Optimal. Leaf size=86 \[ \frac{\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e \sqrt{a+c x^2} (d+e x)}{2 c} \]
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Rubi [A] time = 0.0402104, antiderivative size = 86, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {743, 641, 217, 206} \[ \frac{\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e \sqrt{a+c x^2} (d+e x)}{2 c} \]
Antiderivative was successfully verified.
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Rule 743
Rule 641
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\sqrt{a+c x^2}} \, dx &=\frac{e (d+e x) \sqrt{a+c x^2}}{2 c}+\frac{\int \frac{2 c d^2-a e^2+3 c d e x}{\sqrt{a+c x^2}} \, dx}{2 c}\\ &=\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e (d+e x) \sqrt{a+c x^2}}{2 c}+\frac{\left (2 c d^2-a e^2\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{2 c}\\ &=\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e (d+e x) \sqrt{a+c x^2}}{2 c}+\frac{\left (2 c d^2-a e^2\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{2 c}\\ &=\frac{3 d e \sqrt{a+c x^2}}{2 c}+\frac{e (d+e x) \sqrt{a+c x^2}}{2 c}+\frac{\left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0424227, size = 71, normalized size = 0.83 \[ \frac{\left (2 c d^2-a e^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )+\sqrt{c} e \sqrt{a+c x^2} (4 d+e x)}{2 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.047, size = 84, normalized size = 1. \begin{align*}{\frac{{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+a}}-{\frac{a{e}^{2}}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+2\,{\frac{de\sqrt{c{x}^{2}+a}}{c}}+{{d}^{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){\frac{1}{\sqrt{c}}}} \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 = 1.92376, size = 327, normalized size = 3.8 \begin{align*} \left [-\frac{{\left (2 \, c d^{2} - a e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (c e^{2} x + 4 \, c d e\right )} \sqrt{c x^{2} + a}}{4 \, c^{2}}, -\frac{{\left (2 \, c d^{2} - a e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (c e^{2} x + 4 \, c d e\right )} \sqrt{c x^{2} + a}}{2 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 4.90411, size = 158, normalized size = 1.84 \begin{align*} \frac{\sqrt{a} e^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2 c} - \frac{a e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}}} + d^{2} \left (\begin{cases} \frac{\sqrt{- \frac{a}{c}} \operatorname{asin}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c < 0 \\\frac{\sqrt{\frac{a}{c}} \operatorname{asinh}{\left (x \sqrt{\frac{c}{a}} \right )}}{\sqrt{a}} & \text{for}\: a > 0 \wedge c > 0 \\\frac{\sqrt{- \frac{a}{c}} \operatorname{acosh}{\left (x \sqrt{- \frac{c}{a}} \right )}}{\sqrt{- a}} & \text{for}\: c > 0 \wedge a < 0 \end{cases}\right ) + 2 d e \left (\begin{cases} \frac{x^{2}}{2 \sqrt{a}} & \text{for}\: c = 0 \\\frac{\sqrt{a + c x^{2}}}{c} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37338, size = 85, normalized size = 0.99 \begin{align*} \frac{1}{2} \, \sqrt{c x^{2} + a}{\left (\frac{x e^{2}}{c} + \frac{4 \, d e}{c}\right )} - \frac{{\left (2 \, c d^{2} - a e^{2}\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{2 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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