Optimal. Leaf size=180 \[ \frac{3 e \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^2 d^2}+\frac{3 \sqrt{e} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{5/2} d^{5/2}}-\frac{2 (d+e x)^2}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rubi [A] time = 0.0991254, antiderivative size = 180, 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 = {668, 640, 621, 206} \[ \frac{3 e \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c^2 d^2}+\frac{3 \sqrt{e} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{5/2} d^{5/2}}-\frac{2 (d+e x)^2}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
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Rule 668
Rule 640
Rule 621
Rule 206
Rubi steps
\begin{align*} \int \frac{(d+e x)^3}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx &=-\frac{2 (d+e x)^2}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{(3 e) \int \frac{d+e x}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d}\\ &=-\frac{2 (d+e x)^2}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3 e \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2}+\frac{\left (3 e \left (c d^2-a e^2\right )\right ) \int \frac{1}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{2 c^2 d^2}\\ &=-\frac{2 (d+e x)^2}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3 e \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2}+\frac{\left (3 e \left (c d^2-a e^2\right )\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 )}{c^2 d^2}\\ &=-\frac{2 (d+e x)^2}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{3 e \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c^2 d^2}+\frac{3 \sqrt{e} \left (c d^2-a e^2\right ) \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 )}{2 c^{5/2} d^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0510216, size = 93, normalized size = 0.52 \[ -\frac{2 (d+e x)^2 \, _2F_1\left (-\frac{3}{2},-\frac{1}{2};\frac{1}{2};\frac{e (a e+c d x)}{a e^2-c d^2}\right )}{c d \left (\frac{c d (d+e x)}{c d^2-a e^2}\right )^{3/2} \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.056, size = 1047, normalized size = 5.8 \begin{align*} \text{result too large to display} \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 = 2.82326, size = 932, normalized size = 5.18 \begin{align*} \left [\frac{3 \,{\left (a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt{\frac{e}{c d}} \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} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x + 4 \,{\left (2 \, c^{2} d^{2} e x + c^{2} d^{3} + a c d e^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{e}{c d}}\right ) + 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d e x - 2 \, c d^{2} + 3 \, a e^{2}\right )}}{4 \,{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )}}, -\frac{3 \,{\left (a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )} \sqrt{-\frac{e}{c d}} \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{-\frac{e}{c d}}}{2 \,{\left (c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x\right )}}\right ) - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d e x - 2 \, c d^{2} + 3 \, a e^{2}\right )}}{2 \,{\left (c^{3} d^{3} x + a c^{2} d^{2} e\right )}}\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 )^{3}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.28745, size = 478, normalized size = 2.66 \begin{align*} \frac{{\left (\frac{{\left (c^{3} d^{5} e^{3} - 2 \, a c^{2} d^{3} e^{5} + a^{2} c d e^{7}\right )} x}{c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}} - \frac{c^{3} d^{6} e^{2} - 5 \, a c^{2} d^{4} e^{4} + 7 \, a^{2} c d^{2} e^{6} - 3 \, a^{3} e^{8}}{c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}}\right )} x - \frac{2 \, c^{3} d^{7} e - 7 \, a c^{2} d^{5} e^{3} + 8 \, a^{2} c d^{3} e^{5} - 3 \, a^{3} d e^{7}}{c^{4} d^{6} e - 2 \, a c^{3} d^{4} e^{3} + a^{2} c^{2} d^{2} e^{5}}}{\sqrt{c d x^{2} e + a d e +{\left (c d^{2} + a e^{2}\right )} x}} - \frac{3 \,{\left (c d^{2} e - a e^{3}\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 )}{2 \, c^{3} d^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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