3.2463 \(\int \frac{1}{(d+e x)^{3/2} \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=607 \[ -\frac{4 e \sqrt{a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]) - (4*e*(c^2*d^2 + b^2*e^2 - c*e*(
b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*Sq
rt[d + e*x]) + (2*Sqrt[2]*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*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]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d
- (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)^2*Sqrt
[(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]*(2*c*d - b*e)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sq
rt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[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)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*Sqrt[d +
e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 1.64154, antiderivative size = 607, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ -\frac{4 e \sqrt{a+b x+c x^2} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \sqrt{2} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )^2 \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{2 \left (2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d\right )}{\left (b^2-4 a c\right ) \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{2 \sqrt{2} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{\sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(-2*(b*c*d - b^2*e + 2*a*c*e + c*(2*c*d - b*e)*x))/((b^2 - 4*a*c)*(c*d^2 - b*d*e
+ a*e^2)*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2]) - (4*e*(c^2*d^2 + b^2*e^2 - c*e*(
b*d + 3*a*e))*Sqrt[a + b*x + c*x^2])/((b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)^2*Sq
rt[d + e*x]) + (2*Sqrt[2]*(c^2*d^2 + b^2*e^2 - c*e*(b*d + 3*a*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]]/Sqrt[2]], (-2*Sqrt[b^2 - 4*a*c]*e)/(2*c*d
- (b + Sqrt[b^2 - 4*a*c])*e)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)^2*Sqrt
[(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]*(2*c*d - b*e)*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sq
rt[-((c*(a + b*x + c*x^2))/(b^2 - 4*a*c))]*EllipticF[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)])/(Sqrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)*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(1/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 13.1457, size = 3554, normalized size = 5.86 \[ \text{Result too large to show} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(a + b*x + c*x^2)^(3/2)),x]

[Out]

(Sqrt[d + e*x]*(a + b*x + c*x^2)^2*((-2*e^3)/((c*d^2 - b*d*e + a*e^2)^2*(d + e*x
)) - (2*(-(b*c^2*d^2) + 2*b^2*c*d*e - 4*a*c^2*d*e - b^3*e^2 + 3*a*b*c*e^2 - 2*c^
3*d^2*x + 2*b*c^2*d*e*x - b^2*c*e^2*x + 2*a*c^2*e^2*x))/((-b^2 + 4*a*c)*(c*d^2 -
b*d*e + a*e^2)^2*(a + b*x + c*x^2))))/(a + x*(b + c*x))^(3/2) - (2*c*(a + b*x +
c*x^2)^(3/2)*((2*(c^2*d^2 - b*c*d*e + b^2*e^2 - 3*a*c*e^2)*(d + e*x)^(3/2)*(c +
(c*d^2)/(d + e*x)^2 - (b*d*e)/(d + e*x)^2 + (a*e^2)/(d + e*x)^2 - (2*c*d)/(d +
e*x) + (b*e)/(d + e*x)))/(c*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]) - ((c*d^2 - b*d*e + a*e^2)
*(d + e*x)*Sqrt[c + (c*d^2)/(d + e*x)^2 - (b*d*e)/(d + e*x)^2 + (a*e^2)/(d + e*x
)^2 - (2*c*d)/(d + e*x) + (b*e)/(d + e*x)]*((I*c^2*d^2*(2*c*d - b*e + Sqrt[b^2*e
^2 - 4*a*c*e^2])*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e
^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e
+ Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c
*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]
], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c
*e^2])] - EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b
*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 -
4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])]))/(Sqrt[2]*(c*d^2 - b*d*
e + a*e^2)*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^
2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)])
- (I*b*c*d*e*(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*Sqrt[1 - (2*(c*d^2 - b*d*
e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(
c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*(
EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[
b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2
])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])] - EllipticF[I*ArcSinh[(Sqrt[2]*Sqr
t[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d
+ e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 -
4*a*c*e^2])]))/(Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*Sqrt[-((c*d^2 - b*d*e + a*e^2)/
(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d
+ e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) + (I*b^2*e^2*(2*c*d - b*e + Sqrt[b^2*e^2 -
4*a*c*e^2])*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 -
4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sq
rt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*(EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2
- b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (
2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2
])] - EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e -
Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a
*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])]))/(Sqrt[2]*(c*d^2 - b*d*e +
a*e^2)*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))
]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) - ((
3*I)*a*c*e^2*(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*Sqrt[1 - (2*(c*d^2 - b*d*
e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(
c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*(
EllipticE[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[
b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2
])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])] - EllipticF[I*ArcSinh[(Sqrt[2]*Sqr
t[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d
+ e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c*d - b*e + Sqrt[b^2*e^2 -
4*a*c*e^2])]))/(Sqrt[2]*(c*d^2 - b*d*e + a*e^2)*Sqrt[-((c*d^2 - b*d*e + a*e^2)/
(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d
+ e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) + (I*Sqrt[2]*c^2*d*Sqrt[1 - (2*(c*d^2 - b*
d*e + a*e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2
*(c*d^2 - b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]
*EllipticF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt
[b^2*e^2 - 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^
2])/(2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])])/(Sqrt[-((c*d^2 - b*d*e + a*e^2)/
(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d
+ e*x)^2 + (-2*c*d + b*e)/(d + e*x)]) - (I*b*c*e*Sqrt[1 - (2*(c*d^2 - b*d*e + a*
e^2))/((2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Sqrt[1 - (2*(c*d^2
- b*d*e + a*e^2))/((2*c*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*Ellipti
cF[I*ArcSinh[(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/(2*c*d - b*e - Sqrt[b^2*e^2
- 4*a*c*e^2]))])/Sqrt[d + e*x]], (2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2])/(2*c
*d - b*e + Sqrt[b^2*e^2 - 4*a*c*e^2])])/(Sqrt[2]*Sqrt[-((c*d^2 - b*d*e + a*e^2)/
(2*c*d - b*e - Sqrt[b^2*e^2 - 4*a*c*e^2]))]*Sqrt[c + (c*d^2 - b*d*e + a*e^2)/(d
+ e*x)^2 + (-2*c*d + b*e)/(d + e*x)])))/(c*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])))/((-b^2 +
4*a*c)*e*(c*d^2 - b*d*e + a*e^2)^2*(a + x*(b + c*x))^(3/2))

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Maple [B]  time = 0.131, size = 4415, normalized size = 7.3 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(c*x^2+b*x+a)^(3/2),x)

[Out]

(4*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*
a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)
^(1/2))/(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^3*d^4+8*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^
(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^
2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(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*c
*d^2*e^2+4*x*b^3*e^4+2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)
*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+
2*c*x+(-4*a*c+b^2)^(1/2))/(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*b*e^4-3
*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*
c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(
1/2))/(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*c*d^2*e^2-8*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b
^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*
c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(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^2*d^2*e^2-2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-
2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-
4*a*c+b^2)^(1/2))/(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^2*d*e^3-2*2^(1/
2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)
^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/
(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)*c^2*d^3*e+12*2^(1/2)*(-(e*x+d)*c/(e
*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*
e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(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^2*d^2*e^2-8*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(
1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e
*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(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^2*d^3*e+2*d^3*e*b*c^
2-3*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4
*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2
)^(1/2))/(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^2*e^4+4*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^
2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c
+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(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^2*e^4-4*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c
*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a
*c+b^2)^(1/2))/(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^3*d*e^3+3*2^(1/2)*(-(e*x+d)*c/(e*(-4*
a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(
-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(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^3*d*e^3-12*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*
(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*
x+(-4*a*c+b^2)^(1/2))/(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^2*c*e^4+8*a*c^2*d^2*e^2-14*x*a
*b*c*e^4+4*x*a*c^2*d*e^3-2*x*b^2*c*d*e^3-2*x*b*c^2*d^2*e^2-8*a^2*c*e^4-4*x^2*b*c
^2*d*e^3-4*b^2*c*d^2*e^2+12*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d)
)^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)
*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*Ellipti
cF(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^2*c*e^4+4*x*c^3*d^3
*e-6*a*b*c*d*e^3-12*x^2*a*c^2*e^4+4*x^2*b^2*c*e^4+4*x^2*c^3*d^2*e^2-12*2^(1/2)*(
-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/
2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(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*c*d*e^3+8*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*
e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)
))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(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*c*d*e^3-2
*2^(1/2)*(-(e*x+d)*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*
c+b^2)^(1/2))/(2*c*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(
1/2))/(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*c*d*e^3+3*2^(1/2)*(-(e*x+d)
*c/(e*(-4*a*c+b^2)^(1/2)+b*e-2*c*d))^(1/2)*(e*(-b-2*c*x+(-4*a*c+b^2)^(1/2))/(2*c
*d-b*e+e*(-4*a*c+b^2)^(1/2)))^(1/2)*(e*(b+2*c*x+(-4*a*c+b^2)^(1/2))/(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*c*d^2*e^2+2*a*b^2*e^4+2*b^3*d*e^3)*(c*x^2+b*x
+a)^(1/2)*(e*x+d)^(1/2)/e/(4*a*c-b^2)/(a*e^2-b*d*e+c*d^2)^2/(c*e*x^3+b*e*x^2+c*d
*x^2+a*e*x+b*d*x+a*d)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}{\left (e x + d\right )}^{\frac{3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^(3/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{1}{{\left (c e x^{3} +{\left (c d + b e\right )} x^{2} + a d +{\left (b d + a e\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{e x + d}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

integral(1/((c*e*x^3 + (c*d + b*e)*x^2 + a*d + (b*d + a*e)*x)*sqrt(c*x^2 + b*x +
a)*sqrt(e*x + d)), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right )^{\frac{3}{2}} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(c*x**2+b*x+a)**(3/2),x)

[Out]

Integral(1/((d + e*x)**(3/2)*(a + b*x + c*x**2)**(3/2)), 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(1/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Timed out