Optimal. Leaf size=137 \[ \frac{a^2 (6 c d-a f) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}+\frac{x \left (a+c x^2\right )^{3/2} (6 c d-a f)}{24 c}+\frac{a x \sqrt{a+c x^2} (6 c d-a f)}{16 c}+\frac{e \left (a+c x^2\right )^{5/2}}{5 c}+\frac{f x \left (a+c x^2\right )^{5/2}}{6 c} \]
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Rubi [A] time = 0.0834329, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {1815, 641, 195, 217, 206} \[ \frac{a^2 (6 c d-a f) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}+\frac{x \left (a+c x^2\right )^{3/2} (6 c d-a f)}{24 c}+\frac{a x \sqrt{a+c x^2} (6 c d-a f)}{16 c}+\frac{e \left (a+c x^2\right )^{5/2}}{5 c}+\frac{f x \left (a+c x^2\right )^{5/2}}{6 c} \]
Antiderivative was successfully verified.
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Rule 1815
Rule 641
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \left (a+c x^2\right )^{3/2} \left (d+e x+f x^2\right ) \, dx &=\frac{f x \left (a+c x^2\right )^{5/2}}{6 c}+\frac{\int (6 c d-a f+6 c e x) \left (a+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac{e \left (a+c x^2\right )^{5/2}}{5 c}+\frac{f x \left (a+c x^2\right )^{5/2}}{6 c}+\frac{(6 c d-a f) \int \left (a+c x^2\right )^{3/2} \, dx}{6 c}\\ &=\frac{(6 c d-a f) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{e \left (a+c x^2\right )^{5/2}}{5 c}+\frac{f x \left (a+c x^2\right )^{5/2}}{6 c}+\frac{(a (6 c d-a f)) \int \sqrt{a+c x^2} \, dx}{8 c}\\ &=\frac{a (6 c d-a f) x \sqrt{a+c x^2}}{16 c}+\frac{(6 c d-a f) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{e \left (a+c x^2\right )^{5/2}}{5 c}+\frac{f x \left (a+c x^2\right )^{5/2}}{6 c}+\frac{\left (a^2 (6 c d-a f)\right ) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{16 c}\\ &=\frac{a (6 c d-a f) x \sqrt{a+c x^2}}{16 c}+\frac{(6 c d-a f) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{e \left (a+c x^2\right )^{5/2}}{5 c}+\frac{f x \left (a+c x^2\right )^{5/2}}{6 c}+\frac{\left (a^2 (6 c d-a f)\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 (6 c d-a f) x \sqrt{a+c x^2}}{16 c}+\frac{(6 c d-a f) x \left (a+c x^2\right )^{3/2}}{24 c}+\frac{e \left (a+c x^2\right )^{5/2}}{5 c}+\frac{f x \left (a+c x^2\right )^{5/2}}{6 c}+\frac{a^2 (6 c d-a f) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{16 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.268972, size = 125, normalized size = 0.91 \[ \frac{\sqrt{a+c x^2} \left (\sqrt{c} \left (3 a^2 (16 e+5 f x)+2 a c x (75 d+x (48 e+35 f x))+4 c^2 x^3 (15 d+2 x (6 e+5 f x))\right )-\frac{15 a^{3/2} (a f-6 c d) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{\frac{c x^2}{a}+1}}\right )}{240 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 146, normalized size = 1.1 \begin{align*}{\frac{fx}{6\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{afx}{24\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{{a}^{2}fx}{16\,c}\sqrt{c{x}^{2}+a}}-{\frac{{a}^{3}f}{16}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{e}{5\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{dx}{4} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{3\,adx}{8}\sqrt{c{x}^{2}+a}}+{\frac{3\,{a}^{2}d}{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 = 1.39463, size = 618, normalized size = 4.51 \begin{align*} \left [-\frac{15 \,{\left (6 \, a^{2} c d - a^{3} f\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} f x^{5} + 48 \, c^{3} e x^{4} + 96 \, a c^{2} e x^{2} + 48 \, a^{2} c e + 10 \,{\left (6 \, c^{3} d + 7 \, a c^{2} f\right )} x^{3} + 15 \,{\left (10 \, a c^{2} d + a^{2} c f\right )} x\right )} \sqrt{c x^{2} + a}}{480 \, c^{2}}, -\frac{15 \,{\left (6 \, a^{2} c d - a^{3} f\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (40 \, c^{3} f x^{5} + 48 \, c^{3} e x^{4} + 96 \, a c^{2} e x^{2} + 48 \, a^{2} c e + 10 \,{\left (6 \, c^{3} d + 7 \, a c^{2} f\right )} x^{3} + 15 \,{\left (10 \, a c^{2} d + a^{2} c f\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 = 14.9926, size = 348, normalized size = 2.54 \begin{align*} \frac{a^{\frac{5}{2}} f x}{16 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{a^{\frac{3}{2}} d x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{a^{\frac{3}{2}} d x}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{17 a^{\frac{3}{2}} f x^{3}}{48 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{3 \sqrt{a} c d x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{11 \sqrt{a} c f x^{5}}{24 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{a^{3} f \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{16 c^{\frac{3}{2}}} + \frac{3 a^{2} d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 \sqrt{c}} + a 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 ) + c 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 x^{5}}{4 \sqrt{a} \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{c^{2} f 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 = 1.19338, size = 174, normalized size = 1.27 \begin{align*} \frac{1}{240} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left ({\left (4 \,{\left (5 \, c f x + 6 \, c e\right )} x + \frac{5 \,{\left (6 \, c^{5} d + 7 \, a c^{4} f\right )}}{c^{4}}\right )} x + 48 \, a e\right )} x + \frac{15 \,{\left (10 \, a c^{4} d + a^{2} c^{3} f\right )}}{c^{4}}\right )} x + \frac{48 \, a^{2} e}{c}\right )} - \frac{{\left (6 \, a^{2} c d - a^{3} f\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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