3.2630 \(\int \frac{(A+B x) \sqrt{d+e x}}{\sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=452 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (3 A c e-2 b B e+B c d) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^2 e \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{2} B \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^2 e \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 B \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 c} \]

[Out]

(2*B*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(B*
c*d - 2*b*B*e + 3*A*c*e)*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c
))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqr
t[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*c^2*e*S
qrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) -
(2*Sqrt[2]*B*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*Sqrt[(c*(d + e*x))/(2*c*d
- (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Elli
pticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (
-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*c^2*e*Sqrt[d +
e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 1.19958, antiderivative size = 452, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (3 A c e-2 b B e+B c d) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^2 e \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \sqrt{2} B \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{3 c^2 e \sqrt{d+e x} \sqrt{a+b x+c x^2}}+\frac{2 B \sqrt{d+e x} \sqrt{a+b x+c x^2}}{3 c} \]

Antiderivative was successfully verified.

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

[Out]

(2*B*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])/(3*c) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(B*
c*d - 2*b*B*e + 3*A*c*e)*Sqrt[d + e*x]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c
))]*EllipticE[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqr
t[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*c^2*e*S
qrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*x + c*x^2]) -
(2*Sqrt[2]*B*Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*Sqrt[(c*(d + e*x))/(2*c*d
- (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*Elli
pticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqrt[b^2 - 4*a*c]]/Sqrt[2]], (
-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)])/(3*c^2*e*Sqrt[d +
e*x]*Sqrt[a + b*x + c*x^2])

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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((B*x+A)*(e*x+d)**(1/2)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Timed out

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Mathematica [C]  time = 10.9618, size = 781, normalized size = 1.73 \[ \frac{2 B \sqrt{d+e x} \left (a+b x+c x^2\right )}{3 c \sqrt{a+x (b+c x)}}+\frac{2 (d+e x)^{3/2} \sqrt{a+b x+c x^2} \left (\left (\frac{e \left (\frac{a e}{d+e x}-\frac{b d}{d+e x}+b\right )}{d+e x}+c \left (\frac{d}{d+e x}-1\right )^2\right ) (3 A c e-2 b B e+B c d)+\frac{i \sqrt{1-\frac{2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right )}} \sqrt{\frac{2 \left (e (a e-b d)+c d^2\right )}{(d+e x) \left (\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d\right )}+1} \left (\left (-b e \left (2 B \sqrt{e^2 \left (b^2-4 a c\right )}+3 A c e+3 B c d\right )+c \left (3 A e \left (\sqrt{e^2 \left (b^2-4 a c\right )}+2 c d\right )+B d \sqrt{e^2 \left (b^2-4 a c\right )}-2 a B e^2\right )+2 b^2 B e^2\right ) F\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )+\left (\sqrt{e^2 \left (b^2-4 a c\right )}-b e+2 c d\right ) (-3 A c e+2 b B e-B c d) E\left (i \sinh ^{-1}\left (\frac{\sqrt{2} \sqrt{\frac{c d^2-b e d+a e^2}{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}}}{\sqrt{d+e x}}\right )|-\frac{-2 c d+b e+\sqrt{\left (b^2-4 a c\right ) e^2}}{2 c d-b e+\sqrt{\left (b^2-4 a c\right ) e^2}}\right )\right )}{2 \sqrt{2} \sqrt{d+e x} \sqrt{\frac{e (a e-b d)+c d^2}{\sqrt{e^2 \left (b^2-4 a c\right )}+b e-2 c d}}}\right )}{3 c^2 e^2 \sqrt{a+x (b+c x)} \sqrt{\frac{(d+e x)^2 \left (\frac{e \left (\frac{a e}{d+e x}-\frac{b d}{d+e x}+b\right )}{d+e x}+c \left (\frac{d}{d+e x}-1\right )^2\right )}{e^2}}} \]

Antiderivative was successfully verified.

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

[Out]

(2*B*Sqrt[d + e*x]*(a + b*x + c*x^2))/(3*c*Sqrt[a + x*(b + c*x)]) + (2*(d + e*x)
^(3/2)*Sqrt[a + b*x + c*x^2]*((B*c*d - 2*b*B*e + 3*A*c*e)*(c*(-1 + d/(d + e*x))^
2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d + e*x)) + ((I/2)*Sqrt[1 - (2*
(c*d^2 + e*(-(b*d) + a*e)))/((2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2])*(d + e*x))]
*Sqrt[1 + (2*(c*d^2 + e*(-(b*d) + a*e)))/((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2
])*(d + e*x))]*((-(B*c*d) + 2*b*B*e - 3*A*c*e)*(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)
*e^2])*EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e +
Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d + e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)
*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a*c)*e^2]))] + (2*b^2*B*e^2 - b*e*(3*B*c*d +
3*A*c*e + 2*B*Sqrt[(b^2 - 4*a*c)*e^2]) + c*(-2*a*B*e^2 + B*d*Sqrt[(b^2 - 4*a*c)
*e^2] + 3*A*e*(2*c*d + Sqrt[(b^2 - 4*a*c)*e^2])))*EllipticF[I*ArcSinh[(Sqrt[2]*S
qrt[(c*d^2 - b*d*e + a*e^2)/(-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])])/Sqrt[d +
e*x]], -((-2*c*d + b*e + Sqrt[(b^2 - 4*a*c)*e^2])/(2*c*d - b*e + Sqrt[(b^2 - 4*a
*c)*e^2]))]))/(Sqrt[2]*Sqrt[(c*d^2 + e*(-(b*d) + a*e))/(-2*c*d + b*e + Sqrt[(b^2
- 4*a*c)*e^2])]*Sqrt[d + e*x])))/(3*c^2*e^2*Sqrt[a + x*(b + c*x)]*Sqrt[((d + e*
x)^2*(c*(-1 + d/(d + e*x))^2 + (e*(b - (b*d)/(d + e*x) + (a*e)/(d + e*x)))/(d +
e*x)))/e^2])

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Maple [B]  time = 0.049, size = 3804, normalized size = 8.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(e*x+d)^(1/2)*(c*x^2+b*x+a)^(1/2)/c^2*(6*A*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^
2)^(1/2)+b*e-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c
+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*
c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2
),(-(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)))^(1/2))*a*
c*e^3-6*A*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-2*c*x+(
-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b
^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+
d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(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)))^(1/2))*b*c*d*e^2+6*A*2^(1/2)*(-(e*x+d)*c/(e*(-
4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e
*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2
)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*
d))^(1/2),(-(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)))^(
1/2))*c^2*d^2*e-6*A*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*
((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x
+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/
2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(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)))^(1/2))*a*c*e^3+6*A*2^(1/2)*(-(e*x+d)
*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c
*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b
^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+
b*e-2*c*d))^(1/2),(-(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)))^(1/2))*b*c*d*e^2-6*A*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)
)^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)
*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*Ellipti
cE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(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)))^(1/2))*c^2*d^2*e+B*2^(1/2)*(
-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b
)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(
-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2
)^(1/2)+b*e-2*c*d))^(1/2),(-(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)))^(1/2))*(-4*a*c+b^2)^(1/2)*a*e^3-B*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+
b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a
*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-
2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1
/2),(-(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)))^(1/2))*
(-4*a*c+b^2)^(1/2)*b*d*e^2+B*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d
))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2
)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*Ellipt
icF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(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)))^(1/2))*(-4*a*c+b^2)^(1/2)*c
*d^2*e-3*B*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-2*c*x+
(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+
b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x
+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(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)))^(1/2))*a*b*e^3+3*B*2^(1/2)*(-(e*x+d)*c/(e*(-4
*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*
(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)
+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d
))^(1/2),(-(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)))^(1
/2))*b^2*d*e^2-3*B*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(
(-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+
(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticF(2^(1/2
)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(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)))^(1/2))*b*c*d^2*e+4*B*2^(1/2)*(-(e*x+d
)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*
c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+
b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)
+b*e-2*c*d))^(1/2),(-(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)))^(1/2))*a*b*e^3-2*B*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))
^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*
((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*Elliptic
E(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(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)))^(1/2))*a*c*d*e^2-4*B*2^(1/2)*
(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-
b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*
(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^
2)^(1/2)+b*e-2*c*d))^(1/2),(-(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)))^(1/2))*b^2*d*e^2+6*B*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b
*e-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2
)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2
)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(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)))^(1/2))*b*c*d^2*e-2*
B*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*((-2*c*x+(-4*a*c+b
^2)^(1/2)-b)*e/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*((2*c*x+(-4*a*c+b^2)^(1/2
)+b)*e/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*EllipticE(2^(1/2)*(-(e*x+d)*c/(e*
(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2),(-(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)))^(1/2))*c^2*d^3+2*B*x^3*c^2*e^3+2*B*x^2*b*c*e^3+2*B*x^2
*c^2*d*e^2+2*B*x*a*c*e^3+2*B*x*b*c*d*e^2+2*B*a*c*d*e^2)/(c*e*x^3+b*e*x^2+c*d*x^2
+a*e*x+b*d*x+a*d)/e^2

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x + A\right )} \sqrt{e x + d}}{\sqrt{c x^{2} + b x + a}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((B*x + A)*sqrt(e*x + d)/sqrt(c*x^2 + b*x + a), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \sqrt{d + e x}}{\sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((A + B*x)*sqrt(d + e*x)/sqrt(a + b*x + c*x**2), x)

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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((B*x + A)*sqrt(e*x + d)/sqrt(c*x^2 + b*x + a),x, algorithm="giac")

[Out]

Timed out