3.635 \(\int x^2 \left (a+8 x-8 x^2+4 x^3-x^4\right )^{3/2} \, dx\)

Optimal. Leaf size=585 \[ \frac{4 \left (21 a^2+111 a+140\right ) \left (1-\sqrt{a+4}\right ) (x-1) \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right )}{315 \sqrt{a-(x-1)^4-2 (x-1)^2+3}}-\frac{4 \left (21 a^2+111 a+140\right ) \left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{315 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}}+\frac{3}{8} (a+4) \left ((x-1)^2+1\right ) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{1}{4} \left ((x-1)^2+1\right ) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{1}{63} \left (7 (x-1)^2+15\right ) (x-1) \left (a-(x-1)^4-2 (x-1)^2+3\right )^{3/2}+\frac{2}{315} (x-1) \left (3 (7 a+20) (x-1)^2+2 (27 a+80)\right ) \sqrt{a-(x-1)^4-2 (x-1)^2+3}+\frac{3}{8} (a+4)^2 \tan ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a-(x-1)^4-2 (x-1)^2+3}}\right )+\frac{4 (a+3) (33 a+100) \sqrt{\sqrt{a+4}+1} \left (\frac{(x-1)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{x-1}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{315 \sqrt{\frac{\frac{(x-1)^2}{1-\sqrt{a+4}}+1}{\frac{(x-1)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(x-1)^4-2 (x-1)^2+3}} \]

[Out]

(3*(4 + a)*(1 + (-1 + x)^2)*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4])/8 + ((1 + (
-1 + x)^2)*(3 + a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2))/4 + (4*(140 + 111*a + 21*a
^2)*(1 - Sqrt[4 + a])*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*(-1 + x))/(315*Sqrt[3 +
a - 2*(-1 + x)^2 - (-1 + x)^4]) + (2*(2*(80 + 27*a) + 3*(20 + 7*a)*(-1 + x)^2)*
Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]*(-1 + x))/315 + ((15 + 7*(-1 + x)^2)*(3
+ a - 2*(-1 + x)^2 - (-1 + x)^4)^(3/2)*(-1 + x))/63 + (3*(4 + a)^2*ArcTan[(1 + (
-1 + x)^2)/Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]])/8 - (4*(140 + 111*a + 21*a^
2)*(1 - Sqrt[4 + a])*Sqrt[1 + Sqrt[4 + a]]*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*El
lipticE[ArcTan[(-1 + x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a
])])/(315*Sqrt[(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 +
a]))]*Sqrt[3 + a - 2*(-1 + x)^2 - (-1 + x)^4]) + (4*(3 + a)*(100 + 33*a)*Sqrt[1
+ Sqrt[4 + a]]*(1 + (-1 + x)^2/(1 - Sqrt[4 + a]))*EllipticF[ArcTan[(-1 + x)/Sqrt
[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(315*Sqrt[(1 + (-1 + x)
^2/(1 - Sqrt[4 + a]))/(1 + (-1 + x)^2/(1 + Sqrt[4 + a]))]*Sqrt[3 + a - 2*(-1 + x
)^2 - (-1 + x)^4])

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Rubi [A]  time = 1.63525, antiderivative size = 585, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 13, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.464 \[ -\frac{4 \left (21 a^2+111 a+140\right ) \left (1-\sqrt{a+4}\right ) (1-x) \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right )}{315 \sqrt{a-(1-x)^4-2 (1-x)^2+3}}+\frac{4 \left (21 a^2+111 a+140\right ) \left (1-\sqrt{a+4}\right ) \sqrt{\sqrt{a+4}+1} \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right ) E\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{315 \sqrt{\frac{\frac{(1-x)^2}{1-\sqrt{a+4}}+1}{\frac{(1-x)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(1-x)^4-2 (1-x)^2+3}}+\frac{3}{8} (a+4) \left ((x-1)^2+1\right ) \sqrt{a-(1-x)^4-2 (1-x)^2+3}+\frac{1}{4} \left ((x-1)^2+1\right ) \left (a-(1-x)^4-2 (1-x)^2+3\right )^{3/2}-\frac{1}{63} \left (7 (1-x)^2+15\right ) (1-x) \left (a-(1-x)^4-2 (1-x)^2+3\right )^{3/2}-\frac{2}{315} (1-x) \left (3 (7 a+20) (1-x)^2+2 (27 a+80)\right ) \sqrt{a-(1-x)^4-2 (1-x)^2+3}+\frac{3}{8} (a+4)^2 \tan ^{-1}\left (\frac{(x-1)^2+1}{\sqrt{a-(1-x)^4-2 (1-x)^2+3}}\right )-\frac{4 (a+3) (33 a+100) \sqrt{\sqrt{a+4}+1} \left (\frac{(1-x)^2}{1-\sqrt{a+4}}+1\right ) F\left (\tan ^{-1}\left (\frac{1-x}{\sqrt{\sqrt{a+4}+1}}\right )|-\frac{2 \sqrt{a+4}}{1-\sqrt{a+4}}\right )}{315 \sqrt{\frac{\frac{(1-x)^2}{1-\sqrt{a+4}}+1}{\frac{(1-x)^2}{\sqrt{a+4}+1}+1}} \sqrt{a-(1-x)^4-2 (1-x)^2+3}} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]

[Out]

(3*(4 + a)*Sqrt[3 + a - 2*(1 - x)^2 - (1 - x)^4]*(1 + (-1 + x)^2))/8 + ((3 + a -
2*(1 - x)^2 - (1 - x)^4)^(3/2)*(1 + (-1 + x)^2))/4 - (4*(140 + 111*a + 21*a^2)*
(1 - Sqrt[4 + a])*(1 + (1 - x)^2/(1 - Sqrt[4 + a]))*(1 - x))/(315*Sqrt[3 + a - 2
*(1 - x)^2 - (1 - x)^4]) - (2*(2*(80 + 27*a) + 3*(20 + 7*a)*(1 - x)^2)*Sqrt[3 +
a - 2*(1 - x)^2 - (1 - x)^4]*(1 - x))/315 - ((15 + 7*(1 - x)^2)*(3 + a - 2*(1 -
x)^2 - (1 - x)^4)^(3/2)*(1 - x))/63 + (3*(4 + a)^2*ArcTan[(1 + (-1 + x)^2)/Sqrt[
3 + a - 2*(1 - x)^2 - (1 - x)^4]])/8 + (4*(140 + 111*a + 21*a^2)*(1 - Sqrt[4 + a
])*Sqrt[1 + Sqrt[4 + a]]*(1 + (1 - x)^2/(1 - Sqrt[4 + a]))*EllipticE[ArcTan[(1 -
x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sqrt[4 + a])/(1 - Sqrt[4 + a])])/(315*Sqrt[(1 +
(1 - x)^2/(1 - Sqrt[4 + a]))/(1 + (1 - x)^2/(1 + Sqrt[4 + a]))]*Sqrt[3 + a - 2*(
1 - x)^2 - (1 - x)^4]) - (4*(3 + a)*(100 + 33*a)*Sqrt[1 + Sqrt[4 + a]]*(1 + (1 -
x)^2/(1 - Sqrt[4 + a]))*EllipticF[ArcTan[(1 - x)/Sqrt[1 + Sqrt[4 + a]]], (-2*Sq
rt[4 + a])/(1 - Sqrt[4 + a])])/(315*Sqrt[(1 + (1 - x)^2/(1 - Sqrt[4 + a]))/(1 +
(1 - x)^2/(1 + Sqrt[4 + a]))]*Sqrt[3 + a - 2*(1 - x)^2 - (1 - x)^4])

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Rubi in Sympy [A]  time = 84.6116, size = 498, normalized size = 0.85 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(-x**4+4*x**3-8*x**2+a+8*x)**(3/2),x)

[Out]

(3*a/16 + 3/4)*(2*(x - 1)**2 + 2)*sqrt(a - (x - 1)**4 - 2*(x - 1)**2 + 3) + 3*(a
+ 4)**2*atan(-(-2*(x - 1)**2 - 2)/(2*sqrt(a - (x - 1)**4 - 2*(x - 1)**2 + 3)))/
8 + 4*(x - 1)*((x - 1)**2/(-sqrt(a + 4) + 1) + 1)*(-sqrt(a + 4) + 1)*(21*a**2 +
111*a + 140)/(315*sqrt(a - (x - 1)**4 - 2*(x - 1)**2 + 3)) + (x - 1)*(7*(x - 1)*
*2 + 15)*(a - (x - 1)**4 - 2*(x - 1)**2 + 3)**(3/2)/63 + (x - 1)*(108*a + (42*a
+ 120)*(x - 1)**2 + 320)*sqrt(a - (x - 1)**4 - 2*(x - 1)**2 + 3)/315 + (2*(x - 1
)**2 + 2)*(a - (x - 1)**4 - 2*(x - 1)**2 + 3)**(3/2)/8 + 4*(a + 3)*(33*a + 100)*
((x - 1)**2/(-sqrt(a + 4) + 1) + 1)*sqrt(sqrt(a + 4) + 1)*elliptic_f(atan((x - 1
)/sqrt(sqrt(a + 4) + 1)), 2*sqrt(a + 4)/(sqrt(a + 4) - 1))/(315*sqrt((-(x - 1)**
2/(sqrt(a + 4) - 1) + 1)/((x - 1)**2/(sqrt(a + 4) + 1) + 1))*sqrt(a - (x - 1)**4
- 2*(x - 1)**2 + 3)) - 4*((x - 1)**2/(-sqrt(a + 4) + 1) + 1)*(-sqrt(a + 4) + 1)
*sqrt(sqrt(a + 4) + 1)*(21*a**2 + 111*a + 140)*elliptic_e(atan((x - 1)/sqrt(sqrt
(a + 4) + 1)), 2*sqrt(a + 4)/(sqrt(a + 4) - 1))/(315*sqrt((-(x - 1)**2/(sqrt(a +
4) - 1) + 1)/((x - 1)**2/(sqrt(a + 4) + 1) + 1))*sqrt(a - (x - 1)**4 - 2*(x - 1
)**2 + 3))

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Mathematica [B]  time = 6.24083, size = 8500, normalized size = 14.53 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^2*(a + 8*x - 8*x^2 + 4*x^3 - x^4)^(3/2),x]

[Out]

Result too large to show

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Maple [B]  time = 0.033, size = 2733, normalized size = 4.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(-x^4+4*x^3-8*x^2+a+8*x)^(3/2),x)

[Out]

-1/9*x^7*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+19/36*x^6*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)
-163/126*x^5*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+71/42*x^4*(-x^4+4*x^3-8*x^2+a+8*x)^(
1/2)+(11/45*a-16/63)*x^3*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(-13/120*a-5/18)*x^2*(-x
^4+4*x^3-8*x^2+a+8*x)^(1/2)+(9/140*a+23/63)*x*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)+(10
7/252*a+101/63)*(-x^4+4*x^3-8*x^2+a+8*x)^(1/2)-(-(9/140*a+23/63)*a-107/63*a-404/
63)*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*((-(-1-(4+a)^(1/2))^(1/2)+(-
1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+
(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(x-1+(-1+(4+a)^(1/2))^(1
/2))^2*(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1-(-1-(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))
^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(-2*(-1+(4+a)
^(1/2))^(1/2)*(x-1+(-1-(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1
/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+
a)^(1/2))^(1/2))/(-1+(4+a)^(1/2))^(1/2)/(-(x-1-(-1+(4+a)^(1/2))^(1/2))*(x-1+(-1+
(4+a)^(1/2))^(1/2))*(x-1-(-1-(4+a)^(1/2))^(1/2))*(x-1+(-1-(4+a)^(1/2))^(1/2)))^(
1/2)*EllipticF(((-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^
(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1
/2))^(1/2)))^(1/2),((-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))*((-1-(4+a)^
(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(
1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)))^(1/2))-(-2*(-13/120*a-5/1
8)*a+827/315*a+76/9)*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*((-(-1-(4+a
)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^
(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(x-1+(-
1+(4+a)^(1/2))^(1/2))^2*(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1-(-1-(4+a)^(1/2))^(1/2))/
((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1
/2)*(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1+(-1-(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(
1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)/(-(-1-(4+a)^(1/
2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-1+(4+a)^(1/2))^(1/2)/(-(x-1-(-1+(4+a)^(1/2))
^(1/2))*(x-1+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1-(4+a)^(1/2))^(1/2))*(x-1+(-1-(4+a)
^(1/2))^(1/2)))^(1/2)*((1-(-1+(4+a)^(1/2))^(1/2))*EllipticF(((-(-1-(4+a)^(1/2))^
(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1
/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2),((-(-1-(4+a)^(1/
2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2)
)/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(
4+a)^(1/2))^(1/2)))^(1/2))+2*(-1+(4+a)^(1/2))^(1/2)*EllipticPi(((-(-1-(4+a)^(1/2
))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))
^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2),(-(-1-(4+a)^(
1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1
/2)),((-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(
-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+
a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)))^(1/2)))+(a^2-3*(11/45*a-16/63)*a+68/105
*a+16/9)*((x-1-(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1-(4+a)^(1/2))^(1/2))*(x-1+(-1-(4+
a)^(1/2))^(1/2))+((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*((-(-1-(4+a)^(1
/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2
))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(x-1+(-1+(4
+a)^(1/2))^(1/2))^2*(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1-(-1-(4+a)^(1/2))^(1/2))/((-1
-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*
(-2*(-1+(4+a)^(1/2))^(1/2)*(x-1+(-1-(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)
-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2)*(-1/2*((1-(-1+(4+a)
^(1/2))^(1/2))*(1+(-1+(4+a)^(1/2))^(1/2))-(1-(-1-(4+a)^(1/2))^(1/2))*(1+(-1+(4+a
)^(1/2))^(1/2))+(1-(-1-(4+a)^(1/2))^(1/2))*(1-(-1+(4+a)^(1/2))^(1/2))+(1-(-1+(4+
a)^(1/2))^(1/2))^2)/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-1+(4+a)^(
1/2))^(1/2)*EllipticF(((-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1
+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1+(
4+a)^(1/2))^(1/2)))^(1/2),((-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))*((-1
-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(
1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)))^(1/2))-1/2*(-(-1-(
4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*EllipticE(((-(-1-(4+a)^(1/2))^(1/2)+(-
1+(4+a)^(1/2))^(1/2))*(x-1-(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+
(4+a)^(1/2))^(1/2))/(x-1+(-1+(4+a)^(1/2))^(1/2)))^(1/2),((-(-1-(4+a)^(1/2))^(1/2
)-(-1+(4+a)^(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-(-1-
(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/
2))^(1/2)))^(1/2))/(-1+(4+a)^(1/2))^(1/2)-4/(-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(
1/2))^(1/2))*EllipticPi(((-(-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))*(x-1-(
-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2))/(x-1+(-1
+(4+a)^(1/2))^(1/2)))^(1/2),((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1
-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)),((-(-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^
(1/2))^(1/2))*((-1-(4+a)^(1/2))^(1/2)+(-1+(4+a)^(1/2))^(1/2))/(-(-1-(4+a)^(1/2))
^(1/2)+(-1+(4+a)^(1/2))^(1/2))/((-1-(4+a)^(1/2))^(1/2)-(-1+(4+a)^(1/2))^(1/2)))^
(1/2))))/(-(x-1-(-1+(4+a)^(1/2))^(1/2))*(x-1+(-1+(4+a)^(1/2))^(1/2))*(x-1-(-1-(4
+a)^(1/2))^(1/2))*(x-1+(-1-(4+a)^(1/2))^(1/2)))^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac{3}{2}} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(3/2)*x^2,x, algorithm="maxima")

[Out]

integrate((-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(3/2)*x^2, x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (-{\left (x^{6} - 4 \, x^{5} + 8 \, x^{4} - a x^{2} - 8 \, x^{3}\right )} \sqrt{-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(3/2)*x^2,x, algorithm="fricas")

[Out]

integral(-(x^6 - 4*x^5 + 8*x^4 - a*x^2 - 8*x^3)*sqrt(-x^4 + 4*x^3 - 8*x^2 + a +
8*x), x)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x^{2} \left (a - x^{4} + 4 x^{3} - 8 x^{2} + 8 x\right )^{\frac{3}{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(-x**4+4*x**3-8*x**2+a+8*x)**(3/2),x)

[Out]

Integral(x**2*(a - x**4 + 4*x**3 - 8*x**2 + 8*x)**(3/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int{\left (-x^{4} + 4 \, x^{3} - 8 \, x^{2} + a + 8 \, x\right )}^{\frac{3}{2}} x^{2}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(3/2)*x^2,x, algorithm="giac")

[Out]

integrate((-x^4 + 4*x^3 - 8*x^2 + a + 8*x)^(3/2)*x^2, x)