Optimal. Leaf size=159 \[ \frac{\left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac{\left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}} \]
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Rubi [A] time = 0.0502793, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {612, 621, 206} \[ \frac{\left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c d e}-\frac{\left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}} \]
Antiderivative was successfully verified.
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Rule 612
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\frac{\left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c d e}-\frac{\left (c d^2-a e^2\right )^2 \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 c d e}\\ &=\frac{\left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c d e}-\frac{\left (c d^2-a e^2\right )^2 \operatorname{Subst}\left (\int \frac{1}{4 c d e-x^2} \, dx,x,\frac{c d^2+a e^2+2 c d e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{4 c d e}\\ &=\frac{\left (c d^2+a e^2+2 c d e x\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c d e}-\frac{\left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{c d^2+a e^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\right )}{8 c^{3/2} d^{3/2} e^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.487997, size = 214, normalized size = 1.35 \[ \frac{\sqrt{c} \sqrt{d} \left (\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{c d} (d+e x) \left (a^2 e^3+a c d e (d+3 e x)+c^2 d^2 x (d+2 e x)\right )-\left (c d^2-a e^2\right )^{5/2} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )\right )}{4 e^{3/2} (c d)^{5/2} \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.044, size = 265, normalized size = 1.7 \begin{align*}{\frac{2\,cdex+a{e}^{2}+c{d}^{2}}{4\,dec}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}-{\frac{{a}^{2}{e}^{3}}{8\,cd}\ln \left ({ \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}}+{\frac{ade}{4}\ln \left ({ \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}}-{\frac{c{d}^{3}}{8\,e}\ln \left ({ \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{dec}}}} \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 = 1.79373, size = 888, normalized size = 5.58 \begin{align*} \left [\frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{c d e} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{c d e} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{16 \, c^{2} d^{2} e^{2}}, \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{-c d e} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-c d e}}{2 \,{\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} +{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )}}\right ) + 2 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{8 \, c^{2} d^{2} e^{2}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.35217, size = 219, normalized size = 1.38 \begin{align*} \frac{1}{4} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \, x + \frac{{\left (c d^{2} + a e^{2}\right )} e^{\left (-1\right )}}{c d}\right )} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{c d} e^{\left (-\frac{3}{2}\right )} \log \left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{8 \, c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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