3.2231 \(\int \sqrt{a+b \sqrt{x}} x^2 \, dx\)

Optimal. Leaf size=132 \[ -\frac{4 a^5 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^6}+\frac{4 a^4 \left (a+b \sqrt{x}\right )^{5/2}}{b^6}-\frac{40 a^3 \left (a+b \sqrt{x}\right )^{7/2}}{7 b^6}+\frac{40 a^2 \left (a+b \sqrt{x}\right )^{9/2}}{9 b^6}+\frac{4 \left (a+b \sqrt{x}\right )^{13/2}}{13 b^6}-\frac{20 a \left (a+b \sqrt{x}\right )^{11/2}}{11 b^6} \]

[Out]

(-4*a^5*(a + b*Sqrt[x])^(3/2))/(3*b^6) + (4*a^4*(a + b*Sqrt[x])^(5/2))/b^6 - (40
*a^3*(a + b*Sqrt[x])^(7/2))/(7*b^6) + (40*a^2*(a + b*Sqrt[x])^(9/2))/(9*b^6) - (
20*a*(a + b*Sqrt[x])^(11/2))/(11*b^6) + (4*(a + b*Sqrt[x])^(13/2))/(13*b^6)

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Rubi [A]  time = 0.148104, antiderivative size = 132, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{4 a^5 \left (a+b \sqrt{x}\right )^{3/2}}{3 b^6}+\frac{4 a^4 \left (a+b \sqrt{x}\right )^{5/2}}{b^6}-\frac{40 a^3 \left (a+b \sqrt{x}\right )^{7/2}}{7 b^6}+\frac{40 a^2 \left (a+b \sqrt{x}\right )^{9/2}}{9 b^6}+\frac{4 \left (a+b \sqrt{x}\right )^{13/2}}{13 b^6}-\frac{20 a \left (a+b \sqrt{x}\right )^{11/2}}{11 b^6} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a + b*Sqrt[x]]*x^2,x]

[Out]

(-4*a^5*(a + b*Sqrt[x])^(3/2))/(3*b^6) + (4*a^4*(a + b*Sqrt[x])^(5/2))/b^6 - (40
*a^3*(a + b*Sqrt[x])^(7/2))/(7*b^6) + (40*a^2*(a + b*Sqrt[x])^(9/2))/(9*b^6) - (
20*a*(a + b*Sqrt[x])^(11/2))/(11*b^6) + (4*(a + b*Sqrt[x])^(13/2))/(13*b^6)

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Rubi in Sympy [A]  time = 22.0667, size = 124, normalized size = 0.94 \[ - \frac{4 a^{5} \left (a + b \sqrt{x}\right )^{\frac{3}{2}}}{3 b^{6}} + \frac{4 a^{4} \left (a + b \sqrt{x}\right )^{\frac{5}{2}}}{b^{6}} - \frac{40 a^{3} \left (a + b \sqrt{x}\right )^{\frac{7}{2}}}{7 b^{6}} + \frac{40 a^{2} \left (a + b \sqrt{x}\right )^{\frac{9}{2}}}{9 b^{6}} - \frac{20 a \left (a + b \sqrt{x}\right )^{\frac{11}{2}}}{11 b^{6}} + \frac{4 \left (a + b \sqrt{x}\right )^{\frac{13}{2}}}{13 b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4*a**5*(a + b*sqrt(x))**(3/2)/(3*b**6) + 4*a**4*(a + b*sqrt(x))**(5/2)/b**6 - 4
0*a**3*(a + b*sqrt(x))**(7/2)/(7*b**6) + 40*a**2*(a + b*sqrt(x))**(9/2)/(9*b**6)
- 20*a*(a + b*sqrt(x))**(11/2)/(11*b**6) + 4*(a + b*sqrt(x))**(13/2)/(13*b**6)

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Mathematica [A]  time = 0.0329966, size = 89, normalized size = 0.67 \[ \frac{4 \sqrt{a+b \sqrt{x}} \left (-256 a^6+128 a^5 b \sqrt{x}-96 a^4 b^2 x+80 a^3 b^3 x^{3/2}-70 a^2 b^4 x^2+63 a b^5 x^{5/2}+693 b^6 x^3\right )}{9009 b^6} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a + b*Sqrt[x]]*x^2,x]

[Out]

(4*Sqrt[a + b*Sqrt[x]]*(-256*a^6 + 128*a^5*b*Sqrt[x] - 96*a^4*b^2*x + 80*a^3*b^3
*x^(3/2) - 70*a^2*b^4*x^2 + 63*a*b^5*x^(5/2) + 693*b^6*x^3))/(9009*b^6)

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Maple [A]  time = 0.008, size = 85, normalized size = 0.6 \[ 4\,{\frac{1}{{b}^{6}} \left ( 1/13\, \left ( a+b\sqrt{x} \right ) ^{13/2}-{\frac{5\,a \left ( a+b\sqrt{x} \right ) ^{11/2}}{11}}+{\frac{10\,{a}^{2} \left ( a+b\sqrt{x} \right ) ^{9/2}}{9}}-{\frac{10\,{a}^{3} \left ( a+b\sqrt{x} \right ) ^{7/2}}{7}}+ \left ( a+b\sqrt{x} \right ) ^{5/2}{a}^{4}-1/3\, \left ( a+b\sqrt{x} \right ) ^{3/2}{a}^{5} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(a+b*x^(1/2))^(1/2),x)

[Out]

4/b^6*(1/13*(a+b*x^(1/2))^(13/2)-5/11*a*(a+b*x^(1/2))^(11/2)+10/9*a^2*(a+b*x^(1/
2))^(9/2)-10/7*a^3*(a+b*x^(1/2))^(7/2)+(a+b*x^(1/2))^(5/2)*a^4-1/3*(a+b*x^(1/2))
^(3/2)*a^5)

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Maxima [A]  time = 1.44599, size = 132, normalized size = 1. \[ \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{13}{2}}}{13 \, b^{6}} - \frac{20 \,{\left (b \sqrt{x} + a\right )}^{\frac{11}{2}} a}{11 \, b^{6}} + \frac{40 \,{\left (b \sqrt{x} + a\right )}^{\frac{9}{2}} a^{2}}{9 \, b^{6}} - \frac{40 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} a^{3}}{7 \, b^{6}} + \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a^{4}}{b^{6}} - \frac{4 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{5}}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/13*(b*sqrt(x) + a)^(13/2)/b^6 - 20/11*(b*sqrt(x) + a)^(11/2)*a/b^6 + 40/9*(b*s
qrt(x) + a)^(9/2)*a^2/b^6 - 40/7*(b*sqrt(x) + a)^(7/2)*a^3/b^6 + 4*(b*sqrt(x) +
a)^(5/2)*a^4/b^6 - 4/3*(b*sqrt(x) + a)^(3/2)*a^5/b^6

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Fricas [A]  time = 0.251863, size = 104, normalized size = 0.79 \[ \frac{4 \,{\left (693 \, b^{6} x^{3} - 70 \, a^{2} b^{4} x^{2} - 96 \, a^{4} b^{2} x - 256 \, a^{6} +{\left (63 \, a b^{5} x^{2} + 80 \, a^{3} b^{3} x + 128 \, a^{5} b\right )} \sqrt{x}\right )} \sqrt{b \sqrt{x} + a}}{9009 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/9009*(693*b^6*x^3 - 70*a^2*b^4*x^2 - 96*a^4*b^2*x - 256*a^6 + (63*a*b^5*x^2 +
80*a^3*b^3*x + 128*a^5*b)*sqrt(x))*sqrt(b*sqrt(x) + a)/b^6

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Sympy [A]  time = 30.3945, size = 8588, normalized size = 65.06 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1024*a**(153/2)*x**18*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**
69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 122
97285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*
x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a
**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/
2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**
20*x**25 + 9009*a**55*b**21*x**(51/2)) + 1024*a**(153/2)*x**18/(9009*a**70*b**6*
x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b*
*9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 450
90045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*
x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a
**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2)
+ 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) - 14848*a**(151/2)*b*x*
*(37/2)*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/
2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**
10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 5797291
5*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(
45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a*
*58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009
*a**55*b**21*x**(51/2)) + 15360*a**(151/2)*b*x**(37/2)/(9009*a**70*b**6*x**18 +
135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(3
9/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a*
*64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 +
45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**
17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135
*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) - 100224*a**(149/2)*b**2*x**19*
sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945
945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20
+ 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*
b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) +
27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**1
8*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b
**21*x**(51/2)) + 107520*a**(149/2)*b**2*x**19/(9009*a**70*b**6*x**18 + 135135*a
**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 1
2297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**1
2*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045
*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(4
7/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b
**20*x**25 + 9009*a**55*b**21*x**(51/2)) - 417600*a**(147/2)*b**3*x**(39/2)*sqrt
(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*
a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 2
7054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**1
3*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 2705
4027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x*
*24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21
*x**(51/2)) + 465920*a**(147/2)*b**3*x**(39/2)/(9009*a**70*b**6*x**18 + 135135*a
**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 1
2297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**1
2*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045
*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(4
7/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b
**20*x**25 + 9009*a**55*b**21*x**(51/2)) - 1200600*a**(145/2)*b**4*x**20*sqrt(1
+ b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**
68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 2705
4027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x
**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 2705402
7*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24
+ 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x*
*(51/2)) + 1397760*a**(145/2)*b**4*x**20/(9009*a**70*b**6*x**18 + 135135*a**69*b
**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 1229728
5*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**2
1 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61
*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) +
4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x
**25 + 9009*a**55*b**21*x**(51/2)) - 2521260*a**(143/2)*b**5*x**(41/2)*sqrt(1 +
b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68
*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 270540
27*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**
(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*
a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 +
945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(
51/2)) + 3075072*a**(143/2)*b**5*x**(41/2)/(9009*a**70*b**6*x**18 + 135135*a**69
*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297
285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x*
*21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**
61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2)
+ 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20
*x**25 + 9009*a**55*b**21*x**(51/2)) - 3988992*a**(141/2)*b**6*x**21*sqrt(1 + b*
sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b
**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027
*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(4
3/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a*
*60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 9
45945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51
/2)) + 5125120*a**(141/2)*b**6*x**21/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*
x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a*
*66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 +
57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**
15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 409
9095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25
+ 9009*a**55*b**21*x**(51/2)) - 4802592*a**(139/2)*b**7*x**(43/2)*sqrt(1 + b*sq
rt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**
8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a
**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/
2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**6
0*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945
945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2
)) + 6589440*a**(139/2)*b**7*x**(43/2)/(9009*a**70*b**6*x**18 + 135135*a**69*b**
7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*
a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21
+ 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b
**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4
099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**
25 + 9009*a**55*b**21*x**(51/2)) - 4232160*a**(137/2)*b**8*x**22*sqrt(1 + b*sqrt
(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*
x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**
65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2)
+ 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*
b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 94594
5*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2))
+ 6589440*a**(137/2)*b**8*x**22/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(
37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*
b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 5797
2915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x
**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095
*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9
009*a**55*b**21*x**(51/2)) - 1935440*a**(135/2)*b**9*x**(45/2)*sqrt(1 + b*sqrt(x
)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x*
*19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65
*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) +
57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b*
*16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*
a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) +
5125120*a**(135/2)*b**9*x**(45/2)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x*
*(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**6
6*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57
972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15
*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 40990
95*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 +
9009*a**55*b**21*x**(51/2)) + 2214784*a**(133/2)*b**10*x**23*sqrt(1 + b*sqrt(x)
/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**
19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*
b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) +
57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**
16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a
**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) +
3075072*a**(133/2)*b**10*x**23/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37
/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b*
*10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 579729
15*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**
(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a
**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 900
9*a**55*b**21*x**(51/2)) + 8060832*a**(131/2)*b**11*x**(47/2)*sqrt(1 + b*sqrt(x)
/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**
19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*
b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) +
57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**
16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a
**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) +
1397760*a**(131/2)*b**11*x**(47/2)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x*
*(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**6
6*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57
972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15
*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 40990
95*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 +
9009*a**55*b**21*x**(51/2)) + 14375088*a**(129/2)*b**12*x**24*sqrt(1 + b*sqrt(x
)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x*
*19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65
*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) +
57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b*
*16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*
a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) +
465920*a**(129/2)*b**12*x**24/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37
/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b*
*10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 579729
15*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**
(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a
**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 900
9*a**55*b**21*x**(51/2)) + 18620280*a**(127/2)*b**13*x**(49/2)*sqrt(1 + b*sqrt(x
)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x*
*19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65
*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) +
57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b*
*16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*
a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) +
107520*a**(127/2)*b**13*x**(49/2)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x*
*(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**6
6*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57
972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15
*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 40990
95*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 +
9009*a**55*b**21*x**(51/2)) + 18558720*a**(125/2)*b**14*x**25*sqrt(1 + b*sqrt(x
)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x*
*19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65
*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) +
57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b*
*16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*
a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) +
15360*a**(125/2)*b**14*x**25/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/
2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**
10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 5797291
5*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(
45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a*
*58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009
*a**55*b**21*x**(51/2)) + 14360352*a**(123/2)*b**15*x**(51/2)*sqrt(1 + b*sqrt(x)
/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**
19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*
b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) +
57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**
16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a
**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) +
1024*a**(123/2)*b**15*x**(51/2)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(3
7/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b
**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972
915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x*
*(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*
a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 90
09*a**55*b**21*x**(51/2)) + 8569248*a**(121/2)*b**16*x**26*sqrt(1 + b*sqrt(x)/a)
/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19
+ 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**
11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 579
72915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*
x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**5
7*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 387
3456*a**(119/2)*b**17*x**(53/2)*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 1
35135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39
/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**
64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 4
5090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**1
7*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*
a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 1283840*a**(117/2)*b**18*x**27
*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 94
5945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**2
0 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63
*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) +
27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**
18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*
b**21*x**(51/2)) + 294560*a**(115/2)*b**19*x**(55/2)*sqrt(1 + b*sqrt(x)/a)/(9009
*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099
095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**
(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(43/2) + 57972915*
a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a**60*b**16*x**23
+ 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 + 945945*a**57*b**1
9*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(51/2)) + 41832*a**
(113/2)*b**20*x**28*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*
b**7*x**(37/2) + 945945*a**68*b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 122972
85*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**
21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**6
1*b**15*x**(45/2) + 27054027*a**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2)
+ 4099095*a**58*b**18*x**24 + 945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*
x**25 + 9009*a**55*b**21*x**(51/2)) + 2772*a**(111/2)*b**21*x**(57/2)*sqrt(1 + b
*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 135135*a**69*b**7*x**(37/2) + 945945*a**68*
b**8*x**19 + 4099095*a**67*b**9*x**(39/2) + 12297285*a**66*b**10*x**20 + 2705402
7*a**65*b**11*x**(41/2) + 45090045*a**64*b**12*x**21 + 57972915*a**63*b**13*x**(
43/2) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 27054027*a
**60*b**16*x**23 + 12297285*a**59*b**17*x**(47/2) + 4099095*a**58*b**18*x**24 +
945945*a**57*b**19*x**(49/2) + 135135*a**56*b**20*x**25 + 9009*a**55*b**21*x**(5
1/2))

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GIAC/XCAS [A]  time = 0.275925, size = 115, normalized size = 0.87 \[ \frac{4 \,{\left (693 \,{\left (b \sqrt{x} + a\right )}^{\frac{13}{2}} - 4095 \,{\left (b \sqrt{x} + a\right )}^{\frac{11}{2}} a + 10010 \,{\left (b \sqrt{x} + a\right )}^{\frac{9}{2}} a^{2} - 12870 \,{\left (b \sqrt{x} + a\right )}^{\frac{7}{2}} a^{3} + 9009 \,{\left (b \sqrt{x} + a\right )}^{\frac{5}{2}} a^{4} - 3003 \,{\left (b \sqrt{x} + a\right )}^{\frac{3}{2}} a^{5}\right )}}{9009 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/9009*(693*(b*sqrt(x) + a)^(13/2) - 4095*(b*sqrt(x) + a)^(11/2)*a + 10010*(b*sq
rt(x) + a)^(9/2)*a^2 - 12870*(b*sqrt(x) + a)^(7/2)*a^3 + 9009*(b*sqrt(x) + a)^(5
/2)*a^4 - 3003*(b*sqrt(x) + a)^(3/2)*a^5)/b^6