Optimal. Leaf size=588 \[ -\frac{(d+e x) \sqrt [4]{c f^2-g (b f-a g)} \sqrt{\frac{\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \left (\frac{(f+g x) \sqrt{a e^2-b d e+c d^2}}{(d+e x) \sqrt{c f^2-g (b f-a g)}}+1\right ) \sqrt{\frac{\frac{(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac{(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}{\left (\frac{(f+g x) \sqrt{a e^2-b d e+c d^2}}{(d+e x) \sqrt{c f^2-g (b f-a g)}}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c d^2-b e d+a e^2} \sqrt{f+g x}}{\sqrt [4]{c f^2-b g f+a g^2} \sqrt{d+e x}}\right )|\frac{1}{4} \left (\frac{2 c d f+2 a e g-b (e f+d g)}{\sqrt{c d^2-e (b d-a e)} \sqrt{c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt{a+b x+c x^2} (e f-d g) \sqrt [4]{a e^2-b d e+c d^2} \sqrt{\frac{(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac{(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}} \]
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Rubi [A] time = 2.44815, antiderivative size = 589, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ -\frac{(d+e x) \sqrt [4]{c f^2-g (b f-a g)} \sqrt{\frac{\left (a+b x+c x^2\right ) (e f-d g)^2}{(d+e x)^2 \left (a g^2-b f g+c f^2\right )}} \left (\frac{(f+g x) \sqrt{a e^2-b d e+c d^2}}{(d+e x) \sqrt{c f^2-g (b f-a g)}}+1\right ) \sqrt{\frac{\frac{(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac{(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}{\left (\frac{(f+g x) \sqrt{a e^2-b d e+c d^2}}{(d+e x) \sqrt{c f^2-g (b f-a g)}}+1\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c d^2-b e d+a e^2} \sqrt{f+g x}}{\sqrt [4]{c f^2-g (b f-a g)} \sqrt{d+e x}}\right )|\frac{1}{4} \left (\frac{2 c d f+2 a e g-b (e f+d g)}{\sqrt{c d^2-e (b d-a e)} \sqrt{c f^2-g (b f-a g)}}+2\right )\right )}{\sqrt{a+b x+c x^2} (e f-d g) \sqrt [4]{a e^2-b d e+c d^2} \sqrt{\frac{(f+g x)^2 \left (a e^2-b d e+c d^2\right )}{(d+e x)^2 \left (c f^2-g (b f-a g)\right )}-\frac{(f+g x) (2 a e g-b (d g+e f)+2 c d f)}{(d+e x) \left (a g^2-b f g+c f^2\right )}+1}} \]
Warning: Unable to verify antiderivative.
[In] Int[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(e*x+d)**(1/2)/(g*x+f)**(1/2)/(c*x**2+b*x+a)**(1/2),x)
[Out]
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Mathematica [A] time = 6.05506, size = 375, normalized size = 0.64 \[ \frac{2 \sqrt{2} e \sqrt{a+x (b+c x)} \sqrt{-\frac{e (f+g x) \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (-d g \sqrt{e^2 \left (b^2-4 a c\right )}+e f \sqrt{e^2 \left (b^2-4 a c\right )}-2 a e^2 g+b e (d g+e f)-2 c d e f\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{2 a e^2-2 c d x e+b (e x-d) e+\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}{\sqrt{\left (b^2-4 a c\right ) e^2} (d+e x)}}}{\sqrt{2}}\right )|\frac{2 \sqrt{\left (b^2-4 a c\right ) e^2} (e f-d g)}{-2 a g e^2-2 c d f e+\sqrt{\left (b^2-4 a c\right ) e^2} f e+b (e f+d g) e-d \sqrt{\left (b^2-4 a c\right ) e^2} g}\right )}{\sqrt{d+e x} \sqrt{f+g x} \sqrt{e^2 \left (b^2-4 a c\right )} \sqrt{-\frac{(a+x (b+c x)) \left (e (a e-b d)+c d^2\right )}{\left (b^2-4 a c\right ) (d+e x)^2}}} \]
Antiderivative was successfully verified.
[In] Integrate[1/(Sqrt[d + e*x]*Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]
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Maple [A] time = 0.114, size = 605, normalized size = 1. \[ 4\,{\frac{ \left ( \sqrt{-4\,ac+{b}^{2}}{x}^{2}{e}^{2}g+b{e}^{2}g{x}^{2}-2\,c{e}^{2}f{x}^{2}+2\,\sqrt{-4\,ac+{b}^{2}}xdeg+2\,xbdeg-4\,xcdef+\sqrt{-4\,ac+{b}^{2}}{d}^{2}g+b{d}^{2}g-2\,c{d}^{2}f \right ) \sqrt{ex+d}\sqrt{gx+f}\sqrt{c{x}^{2}+bx+a}}{ \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( dg-ef \right ) \sqrt{ceg{x}^{4}+beg{x}^{3}+cdg{x}^{3}+cef{x}^{3}+aeg{x}^{2}+bdg{x}^{2}+bef{x}^{2}+cdf{x}^{2}+adgx+aefx+bdfx+adf}}{\it EllipticF} \left ( \sqrt{{\frac{ \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( gx+f \right ) }{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( ex+d \right ) }}},\sqrt{{\frac{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}-be+2\,cd \right ) }{ \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) }}} \right ) \sqrt{{\frac{ \left ( dg-ef \right ) \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) }{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( ex+d \right ) }}}\sqrt{{\frac{ \left ( dg-ef \right ) \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) }{ \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) \left ( ex+d \right ) }}}\sqrt{{\frac{ \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) \left ( gx+f \right ) }{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) \left ( ex+d \right ) }}}{\frac{1}{\sqrt{-{\frac{ \left ( ex+d \right ) \left ( gx+f \right ) \left ( -2\,cx+\sqrt{-4\,ac+{b}^{2}}-b \right ) \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) }{c}}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(e*x+d)^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + b x + a} \sqrt{e x + d} \sqrt{g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)*sqrt(g*x + f)),x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{\sqrt{c x^{2} + b x + a} \sqrt{e x + d} \sqrt{g x + f}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)*sqrt(g*x + f)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{d + e x} \sqrt{f + g x} \sqrt{a + b x + c x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(e*x+d)**(1/2)/(g*x+f)**(1/2)/(c*x**2+b*x+a)**(1/2),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{c x^{2} + b x + a} \sqrt{e x + d} \sqrt{g x + f}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(sqrt(c*x^2 + b*x + a)*sqrt(e*x + d)*sqrt(g*x + f)),x, algorithm="giac")
[Out]