Optimal. Leaf size=109 \[ \frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 c^2 d^2 (d+e x)^{5/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}} \]
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Rubi [A] time = 0.0610188, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.051, Rules used = {656, 648} \[ \frac{4 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{35 c^2 d^2 (d+e x)^{5/2}}+\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 656
Rule 648
Rubi steps
\begin{align*} \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{\sqrt{d+e x}} \, dx &=\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}}+\frac{\left (2 \left (d^2-\frac{a e^2}{c}\right )\right ) \int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx}{7 d}\\ &=\frac{4 \left (c d^2-a e^2\right ) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{35 c^2 d^2 (d+e x)^{5/2}}+\frac{2 \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{5/2}}{7 c d (d+e x)^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.0630417, size = 55, normalized size = 0.5 \[ \frac{2 ((d+e x) (a e+c d x))^{5/2} \left (c d (7 d+5 e x)-2 a e^2\right )}{35 c^2 d^2 (d+e x)^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.043, size = 69, normalized size = 0.6 \begin{align*} -{\frac{ \left ( 2\,cdx+2\,ae \right ) \left ( -5\,cdex+2\,a{e}^{2}-7\,c{d}^{2} \right ) }{35\,{c}^{2}{d}^{2}} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{{\frac{3}{2}}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.06772, size = 132, normalized size = 1.21 \begin{align*} \frac{2 \,{\left (5 \, c^{3} d^{3} e x^{3} + 7 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4} +{\left (7 \, c^{3} d^{4} + 8 \, a c^{2} d^{2} e^{2}\right )} x^{2} +{\left (14 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x\right )} \sqrt{c d x + a e}}{35 \, c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.82388, size = 279, normalized size = 2.56 \begin{align*} \frac{2 \,{\left (5 \, c^{3} d^{3} e x^{3} + 7 \, a^{2} c d^{2} e^{2} - 2 \, a^{3} e^{4} +{\left (7 \, c^{3} d^{4} + 8 \, a c^{2} d^{2} e^{2}\right )} x^{2} +{\left (14 \, a c^{2} d^{3} e + a^{2} c d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{35 \,{\left (c^{2} d^{2} e x + c^{2} d^{3}\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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