Optimal. Leaf size=146 \[ \frac{3 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{\sqrt{d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{5/2}} \]
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Rubi [A] time = 0.0856611, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108, Rules used = {626, 51, 63, 208} \[ \frac{3 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{\sqrt{d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 626
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac{1}{(a e+c d x)^3 \sqrt{d+e x}} \, dx\\ &=-\frac{\sqrt{d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac{(3 e) \int \frac{1}{(a e+c d x)^2 \sqrt{d+e x}} \, dx}{4 \left (c d^2-a e^2\right )}\\ &=-\frac{\sqrt{d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac{3 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac{\left (3 e^2\right ) \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{8 \left (c d^2-a e^2\right )^2}\\ &=-\frac{\sqrt{d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac{3 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}+\frac{(3 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 \left (c d^2-a e^2\right )^2}\\ &=-\frac{\sqrt{d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}+\frac{3 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.0129869, size = 59, normalized size = 0.4 \[ \frac{2 e^2 \sqrt{d+e x} \, _2F_1\left (\frac{1}{2},3;\frac{3}{2};-\frac{c d (d+e x)}{a e^2-c d^2}\right )}{\left (a e^2-c d^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.199, size = 144, normalized size = 1. \begin{align*}{\frac{{e}^{2}}{ \left ( 2\,a{e}^{2}-2\,c{d}^{2} \right ) \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{3\,{e}^{2}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+{\frac{3\,{e}^{2}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}\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 = 1.96957, size = 1295, normalized size = 8.87 \begin{align*} \left [\frac{3 \,{\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} - 7 \, a c^{2} d^{3} e^{2} + 5 \, a^{2} c d e^{4} - 3 \,{\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^{7} e^{2} - 3 \, a^{3} c^{3} d^{5} e^{4} + 3 \, a^{4} c^{2} d^{3} e^{6} - a^{5} c d e^{8} +{\left (c^{6} d^{9} - 3 \, a c^{5} d^{7} e^{2} + 3 \, a^{2} c^{4} d^{5} e^{4} - a^{3} c^{3} d^{3} e^{6}\right )} x^{2} + 2 \,{\left (a c^{5} d^{8} e - 3 \, a^{2} c^{4} d^{6} e^{3} + 3 \, a^{3} c^{3} d^{4} e^{5} - a^{4} c^{2} d^{2} e^{7}\right )} x\right )}}, \frac{3 \,{\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} - 7 \, a c^{2} d^{3} e^{2} + 5 \, a^{2} c d e^{4} - 3 \,{\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^{7} e^{2} - 3 \, a^{3} c^{3} d^{5} e^{4} + 3 \, a^{4} c^{2} d^{3} e^{6} - a^{5} c d e^{8} +{\left (c^{6} d^{9} - 3 \, a c^{5} d^{7} e^{2} + 3 \, a^{2} c^{4} d^{5} e^{4} - a^{3} c^{3} d^{3} e^{6}\right )} x^{2} + 2 \,{\left (a c^{5} d^{8} e - 3 \, a^{2} c^{4} d^{6} e^{3} + 3 \, a^{3} c^{3} d^{4} e^{5} - a^{4} c^{2} d^{2} e^{7}\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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