Optimal. Leaf size=110 \[ \frac{e \sqrt{a+c x^2} \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right )}{6 c^2}+\frac{d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{e \sqrt{a+c x^2} (d+e x)^2}{3 c} \]
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Rubi [A] time = 0.0753626, antiderivative size = 110, 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, 780, 217, 206} \[ \frac{e \sqrt{a+c x^2} \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right )}{6 c^2}+\frac{d \left (2 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{2 c^{3/2}}+\frac{e \sqrt{a+c x^2} (d+e x)^2}{3 c} \]
Antiderivative was successfully verified.
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Rule 743
Rule 780
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\sqrt{a+c x^2}} \, dx &=\frac{e (d+e x)^2 \sqrt{a+c x^2}}{3 c}+\frac{\int \frac{(d+e x) \left (3 c d^2-2 a e^2+5 c d e x\right )}{\sqrt{a+c x^2}} \, dx}{3 c}\\ &=\frac{e (d+e x)^2 \sqrt{a+c x^2}}{3 c}+\frac{e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt{a+c x^2}}{6 c^2}+\frac{\left (d \left (2 c d^2-3 a e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{2 c}\\ &=\frac{e (d+e x)^2 \sqrt{a+c x^2}}{3 c}+\frac{e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt{a+c x^2}}{6 c^2}+\frac{\left (d \left (2 c d^2-3 a e^2\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{2 c}\\ &=\frac{e (d+e x)^2 \sqrt{a+c x^2}}{3 c}+\frac{e \left (4 \left (4 c d^2-a e^2\right )+5 c d e x\right ) \sqrt{a+c x^2}}{6 c^2}+\frac{d \left (2 c d^2-3 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 = 0.0623585, size = 92, normalized size = 0.84 \[ \frac{e \sqrt{a+c x^2} \left (c \left (18 d^2+9 d e x+2 e^2 x^2\right )-4 a e^2\right )+3 \sqrt{c} d \left (2 c d^2-3 a e^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{6 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.048, size = 126, normalized size = 1.2 \begin{align*}{\frac{{e}^{3}{x}^{2}}{3\,c}\sqrt{c{x}^{2}+a}}-{\frac{2\,a{e}^{3}}{3\,{c}^{2}}\sqrt{c{x}^{2}+a}}+{\frac{3\,d{e}^{2}x}{2\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,ad{e}^{2}}{2}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+3\,{\frac{{d}^{2}e\sqrt{c{x}^{2}+a}}{c}}+{{d}^{3}\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 = 1.90701, size = 428, normalized size = 3.89 \begin{align*} \left [-\frac{3 \,{\left (2 \, c d^{3} - 3 \, a d e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (2 \, c e^{3} x^{2} + 9 \, c d e^{2} x + 18 \, c d^{2} e - 4 \, a e^{3}\right )} \sqrt{c x^{2} + a}}{12 \, c^{2}}, -\frac{3 \,{\left (2 \, c d^{3} - 3 \, a d e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (2 \, c e^{3} x^{2} + 9 \, c d e^{2} x + 18 \, c d^{2} e - 4 \, a e^{3}\right )} \sqrt{c x^{2} + a}}{6 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 5.99054, size = 216, normalized size = 1.96 \begin{align*} \frac{3 \sqrt{a} d e^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2 c} - \frac{3 a d e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}}} + d^{3} \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 ) + 3 d^{2} 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 ) + e^{3} \left (\begin{cases} - \frac{2 a \sqrt{a + c x^{2}}}{3 c^{2}} + \frac{x^{2} \sqrt{a + c x^{2}}}{3 c} & \text{for}\: c \neq 0 \\\frac{x^{4}}{4 \sqrt{a}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.36014, size = 122, normalized size = 1.11 \begin{align*} \frac{1}{6} \, \sqrt{c x^{2} + a}{\left (x{\left (\frac{2 \, x e^{3}}{c} + \frac{9 \, d e^{2}}{c}\right )} + \frac{2 \,{\left (9 \, c^{2} d^{2} e - 2 \, a c e^{3}\right )}}{c^{3}}\right )} - \frac{{\left (2 \, c d^{3} - 3 \, a d 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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