Integrand size = 17, antiderivative size = 130 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {x}}} \, dx=-\frac {4 a^5 \sqrt {a+b \sqrt {x}}}{b^6}+\frac {20 a^4 \left (a+b \sqrt {x}\right )^{3/2}}{3 b^6}-\frac {8 a^3 \left (a+b \sqrt {x}\right )^{5/2}}{b^6}+\frac {40 a^2 \left (a+b \sqrt {x}\right )^{7/2}}{7 b^6}-\frac {20 a \left (a+b \sqrt {x}\right )^{9/2}}{9 b^6}+\frac {4 \left (a+b \sqrt {x}\right )^{11/2}}{11 b^6} \] Output:
-4*a^5*(a+b*x^(1/2))^(1/2)/b^6+20/3*a^4*(a+b*x^(1/2))^(3/2)/b^6-8*a^3*(a+b *x^(1/2))^(5/2)/b^6+40/7*a^2*(a+b*x^(1/2))^(7/2)/b^6-20/9*a*(a+b*x^(1/2))^ (9/2)/b^6+4/11*(a+b*x^(1/2))^(11/2)/b^6
Time = 0.05 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.60 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {x}}} \, dx=\frac {4 \sqrt {a+b \sqrt {x}} \left (-256 a^5+128 a^4 b \sqrt {x}-96 a^3 b^2 x+80 a^2 b^3 x^{3/2}-70 a b^4 x^2+63 b^5 x^{5/2}\right )}{693 b^6} \] Input:
Integrate[x^2/Sqrt[a + b*Sqrt[x]],x]
Output:
(4*Sqrt[a + b*Sqrt[x]]*(-256*a^5 + 128*a^4*b*Sqrt[x] - 96*a^3*b^2*x + 80*a ^2*b^3*x^(3/2) - 70*a*b^4*x^2 + 63*b^5*x^(5/2)))/(693*b^6)
Time = 0.39 (sec) , antiderivative size = 132, 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.
\(\displaystyle \int \frac {x^2}{\sqrt {a+b \sqrt {x}}} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \frac {x^{5/2}}{\sqrt {a+b \sqrt {x}}}d\sqrt {x}\) |
\(\Big \downarrow \) 53 |
\(\displaystyle 2 \int \left (-\frac {a^5}{b^5 \sqrt {a+b \sqrt {x}}}+\frac {5 \sqrt {a+b \sqrt {x}} a^4}{b^5}-\frac {10 \left (a+b \sqrt {x}\right )^{3/2} a^3}{b^5}+\frac {10 \left (a+b \sqrt {x}\right )^{5/2} a^2}{b^5}-\frac {5 \left (a+b \sqrt {x}\right )^{7/2} a}{b^5}+\frac {\left (a+b \sqrt {x}\right )^{9/2}}{b^5}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {2 a^5 \sqrt {a+b \sqrt {x}}}{b^6}+\frac {10 a^4 \left (a+b \sqrt {x}\right )^{3/2}}{3 b^6}-\frac {4 a^3 \left (a+b \sqrt {x}\right )^{5/2}}{b^6}+\frac {20 a^2 \left (a+b \sqrt {x}\right )^{7/2}}{7 b^6}+\frac {2 \left (a+b \sqrt {x}\right )^{11/2}}{11 b^6}-\frac {10 a \left (a+b \sqrt {x}\right )^{9/2}}{9 b^6}\right )\) |
Input:
Int[x^2/Sqrt[a + b*Sqrt[x]],x]
Output:
2*((-2*a^5*Sqrt[a + b*Sqrt[x]])/b^6 + (10*a^4*(a + b*Sqrt[x])^(3/2))/(3*b^ 6) - (4*a^3*(a + b*Sqrt[x])^(5/2))/b^6 + (20*a^2*(a + b*Sqrt[x])^(7/2))/(7 *b^6) - (10*a*(a + b*Sqrt[x])^(9/2))/(9*b^6) + (2*(a + b*Sqrt[x])^(11/2))/ (11*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 = 0.45 (sec) , antiderivative size = 86, normalized size of antiderivative = 0.66
method | result | size |
derivativedivides | \(\frac {\frac {4 \left (a +b \sqrt {x}\right )^{\frac {11}{2}}}{11}-\frac {20 a \left (a +b \sqrt {x}\right )^{\frac {9}{2}}}{9}+\frac {40 a^{2} \left (a +b \sqrt {x}\right )^{\frac {7}{2}}}{7}-8 a^{3} \left (a +b \sqrt {x}\right )^{\frac {5}{2}}+\frac {20 a^{4} \left (a +b \sqrt {x}\right )^{\frac {3}{2}}}{3}-4 a^{5} \sqrt {a +b \sqrt {x}}}{b^{6}}\) | \(86\) |
default | \(\frac {\frac {4 \left (a +b \sqrt {x}\right )^{\frac {11}{2}}}{11}-\frac {20 a \left (a +b \sqrt {x}\right )^{\frac {9}{2}}}{9}+\frac {40 a^{2} \left (a +b \sqrt {x}\right )^{\frac {7}{2}}}{7}-8 a^{3} \left (a +b \sqrt {x}\right )^{\frac {5}{2}}+\frac {20 a^{4} \left (a +b \sqrt {x}\right )^{\frac {3}{2}}}{3}-4 a^{5} \sqrt {a +b \sqrt {x}}}{b^{6}}\) | \(86\) |
Input:
int(x^2/(a+b*x^(1/2))^(1/2),x,method=_RETURNVERBOSE)
Output:
4/b^6*(1/11*(a+b*x^(1/2))^(11/2)-5/9*a*(a+b*x^(1/2))^(9/2)+10/7*a^2*(a+b*x ^(1/2))^(7/2)-2*a^3*(a+b*x^(1/2))^(5/2)+5/3*a^4*(a+b*x^(1/2))^(3/2)-a^5*(a +b*x^(1/2))^(1/2))
Time = 0.08 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.52 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {x}}} \, dx=-\frac {4 \, {\left (70 \, a b^{4} x^{2} + 96 \, a^{3} b^{2} x + 256 \, a^{5} - {\left (63 \, b^{5} x^{2} + 80 \, a^{2} b^{3} x + 128 \, a^{4} b\right )} \sqrt {x}\right )} \sqrt {b \sqrt {x} + a}}{693 \, b^{6}} \] Input:
integrate(x^2/(a+b*x^(1/2))^(1/2),x, algorithm="fricas")
Output:
-4/693*(70*a*b^4*x^2 + 96*a^3*b^2*x + 256*a^5 - (63*b^5*x^2 + 80*a^2*b^3*x + 128*a^4*b)*sqrt(x))*sqrt(b*sqrt(x) + a)/b^6
Leaf count of result is larger than twice the leaf count of optimal. 8356 vs. \(2 (122) = 244\).
Time = 5.72 (sec) , antiderivative size = 8356, normalized size of antiderivative = 64.28 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {x}}} \, dx=\text {Too large to display} \] Input:
integrate(x**2/(a+b*x**(1/2))**(1/2),x)
Output:
-1024*a**(151/2)*x**18*sqrt(1 + b*sqrt(x)/a)/(693*a**70*b**6*x**18 + 10395 *a**69*b**7*x**(37/2) + 72765*a**68*b**8*x**19 + 315315*a**67*b**9*x**(39/ 2) + 945945*a**66*b**10*x**20 + 2081079*a**65*b**11*x**(41/2) + 3468465*a* *64*b**12*x**21 + 4459455*a**63*b**13*x**(43/2) + 4459455*a**62*b**14*x**2 2 + 3468465*a**61*b**15*x**(45/2) + 2081079*a**60*b**16*x**23 + 945945*a** 59*b**17*x**(47/2) + 315315*a**58*b**18*x**24 + 72765*a**57*b**19*x**(49/2 ) + 10395*a**56*b**20*x**25 + 693*a**55*b**21*x**(51/2)) + 1024*a**(151/2) *x**18/(693*a**70*b**6*x**18 + 10395*a**69*b**7*x**(37/2) + 72765*a**68*b* *8*x**19 + 315315*a**67*b**9*x**(39/2) + 945945*a**66*b**10*x**20 + 208107 9*a**65*b**11*x**(41/2) + 3468465*a**64*b**12*x**21 + 4459455*a**63*b**13* x**(43/2) + 4459455*a**62*b**14*x**22 + 3468465*a**61*b**15*x**(45/2) + 20 81079*a**60*b**16*x**23 + 945945*a**59*b**17*x**(47/2) + 315315*a**58*b**1 8*x**24 + 72765*a**57*b**19*x**(49/2) + 10395*a**56*b**20*x**25 + 693*a**5 5*b**21*x**(51/2)) - 14848*a**(149/2)*b*x**(37/2)*sqrt(1 + b*sqrt(x)/a)/(6 93*a**70*b**6*x**18 + 10395*a**69*b**7*x**(37/2) + 72765*a**68*b**8*x**19 + 315315*a**67*b**9*x**(39/2) + 945945*a**66*b**10*x**20 + 2081079*a**65*b **11*x**(41/2) + 3468465*a**64*b**12*x**21 + 4459455*a**63*b**13*x**(43/2) + 4459455*a**62*b**14*x**22 + 3468465*a**61*b**15*x**(45/2) + 2081079*a** 60*b**16*x**23 + 945945*a**59*b**17*x**(47/2) + 315315*a**58*b**18*x**24 + 72765*a**57*b**19*x**(49/2) + 10395*a**56*b**20*x**25 + 693*a**55*b**2...
Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.75 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {x}}} \, dx=\frac {4 \, {\left (b \sqrt {x} + a\right )}^{\frac {11}{2}}}{11 \, b^{6}} - \frac {20 \, {\left (b \sqrt {x} + a\right )}^{\frac {9}{2}} a}{9 \, b^{6}} + \frac {40 \, {\left (b \sqrt {x} + a\right )}^{\frac {7}{2}} a^{2}}{7 \, b^{6}} - \frac {8 \, {\left (b \sqrt {x} + a\right )}^{\frac {5}{2}} a^{3}}{b^{6}} + \frac {20 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} a^{4}}{3 \, b^{6}} - \frac {4 \, \sqrt {b \sqrt {x} + a} a^{5}}{b^{6}} \] Input:
integrate(x^2/(a+b*x^(1/2))^(1/2),x, algorithm="maxima")
Output:
4/11*(b*sqrt(x) + a)^(11/2)/b^6 - 20/9*(b*sqrt(x) + a)^(9/2)*a/b^6 + 40/7* (b*sqrt(x) + a)^(7/2)*a^2/b^6 - 8*(b*sqrt(x) + a)^(5/2)*a^3/b^6 + 20/3*(b* sqrt(x) + a)^(3/2)*a^4/b^6 - 4*sqrt(b*sqrt(x) + a)*a^5/b^6
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.65 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {x}}} \, dx=\frac {4 \, {\left (63 \, {\left (b \sqrt {x} + a\right )}^{\frac {11}{2}} - 385 \, {\left (b \sqrt {x} + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b \sqrt {x} + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b \sqrt {x} + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b \sqrt {x} + a\right )}^{\frac {3}{2}} a^{4} - 693 \, \sqrt {b \sqrt {x} + a} a^{5}\right )}}{693 \, b^{6}} \] Input:
integrate(x^2/(a+b*x^(1/2))^(1/2),x, algorithm="giac")
Output:
4/693*(63*(b*sqrt(x) + a)^(11/2) - 385*(b*sqrt(x) + a)^(9/2)*a + 990*(b*sq rt(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)/b^6
Time = 0.71 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.75 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {x}}} \, dx=\frac {4\,{\left (a+b\,\sqrt {x}\right )}^{11/2}}{11\,b^6}-\frac {20\,a\,{\left (a+b\,\sqrt {x}\right )}^{9/2}}{9\,b^6}-\frac {4\,a^5\,\sqrt {a+b\,\sqrt {x}}}{b^6}+\frac {20\,a^4\,{\left (a+b\,\sqrt {x}\right )}^{3/2}}{3\,b^6}-\frac {8\,a^3\,{\left (a+b\,\sqrt {x}\right )}^{5/2}}{b^6}+\frac {40\,a^2\,{\left (a+b\,\sqrt {x}\right )}^{7/2}}{7\,b^6} \] Input:
int(x^2/(a + b*x^(1/2))^(1/2),x)
Output:
(4*(a + b*x^(1/2))^(11/2))/(11*b^6) - (20*a*(a + b*x^(1/2))^(9/2))/(9*b^6) - (4*a^5*(a + b*x^(1/2))^(1/2))/b^6 + (20*a^4*(a + b*x^(1/2))^(3/2))/(3*b ^6) - (8*a^3*(a + b*x^(1/2))^(5/2))/b^6 + (40*a^2*(a + b*x^(1/2))^(7/2))/( 7*b^6)
Time = 0.22 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.50 \[ \int \frac {x^2}{\sqrt {a+b \sqrt {x}}} \, dx=\frac {4 \sqrt {\sqrt {x}\, b +a}\, \left (128 \sqrt {x}\, a^{4} b +80 \sqrt {x}\, a^{2} b^{3} x +63 \sqrt {x}\, b^{5} x^{2}-256 a^{5}-96 a^{3} b^{2} x -70 a \,b^{4} x^{2}\right )}{693 b^{6}} \] Input:
int(x^2/(a+b*x^(1/2))^(1/2),x)
Output:
(4*sqrt(sqrt(x)*b + a)*(128*sqrt(x)*a**4*b + 80*sqrt(x)*a**2*b**3*x + 63*s qrt(x)*b**5*x**2 - 256*a**5 - 96*a**3*b**2*x - 70*a*b**4*x**2))/(693*b**6)