Optimal. Leaf size=147 \[ \frac{\left (8 a e^2 g^2+c \left (3 d^2 g^2+2 d e f g+3 e^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{4 e^{5/2} g^{5/2}}-\frac{c \sqrt{d+e x} \sqrt{f+g x} (5 d g+3 e f)}{4 e^2 g^2}+\frac{c (d+e x)^{3/2} \sqrt{f+g x}}{2 e^2 g} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.323546, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\left (8 a e^2 g^2+c \left (3 d^2 g^2+2 d e f g+3 e^2 f^2\right )\right ) \tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right )}{4 e^{5/2} g^{5/2}}-\frac{c \sqrt{d+e x} \sqrt{f+g x} (5 d g+3 e f)}{4 e^2 g^2}+\frac{c (d+e x)^{3/2} \sqrt{f+g x}}{2 e^2 g} \]
Antiderivative was successfully verified.
[In] Int[(a + c*x^2)/(Sqrt[d + e*x]*Sqrt[f + g*x]),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 30.3004, size = 163, normalized size = 1.11 \[ \frac{2 a \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{f + g x}}{\sqrt{g} \sqrt{d + e x}} \right )}}{\sqrt{e} \sqrt{g}} + \frac{c x \sqrt{d + e x} \sqrt{f + g x}}{2 e g} - \frac{3 c \sqrt{d + e x} \sqrt{f + g x} \left (d g + e f\right )}{4 e^{2} g^{2}} - \frac{c \left (4 d e f g - 3 \left (d g + e f\right )^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{f + g x}}{\sqrt{g} \sqrt{d + e x}} \right )}}{4 e^{\frac{5}{2}} g^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+a)/(e*x+d)**(1/2)/(g*x+f)**(1/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.165486, size = 136, normalized size = 0.93 \[ \frac{\left (8 a e^2 g^2+c \left (3 d^2 g^2+2 d e f g+3 e^2 f^2\right )\right ) \log \left (2 \sqrt{e} \sqrt{g} \sqrt{d+e x} \sqrt{f+g x}+d g+e f+2 e g x\right )}{8 e^{5/2} g^{5/2}}+\frac{c \sqrt{d+e x} \sqrt{f+g x} (-3 d g-3 e f+2 e g x)}{4 e^2 g^2} \]
Antiderivative was successfully verified.
[In] Integrate[(a + c*x^2)/(Sqrt[d + e*x]*Sqrt[f + g*x]),x]
[Out]
_______________________________________________________________________________________
Maple [B] time = 0.044, size = 306, normalized size = 2.1 \[{\frac{1}{8\,{e}^{2}{g}^{2}} \left ( 8\,\ln \left ( 1/2\,{\frac{2\,egx+2\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}+dg+ef}{\sqrt{eg}}} \right ) a{e}^{2}{g}^{2}+3\,\ln \left ( 1/2\,{\frac{2\,egx+2\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}+dg+ef}{\sqrt{eg}}} \right ) c{d}^{2}{g}^{2}+2\,c\ln \left ( 1/2\,{\frac{2\,egx+2\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}+dg+ef}{\sqrt{eg}}} \right ) dfeg+3\,\ln \left ( 1/2\,{\frac{2\,egx+2\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}+dg+ef}{\sqrt{eg}}} \right ) c{e}^{2}{f}^{2}+4\,c\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }xeg\sqrt{eg}-6\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}cdg-6\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}cef \right ) \sqrt{ex+d}\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }}}{\frac{1}{\sqrt{eg}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+a)/(e*x+d)^(1/2)/(g*x+f)^(1/2),x)
[Out]
_______________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(sqrt(e*x + d)*sqrt(g*x + f)),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.608631, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, c e g x - 3 \, c e f - 3 \, c d g\right )} \sqrt{e g} \sqrt{e x + d} \sqrt{g x + f} +{\left (3 \, c e^{2} f^{2} + 2 \, c d e f g +{\left (3 \, c d^{2} + 8 \, a e^{2}\right )} g^{2}\right )} \log \left (4 \,{\left (2 \, e^{2} g^{2} x + e^{2} f g + d e g^{2}\right )} \sqrt{e x + d} \sqrt{g x + f} +{\left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 8 \,{\left (e^{2} f g + d e g^{2}\right )} x\right )} \sqrt{e g}\right )}{16 \, \sqrt{e g} e^{2} g^{2}}, \frac{2 \,{\left (2 \, c e g x - 3 \, c e f - 3 \, c d g\right )} \sqrt{-e g} \sqrt{e x + d} \sqrt{g x + f} +{\left (3 \, c e^{2} f^{2} + 2 \, c d e f g +{\left (3 \, c d^{2} + 8 \, a e^{2}\right )} g^{2}\right )} \arctan \left (\frac{{\left (2 \, e g x + e f + d g\right )} \sqrt{-e g}}{2 \, \sqrt{e x + d} \sqrt{g x + f} e g}\right )}{8 \, \sqrt{-e g} e^{2} g^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(sqrt(e*x + d)*sqrt(g*x + f)),x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{a + c x^{2}}{\sqrt{d + e x} \sqrt{f + g x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2+a)/(e*x+d)**(1/2)/(g*x+f)**(1/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.288106, size = 209, normalized size = 1.42 \[ \frac{1}{4} \, \sqrt{{\left (x e + d\right )} g e - d g e + f e^{2}} \sqrt{x e + d}{\left (\frac{2 \,{\left (x e + d\right )} c e^{\left (-3\right )}}{g} - \frac{{\left (5 \, c d g^{2} e^{5} + 3 \, c f g e^{6}\right )} e^{\left (-8\right )}}{g^{3}}\right )} - \frac{{\left (3 \, c d^{2} g^{2} + 2 \, c d f g e + 3 \, c f^{2} e^{2} + 8 \, a g^{2} e^{2}\right )} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -\sqrt{x e + d} \sqrt{g} e^{\frac{1}{2}} + \sqrt{{\left (x e + d\right )} g e - d g e + f e^{2}} \right |}\right )}{4 \, g^{\frac{5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + a)/(sqrt(e*x + d)*sqrt(g*x + f)),x, algorithm="giac")
[Out]