Optimal. Leaf size=154 \[ \frac{a^2 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}+\frac{x \left (a+c x^2\right )^{3/2} \left (6 c d^2-a e^2\right )}{24 c}+\frac{a x \sqrt{a+c x^2} \left (6 c d^2-a e^2\right )}{16 c}+\frac{7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac{e \left (a+c x^2\right )^{5/2} (d+e x)}{6 c} \]
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Rubi [A] time = 0.0685449, antiderivative size = 154, 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, 641, 195, 217, 206} \[ \frac{a^2 \left (6 c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}+\frac{x \left (a+c x^2\right )^{3/2} \left (6 c d^2-a e^2\right )}{24 c}+\frac{a x \sqrt{a+c x^2} \left (6 c d^2-a e^2\right )}{16 c}+\frac{7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac{e \left (a+c x^2\right )^{5/2} (d+e x)}{6 c} \]
Antiderivative was successfully verified.
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Rule 743
Rule 641
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int (d+e x)^2 \left (a+c x^2\right )^{3/2} \, dx &=\frac{e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac{\int \left (6 c d^2-a e^2+7 c d e x\right ) \left (a+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac{7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac{e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac{\left (6 c d^2-a e^2\right ) \int \left (a+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac{\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac{e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac{\left (a \left (6 c d^2-a e^2\right )\right ) \int \sqrt{a+c x^2} \, dx}{8 c}\\ &=\frac{a \left (6 c d^2-a e^2\right ) x \sqrt{a+c x^2}}{16 c}+\frac{\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac{e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac{\left (a^2 \left (6 c d^2-a e^2\right )\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{16 c}\\ &=\frac{a \left (6 c d^2-a e^2\right ) x \sqrt{a+c x^2}}{16 c}+\frac{\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac{e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac{\left (a^2 \left (6 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{a \left (6 c d^2-a e^2\right ) x \sqrt{a+c x^2}}{16 c}+\frac{\left (6 c d^2-a e^2\right ) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{7 d e \left (a+c x^2\right )^{5/2}}{30 c}+\frac{e (d+e x) \left (a+c x^2\right )^{5/2}}{6 c}+\frac{a^2 \left (6 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 = 0.0930768, size = 132, normalized size = 0.86 \[ \frac{\sqrt{c} \sqrt{a+c x^2} \left (3 a^2 e (32 d+5 e x)+2 a c x \left (75 d^2+96 d e x+35 e^2 x^2\right )+4 c^2 x^3 \left (15 d^2+24 d e x+10 e^2 x^2\right )\right )-15 a^2 \left (a e^2-6 c d^2\right ) \log \left (\sqrt{c} \sqrt{a+c x^2}+c x\right )}{240 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 161, normalized size = 1.1 \begin{align*}{\frac{{e}^{2}x}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{a{e}^{2}x}{24\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}{e}^{2}x}{16\,c}\sqrt{c{x}^{2}+a}}-{\frac{{e}^{2}{a}^{3}}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{2\,de}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{{d}^{2}x}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,a{d}^{2}x}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,{a}^{2}{d}^{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 = 2.13032, size = 675, normalized size = 4.38 \begin{align*} \left [-\frac{15 \,{\left (6 \, a^{2} c d^{2} - a^{3} e^{2}\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (40 \, c^{3} e^{2} x^{5} + 96 \, c^{3} d e x^{4} + 192 \, a c^{2} d e x^{2} + 96 \, a^{2} c d e + 10 \,{\left (6 \, c^{3} d^{2} + 7 \, a c^{2} e^{2}\right )} x^{3} + 15 \,{\left (10 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{480 \, c^{2}}, -\frac{15 \,{\left (6 \, a^{2} c d^{2} - a^{3} e^{2}\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (40 \, c^{3} e^{2} x^{5} + 96 \, c^{3} d e x^{4} + 192 \, a c^{2} d e x^{2} + 96 \, a^{2} c d e + 10 \,{\left (6 \, c^{3} d^{2} + 7 \, a c^{2} e^{2}\right )} x^{3} + 15 \,{\left (10 \, a c^{2} d^{2} + a^{2} c e^{2}\right )} x\right )} \sqrt{c x^{2} + a}}{240 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 15.5335, size = 372, normalized size = 2.42 \begin{align*} \frac{a^{\frac{5}{2}} e^{2} x}{16 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{a^{\frac{3}{2}} d^{2} x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{a^{\frac{3}{2}} d^{2} x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 a^{\frac{3}{2}} e^{2} x^{3}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 \sqrt{a} c d^{2} x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{11 \sqrt{a} c e^{2} x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{a^{3} e^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 c^{\frac{3}{2}}} + \frac{3 a^{2} d^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 \sqrt{c}} + 2 a d 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 ) + 2 c d 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 ) + \frac{c^{2} d^{2} x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{c^{2} e^{2} x^{7}}{6 \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 = 1.37037, size = 192, normalized size = 1.25 \begin{align*} \frac{1}{240} \, \sqrt{c x^{2} + a}{\left (\frac{96 \, a^{2} d e}{c} +{\left (2 \,{\left (96 \, a d e +{\left (4 \,{\left (5 \, c x e^{2} + 12 \, c d e\right )} x + \frac{5 \,{\left (6 \, c^{5} d^{2} + 7 \, a c^{4} e^{2}\right )}}{c^{4}}\right )} x\right )} x + \frac{15 \,{\left (10 \, a c^{4} d^{2} + a^{2} c^{3} e^{2}\right )}}{c^{4}}\right )} x\right )} - \frac{{\left (6 \, a^{2} c d^{2} - a^{3} 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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