Optimal. Leaf size=180 \[ \frac{3 a^2 d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}+\frac{e \left (a+c x^2\right )^{5/2} \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right )}{70 c^2}+\frac{d x \left (a+c x^2\right )^{3/2} \left (2 c d^2-a e^2\right )}{8 c}+\frac{3 a d x \sqrt{a+c x^2} \left (2 c d^2-a e^2\right )}{16 c}+\frac{e \left (a+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]
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Rubi [A] time = 0.145263, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{3 a^2 d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}+\frac{e \left (a+c x^2\right )^{5/2} \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right )}{70 c^2}+\frac{d x \left (a+c x^2\right )^{3/2} \left (2 c d^2-a e^2\right )}{8 c}+\frac{3 a d x \sqrt{a+c x^2} \left (2 c d^2-a e^2\right )}{16 c}+\frac{e \left (a+c x^2\right )^{5/2} (d+e x)^2}{7 c} \]
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 \left (a+c x^2\right )^{3/2} \, dx &=\frac{e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{\int (d+e x) \left (7 c d^2-2 a e^2+9 c d e x\right ) \left (a+c x^2\right )^{3/2} \, dx}{7 c}\\ &=\frac{e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac{\left (d \left (2 c d^2-a e^2\right )\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{2 c}\\ &=\frac{d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac{\left (3 a d \left (2 c d^2-a e^2\right )\right ) \int \sqrt{a+c x^2} \, dx}{8 c}\\ &=\frac{3 a d \left (2 c d^2-a e^2\right ) x \sqrt{a+c x^2}}{16 c}+\frac{d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac{\left (3 a^2 d \left (2 c d^2-a e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{16 c}\\ &=\frac{3 a d \left (2 c d^2-a e^2\right ) x \sqrt{a+c x^2}}{16 c}+\frac{d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac{\left (3 a^2 d \left (2 c d^2-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 )}{16 c}\\ &=\frac{3 a d \left (2 c d^2-a e^2\right ) x \sqrt{a+c x^2}}{16 c}+\frac{d \left (2 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{8 c}+\frac{e (d+e x)^2 \left (a+c x^2\right )^{5/2}}{7 c}+\frac{e \left (4 \left (8 c d^2-a e^2\right )+15 c d e x\right ) \left (a+c x^2\right )^{5/2}}{70 c^2}+\frac{3 a^2 d \left (2 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.121345, size = 174, normalized size = 0.97 \[ \frac{\sqrt{a+c x^2} \left (a^2 c e \left (336 d^2+105 d e x+16 e^2 x^2\right )-32 a^3 e^3+2 a c^2 x \left (336 d^2 e x+175 d^3+245 d e^2 x^2+64 e^3 x^3\right )+4 c^3 x^3 \left (84 d^2 e x+35 d^3+70 d e^2 x^2+20 e^3 x^3\right )\right )-105 a^2 \sqrt{c} d \left (a e^2-2 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{560 c^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 205, normalized size = 1.1 \begin{align*}{\frac{{e}^{3}{x}^{2}}{7\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{2\,a{e}^{3}}{35\,{c}^{2}} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{d{e}^{2}x}{2\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{ad{e}^{2}x}{8\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{3\,d{e}^{2}{a}^{2}x}{16\,c}\sqrt{c{x}^{2}+a}}-{\frac{3\,d{e}^{2}{a}^{3}}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{3\,{d}^{2}e}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{3}x}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,{d}^{3}ax}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,{d}^{3}{a}^{2}}{8}\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 = 2.27818, size = 902, normalized size = 5.01 \begin{align*} \left [\frac{105 \,{\left (2 \, a^{2} c d^{3} - a^{3} 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 (80 \, c^{3} e^{3} x^{6} + 280 \, c^{3} d e^{2} x^{5} + 336 \, a^{2} c d^{2} e - 32 \, a^{3} e^{3} + 16 \,{\left (21 \, c^{3} d^{2} e + 8 \, a c^{2} e^{3}\right )} x^{4} + 70 \,{\left (2 \, c^{3} d^{3} + 7 \, a c^{2} d e^{2}\right )} x^{3} + 16 \,{\left (42 \, a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{2} + 35 \,{\left (10 \, a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{1120 \, c^{2}}, -\frac{105 \,{\left (2 \, a^{2} c d^{3} - a^{3} d e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (80 \, c^{3} e^{3} x^{6} + 280 \, c^{3} d e^{2} x^{5} + 336 \, a^{2} c d^{2} e - 32 \, a^{3} e^{3} + 16 \,{\left (21 \, c^{3} d^{2} e + 8 \, a c^{2} e^{3}\right )} x^{4} + 70 \,{\left (2 \, c^{3} d^{3} + 7 \, a c^{2} d e^{2}\right )} x^{3} + 16 \,{\left (42 \, a c^{2} d^{2} e + a^{2} c e^{3}\right )} x^{2} + 35 \,{\left (10 \, a c^{2} d^{3} + 3 \, a^{2} c d e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{560 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 17.0949, size = 551, normalized size = 3.06 \begin{align*} \frac{3 a^{\frac{5}{2}} d e^{2} x}{16 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{a^{\frac{3}{2}} d^{3} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{a^{\frac{3}{2}} d^{3} x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 a^{\frac{3}{2}} d e^{2} x^{3}}{16 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 \sqrt{a} c d^{3} x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{11 \sqrt{a} c d e^{2} x^{5}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{3 a^{3} d e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 c^{\frac{3}{2}}} + \frac{3 a^{2} d^{3} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 \sqrt{c}} + 3 a 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 ) + a 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 ) + 3 c d^{2} e \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 ) + c e^{3} \left (\begin{cases} \frac{8 a^{3} \sqrt{a + c x^{2}}}{105 c^{3}} - \frac{4 a^{2} x^{2} \sqrt{a + c x^{2}}}{105 c^{2}} + \frac{a x^{4} \sqrt{a + c x^{2}}}{35 c} + \frac{x^{6} \sqrt{a + c x^{2}}}{7} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{6}}{6} & \text{otherwise} \end{cases}\right ) + \frac{c^{2} d^{3} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{c^{2} d e^{2} x^{7}}{2 \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 = 1.29844, size = 286, normalized size = 1.59 \begin{align*} \frac{1}{560} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \,{\left (2 \, c x e^{3} + 7 \, c d e^{2}\right )} x + \frac{2 \,{\left (21 \, c^{6} d^{2} e + 8 \, a c^{5} e^{3}\right )}}{c^{5}}\right )} x + \frac{35 \,{\left (2 \, c^{6} d^{3} + 7 \, a c^{5} d e^{2}\right )}}{c^{5}}\right )} x + \frac{8 \,{\left (42 \, a c^{5} d^{2} e + a^{2} c^{4} e^{3}\right )}}{c^{5}}\right )} x + \frac{35 \,{\left (10 \, a c^{5} d^{3} + 3 \, a^{2} c^{4} d e^{2}\right )}}{c^{5}}\right )} x + \frac{16 \,{\left (21 \, a^{2} c^{4} d^{2} e - 2 \, a^{3} c^{3} e^{3}\right )}}{c^{5}}\right )} - \frac{3 \,{\left (2 \, a^{2} c d^{3} - a^{3} d e^{2}\right )} \log \left ({\left | -\sqrt{c} x + \sqrt{c x^{2} + a} \right |}\right )}{16 \, c^{\frac{3}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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