3.2439 \(\int \frac{\left (a+b x+c x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=515 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{5 c e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{16 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{5 c e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} (-7 b e+8 c d-6 c e x)}{5 e^3}-\frac{2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt{d+e x}} \]

[Out]

(-2*Sqrt[d + e*x]*(8*c*d - 7*b*e - 6*c*e*x)*Sqrt[a + b*x + c*x^2])/(5*e^3) - (2*
(a + b*x + c*x^2)^(3/2))/(e*Sqrt[d + e*x]) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(16*c^2*
d^2 + b^2*e^2 - 4*c*e*(4*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
)])/(5*c*e^4*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*
x + c*x^2]) - (16*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(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))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqr
t[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)])/(5*c*e^4*Sqrt[d + e*x]*Sqrt[a + b*x + c*x^2])

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Rubi [A]  time = 1.51238, antiderivative size = 515, 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 \[ \frac{\sqrt{2} \sqrt{b^2-4 a c} \sqrt{d+e x} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} \left (-4 c e (4 b d-3 a e)+b^2 e^2+16 c^2 d^2\right ) E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{5 c e^4 \sqrt{a+b x+c x^2} \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}}}-\frac{16 \sqrt{2} \sqrt{b^2-4 a c} \sqrt{-\frac{c \left (a+b x+c x^2\right )}{b^2-4 a c}} (2 c d-b e) \left (a e^2-b d e+c d^2\right ) \sqrt{\frac{c (d+e x)}{2 c d-e \left (\sqrt{b^2-4 a c}+b\right )}} F\left (\sin ^{-1}\left (\frac{\sqrt{\frac{b+2 c x+\sqrt{b^2-4 a c}}{\sqrt{b^2-4 a c}}}}{\sqrt{2}}\right )|-\frac{2 \sqrt{b^2-4 a c} e}{2 c d-\left (b+\sqrt{b^2-4 a c}\right ) e}\right )}{5 c e^4 \sqrt{d+e x} \sqrt{a+b x+c x^2}}-\frac{2 \sqrt{d+e x} \sqrt{a+b x+c x^2} (-7 b e+8 c d-6 c e x)}{5 e^3}-\frac{2 \left (a+b x+c x^2\right )^{3/2}}{e \sqrt{d+e x}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[d + e*x]*(8*c*d - 7*b*e - 6*c*e*x)*Sqrt[a + b*x + c*x^2])/(5*e^3) - (2*
(a + b*x + c*x^2)^(3/2))/(e*Sqrt[d + e*x]) + (Sqrt[2]*Sqrt[b^2 - 4*a*c]*(16*c^2*
d^2 + b^2*e^2 - 4*c*e*(4*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
)])/(5*c*e^4*Sqrt[(c*(d + e*x))/(2*c*d - (b + Sqrt[b^2 - 4*a*c])*e)]*Sqrt[a + b*
x + c*x^2]) - (16*Sqrt[2]*Sqrt[b^2 - 4*a*c]*(2*c*d - b*e)*(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))]*EllipticF[ArcSin[Sqrt[(b + Sqrt[b^2 - 4*a*c] + 2*c*x)/Sqr
t[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)])/(5*c*e^4*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((c*x**2+b*x+a)**(3/2)/(e*x+d)**(3/2),x)

[Out]

Timed out

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Mathematica [C]  time = 13.4388, size = 3427, normalized size = 6.65 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

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

[Out]

(Sqrt[d + e*x]*(a + x*(b + c*x))^(3/2)*((2*(-3*c*d + 2*b*e))/(5*e^3) + (2*c*x)/(
5*e^2) - (2*(c*d^2 - b*d*e + a*e^2))/(e^3*(d + e*x))))/(a + b*x + c*x^2) + ((a +
x*(b + c*x))^(3/2)*((2*(16*c^2*d^2 - 16*b*c*d*e + b^2*e^2 + 12*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]) - (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)]*(((4*I)*Sqrt[2]*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]
)]))/((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)]) - ((4*I)*Sqrt[2]*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])] - E
llipticF[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])]))/((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/2)*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 + Sqrt[b^2*e^2 - 4*a*c*e^2])*(d + e*x))]*(EllipticE[I*Arc
Sinh[(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)*Sqrt[2]*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]*Sqrt[-((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]))])/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])]))/((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)]) + ((8*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)]) - ((4*I)*Sq
rt[2]*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 + Sqr
t[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]))]*S
qrt[c + (c*d^2 - b*d*e + a*e^2)/(d + e*x)^2 + (-2*c*d + b*e)/(d + e*x)])))/(c*Sq
rt[((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])))/(5*e^5*(a + b*x + c*x^2)^(3/2))

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Maple [B]  time = 0.053, size = 4364, normalized size = 8.5 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/5*(c*x^2+b*x+a)^(1/2)*(e*x+d)^(1/2)*(-16*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^
4-17*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*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)*Ell
ipticF(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^2-28*
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+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)*b^2*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)*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
^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+32*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-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^4-2^(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+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)*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))*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)*E
llipticE(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^2-3*x*a*b*c*e^4-2*x*a*c^2*d*e^3+7*x*b^2*c*d*e^3-8*x*b*c^2*d^2*e^2-5*a^2
*c*e^4+x^4*c^3*e^4+5*x^2*b*c^2*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)*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^2*c*e^4
+7*a*b*c*d*e^3+3*x^3*b*c^2*e^4-2*x^3*c^3*d*e^3-4*x^2*a*c^2*e^4+2*x^2*b^2*c*e^4-8
*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+28*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+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)*Elli
pticF(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^3-12*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))*(-4*a*c+b^2)^(1/2)*b*c*d^2*e^2)/c/(
c*e*x^3+b*e*x^2+c*d*x^2+a*e*x+b*d*x+a*d)/e^5

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((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. \[{\rm integral}\left (\frac{{\left (c x^{2} + b x + a\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{3}{2}}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Timed out