Optimal. Leaf size=91 \[ \frac{2}{\sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{2 \sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0830422, antiderivative size = 91, 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, 51, 63, 208} \[ \frac{2}{\sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{2 \sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 626
Rule 51
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{1}{\sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx &=\int \frac{1}{(a e+c d x) (d+e x)^{3/2}} \, dx\\ &=\frac{2}{\left (c d^2-a e^2\right ) \sqrt{d+e x}}+\frac{(c d) \int \frac{1}{(a e+c d x) \sqrt{d+e x}} \, dx}{c d^2-a e^2}\\ &=\frac{2}{\left (c d^2-a e^2\right ) \sqrt{d+e x}}+\frac{(2 c d) \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 )}{e \left (c d^2-a e^2\right )}\\ &=\frac{2}{\left (c d^2-a e^2\right ) \sqrt{d+e x}}-\frac{2 \sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}}\\ \end{align*}
Mathematica [C] time = 0.0151336, size = 55, normalized size = 0.6 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};\frac{c d (d+e x)}{c d^2-a e^2}\right )}{\sqrt{d+e x} \left (a e^2-c d^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.195, size = 88, normalized size = 1. \begin{align*} -2\,{\frac{cd}{ \left ( a{e}^{2}-c{d}^{2} \right ) \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{1}{ \left ( a{e}^{2}-c{d}^{2} \right ) \sqrt{ex+d}}} \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 = 1.85558, size = 529, normalized size = 5.81 \begin{align*} \left [-\frac{{\left (e x + d\right )} \sqrt{\frac{c d}{c d^{2} - a e^{2}}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} + 2 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right ) - 2 \, \sqrt{e x + d}}{c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x}, -\frac{2 \,{\left ({\left (e x + d\right )} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac{{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}}}{c d e x + c d^{2}}\right ) - \sqrt{e x + d}\right )}}{c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A] time = 7.51262, size = 82, normalized size = 0.9 \begin{align*} \frac{2 c d \operatorname{atan}{\left (\frac{1}{\sqrt{\frac{c d}{a e^{2} - c d^{2}}} \sqrt{d + e x}} \right )}}{\sqrt{\frac{c d}{a e^{2} - c d^{2}}} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )} \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]