Optimal. Leaf size=83 \[ \frac{2 \sqrt{d+e x}}{c d}-\frac{2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2}} \]
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Rubi [A] time = 0.0546732, antiderivative size = 83, 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 = {626, 50, 63, 208} \[ \frac{2 \sqrt{d+e x}}{c d}-\frac{2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2}} \]
Antiderivative was successfully verified.
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Rule 626
Rule 50
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx &=\int \frac{\sqrt{d+e x}}{a e+c d x} \, dx\\ &=\frac{2 \sqrt{d+e x}}{c d}+\frac{\left (c d^2-a e^2\right ) \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{c d}\\ &=\frac{2 \sqrt{d+e x}}{c d}+\left (2 \left (\frac{d}{e}-\frac{a e}{c d}\right )\right ) \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 )\\ &=\frac{2 \sqrt{d+e x}}{c d}-\frac{2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0340531, size = 83, normalized size = 1. \[ \frac{2 \sqrt{d+e x}}{c d}-\frac{2 \sqrt{c d^2-a e^2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.192, size = 122, normalized size = 1.5 \begin{align*} 2\,{\frac{\sqrt{ex+d}}{cd}}-2\,{\frac{a{e}^{2}}{cd\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }+2\,{\frac{d}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{\sqrt{ex+d}cd}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \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 = 1.97021, size = 385, normalized size = 4.64 \begin{align*} \left [\frac{\sqrt{\frac{c d^{2} - a e^{2}}{c d}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \, \sqrt{e x + d} c d \sqrt{\frac{c d^{2} - a e^{2}}{c d}}}{c d x + a e}\right ) + 2 \, \sqrt{e x + d}}{c d}, -\frac{2 \,{\left (\sqrt{-\frac{c d^{2} - a e^{2}}{c d}} \arctan \left (-\frac{\sqrt{e x + d} c d \sqrt{-\frac{c d^{2} - a e^{2}}{c d}}}{c d^{2} - a e^{2}}\right ) - \sqrt{e x + d}\right )}}{c d}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 12.6772, size = 80, normalized size = 0.96 \begin{align*} \frac{2 \left (\frac{e \sqrt{d + e x}}{c d} - \frac{e \left (a e^{2} - c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{d + e x}}{\sqrt{\frac{a e^{2} - c d^{2}}{c d}}} \right )}}{c^{2} d^{2} \sqrt{\frac{a e^{2} - c d^{2}}{c d}}}\right )}{e} \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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