Integrand size = 17, antiderivative size = 132 \[ \int \sqrt {a+b \sqrt {x}} x^2 \, dx=-\frac {4 a^5 \left (a+b \sqrt {x}\right )^{3/2}}{3 b^6}+\frac {4 a^4 \left (a+b \sqrt {x}\right )^{5/2}}{b^6}-\frac {40 a^3 \left (a+b \sqrt {x}\right )^{7/2}}{7 b^6}+\frac {40 a^2 \left (a+b \sqrt {x}\right )^{9/2}}{9 b^6}-\frac {20 a \left (a+b \sqrt {x}\right )^{11/2}}{11 b^6}+\frac {4 \left (a+b \sqrt {x}\right )^{13/2}}{13 b^6} \] Output:
-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
Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.59 \[ \int \sqrt {a+b \sqrt {x}} x^2 \, dx=\frac {4 \left (a+b \sqrt {x}\right )^{3/2} \left (-256 a^5+384 a^4 b \sqrt {x}-480 a^3 b^2 x+560 a^2 b^3 x^{3/2}-630 a b^4 x^2+693 b^5 x^{5/2}\right )}{9009 b^6} \] Input:
Integrate[Sqrt[a + b*Sqrt[x]]*x^2,x]
Output:
(4*(a + b*Sqrt[x])^(3/2)*(-256*a^5 + 384*a^4*b*Sqrt[x] - 480*a^3*b^2*x + 5 60*a^2*b^3*x^(3/2) - 630*a*b^4*x^2 + 693*b^5*x^(5/2)))/(9009*b^6)
Time = 0.41 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {798, 53, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^2 \sqrt {a+b \sqrt {x}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \sqrt {a+b \sqrt {x}} x^{5/2}d\sqrt {x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle 2 \int \left (\frac {\left (a+b \sqrt {x}\right )^{11/2}}{b^5}-\frac {5 a \left (a+b \sqrt {x}\right )^{9/2}}{b^5}+\frac {10 a^2 \left (a+b \sqrt {x}\right )^{7/2}}{b^5}-\frac {10 a^3 \left (a+b \sqrt {x}\right )^{5/2}}{b^5}+\frac {5 a^4 \left (a+b \sqrt {x}\right )^{3/2}}{b^5}-\frac {a^5 \sqrt {a+b \sqrt {x}}}{b^5}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {2 a^5 \left (a+b \sqrt {x}\right )^{3/2}}{3 b^6}+\frac {2 a^4 \left (a+b \sqrt {x}\right )^{5/2}}{b^6}-\frac {20 a^3 \left (a+b \sqrt {x}\right )^{7/2}}{7 b^6}+\frac {20 a^2 \left (a+b \sqrt {x}\right )^{9/2}}{9 b^6}+\frac {2 \left (a+b \sqrt {x}\right )^{13/2}}{13 b^6}-\frac {10 a \left (a+b \sqrt {x}\right )^{11/2}}{11 b^6}\right )\) |
Input:
Int[Sqrt[a + b*Sqrt[x]]*x^2,x]
Output:
2*((-2*a^5*(a + b*Sqrt[x])^(3/2))/(3*b^6) + (2*a^4*(a + b*Sqrt[x])^(5/2))/ b^6 - (20*a^3*(a + b*Sqrt[x])^(7/2))/(7*b^6) + (20*a^2*(a + b*Sqrt[x])^(9/ 2))/(9*b^6) - (10*a*(a + b*Sqrt[x])^(11/2))/(11*b^6) + (2*(a + b*Sqrt[x])^ (13/2))/(13*b^6))
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] && 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])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[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]]
Time = 0.48 (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 | \(\frac {\frac {4 \left (a +b \sqrt {x}\right )^{\frac {13}{2}}}{13}-\frac {20 a \left (a +b \sqrt {x}\right )^{\frac {11}{2}}}{11}+\frac {40 a^{2} \left (a +b \sqrt {x}\right )^{\frac {9}{2}}}{9}-\frac {40 a^{3} \left (a +b \sqrt {x}\right )^{\frac {7}{2}}}{7}+4 a^{4} \left (a +b \sqrt {x}\right )^{\frac {5}{2}}-\frac {4 a^{5} \left (a +b \sqrt {x}\right )^{\frac {3}{2}}}{3}}{b^{6}}\) | \(85\) |
Input:
int((a+b*x^(1/2))^(1/2)*x^2,x,method=_RETURNVERBOSE)
Output:
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))
Time = 0.08 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.58 \[ \int \sqrt {a+b \sqrt {x}} x^2 \, dx=\frac {4 \, {\left (693 \, b^{6} x^{3} - 70 \, a^{2} b^{4} x^{2} - 96 \, a^{4} b^{2} x - 256 \, a^{6} + {\left (63 \, a b^{5} x^{2} + 80 \, a^{3} b^{3} x + 128 \, a^{5} b\right )} \sqrt {x}\right )} \sqrt {b \sqrt {x} + a}}{9009 \, b^{6}} \] Input:
integrate((a+b*x^(1/2))^(1/2)*x^2,x, algorithm="fricas")
Output:
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
Leaf count of result is larger than twice the leaf count of optimal. 8588 vs. \(2 (124) = 248\).
Time = 5.06 (sec) , antiderivative size = 8588, normalized size of antiderivative = 65.06 \[ \int \sqrt {a+b \sqrt {x}} x^2 \, dx=\text {Too large to display} \] Input:
integrate((a+b*x**(1/2))**(1/2)*x**2,x)
Output:
-1024*a**(153/2)*x**18*sqrt(1 + b*sqrt(x)/a)/(9009*a**70*b**6*x**18 + 1351 35*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**5 7*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) + 45090045*a**64*b**12*x* *21 + 57972915*a**63*b**13*x**(43/2) + 57972915*a**62*b**14*x**22 + 450900 45*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) + 12 297285*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*...
Time = 0.03 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.74 \[ \int \sqrt {a+b \sqrt {x}} x^2 \, dx=\frac {4 \, {\left (b \sqrt {x} + a\right )}^{\frac {13}{2}}}{13 \, b^{6}} - \frac {20 \, {\left (b \sqrt {x} + a\right )}^{\frac {11}{2}} a}{11 \, b^{6}} + \frac {40 \, {\left (b \sqrt {x} + a\right )}^{\frac {9}{2}} a^{2}}{9 \, b^{6}} - \frac {40 \, {\left (b \sqrt {x} + a\right )}^{\frac {7}{2}} a^{3}}{7 \, b^{6}} + \frac {4 \, {\left (b \sqrt {x} + a\right )}^{\frac {5}{2}} a^{4}}{b^{6}} - \frac {4 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} a^{5}}{3 \, b^{6}} \] Input:
integrate((a+b*x^(1/2))^(1/2)*x^2,x, algorithm="maxima")
Output:
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
Time = 0.12 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.45 \[ \int \sqrt {a+b \sqrt {x}} x^2 \, dx=\frac {4 \, {\left (\frac {13 \, {\left (63 \, {\left (b \sqrt {x} + a\right )}^{\frac {11}{2}} - 385 \, {\left (b \sqrt {x} + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b \sqrt {x} + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b \sqrt {x} + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b \sqrt {x} + a} a^{5}\right )} a}{b^{5}} + \frac {3 \, {\left (231 \, {\left (b \sqrt {x} + a\right )}^{\frac {13}{2}} - 1638 \, {\left (b \sqrt {x} + a\right )}^{\frac {11}{2}} a + 5005 \, {\left (b \sqrt {x} + a\right )}^{\frac {9}{2}} a^{2} - 8580 \, {\left (b \sqrt {x} + a\right )}^{\frac {7}{2}} a^{3} + 9009 \, {\left (b \sqrt {x} + a\right )}^{\frac {5}{2}} a^{4} - 6006 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} a^{5} + 3003 \, \sqrt {b \sqrt {x} + a} a^{6}\right )}}{b^{5}}\right )}}{9009 \, b} \] Input:
integrate((a+b*x^(1/2))^(1/2)*x^2,x, algorithm="giac")
Output:
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 - 60 06*(b*sqrt(x) + a)^(3/2)*a^5 + 3003*sqrt(b*sqrt(x) + a)*a^6)/b^5)/b
Time = 1.18 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.74 \[ \int \sqrt {a+b \sqrt {x}} x^2 \, dx=\frac {4\,{\left (a+b\,\sqrt {x}\right )}^{13/2}}{13\,b^6}-\frac {20\,a\,{\left (a+b\,\sqrt {x}\right )}^{11/2}}{11\,b^6}-\frac {4\,a^5\,{\left (a+b\,\sqrt {x}\right )}^{3/2}}{3\,b^6}+\frac {4\,a^4\,{\left (a+b\,\sqrt {x}\right )}^{5/2}}{b^6}-\frac {40\,a^3\,{\left (a+b\,\sqrt {x}\right )}^{7/2}}{7\,b^6}+\frac {40\,a^2\,{\left (a+b\,\sqrt {x}\right )}^{9/2}}{9\,b^6} \] Input:
int(x^2*(a + b*x^(1/2))^(1/2),x)
Output:
(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)
Time = 0.25 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.58 \[ \int \sqrt {a+b \sqrt {x}} x^2 \, dx=\frac {4 \sqrt {\sqrt {x}\, b +a}\, \left (128 \sqrt {x}\, a^{5} b +80 \sqrt {x}\, a^{3} b^{3} x +63 \sqrt {x}\, a \,b^{5} x^{2}-256 a^{6}-96 a^{4} b^{2} x -70 a^{2} b^{4} x^{2}+693 b^{6} x^{3}\right )}{9009 b^{6}} \] Input:
int((a+b*x^(1/2))^(1/2)*x^2,x)
Output:
(4*sqrt(sqrt(x)*b + a)*(128*sqrt(x)*a**5*b + 80*sqrt(x)*a**3*b**3*x + 63*s qrt(x)*a*b**5*x**2 - 256*a**6 - 96*a**4*b**2*x - 70*a**2*b**4*x**2 + 693*b **6*x**3))/(9009*b**6)