Optimal. Leaf size=144 \[ \frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{3/2} d^{3/2} \left (c d^2-a e^2\right )^{3/2}}-\frac{e \sqrt{d+e x}}{4 c d \left (c d^2-a e^2\right ) (a e+c d x)}-\frac{\sqrt{d+e x}}{2 c d (a e+c d x)^2} \]
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Rubi [A] time = 0.0882176, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135, Rules used = {626, 47, 51, 63, 208} \[ \frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{3/2} d^{3/2} \left (c d^2-a e^2\right )^{3/2}}-\frac{e \sqrt{d+e x}}{4 c d \left (c d^2-a e^2\right ) (a e+c d x)}-\frac{\sqrt{d+e x}}{2 c d (a e+c d x)^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 47
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{7/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac{\sqrt{d+e x}}{(a e+c d x)^3} \, dx\\ &=-\frac{\sqrt{d+e x}}{2 c d (a e+c d x)^2}+\frac{e \int \frac{1}{(a e+c d x)^2 \sqrt{d+e x}} \, dx}{4 c d}\\ &=-\frac{\sqrt{d+e x}}{2 c d (a e+c d x)^2}-\frac{e \sqrt{d+e x}}{4 c d \left (c d^2-a e^2\right ) (a e+c d x)}-\frac{e^2 \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{8 c d \left (c d^2-a e^2\right )}\\ &=-\frac{\sqrt{d+e x}}{2 c d (a e+c d x)^2}-\frac{e \sqrt{d+e x}}{4 c d \left (c d^2-a e^2\right ) (a e+c d x)}-\frac{e \operatorname{Subst}\left (\int \frac{1}{-\frac{c d^2}{e}+a e+\frac{c d x^2}{e}} \, dx,x,\sqrt{d+e x}\right )}{4 c d \left (c d^2-a e^2\right )}\\ &=-\frac{\sqrt{d+e x}}{2 c d (a e+c d x)^2}-\frac{e \sqrt{d+e x}}{4 c d \left (c d^2-a e^2\right ) (a e+c d x)}+\frac{e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{3/2} d^{3/2} \left (c d^2-a e^2\right )^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.016056, size = 61, normalized size = 0.42 \[ \frac{2 e^2 (d+e x)^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right )}{3 \left (a e^2-c d^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.201, size = 142, normalized size = 1. \begin{align*}{\frac{{e}^{2}}{4\, \left ( cdex+a{e}^{2} \right ) ^{2} \left ( a{e}^{2}-c{d}^{2} \right ) } \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{{e}^{2}}{4\, \left ( cdex+a{e}^{2} \right ) ^{2}dc}\sqrt{ex+d}}+{\frac{{e}^{2}}{ \left ( 4\,a{e}^{2}-4\,c{d}^{2} \right ) cd}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \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 [B] time = 1.9764, size = 1129, normalized size = 7.84 \begin{align*} \left [\frac{{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \sqrt{c^{2} d^{3} - a c d e^{2}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} + 2 \, \sqrt{c^{2} d^{3} - a c d e^{2}} \sqrt{e x + d}}{c d x + a e}\right ) - 2 \,{\left (2 \, c^{3} d^{5} - 3 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4} +{\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{8 \,{\left (a^{2} c^{4} d^{6} e^{2} - 2 \, a^{3} c^{3} d^{4} e^{4} + a^{4} c^{2} d^{2} e^{6} +{\left (c^{6} d^{8} - 2 \, a c^{5} d^{6} e^{2} + a^{2} c^{4} d^{4} e^{4}\right )} x^{2} + 2 \,{\left (a c^{5} d^{7} e - 2 \, a^{2} c^{4} d^{5} e^{3} + a^{3} c^{3} d^{3} e^{5}\right )} x\right )}}, -\frac{{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \sqrt{-c^{2} d^{3} + a c d e^{2}} \arctan \left (\frac{\sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}{c d e x + c d^{2}}\right ) +{\left (2 \, c^{3} d^{5} - 3 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4} +{\left (c^{3} d^{4} e - a c^{2} d^{2} e^{3}\right )} x\right )} \sqrt{e x + d}}{4 \,{\left (a^{2} c^{4} d^{6} e^{2} - 2 \, a^{3} c^{3} d^{4} e^{4} + a^{4} c^{2} d^{2} e^{6} +{\left (c^{6} d^{8} - 2 \, a c^{5} d^{6} e^{2} + a^{2} c^{4} d^{4} e^{4}\right )} x^{2} + 2 \,{\left (a c^{5} d^{7} e - 2 \, a^{2} c^{4} d^{5} e^{3} + a^{3} c^{3} d^{3} e^{5}\right )} x\right )}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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