3.918 \(\int \frac{\sqrt{d+e x}}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=475 \[ \frac{\sqrt{2} (d+e x) \sqrt{-\sqrt{b^2-4 a c}+b+2 c x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )} \sqrt{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) (e f-d g)}{(d+e x) \left (2 c f-g \left (\sqrt{b^2-4 a c}+b\right )\right )}} \sqrt{\frac{\left (x \left (\sqrt{b^2-4 a c}+b\right )+2 a\right ) (e f-d g)}{(d+e x) \left (f \sqrt{b^2-4 a c}-2 a g+b f\right )}} \Pi \left (\frac{e \left (2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) g};\sin ^{-1}\left (\frac{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \sqrt{f+g x}}{\sqrt{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g} \sqrt{d+e x}}\right )|\frac{\left (b d+\sqrt{b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b f+\sqrt{b^2-4 a c} f-2 a g\right )}\right )}{g \sqrt{\frac{2 a c}{\sqrt{b^2-4 a c}+b}+c x} \sqrt{a+b x+c x^2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \]

[Out]

(Sqrt[2]*Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Sqrt[((e*f - d*g)*(b + Sq
rt[b^2 - 4*a*c] + 2*c*x))/((2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)*(d + e*x))]*Sqrt[((e*f - d*g)*(2*a + (b + Sqrt[
b^2 - 4*a*c])*x))/((b*f + Sqrt[b^2 - 4*a*c]*f - 2*a*g)*(d + e*x))]*(d + e*x)*EllipticPi[(e*(2*c*f - (b + Sqrt[
b^2 - 4*a*c])*g))/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*g), ArcSin[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*Sqr
t[f + g*x])/(Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])], ((b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e)*(2*
c*f - (b + Sqrt[b^2 - 4*a*c])*g))/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(b*f + Sqrt[b^2 - 4*a*c]*f - 2*a*g))])/
(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*g*Sqrt[(2*a*c)/(b + Sqrt[b^2 - 4*a*c]) + c*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 0.419329, antiderivative size = 475, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03, Rules used = {926} \[ \frac{\sqrt{2} (d+e x) \sqrt{-\sqrt{b^2-4 a c}+b+2 c x} \sqrt{2 c f-g \left (\sqrt{b^2-4 a c}+b\right )} \sqrt{\frac{\left (\sqrt{b^2-4 a c}+b+2 c x\right ) (e f-d g)}{(d+e x) \left (2 c f-g \left (\sqrt{b^2-4 a c}+b\right )\right )}} \sqrt{\frac{\left (x \left (\sqrt{b^2-4 a c}+b\right )+2 a\right ) (e f-d g)}{(d+e x) \left (f \sqrt{b^2-4 a c}-2 a g+b f\right )}} \Pi \left (\frac{e \left (2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) g};\sin ^{-1}\left (\frac{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \sqrt{f+g x}}{\sqrt{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g} \sqrt{d+e x}}\right )|\frac{\left (b d+\sqrt{b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b f+\sqrt{b^2-4 a c} f-2 a g\right )}\right )}{g \sqrt{\frac{2 a c}{\sqrt{b^2-4 a c}+b}+c x} \sqrt{a+b x+c x^2} \sqrt{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[d + e*x]/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]

[Out]

(Sqrt[2]*Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[b - Sqrt[b^2 - 4*a*c] + 2*c*x]*Sqrt[((e*f - d*g)*(b + Sq
rt[b^2 - 4*a*c] + 2*c*x))/((2*c*f - (b + Sqrt[b^2 - 4*a*c])*g)*(d + e*x))]*Sqrt[((e*f - d*g)*(2*a + (b + Sqrt[
b^2 - 4*a*c])*x))/((b*f + Sqrt[b^2 - 4*a*c]*f - 2*a*g)*(d + e*x))]*(d + e*x)*EllipticPi[(e*(2*c*f - (b + Sqrt[
b^2 - 4*a*c])*g))/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*g), ArcSin[(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*Sqr
t[f + g*x])/(Sqrt[2*c*f - (b + Sqrt[b^2 - 4*a*c])*g]*Sqrt[d + e*x])], ((b*d + Sqrt[b^2 - 4*a*c]*d - 2*a*e)*(2*
c*f - (b + Sqrt[b^2 - 4*a*c])*g))/((2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)*(b*f + Sqrt[b^2 - 4*a*c]*f - 2*a*g))])/
(Sqrt[2*c*d - (b + Sqrt[b^2 - 4*a*c])*e]*g*Sqrt[(2*a*c)/(b + Sqrt[b^2 - 4*a*c]) + c*x]*Sqrt[a + b*x + c*x^2])

Rule 926

Int[Sqrt[(d_.) + (e_.)*(x_)]/(Sqrt[(f_.) + (g_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Symbol] :>
With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(Sqrt[2]*Sqrt[2*c*f - g*(b + q)]*Sqrt[b - q + 2*c*x]*(d + e*x)*Sqrt[((e*f
- d*g)*(b + q + 2*c*x))/((2*c*f - g*(b + q))*(d + e*x))]*Sqrt[((e*f - d*g)*(2*a + (b + q)*x))/((b*f + q*f - 2*
a*g)*(d + e*x))]*EllipticPi[(e*(2*c*f - g*(b + q)))/(g*(2*c*d - e*(b + q))), ArcSin[(Sqrt[2*c*d - e*(b + q)]*S
qrt[f + g*x])/(Sqrt[2*c*f - g*(b + q)]*Sqrt[d + e*x])], ((b*d + q*d - 2*a*e)*(2*c*f - g*(b + q)))/((b*f + q*f
- 2*a*g)*(2*c*d - e*(b + q)))])/(g*Sqrt[2*c*d - e*(b + q)]*Sqrt[(2*a*c)/(b + q) + c*x]*Sqrt[a + b*x + c*x^2]),
x]] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e
^2, 0]

Rubi steps

\begin{align*} \int \frac{\sqrt{d+e x}}{\sqrt{f+g x} \sqrt{a+b x+c x^2}} \, dx &=\frac{\sqrt{2} \sqrt{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g} \sqrt{b-\sqrt{b^2-4 a c}+2 c x} \sqrt{\frac{(e f-d g) \left (b+\sqrt{b^2-4 a c}+2 c x\right )}{\left (2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g\right ) (d+e x)}} \sqrt{\frac{(e f-d g) \left (2 a+\left (b+\sqrt{b^2-4 a c}\right ) x\right )}{\left (b f+\sqrt{b^2-4 a c} f-2 a g\right ) (d+e x)}} (d+e x) \Pi \left (\frac{e \left (2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) g};\sin ^{-1}\left (\frac{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} \sqrt{f+g x}}{\sqrt{2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g} \sqrt{d+e x}}\right )|\frac{\left (b d+\sqrt{b^2-4 a c} d-2 a e\right ) \left (2 c f-\left (b+\sqrt{b^2-4 a c}\right ) g\right )}{\left (2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e\right ) \left (b f+\sqrt{b^2-4 a c} f-2 a g\right )}\right )}{\sqrt{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e} g \sqrt{\frac{2 a c}{b+\sqrt{b^2-4 a c}}+c x} \sqrt{a+b x+c x^2}}\\ \end{align*}

Mathematica [B]  time = 9.95143, size = 1118, normalized size = 2.35 \[ -\frac{\sqrt{2} \sqrt{-\frac{g \left (c f^2+g (a g-b f)\right ) (d+e x)}{\left (-2 a e g^2-2 c d f g+b (e f+d g) g-d \sqrt{\left (b^2-4 a c\right ) g^2} g+e f \sqrt{\left (b^2-4 a c\right ) g^2}\right ) (f+g x)}} (f+g x)^{3/2} \left (\frac{2 e f \sqrt{\left (b^2-4 a c\right ) g^2} \sqrt{-\frac{\left (c f^2+g (a g-b f)\right ) (a+x (b+c x))}{\left (b^2-4 a c\right ) (f+g x)^2}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{-2 a g^2+2 c f x g+b (f-g x) g+\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}}}{\sqrt{2}}\right ),\frac{2 \sqrt{\left (b^2-4 a c\right ) g^2} (d g-e f)}{2 a e g^2+2 c d f g-b (e f+d g) g+d \sqrt{\left (b^2-4 a c\right ) g^2} g-e f \sqrt{\left (b^2-4 a c\right ) g^2}}\right )}{c f^2+g (a g-b f)}+\frac{d g \left (2 a g^2-2 c f x g-\sqrt{\left (b^2-4 a c\right ) g^2} x g+b (g x-f) g-f \sqrt{\left (b^2-4 a c\right ) g^2}\right ) \sqrt{\frac{2 a g^2-2 c f x g+b (g x-f) g+\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{\frac{-2 a g^2+2 c f x g+b (f-g x) g+\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}}}{\sqrt{2}}\right ),\frac{2 \sqrt{\left (b^2-4 a c\right ) g^2} (d g-e f)}{2 a e g^2+2 c d f g-b (e f+d g) g+d \sqrt{\left (b^2-4 a c\right ) g^2} g-e f \sqrt{\left (b^2-4 a c\right ) g^2}}\right )}{\left (c f^2+g (a g-b f)\right ) (f+g x) \sqrt{\frac{-2 a g^2+2 c f x g+b (f-g x) g+\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}}}-\frac{4 e \sqrt{\left (b^2-4 a c\right ) g^2} \sqrt{-\frac{\left (c f^2+g (a g-b f)\right ) (a+x (b+c x))}{\left (b^2-4 a c\right ) (f+g x)^2}} \Pi \left (\frac{2 \sqrt{\left (b^2-4 a c\right ) g^2}}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}};\sin ^{-1}\left (\frac{\sqrt{\frac{-2 a g^2+2 c f x g+b (f-g x) g+\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}{\sqrt{\left (b^2-4 a c\right ) g^2} (f+g x)}}}{\sqrt{2}}\right )|\frac{2 \sqrt{\left (b^2-4 a c\right ) g^2} (d g-e f)}{2 a e g^2+2 c d f g-b (e f+d g) g+d \sqrt{\left (b^2-4 a c\right ) g^2} g-e f \sqrt{\left (b^2-4 a c\right ) g^2}}\right )}{2 c f-b g+\sqrt{\left (b^2-4 a c\right ) g^2}}\right )}{g^2 \sqrt{d+e x} \sqrt{a+x (b+c x)}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[Sqrt[d + e*x]/(Sqrt[f + g*x]*Sqrt[a + b*x + c*x^2]),x]

[Out]

-((Sqrt[2]*Sqrt[-((g*(c*f^2 + g*(-(b*f) + a*g))*(d + e*x))/((-2*c*d*f*g - 2*a*e*g^2 + e*f*Sqrt[(b^2 - 4*a*c)*g
^2] - d*g*Sqrt[(b^2 - 4*a*c)*g^2] + b*g*(e*f + d*g))*(f + g*x)))]*(f + g*x)^(3/2)*((2*e*f*Sqrt[(b^2 - 4*a*c)*g
^2]*Sqrt[-(((c*f^2 + g*(-(b*f) + a*g))*(a + x*(b + c*x)))/((b^2 - 4*a*c)*(f + g*x)^2))]*EllipticF[ArcSin[Sqrt[
(-2*a*g^2 + 2*c*f*g*x + b*g*(f - g*x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/(Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))
]/Sqrt[2]], (2*Sqrt[(b^2 - 4*a*c)*g^2]*(-(e*f) + d*g))/(2*c*d*f*g + 2*a*e*g^2 - e*f*Sqrt[(b^2 - 4*a*c)*g^2] +
d*g*Sqrt[(b^2 - 4*a*c)*g^2] - b*g*(e*f + d*g))])/(c*f^2 + g*(-(b*f) + a*g)) + (d*g*(2*a*g^2 - f*Sqrt[(b^2 - 4*
a*c)*g^2] - 2*c*f*g*x - g*Sqrt[(b^2 - 4*a*c)*g^2]*x + b*g*(-f + g*x))*Sqrt[(2*a*g^2 - 2*c*f*g*x + b*g*(-f + g*
x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/(Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))]*EllipticF[ArcSin[Sqrt[(-2*a*g^2 +
2*c*f*g*x + b*g*(f - g*x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/(Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))]/Sqrt[2]],
(2*Sqrt[(b^2 - 4*a*c)*g^2]*(-(e*f) + d*g))/(2*c*d*f*g + 2*a*e*g^2 - e*f*Sqrt[(b^2 - 4*a*c)*g^2] + d*g*Sqrt[(b
^2 - 4*a*c)*g^2] - b*g*(e*f + d*g))])/((c*f^2 + g*(-(b*f) + a*g))*(f + g*x)*Sqrt[(-2*a*g^2 + 2*c*f*g*x + b*g*(
f - g*x) + Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/(Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))]) - (4*e*Sqrt[(b^2 - 4*a*c)*
g^2]*Sqrt[-(((c*f^2 + g*(-(b*f) + a*g))*(a + x*(b + c*x)))/((b^2 - 4*a*c)*(f + g*x)^2))]*EllipticPi[(2*Sqrt[(b
^2 - 4*a*c)*g^2])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2]), ArcSin[Sqrt[(-2*a*g^2 + 2*c*f*g*x + b*g*(f - g*x) +
Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))/(Sqrt[(b^2 - 4*a*c)*g^2]*(f + g*x))]/Sqrt[2]], (2*Sqrt[(b^2 - 4*a*c)*g^2]*
(-(e*f) + d*g))/(2*c*d*f*g + 2*a*e*g^2 - e*f*Sqrt[(b^2 - 4*a*c)*g^2] + d*g*Sqrt[(b^2 - 4*a*c)*g^2] - b*g*(e*f
+ d*g))])/(2*c*f - b*g + Sqrt[(b^2 - 4*a*c)*g^2])))/(g^2*Sqrt[d + e*x]*Sqrt[a + x*(b + c*x)]))

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Maple [A]  time = 0.637, size = 645, normalized size = 1.4 \begin{align*} 4\,{\frac{\sqrt{ex+d}\sqrt{gx+f}\sqrt{c{x}^{2}+bx+a} \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 ) }{g \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \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}}\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 ( dg-ef \right ) \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) }{ \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) \left ( ex+d \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 ) }}}{\it EllipticPi} \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 ) }}},{\frac{ \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) e}{g \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) }},\sqrt{{\frac{ \left ( 2\,cd-be+e\sqrt{-4\,ac+{b}^{2}} \right ) \left ( g\sqrt{-4\,ac+{b}^{2}}+bg-2\,cf \right ) }{ \left ( 2\,cf-bg+g\sqrt{-4\,ac+{b}^{2}} \right ) \left ( e\sqrt{-4\,ac+{b}^{2}}+be-2\,cd \right ) }}} \right ){\frac{1}{\sqrt{-{\frac{ \left ( gx+f \right ) \left ( ex+d \right ) \left ( -b-2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) \left ( b+2\,cx+\sqrt{-4\,ac+{b}^{2}} \right ) }{c}}}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x)

[Out]

4*(e*x+d)^(1/2)*(g*x+f)^(1/2)*(c*x^2+b*x+a)^(1/2)/g*((e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)*(g*x+f)/(g*(-4*a*c+b^2)^
(1/2)+b*g-2*c*f)/(e*x+d))^(1/2)*((d*g-e*f)*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2))/(e*x
+d))^(1/2)*((d*g-e*f)*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(e*x+d))^(1/2)*EllipticPi(
((e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)*(g*x+f)/(g*(-4*a*c+b^2)^(1/2)+b*g-2*c*f)/(e*x+d))^(1/2),(g*(-4*a*c+b^2)^(1/2
)+b*g-2*c*f)*e/g/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d),((2*c*d-b*e+e*(-4*a*c+b^2)^(1/2))*(g*(-4*a*c+b^2)^(1/2)+b*g-
2*c*f)/(2*c*f-b*g+g*(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2))*((-4*a*c+b^2)^(1/2)*x^2*e^2*g
+b*e^2*g*x^2-2*c*e^2*f*x^2+2*(-4*a*c+b^2)^(1/2)*x*d*e*g+2*x*b*d*e*g-4*x*c*d*e*f+(-4*a*c+b^2)^(1/2)*d^2*g+b*d^2
*g-2*c*d^2*f)/(-1/c*(g*x+f)*(e*x+d)*(-b-2*c*x+(-4*a*c+b^2)^(1/2))*(b+2*c*x+(-4*a*c+b^2)^(1/2)))^(1/2)/(e*(-4*a
*c+b^2)^(1/2)+b*e-2*c*d)/(c*e*g*x^4+b*e*g*x^3+c*d*g*x^3+c*e*f*x^3+a*e*g*x^2+b*d*g*x^2+b*e*f*x^2+c*d*f*x^2+a*d*
g*x+a*e*f*x+b*d*f*x+a*d*f)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}}{\sqrt{c x^{2} + b x + a} \sqrt{g x + f}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(e*x + d)/(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f)), x)

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Fricas [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]

integrate((e*x+d)^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{d + e x}}{\sqrt{f + g x} \sqrt{a + b x + c x^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**(1/2)/(g*x+f)**(1/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral(sqrt(d + e*x)/(sqrt(f + g*x)*sqrt(a + b*x + c*x**2)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{e x + d}}{\sqrt{c x^{2} + b x + a} \sqrt{g x + f}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^(1/2)/(g*x+f)^(1/2)/(c*x^2+b*x+a)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(e*x + d)/(sqrt(c*x^2 + b*x + a)*sqrt(g*x + f)), x)