Optimal. Leaf size=105 \[ \frac{2 (d+e x)^{3/2}}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{4 \sqrt{d+e x} \left (c d^2-a e^2\right )}{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0581765, antiderivative size = 105, 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} \[ \frac{2 (d+e x)^{3/2}}{c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{4 \sqrt{d+e x} \left (c d^2-a e^2\right )}{c^2 d^2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 656
Rule 648
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 )^{3/2}} \, dx &=\frac{2 (d+e x)^{3/2}}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{\left (2 \left (2 c d^2 e-e \left (c d^2+a e^2\right )\right )\right ) \int \frac{(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx}{c d e}\\ &=-\frac{4 \left (c d^2-a e^2\right ) \sqrt{d+e x}}{c^2 d^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac{2 (d+e x)^{3/2}}{c d \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}\\ \end{align*}
Mathematica [A] time = 0.0339236, size = 51, normalized size = 0.49 \[ -\frac{2 \sqrt{d+e x} \left (c d (d-e x)-2 a e^2\right )}{c^2 d^2 \sqrt{(d+e x) (a e+c d x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 68, normalized size = 0.7 \begin{align*} 2\,{\frac{ \left ( cdx+ae \right ) \left ( cdex+2\,a{e}^{2}-c{d}^{2} \right ) \left ( ex+d \right ) ^{3/2}}{{c}^{2}{d}^{2} \left ( cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade \right ) ^{3/2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.12429, size = 49, normalized size = 0.47 \begin{align*} \frac{2 \,{\left (c d e x - c d^{2} + 2 \, a e^{2}\right )}}{\sqrt{c d x + a e} c^{2} d^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.04646, size = 201, normalized size = 1.91 \begin{align*} \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d e x - c d^{2} + 2 \, a e^{2}\right )} \sqrt{e x + d}}{c^{3} d^{3} e x^{2} + a c^{2} d^{3} e +{\left (c^{3} d^{4} + a c^{2} d^{2} e^{2}\right )} x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]