Optimal. Leaf size=94 \[ -\frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{d+e x}}{c d (a e+c d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0571539, antiderivative size = 94, 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, 47, 63, 208} \[ -\frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{d+e x}}{c d (a e+c d x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 626
Rule 47
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 )^2} \, dx &=\int \frac{\sqrt{d+e x}}{(a e+c d x)^2} \, dx\\ &=-\frac{\sqrt{d+e x}}{c d (a e+c d x)}+\frac{e \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{2 c d}\\ &=-\frac{\sqrt{d+e x}}{c d (a e+c d x)}+\frac{\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 )}{c d}\\ &=-\frac{\sqrt{d+e x}}{c d (a e+c d x)}-\frac{e \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} \sqrt{c d^2-a e^2}}\\ \end{align*}
Mathematica [A] time = 0.109552, size = 93, normalized size = 0.99 \[ \frac{e \tan ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{a e^2-c d^2}}\right )}{c^{3/2} d^{3/2} \sqrt{a e^2-c d^2}}-\frac{\sqrt{d+e x}}{a c d e+c^2 d^2 x} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.201, size = 84, normalized size = 0.9 \begin{align*} -{\frac{e}{cd \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+{\frac{e}{cd}\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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.99687, size = 625, normalized size = 6.65 \begin{align*} \left [\frac{\sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c d e x + a e^{2}\right )} \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 (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{2 \,{\left (a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3} +{\left (c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x\right )}}, \frac{\sqrt{-c^{2} d^{3} + a c d e^{2}}{\left (c d e x + a e^{2}\right )} \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 (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{a c^{3} d^{4} e - a^{2} c^{2} d^{2} e^{3} +{\left (c^{4} d^{5} - a c^{3} d^{3} e^{2}\right )} x}\right ] \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(-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]