Optimal. Leaf size=59 \[ \frac{\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{d e \log \left (a+c x^2\right )}{c}+\frac{e^2 x}{c} \]
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Rubi [A] time = 0.0509034, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235, Rules used = {702, 635, 205, 260} \[ \frac{\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{d e \log \left (a+c x^2\right )}{c}+\frac{e^2 x}{c} \]
Antiderivative was successfully verified.
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Rule 702
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{a+c x^2} \, dx &=\int \left (\frac{e^2}{c}+\frac{c d^2-a e^2+2 c d e x}{c \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{e^2 x}{c}+\frac{\int \frac{c d^2-a e^2+2 c d e x}{a+c x^2} \, dx}{c}\\ &=\frac{e^2 x}{c}+(2 d e) \int \frac{x}{a+c x^2} \, dx+\frac{\left (c d^2-a e^2\right ) \int \frac{1}{a+c x^2} \, dx}{c}\\ &=\frac{e^2 x}{c}+\frac{\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{d e \log \left (a+c x^2\right )}{c}\\ \end{align*}
Mathematica [A] time = 0.0400094, size = 56, normalized size = 0.95 \[ \frac{\left (c d^2-a e^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{3/2}}+\frac{e \left (d \log \left (a+c x^2\right )+e x\right )}{c} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 65, normalized size = 1.1 \begin{align*}{\frac{{e}^{2}x}{c}}+{\frac{de\ln \left ( c{x}^{2}+a \right ) }{c}}-{\frac{a{e}^{2}}{c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{{d}^{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 = 1.91244, size = 309, normalized size = 5.24 \begin{align*} \left [\frac{2 \, a c e^{2} x + 2 \, a c d e \log \left (c x^{2} + a\right ) +{\left (c d^{2} - a e^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} + 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right )}{2 \, a c^{2}}, \frac{a c e^{2} x + a c d e \log \left (c x^{2} + a\right ) +{\left (c d^{2} - a e^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right )}{a c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 0.700622, size = 185, normalized size = 3.14 \begin{align*} \left (\frac{d e}{c} - \frac{\sqrt{- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) \log{\left (x + \frac{- 2 a c \left (\frac{d e}{c} - \frac{\sqrt{- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) + 2 a d e}{a e^{2} - c d^{2}} \right )} + \left (\frac{d e}{c} + \frac{\sqrt{- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) \log{\left (x + \frac{- 2 a c \left (\frac{d e}{c} + \frac{\sqrt{- a c^{3}} \left (a e^{2} - c d^{2}\right )}{2 a c^{3}}\right ) + 2 a d e}{a e^{2} - c d^{2}} \right )} + \frac{e^{2} x}{c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.42494, size = 70, normalized size = 1.19 \begin{align*} \frac{d e \log \left (c x^{2} + a\right )}{c} + \frac{x e^{2}}{c} + \frac{{\left (c d^{2} - a e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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