Optimal. Leaf size=139 \[ \frac{2 e \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d g \sqrt{d+e x}}-\frac{2 (e f-d g) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{3/2} \sqrt{c d f-a e g}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.19195, antiderivative size = 139, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.065, Rules used = {880, 874, 205} \[ \frac{2 e \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d g \sqrt{d+e x}}-\frac{2 (e f-d g) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d f-a e g}}\right )}{g^{3/2} \sqrt{c d f-a e g}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 880
Rule 874
Rule 205
Rubi steps
\begin{align*} \int \frac{(d+e x)^{3/2}}{(f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx &=\frac{2 e \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g \sqrt{d+e x}}-\frac{\left (2 \left (\frac{1}{2} c d e^2 f-\frac{1}{2} c d^2 e g\right )\right ) \int \frac{\sqrt{d+e x}}{(f+g x) \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{c d e g}\\ &=\frac{2 e \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g \sqrt{d+e x}}-\frac{\left (2 e^2 (e f-d g)\right ) \operatorname{Subst}\left (\int \frac{1}{-e \left (c d^2+a e^2\right ) g+c d e (e f+d g)+e^2 g x^2} \, dx,x,\frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{d+e x}}\right )}{g}\\ &=\frac{2 e \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{c d g \sqrt{d+e x}}-\frac{2 (e f-d g) \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{\sqrt{c d f-a e g} \sqrt{d+e x}}\right )}{g^{3/2} \sqrt{c d f-a e g}}\\ \end{align*}
Mathematica [A] time = 0.110902, size = 140, normalized size = 1.01 \[ \frac{2 \sqrt{d+e x} \left (e \sqrt{g} (a e+c d x) \sqrt{c d f-a e g}+c d (d g-e f) \sqrt{a e+c d x} \tan ^{-1}\left (\frac{\sqrt{g} \sqrt{a e+c d x}}{\sqrt{c d f-a e g}}\right )\right )}{c d g^{3/2} \sqrt{(d+e x) (a e+c d x)} \sqrt{c d f-a e g}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.329, size = 163, normalized size = 1.2 \begin{align*} -2\,{\frac{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+ade}}{\sqrt{ex+d}\sqrt{cdx+ae}dcg\sqrt{ \left ( aeg-cdf \right ) g}} \left ({\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) c{d}^{2}g-{\it Artanh} \left ({\frac{\sqrt{cdx+ae}g}{\sqrt{ \left ( aeg-cdf \right ) g}}} \right ) cdef-e\sqrt{cdx+ae}\sqrt{ \left ( aeg-cdf \right ) g} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (g x + f\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.62759, size = 1085, normalized size = 7.81 \begin{align*} \left [\frac{{\left (c d^{2} e f - c d^{3} g +{\left (c d e^{2} f - c d^{2} e g\right )} x\right )} \sqrt{-c d f g + a e g^{2}} \log \left (-\frac{c d e g x^{2} - c d^{2} f + 2 \, a d e g -{\left (c d e f -{\left (c d^{2} + 2 \, a e^{2}\right )} g\right )} x - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d f g + a e g^{2}} \sqrt{e x + d}}{e g x^{2} + d f +{\left (e f + d g\right )} x}\right ) + 2 \,{\left (c d e f g - a e^{2} g^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}}{c^{2} d^{3} f g^{2} - a c d^{2} e g^{3} +{\left (c^{2} d^{2} e f g^{2} - a c d e^{2} g^{3}\right )} x}, \frac{2 \,{\left ({\left (c d^{2} e f - c d^{3} g +{\left (c d e^{2} f - c d^{2} e g\right )} x\right )} \sqrt{c d f g - a e g^{2}} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d f g - a e g^{2}} \sqrt{e x + d}}{c d e g x^{2} + a d e g +{\left (c d^{2} + a e^{2}\right )} g x}\right ) +{\left (c d e f g - a e^{2} g^{2}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d}\right )}}{c^{2} d^{3} f g^{2} - a c d^{2} e g^{3} +{\left (c^{2} d^{2} e f g^{2} - a c d e^{2} g^{3}\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]