Optimal. Leaf size=136 \[ -\frac{a d^2 f^2}{2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^3}{6 e}+\frac{a d f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{e}+\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 e} \]
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Rubi [A] time = 0.103329, antiderivative size = 136, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2117, 893} \[ -\frac{a d^2 f^2}{2 e \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}+\frac{\left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^3}{6 e}+\frac{a d f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{e}+\frac{a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 e} \]
Antiderivative was successfully verified.
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Rule 2117
Rule 893
Rubi steps
\begin{align*} \int \left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^2 \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x^2 \left (d^2+a f^2-2 d x+x^2\right )}{(d-x)^2} \, dx,x,d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}{2 e}\\ &=\frac{\operatorname{Subst}\left (\int \left (a f^2+\frac{a d^2 f^2}{(d-x)^2}-\frac{2 a d f^2}{d-x}+x^2\right ) \, dx,x,d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}{2 e}\\ &=-\frac{a d^2 f^2}{2 e \left (e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}+\frac{a f^2 \left (e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}{2 e}+\frac{\left (d+e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )^3}{6 e}+\frac{a d f^2 \log \left (e x+f \sqrt{a+\frac{e^2 x^2}{f^2}}\right )}{e}\\ \end{align*}
Mathematica [A] time = 0.201769, size = 128, normalized size = 0.94 \[ \frac{\frac{a d^2 f^2}{f \left (-\sqrt{a+\frac{e^2 x^2}{f^2}}\right )-e x}+\frac{1}{3} \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+d+e x\right )^3+2 a d f^2 \log \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )+a f^2 \left (f \sqrt{a+\frac{e^2 x^2}{f^2}}+e x\right )}{2 e} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.003, size = 126, normalized size = 0.9 \begin{align*}{f}^{2}ax+{\frac{2\,{e}^{2}{x}^{3}}{3}}+{\frac{2\,{f}^{3}}{3\,e} \left ({\frac{{e}^{2}{x}^{2}+a{f}^{2}}{{f}^{2}}} \right ) ^{{\frac{3}{2}}}}+fdx\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}}+{dfa\ln \left ({\frac{{e}^{2}x}{{f}^{2}}{\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+\sqrt{a+{\frac{{e}^{2}{x}^{2}}{{f}^{2}}}} \right ){\frac{1}{\sqrt{{\frac{{e}^{2}}{{f}^{2}}}}}}}+e{x}^{2}d+x{d}^{2}+{\frac{{d}^{3}}{3\,e}} \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 = 1.01359, size = 239, normalized size = 1.76 \begin{align*} \frac{2 \, e^{3} x^{3} + 3 \, d e^{2} x^{2} - 3 \, a d f^{2} \log \left (-e x + f \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}\right ) + 3 \,{\left (a e f^{2} + d^{2} e\right )} x +{\left (2 \, e^{2} f x^{2} + 2 \, a f^{3} + 3 \, d e f x\right )} \sqrt{\frac{e^{2} x^{2} + a f^{2}}{f^{2}}}}{3 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 3.70792, size = 116, normalized size = 0.85 \begin{align*} \sqrt{a} d f x \sqrt{1 + \frac{e^{2} x^{2}}{a f^{2}}} + \frac{a d f^{2} \operatorname{asinh}{\left (\frac{e x}{\sqrt{a} f} \right )}}{e} + a f^{2} x + d^{2} x + d e x^{2} + \frac{2 e^{2} x^{3}}{3} + 2 e f \left (\begin{cases} \frac{\sqrt{a} x^{2}}{2} & \text{for}\: e^{2} = 0 \\\frac{f^{2} \left (a + \frac{e^{2} x^{2}}{f^{2}}\right )^{\frac{3}{2}}}{3 e^{2}} & \text{otherwise} \end{cases}\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.17823, size = 139, normalized size = 1.02 \begin{align*} -a d f{\left | f \right |} e^{\left (-1\right )} \log \left ({\left | -x e + \sqrt{a f^{2} + x^{2} e^{2}} \right |}\right ) + a f^{2} x + \frac{2}{3} \, x^{3} e^{2} + d x^{2} e + d^{2} x + \frac{1}{3} \,{\left (2 \, a f{\left | f \right |} e^{\left (-1\right )} +{\left (\frac{2 \, x{\left | f \right |} e}{f} + \frac{3 \, d{\left | f \right |}}{f}\right )} x\right )} \sqrt{a f^{2} + x^{2} e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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