Optimal. Leaf size=106 \[ \frac{a (4 c d-a f) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2}}+\frac{x \sqrt{a+c x^2} (4 c d-a f)}{8 c}+\frac{e \left (a+c x^2\right )^{3/2}}{3 c}+\frac{f x \left (a+c x^2\right )^{3/2}}{4 c} \]
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Rubi [A] time = 0.0642928, antiderivative size = 106, normalized size of antiderivative = 1., number of steps used = 5, 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} \[ \frac{a (4 c d-a f) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2}}+\frac{x \sqrt{a+c x^2} (4 c d-a f)}{8 c}+\frac{e \left (a+c x^2\right )^{3/2}}{3 c}+\frac{f x \left (a+c x^2\right )^{3/2}}{4 c} \]
Antiderivative was successfully verified.
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Rule 1815
Rule 641
Rule 195
Rule 217
Rule 206
Rubi steps
\begin{align*} \int \sqrt{a+c x^2} \left (d+e x+f x^2\right ) \, dx &=\frac{f x \left (a+c x^2\right )^{3/2}}{4 c}+\frac{\int (4 c d-a f+4 c e x) \sqrt{a+c x^2} \, dx}{4 c}\\ &=\frac{e \left (a+c x^2\right )^{3/2}}{3 c}+\frac{f x \left (a+c x^2\right )^{3/2}}{4 c}+\frac{(4 c d-a f) \int \sqrt{a+c x^2} \, dx}{4 c}\\ &=\frac{(4 c d-a f) x \sqrt{a+c x^2}}{8 c}+\frac{e \left (a+c x^2\right )^{3/2}}{3 c}+\frac{f x \left (a+c x^2\right )^{3/2}}{4 c}+\frac{(a (4 c d-a f)) \int \frac{1}{\sqrt{a+c x^2}} \, dx}{8 c}\\ &=\frac{(4 c d-a f) x \sqrt{a+c x^2}}{8 c}+\frac{e \left (a+c x^2\right )^{3/2}}{3 c}+\frac{f x \left (a+c x^2\right )^{3/2}}{4 c}+\frac{(a (4 c d-a f)) \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x}{\sqrt{a+c x^2}}\right )}{8 c}\\ &=\frac{(4 c d-a f) x \sqrt{a+c x^2}}{8 c}+\frac{e \left (a+c x^2\right )^{3/2}}{3 c}+\frac{f x \left (a+c x^2\right )^{3/2}}{4 c}+\frac{a (4 c d-a f) \tanh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a+c x^2}}\right )}{8 c^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.199741, size = 98, normalized size = 0.92 \[ \frac{\sqrt{a+c x^2} \left (\sqrt{c} (a (8 e+3 f x)+2 c x (6 d+x (4 e+3 f x)))-\frac{3 \sqrt{a} (a f-4 c d) \sinh ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{\frac{c x^2}{a}+1}}\right )}{24 c^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.05, size = 111, normalized size = 1.1 \begin{align*}{\frac{fx}{4\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{afx}{8\,c}\sqrt{c{x}^{2}+a}}-{\frac{{a}^{2}f}{8}\ln \left ( x\sqrt{c}+\sqrt{c{x}^{2}+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{e}{3\,c} \left ( c{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{dx}{2}\sqrt{c{x}^{2}+a}}+{\frac{ad}{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 = 1.35248, size = 448, normalized size = 4.23 \begin{align*} \left [-\frac{3 \,{\left (4 \, a c d - a^{2} f\right )} \sqrt{c} \log \left (-2 \, c x^{2} + 2 \, \sqrt{c x^{2} + a} \sqrt{c} x - a\right ) - 2 \,{\left (6 \, c^{2} f x^{3} + 8 \, c^{2} e x^{2} + 8 \, a c e + 3 \,{\left (4 \, c^{2} d + a c f\right )} x\right )} \sqrt{c x^{2} + a}}{48 \, c^{2}}, -\frac{3 \,{\left (4 \, a c d - a^{2} f\right )} \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x}{\sqrt{c x^{2} + a}}\right ) -{\left (6 \, c^{2} f x^{3} + 8 \, c^{2} e x^{2} + 8 \, a c e + 3 \,{\left (4 \, c^{2} d + a c f\right )} x\right )} \sqrt{c x^{2} + a}}{24 \, c^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 6.02397, size = 170, normalized size = 1.6 \begin{align*} \frac{a^{\frac{3}{2}} f x}{8 c \sqrt{1 + \frac{c x^{2}}{a}}} + \frac{\sqrt{a} d x \sqrt{1 + \frac{c x^{2}}{a}}}{2} + \frac{3 \sqrt{a} f x^{3}}{8 \sqrt{1 + \frac{c x^{2}}{a}}} - \frac{a^{2} f \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{8 c^{\frac{3}{2}}} + \frac{a d \operatorname{asinh}{\left (\frac{\sqrt{c} x}{\sqrt{a}} \right )}}{2 \sqrt{c}} + 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 ) + \frac{c f 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 = 1.15261, size = 117, normalized size = 1.1 \begin{align*} \frac{1}{24} \, \sqrt{c x^{2} + a}{\left ({\left (2 \,{\left (3 \, f x + 4 \, e\right )} x + \frac{3 \,{\left (4 \, c^{2} d + a c f\right )}}{c^{2}}\right )} x + \frac{8 \, a e}{c}\right )} - \frac{{\left (4 \, a c d - a^{2} f\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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