Optimal. Leaf size=130 \[ -\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{5/2} d^{5/2} \sqrt{c d^2-a e^2}}-\frac{3 e \sqrt{d+e x}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{3/2}}{2 c d (a e+c d x)^2} \]
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Rubi [A] time = 0.0809901, antiderivative size = 130, 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, 47, 63, 208} \[ -\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{5/2} d^{5/2} \sqrt{c d^2-a e^2}}-\frac{3 e \sqrt{d+e x}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{3/2}}{2 c d (a e+c d x)^2} \]
Antiderivative was successfully verified.
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Rule 626
Rule 47
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx &=\int \frac{(d+e x)^{3/2}}{(a e+c d x)^3} \, dx\\ &=-\frac{(d+e x)^{3/2}}{2 c d (a e+c d x)^2}+\frac{(3 e) \int \frac{\sqrt{d+e x}}{(a e+c d x)^2} \, dx}{4 c d}\\ &=-\frac{3 e \sqrt{d+e x}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{3/2}}{2 c d (a e+c d x)^2}+\frac{\left (3 e^2\right ) \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{8 c^2 d^2}\\ &=-\frac{3 e \sqrt{d+e x}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{3/2}}{2 c d (a e+c d x)^2}+\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 c^2 d^2}\\ &=-\frac{3 e \sqrt{d+e x}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{3/2}}{2 c d (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 c^{5/2} d^{5/2} \sqrt{c d^2-a e^2}}\\ \end{align*}
Mathematica [A] time = 0.137595, size = 118, normalized size = 0.91 \[ \frac{3 e^2 \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{a e^2-c d^2}}\right )}{4 c^{5/2} d^{5/2} \sqrt{a e^2-c d^2}}-\frac{\sqrt{d+e x} \left (3 a e^2+c d (2 d+5 e x)\right )}{4 c^2 d^2 (a e+c d x)^2} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.201, size = 149, normalized size = 1.2 \begin{align*} -{\frac{5\,{e}^{2}}{4\, \left ( cdex+a{e}^{2} \right ) ^{2}dc} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{4}a}{4\, \left ( cdex+a{e}^{2} \right ) ^{2}{c}^{2}{d}^{2}}\sqrt{ex+d}}+{\frac{3\,{e}^{2}}{4\, \left ( cdex+a{e}^{2} \right ) ^{2}c}\sqrt{ex+d}}+{\frac{3\,{e}^{2}}{4\,{c}^{2}{d}^{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 = 2.06872, size = 976, normalized size = 7.51 \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} + a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4} + 5 \,{\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^{5} e^{2} - a^{3} c^{3} d^{3} e^{4} +{\left (c^{6} d^{7} - a c^{5} d^{5} e^{2}\right )} x^{2} + 2 \,{\left (a c^{5} d^{6} e - a^{2} c^{4} d^{4} e^{3}\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} + a c^{2} d^{3} e^{2} - 3 \, a^{2} c d e^{4} + 5 \,{\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^{5} e^{2} - a^{3} c^{3} d^{3} e^{4} +{\left (c^{6} d^{7} - a c^{5} d^{5} e^{2}\right )} x^{2} + 2 \,{\left (a c^{5} d^{6} e - a^{2} c^{4} d^{4} e^{3}\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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