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

Optimal. Leaf size=955 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a e \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a e \sqrt{d+e x} \left (a x^2+b x+c\right )}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} (b d+c e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{\sqrt{2} c \sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \Pi \left (\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right ) x}{\sqrt{a} \left (a x^2+b x+c\right )}+\frac{2}{3} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x \]

[Out]

(2*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x])/3 + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(a*d +
b*e)*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4
*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]
/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*a*e
*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*(c + b*x + a*x^2)) - (2
*Sqrt[2]*Sqrt[b^2 - 4*a*c]*d*(a*d + b*e)*Sqrt[a + c/x^2 + b/x]*x*Sqrt[(a*(d + e*
x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a
*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/S
qrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*a*e*S
qrt[d + e*x]*(c + b*x + a*x^2)) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(b*d + c*e)*Sqrt[
a + c/x^2 + b/x]*x*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[
-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*
a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b
+ Sqrt[b^2 - 4*a*c])*e)])/(3*a*Sqrt[d + e*x]*(c + b*x + a*x^2)) - (Sqrt[2]*c*Sqr
t[2*a*d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[a + c/x^2 + b/x]*x*Sqrt[1 - (2*a*(d +
e*x))/(2*a*d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*a*(d + e*x))/(2*a*d - (b
+ Sqrt[b^2 - 4*a*c])*e)]*EllipticPi[(2*a*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*a*d),
ArcSin[(Sqrt[2]*Sqrt[a]*Sqrt[d + e*x])/Sqrt[2*a*d - (b - Sqrt[b^2 - 4*a*c])*e]]
, (b - Sqrt[b^2 - 4*a*c] - (2*a*d)/e)/(b + Sqrt[b^2 - 4*a*c] - (2*a*d)/e)])/(Sqr
t[a]*(c + b*x + a*x^2))

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Rubi [A]  time = 7.74253, antiderivative size = 955, normalized size of antiderivative = 1., number of steps used = 16, number of rules used = 11, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.423 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a e \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \left (a x^2+b x+c\right )}-\frac{2 \sqrt{2} \sqrt{b^2-4 a c} d (a d+b e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a e \sqrt{d+e x} \left (a x^2+b x+c\right )}+\frac{4 \sqrt{2} \sqrt{b^2-4 a c} (b d+c e) \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{\frac{a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \sqrt{-\frac{a \left (a x^2+b x+c\right )}{b^2-4 a c}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 a x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right ) x}{3 a \sqrt{d+e x} \left (a x^2+b x+c\right )}-\frac{\sqrt{2} c \sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}} \sqrt{1-\frac{2 a (d+e x)}{2 a d-\left (b+\sqrt{b^2-4 a c}\right ) e}} \Pi \left (\frac{2 a d-b e+\sqrt{b^2-4 a c} e}{2 a d};\sin ^{-1}\left (\frac{\sqrt{2} \sqrt{a} \sqrt{d+e x}}{\sqrt{2 a d-\left (b-\sqrt{b^2-4 a c}\right ) e}}\right )|\frac{b-\sqrt{b^2-4 a c}-\frac{2 a d}{e}}{b+\sqrt{b^2-4 a c}-\frac{2 a d}{e}}\right ) x}{\sqrt{a} \left (a x^2+b x+c\right )}+\frac{2}{3} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{d+e x} x \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x],x]

[Out]

(2*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x])/3 + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(a*d +
b*e)*Sqrt[a + c/x^2 + b/x]*x*Sqrt[d + e*x]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4
*a*c))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]
/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*a*e
*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*(c + b*x + a*x^2)) - (2
*Sqrt[2]*Sqrt[b^2 - 4*a*c]*d*(a*d + b*e)*Sqrt[a + c/x^2 + b/x]*x*Sqrt[(a*(d + e*
x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((a*(c + b*x + a*x^2))/(b^2 - 4*a
*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/S
qrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*a*e*S
qrt[d + e*x]*(c + b*x + a*x^2)) + (4*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(b*d + c*e)*Sqrt[
a + c/x^2 + b/x]*x*Sqrt[(a*(d + e*x))/(2*a*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[
-((a*(c + b*x + a*x^2))/(b^2 - 4*a*c))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*
a*c] + 2*a*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*a*d - (b
+ Sqrt[b^2 - 4*a*c])*e)])/(3*a*Sqrt[d + e*x]*(c + b*x + a*x^2)) - (Sqrt[2]*c*Sqr
t[2*a*d - (b - Sqrt[b^2 - 4*a*c])*e]*Sqrt[a + c/x^2 + b/x]*x*Sqrt[1 - (2*a*(d +
e*x))/(2*a*d - (b - Sqrt[b^2 - 4*a*c])*e)]*Sqrt[1 - (2*a*(d + e*x))/(2*a*d - (b
+ Sqrt[b^2 - 4*a*c])*e)]*EllipticPi[(2*a*d - b*e + Sqrt[b^2 - 4*a*c]*e)/(2*a*d),
ArcSin[(Sqrt[2]*Sqrt[a]*Sqrt[d + e*x])/Sqrt[2*a*d - (b - Sqrt[b^2 - 4*a*c])*e]]
, (b - Sqrt[b^2 - 4*a*c] - (2*a*d)/e)/(b + Sqrt[b^2 - 4*a*c] - (2*a*d)/e)])/(Sqr
t[a]*(c + b*x + a*x^2))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{x \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}} \int \frac{\sqrt{d + e x} \sqrt{a x^{2} + b x + c}}{x}\, dx}{\sqrt{a x^{2} + b x + c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a+c/x**2+b/x)**(1/2)*(e*x+d)**(1/2),x)

[Out]

x*sqrt(a + b/x + c/x**2)*Integral(sqrt(d + e*x)*sqrt(a*x**2 + b*x + c)/x, x)/sqr
t(a*x**2 + b*x + c)

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Mathematica [C]  time = 13.1125, size = 758, normalized size = 0.79 \[ \frac{2}{3} x \sqrt{a+\frac{b x+c}{x^2}} \left (-\frac{i (d+e x) \sqrt{1-\frac{2 \left (a d^2+e (c e-b d)\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}+2 a d-b e\right )}} \sqrt{\frac{2 \left (a d^2+e (c e-b d)\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}-2 a d+b e\right )}+1} \left (-\left (a \left (d \sqrt{e^2 \left (b^2-4 a c\right )}+3 b d e-2 c e^2\right )+b e \left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e\right )\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )+(a d+b e) \left (\sqrt{e^2 \left (b^2-4 a c\right )}+2 a d-b e\right ) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )-6 a c e^2 \Pi \left (\frac{d \left (2 a d-b e-\sqrt{\left (b^2-4 a c\right ) e^2}\right )}{2 \left (a d^2+e (c e-b d)\right )};i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{a d^2-b e d+c e^2}{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 a d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 a d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt{2} a e^2 (x (a x+b)+c) \sqrt{\frac{a d^2+e (c e-b d)}{\sqrt{e^2 \left (b^2-4 a c\right )}-2 a d+b e}}}+\frac{a d+b e}{a \sqrt{d+e x}}+\sqrt{d+e x}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + c/x^2 + b/x]*Sqrt[d + e*x],x]

[Out]

(2*x*Sqrt[a + (c + b*x)/x^2]*((a*d + b*e)/(a*Sqrt[d + e*x]) + Sqrt[d + e*x] - ((
I/2)*(d + e*x)*Sqrt[1 - (2*(a*d^2 + e*(-(b*d) + c*e)))/((2*a*d - b*e + Sqrt[(b^2
- 4*a*c)*e^2])*(d + e*x))]*Sqrt[1 + (2*(a*d^2 + e*(-(b*d) + c*e)))/((-2*a*d + b
*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]*((a*d + b*e)*(2*a*d - b*e + Sqrt[(b^2
- 4*a*c)*e^2])*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d
+ b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2
- 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] - (b*e*(-(b*e) + Sqrt[(
b^2 - 4*a*c)*e^2]) + a*(3*b*d*e - 2*c*e^2 + d*Sqrt[(b^2 - 4*a*c)*e^2]))*Elliptic
F[I*ArcSinh[(Sqrt[2]*Sqrt[(a*d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*
a*c)*e^2])])/Sqrt[d + e*x]], -((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d -
b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] - 6*a*c*e^2*EllipticPi[(d*(2*a*d - b*e - Sqrt[
(b^2 - 4*a*c)*e^2]))/(2*(a*d^2 + e*(-(b*d) + c*e))), I*ArcSinh[(Sqrt[2]*Sqrt[(a*
d^2 - b*d*e + c*e^2)/(-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]],
-((-2*a*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*a*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2
]))]))/(Sqrt[2]*a*e^2*Sqrt[(a*d^2 + e*(-(b*d) + c*e))/(-2*a*d + b*e + Sqrt[(b^2
- 4*a*c)*e^2])]*(c + x*(b + a*x)))))/3

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Maple [B]  time = 0.06, size = 3023, normalized size = 3.2 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a+c/x^2+b/x)^(1/2)*(e*x+d)^(1/2),x)

[Out]

1/3*((a*x^2+b*x+c)/x^2)^(1/2)*x*(e*x+d)^(1/2)*(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^
2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2
)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+
b*e))^(1/2)*EllipticF(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2
),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*(-
4*a*c+b^2)^(1/2)*a*d^2*e-2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(
1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e
*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticF(
2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1
/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*(-4*a*c+b^2)^(1/2)*b*d*e
^2-2*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*
a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^
2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticF(2^(1/2)*(-a*(e*x+d)/
(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-
4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*(-4*a*c+b^2)^(1/2)*c*e^3+3*2^(1/2)*(-a*(e*x+
d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(
-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b
^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticF(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-
2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d
-b*e))^(1/2))*a*b*d^2*e+6*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^
(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(
e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticF
(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(
1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*a*c*d*e^2-3*2^(1/2)*(-a
*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b
)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4
*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticF(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^
(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)
+2*a*d-b*e))^(1/2))*b^2*d*e^2-2*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+
b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(
1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*Ell
ipticE(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+
b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*a^2*d^3-2*2^(1/2)
*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/
2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e
*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticE(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b
^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(
1/2)+2*a*d-b*e))^(1/2))*a*c*d*e^2+2*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*
a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e
))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)
*EllipticE(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*
a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*b^2*d*e^2-2*2
^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^
2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/
2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticE(2^(1/2)*(-a*(e*x+d)/(e*(-4
*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+
b^2)^(1/2)+2*a*d-b*e))^(1/2))*b*c*e^3+3*2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2
)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4*a*c+b^2)^(1/2)+2*a*d
-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(
1/2)*EllipticPi(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2),-1/2
*(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/a/d,(-(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/(e*(-4*
a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*(-4*a*c+b^2)^(1/2)*c*e^3-6*2^(1/2)*(-a*(e*x+d)
/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/(e*(-4
*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2
)^(1/2)-2*a*d+b*e))^(1/2)*EllipticPi(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2
*a*d+b*e))^(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/a/d,(-(e*(-4*a*c+b^2)^(1/
2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*a*c*d*e^2+3*2^(1/2)*(-a*(
e*x+d)/(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*(e*(-2*a*x+(-4*a*c+b^2)^(1/2)-b)/
(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2)*(e*(b+2*a*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a
*c+b^2)^(1/2)-2*a*d+b*e))^(1/2)*EllipticPi(2^(1/2)*(-a*(e*x+d)/(e*(-4*a*c+b^2)^(
1/2)-2*a*d+b*e))^(1/2),-1/2*(e*(-4*a*c+b^2)^(1/2)-2*a*d+b*e)/a/d,(-(e*(-4*a*c+b^
2)^(1/2)-2*a*d+b*e)/(e*(-4*a*c+b^2)^(1/2)+2*a*d-b*e))^(1/2))*b*c*e^3+2*x^3*a^2*e
^3+2*x^2*a^2*d*e^2+2*x^2*a*b*e^3+2*x*a*b*d*e^2+2*x*a*c*e^3+2*a*c*d*e^2)/a/e^2/(a
*e*x^3+a*d*x^2+b*e*x^2+b*d*x+c*e*x+c*d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)*sqrt(a + b/x + c/x^2),x, algorithm="maxima")

[Out]

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

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)*sqrt(a + b/x + c/x^2),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a+c/x**2+b/x)**(1/2)*(e*x+d)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(e*x + d)*sqrt(a + b/x + c/x^2),x, algorithm="giac")

[Out]

Timed out