Optimal. Leaf size=144 \[ \frac{e \left (a+c x^2\right )^{3/2} \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right )}{60 c^2}+\frac{a d \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2}}+\frac{d x \sqrt{a+c x^2} \left (4 c d^2-3 a e^2\right )}{8 c}+\frac{e \left (a+c x^2\right )^{3/2} (d+e x)^2}{5 c} \]
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Rubi [A] time = 0.114714, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.263, Rules used = {743, 780, 195, 217, 206} \[ \frac{e \left (a+c x^2\right )^{3/2} \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right )}{60 c^2}+\frac{a d \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2}}+\frac{d x \sqrt{a+c x^2} \left (4 c d^2-3 a e^2\right )}{8 c}+\frac{e \left (a+c x^2\right )^{3/2} (d+e x)^2}{5 c} \]
Antiderivative was successfully verified.
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Rule 743
Rule 780
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int (d+e x)^3 \sqrt{a+c x^2} \, dx &=\frac{e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac{\int (d+e x) \left (5 c d^2-2 a e^2+7 c d e x\right ) \sqrt{a+c x^2} \, dx}{5 c}\\ &=\frac{e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac{e \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right ) \left (a+c x^2\right )^{3/2}}{60 c^2}+\frac{\left (d \left (4 c d^2-3 a e^2\right )\right ) \int \sqrt{a+c x^2} \, dx}{4 c}\\ &=\frac{d \left (4 c d^2-3 a e^2\right ) x \sqrt{a+c x^2}}{8 c}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac{e \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right ) \left (a+c x^2\right )^{3/2}}{60 c^2}+\frac{\left (a d \left (4 c d^2-3 a e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{8 c}\\ &=\frac{d \left (4 c d^2-3 a e^2\right ) x \sqrt{a+c x^2}}{8 c}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac{e \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right ) \left (a+c x^2\right )^{3/2}}{60 c^2}+\frac{\left (a d \left (4 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 )}{8 c}\\ &=\frac{d \left (4 c d^2-3 a e^2\right ) x \sqrt{a+c x^2}}{8 c}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{3/2}}{5 c}+\frac{e \left (8 \left (6 c d^2-a e^2\right )+21 c d e x\right ) \left (a+c x^2\right )^{3/2}}{60 c^2}+\frac{a d \left (4 c d^2-3 a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0835308, size = 132, normalized size = 0.92 \[ \frac{\sqrt{a+c x^2} \left (-16 a^2 e^3+a c e \left (120 d^2+45 d e x+8 e^2 x^2\right )+6 c^2 x \left (20 d^2 e x+10 d^3+15 d e^2 x^2+4 e^3 x^3\right )\right )+15 a \sqrt{c} d \left (4 c d^2-3 a e^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{120 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 164, normalized size = 1.1 \begin{align*}{\frac{{e}^{3}{x}^{2}}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{2\,a{e}^{3}}{15\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,d{e}^{2}x}{4\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,ad{e}^{2}x}{8\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,d{e}^{2}{a}^{2}}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{d}^{2}e}{c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{{d}^{3}x}{2}\sqrt{c{x}^{2}+a}}+{\frac{{d}^{3}a}{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 = 2.28943, size = 662, normalized size = 4.6 \begin{align*} \left [-\frac{15 \,{\left (4 \, a c d^{3} - 3 \, a^{2} 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 (24 \, c^{2} e^{3} x^{4} + 90 \, c^{2} d e^{2} x^{3} + 120 \, a c d^{2} e - 16 \, a^{2} e^{3} + 8 \,{\left (15 \, c^{2} d^{2} e + a c e^{3}\right )} x^{2} + 15 \,{\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{240 \, c^{2}}, -\frac{15 \,{\left (4 \, a c d^{3} - 3 \, a^{2} d e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (24 \, c^{2} e^{3} x^{4} + 90 \, c^{2} d e^{2} x^{3} + 120 \, a c d^{2} e - 16 \, a^{2} e^{3} + 8 \,{\left (15 \, c^{2} d^{2} e + a c e^{3}\right )} x^{2} + 15 \,{\left (4 \, c^{2} d^{3} + 3 \, a c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{120 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.52748, size = 265, normalized size = 1.84 \begin{align*} \frac{3 a^{\frac{3}{2}} d e^{2} x}{8 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{\sqrt{a} d^{3} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{9 \sqrt{a} d e^{2} x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{3 a^{2} d e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 c^{\frac{3}{2}}} + \frac{a d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{c}} + 3 d^{2} e \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\left (a + c x^{2}\right )^{\frac{3}{2}}}{3 c} & \text{otherwise} \end{cases}\right ) + e^{3} \left (\begin{cases} - \frac{2 a^{2} \sqrt{a + c x^{2}}}{15 c^{2}} + \frac{a x^{2} \sqrt{a + c x^{2}}}{15 c} + \frac{x^{4} \sqrt{a + c x^{2}}}{5} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases}\right ) + \frac{3 c d e^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.40938, size = 194, normalized size = 1.35 \begin{align*} \frac{1}{120} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left (3 \,{\left (4 \, x e^{3} + 15 \, d e^{2}\right )} x + \frac{4 \,{\left (15 \, c^{3} d^{2} e + a c^{2} e^{3}\right )}}{c^{3}}\right )} x + \frac{15 \,{\left (4 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )}}{c^{3}}\right )} x + \frac{8 \,{\left (15 \, a c^{2} d^{2} e - 2 \, a^{2} c e^{3}\right )}}{c^{3}}\right )} - \frac{{\left (4 \, a c d^{3} - 3 \, a^{2} d e^{2}\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{8 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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