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]

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

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Rubi [A]  time = 0.42, antiderivative size = 475, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, 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*}

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Mathematica [B]  time = 9.47, 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}} F\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)}} F\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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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}}\,{d x} \]

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)

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maple [A]  time = 0.20, size = 645, normalized size = 1.36 \[ \frac {4 \sqrt {e x +d}\, \sqrt {g x +f}\, \sqrt {c \,x^{2}+b x +a}\, \sqrt {\frac {\left (b e -2 c d +\sqrt {-4 a c +b^{2}}\, e \right ) \left (g x +f \right )}{\left (b g -2 c f +\sqrt {-4 a c +b^{2}}\, g \right ) \left (e x +d \right )}}\, \sqrt {\frac {\left (d g -e f \right ) \left (-2 c x -b +\sqrt {-4 a c +b^{2}}\right )}{\left (-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g \right ) \left (e x +d \right )}}\, \sqrt {\frac {\left (d g -e f \right ) \left (2 c x +b +\sqrt {-4 a c +b^{2}}\right )}{\left (b g -2 c f +\sqrt {-4 a c +b^{2}}\, g \right ) \left (e x +d \right )}}\, \left (b \,e^{2} g \,x^{2}-2 c \,e^{2} f \,x^{2}+2 b d e g x -4 c d e f x +\sqrt {-4 a c +b^{2}}\, e^{2} g \,x^{2}+b \,d^{2} g -2 c \,d^{2} f +2 \sqrt {-4 a c +b^{2}}\, d e g x +\sqrt {-4 a c +b^{2}}\, d^{2} g \right ) \EllipticPi \left (\sqrt {\frac {\left (b e -2 c d +\sqrt {-4 a c +b^{2}}\, e \right ) \left (g x +f \right )}{\left (b g -2 c f +\sqrt {-4 a c +b^{2}}\, g \right ) \left (e x +d \right )}}, \frac {\left (b g -2 c f +\sqrt {-4 a c +b^{2}}\, g \right ) e}{\left (b e -2 c d +\sqrt {-4 a c +b^{2}}\, e \right ) g}, \sqrt {\frac {\left (-b e +2 c d +\sqrt {-4 a c +b^{2}}\, e \right ) \left (b g -2 c f +\sqrt {-4 a c +b^{2}}\, g \right )}{\left (-b g +2 c f +\sqrt {-4 a c +b^{2}}\, g \right ) \left (b e -2 c d +\sqrt {-4 a c +b^{2}}\, e \right )}}\right )}{\sqrt {-\frac {\left (g x +f \right ) \left (e x +d \right ) \left (-2 c x -b +\sqrt {-4 a c +b^{2}}\right ) \left (2 c x +b +\sqrt {-4 a c +b^{2}}\right )}{c}}\, \left (b e -2 c d +\sqrt {-4 a c +b^{2}}\, e \right ) \sqrt {c e g \,x^{4}+b e g \,x^{3}+c d g \,x^{3}+c e f \,x^{3}+a e g \,x^{2}+b d g \,x^{2}+b e f \,x^{2}+c d f \,x^{2}+a d g x +a e f x +b d f x +a d f}\, g} \]

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)/(b*g-2*c*f+(-4*a
*c+b^2)^(1/2)*g)/(e*x+d))^(1/2)*((d*g-e*f)*(-2*c*x-b+(-4*a*c+b^2)^(1/2))/(-b*g+2*c*f+(-4*a*c+b^2)^(1/2)*g)/(e*
x+d))^(1/2)*((d*g-e*f)*(2*c*x+b+(-4*a*c+b^2)^(1/2))/(b*g-2*c*f+(-4*a*c+b^2)^(1/2)*g)/(e*x+d))^(1/2)*EllipticPi
(((e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)*(g*x+f)/(b*g-2*c*f+(-4*a*c+b^2)^(1/2)*g)/(e*x+d))^(1/2),(b*g-2*c*f+(-4*a*c+
b^2)^(1/2)*g)*e/g/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d),((e*(-4*a*c+b^2)^(1/2)-b*e+2*c*d)*(b*g-2*c*f+(-4*a*c+b^2)^(
1/2)*g)/(-b*g+2*c*f+(-4*a*c+b^2)^(1/2)*g)/(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*b*d*e*g*x-4*c*d*e*f*x+(-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)*(-2*c*x-b+(-4*a*c+b^2)^(1/2))*(2*c*x+b+(-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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e x + d}}{\sqrt {c x^{2} + b x + a} \sqrt {g x + f}}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {d+e\,x}}{\sqrt {f+g\,x}\,\sqrt {c\,x^2+b\,x+a}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {d + e x}}{\sqrt {f + g x} \sqrt {a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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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