Optimal. Leaf size=195 \[ \frac{3 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2}+\frac{3 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} \sqrt{e}}+\frac{(d+e x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d} \]
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Rubi [A] time = 0.115318, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {670, 640, 621, 206} \[ \frac{3 \left (c d^2-a e^2\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 c^2 d^2}+\frac{3 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 c^{5/2} d^{5/2} \sqrt{e}}+\frac{(d+e x) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 c d} \]
Antiderivative was successfully verified.
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Rule 670
Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^2}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{(d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c d}+\frac{\left (3 \left (d^2-\frac{a e^2}{c}\right )\right ) \int \frac{d+e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{4 d}\\ &=\frac{3 \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2}+\frac{(d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c d}+\frac{\left (3 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{8 d^2}\\ &=\frac{3 \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2}+\frac{(d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c d}+\frac{\left (3 \left (d^2-\frac{a e^2}{c}\right )^2\right ) \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 d^2}\\ &=\frac{3 \left (c d^2-a e^2\right ) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{4 c^2 d^2}+\frac{(d+e x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{2 c d}+\frac{3 \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^{5/2} d^{5/2} \sqrt{e}}\\ \end{align*}
Mathematica [A] time = 0.546682, size = 182, normalized size = 0.93 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{3 \sqrt{c d} \left (c d^2-a e^2\right )^{3/2} \sinh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a e+c d x}}{\sqrt{c d} \sqrt{c d^2-a e^2}}\right )}{\sqrt{e} \sqrt{a e+c d x} \sqrt{\frac{c d (d+e x)}{c d^2-a e^2}}}+\sqrt{c} \sqrt{d} \left (c d (5 d+2 e x)-3 a e^2\right )\right )}{4 c^{5/2} d^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.051, size = 318, normalized size = 1.6 \begin{align*}{\frac{ex}{2\,cd}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}-{\frac{3\,a{e}^{2}}{4\,{c}^{2}{d}^{2}}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{5}{4\,c}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{3\,{a}^{2}{e}^{4}}{8\,{c}^{2}{d}^{2}}\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{3\,a{e}^{2}}{4\,c}\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{3\,{d}^{2}}{8}\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 = 2.1435, size = 900, normalized size = 4.62 \begin{align*} \left [\frac{3 \,{\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 + 5 \, c^{2} d^{3} e - 3 \, 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^{3} d^{3} e}, -\frac{3 \,{\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 + 5 \, c^{2} d^{3} e - 3 \, 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^{3} d^{3} e}\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 \frac{\left (d + e x\right )^{2}}{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.37125, size = 238, normalized size = 1.22 \begin{align*} \frac{1}{4} \, \sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (\frac{2 \, x e}{c d} + \frac{{\left (5 \, c d^{2} e - 3 \, a e^{3}\right )} e^{\left (-1\right )}}{c^{2} d^{2}}\right )} - \frac{3 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \sqrt{c d} e^{\left (-\frac{1}{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 + a d e +{\left (c d^{2} + a e^{2}\right )} x}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{8 \, c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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