\(\int \sqrt {a+b \sqrt {x}} x^2 \, dx\) [2233]
Optimal result
Integrand size = 17, antiderivative size = 132 \[
\int \sqrt {a+b \sqrt {x}} x^2 \, dx=-\frac {4 a^5 \left (a+b \sqrt {x}\right )^{3/2}}{3 b^6}+\frac {4 a^4 \left (a+b \sqrt {x}\right )^{5/2}}{b^6}-\frac {40 a^3 \left (a+b \sqrt {x}\right )^{7/2}}{7 b^6}+\frac {40 a^2 \left (a+b \sqrt {x}\right )^{9/2}}{9 b^6}-\frac {20 a \left (a+b \sqrt {x}\right )^{11/2}}{11 b^6}+\frac {4 \left (a+b \sqrt {x}\right )^{13/2}}{13 b^6}
\]
[Out]
-4/3*a^5*(a+b*x^(1/2))^(3/2)/b^6+4*a^4*(a+b*x^(1/2))^(5/2)/b^6-40/7*a^3*(a+b*x^(1/2))^(7/2)/b^6+40/9*a^2*(a+b*
x^(1/2))^(9/2)/b^6-20/11*a*(a+b*x^(1/2))^(11/2)/b^6+4/13*(a+b*x^(1/2))^(13/2)/b^6
Rubi [A] (verified)
Time = 0.05 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.00,
number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {272, 45}
\[
\int \sqrt {a+b \sqrt {x}} x^2 \, dx=-\frac {4 a^5 \left (a+b \sqrt {x}\right )^{3/2}}{3 b^6}+\frac {4 a^4 \left (a+b \sqrt {x}\right )^{5/2}}{b^6}-\frac {40 a^3 \left (a+b \sqrt {x}\right )^{7/2}}{7 b^6}+\frac {40 a^2 \left (a+b \sqrt {x}\right )^{9/2}}{9 b^6}+\frac {4 \left (a+b \sqrt {x}\right )^{13/2}}{13 b^6}-\frac {20 a \left (a+b \sqrt {x}\right )^{11/2}}{11 b^6}
\]
[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)
Rule 45
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Rule 272
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]
Rubi steps \begin{align*}
\text {integral}& = 2 \text {Subst}\left (\int x^5 \sqrt {a+b x} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {Subst}\left (\int \left (-\frac {a^5 \sqrt {a+b x}}{b^5}+\frac {5 a^4 (a+b x)^{3/2}}{b^5}-\frac {10 a^3 (a+b x)^{5/2}}{b^5}+\frac {10 a^2 (a+b x)^{7/2}}{b^5}-\frac {5 a (a+b x)^{9/2}}{b^5}+\frac {(a+b x)^{11/2}}{b^5}\right ) \, dx,x,\sqrt {x}\right ) \\ & = -\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 {20 a \left (a+b \sqrt {x}\right )^{11/2}}{11 b^6}+\frac {4 \left (a+b \sqrt {x}\right )^{13/2}}{13 b^6} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.59
\[
\int \sqrt {a+b \sqrt {x}} x^2 \, dx=\frac {4 \left (a+b \sqrt {x}\right )^{3/2} \left (-256 a^5+384 a^4 b \sqrt {x}-480 a^3 b^2 x+560 a^2 b^3 x^{3/2}-630 a b^4 x^2+693 b^5 x^{5/2}\right )}{9009 b^6}
\]
[In]
Integrate[Sqrt[a + b*Sqrt[x]]*x^2,x]
[Out]
(4*(a + b*Sqrt[x])^(3/2)*(-256*a^5 + 384*a^4*b*Sqrt[x] - 480*a^3*b^2*x + 560*a^2*b^3*x^(3/2) - 630*a*b^4*x^2 +
693*b^5*x^(5/2)))/(9009*b^6)
Maple [A] (verified)
Time = 3.67 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.64
| | |
method | result | size |
| | |
derivativedivides |
\(\frac {\frac {4 \left (a +b \sqrt {x}\right )^{\frac {13}{2}}}{13}-\frac {20 a \left (a +b \sqrt {x}\right )^{\frac {11}{2}}}{11}+\frac {40 a^{2} \left (a +b \sqrt {x}\right )^{\frac {9}{2}}}{9}-\frac {40 a^{3} \left (a +b \sqrt {x}\right )^{\frac {7}{2}}}{7}+4 a^{4} \left (a +b \sqrt {x}\right )^{\frac {5}{2}}-\frac {4 a^{5} \left (a +b \sqrt {x}\right )^{\frac {3}{2}}}{3}}{b^{6}}\) |
\(85\) |
default |
\(\frac {\frac {4 \left (a +b \sqrt {x}\right )^{\frac {13}{2}}}{13}-\frac {20 a \left (a +b \sqrt {x}\right )^{\frac {11}{2}}}{11}+\frac {40 a^{2} \left (a +b \sqrt {x}\right )^{\frac {9}{2}}}{9}-\frac {40 a^{3} \left (a +b \sqrt {x}\right )^{\frac {7}{2}}}{7}+4 a^{4} \left (a +b \sqrt {x}\right )^{\frac {5}{2}}-\frac {4 a^{5} \left (a +b \sqrt {x}\right )^{\frac {3}{2}}}{3}}{b^{6}}\) |
\(85\) |
| | |
|
|
|
[In]
int(x^2*(a+b*x^(1/2))^(1/2),x,method=_RETURNVERBOSE)
[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^4*(a+b*x^(1/2))^(5/2)-1/3*a^5*(a+b*x^(1/2))^(3/2))
Fricas [A] (verification not implemented)
none
Time = 0.27 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.58
\[
\int \sqrt {a+b \sqrt {x}} x^2 \, dx=\frac {4 \, {\left (693 \, b^{6} x^{3} - 70 \, a^{2} b^{4} x^{2} - 96 \, a^{4} b^{2} x - 256 \, a^{6} + {\left (63 \, a b^{5} x^{2} + 80 \, a^{3} b^{3} x + 128 \, a^{5} b\right )} \sqrt {x}\right )} \sqrt {b \sqrt {x} + a}}{9009 \, b^{6}}
\]
[In]
integrate(x^2*(a+b*x^(1/2))^(1/2),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
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 8588 vs. \(2 (124) = 248\).
Time = 8.72 (sec) , antiderivative size = 8588, normalized size of antiderivative = 65.06
\[
\int \sqrt {a+b \sqrt {x}} x^2 \, dx=\text {Too large to display}
\]
[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) + 12297285*a**66*b**10*x**20 + 27054027*a**65*b**11*x**(41/2) + 4
5090045*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) + 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 + 9
009*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 + 13513
5*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**(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**(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)) - 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) + 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 + 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) + 12297285*a**66*b**10*x**20 + 27
054027*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))
- 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 + 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 + 4509004
5*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**(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**6
7*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) + 1351
35*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) + 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)) + 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) + 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 + 4509
0045*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**(1
43/2)*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 + 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)) + 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) + 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)) - 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**(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**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) + 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)) - 4802592*a**(139/2)*b**7*x**
(43/2)*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) + 270
54027*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)) + 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) + 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)) - 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 + 27
054027*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))
+ 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**1
9 + 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**6
4*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)) - 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 + 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) + 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**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)) + 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 + 4
5090045*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**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)) + 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**6
3*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**2
0*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**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**1
4*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 + 9009*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) + 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)) + 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**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)) + 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) + 4
5090045*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 + 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 + 122
97285*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**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)) + 1024*a**(123/2)*b**15*x**(51/2)/(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**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)) + 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) + 57972915*a**62*b**14*x**22 + 45090045*a**61*b**15*x**(45/2) + 2
7054027*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)) + 3873456*a**(119/2)*b**17*x**(53/2)*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 + 579
72915*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)) + 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) + 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 + 9
009*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 + 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)) + 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**(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 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 450
90045*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)) + 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 + 27054027*a**65*b**11*x**(41/2) + 4509
0045*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))
Maxima [A] (verification not implemented)
none
Time = 0.19 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.74
\[
\int \sqrt {a+b \sqrt {x}} x^2 \, dx=\frac {4 \, {\left (b \sqrt {x} + a\right )}^{\frac {13}{2}}}{13 \, b^{6}} - \frac {20 \, {\left (b \sqrt {x} + a\right )}^{\frac {11}{2}} a}{11 \, b^{6}} + \frac {40 \, {\left (b \sqrt {x} + a\right )}^{\frac {9}{2}} a^{2}}{9 \, b^{6}} - \frac {40 \, {\left (b \sqrt {x} + a\right )}^{\frac {7}{2}} a^{3}}{7 \, b^{6}} + \frac {4 \, {\left (b \sqrt {x} + a\right )}^{\frac {5}{2}} a^{4}}{b^{6}} - \frac {4 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} a^{5}}{3 \, b^{6}}
\]
[In]
integrate(x^2*(a+b*x^(1/2))^(1/2),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*sqrt(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
Giac [A] (verification not implemented)
none
Time = 0.29 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.45
\[
\int \sqrt {a+b \sqrt {x}} x^2 \, dx=\frac {4 \, {\left (\frac {13 \, {\left (63 \, {\left (b \sqrt {x} + a\right )}^{\frac {11}{2}} - 385 \, {\left (b \sqrt {x} + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b \sqrt {x} + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b \sqrt {x} + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b \sqrt {x} + a} a^{5}\right )} a}{b^{5}} + \frac {3 \, {\left (231 \, {\left (b \sqrt {x} + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b \sqrt {x} + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b \sqrt {x} + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b \sqrt {x} + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b \sqrt {x} + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b \sqrt {x} + a} a^{6}\right )}}{b^{5}}\right )}}{9009 \, b}
\]
[In]
integrate(x^2*(a+b*x^(1/2))^(1/2),x, algorithm="giac")
[Out]
4/9009*(13*(63*(b*sqrt(x) + a)^(11/2) - 385*(b*sqrt(x) + a)^(9/2)*a + 990*(b*sqrt(x) + a)^(7/2)*a^2 - 1386*(b*
sqrt(x) + a)^(5/2)*a^3 + 1155*(b*sqrt(x) + a)^(3/2)*a^4 - 693*sqrt(b*sqrt(x) + a)*a^5)*a/b^5 + 3*(231*(b*sqrt(
x) + a)^(13/2) - 1638*(b*sqrt(x) + a)^(11/2)*a + 5005*(b*sqrt(x) + a)^(9/2)*a^2 - 8580*(b*sqrt(x) + a)^(7/2)*a
^3 + 9009*(b*sqrt(x) + a)^(5/2)*a^4 - 6006*(b*sqrt(x) + a)^(3/2)*a^5 + 3003*sqrt(b*sqrt(x) + a)*a^6)/b^5)/b
Mupad [B] (verification not implemented)
Time = 6.60 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.74
\[
\int \sqrt {a+b \sqrt {x}} x^2 \, dx=\frac {4\,{\left (a+b\,\sqrt {x}\right )}^{13/2}}{13\,b^6}-\frac {20\,a\,{\left (a+b\,\sqrt {x}\right )}^{11/2}}{11\,b^6}-\frac {4\,a^5\,{\left (a+b\,\sqrt {x}\right )}^{3/2}}{3\,b^6}+\frac {4\,a^4\,{\left (a+b\,\sqrt {x}\right )}^{5/2}}{b^6}-\frac {40\,a^3\,{\left (a+b\,\sqrt {x}\right )}^{7/2}}{7\,b^6}+\frac {40\,a^2\,{\left (a+b\,\sqrt {x}\right )}^{9/2}}{9\,b^6}
\]
[In]
int(x^2*(a + b*x^(1/2))^(1/2),x)
[Out]
(4*(a + b*x^(1/2))^(13/2))/(13*b^6) - (20*a*(a + b*x^(1/2))^(11/2))/(11*b^6) - (4*a^5*(a + b*x^(1/2))^(3/2))/(
3*b^6) + (4*a^4*(a + b*x^(1/2))^(5/2))/b^6 - (40*a^3*(a + b*x^(1/2))^(7/2))/(7*b^6) + (40*a^2*(a + b*x^(1/2))^
(9/2))/(9*b^6)