{c d} \right )} + \frac{d x}{c} + \frac{e x^{2}}{2 c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.13193, size = 76, normalized size = 1.25 \begin{align*} -\frac{a d \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c} - \frac{a e \log \left (c x^{2} + a\right )}{2 \, c^{2}} + \frac{c x^{2} e + 2 \, c d x}{2 \, c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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