Integrand size = 15, antiderivative size = 140 \[ \int \left (a+b \sqrt {x}\right )^{10} x^4 \, dx=\frac {a^{10} x^5}{5}+\frac {20}{11} a^9 b x^{11/2}+\frac {15}{2} a^8 b^2 x^6+\frac {240}{13} a^7 b^3 x^{13/2}+30 a^6 b^4 x^7+\frac {168}{5} a^5 b^5 x^{15/2}+\frac {105}{4} a^4 b^6 x^8+\frac {240}{17} a^3 b^7 x^{17/2}+5 a^2 b^8 x^9+\frac {20}{19} a b^9 x^{19/2}+\frac {b^{10} x^{10}}{10} \] Output:
1/5*a^10*x^5+20/11*a^9*b*x^(11/2)+15/2*a^8*b^2*x^6+240/13*a^7*b^3*x^(13/2) +30*a^6*b^4*x^7+168/5*a^5*b^5*x^(15/2)+105/4*a^4*b^6*x^8+240/17*a^3*b^7*x^ (17/2)+5*a^2*b^8*x^9+20/19*a*b^9*x^(19/2)+1/10*b^10*x^10
Time = 0.03 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.90 \[ \int \left (a+b \sqrt {x}\right )^{10} x^4 \, dx=\frac {184756 a^{10} x^5+1679600 a^9 b x^{11/2}+6928350 a^8 b^2 x^6+17054400 a^7 b^3 x^{13/2}+27713400 a^6 b^4 x^7+31039008 a^5 b^5 x^{15/2}+24249225 a^4 b^6 x^8+13041600 a^3 b^7 x^{17/2}+4618900 a^2 b^8 x^9+972400 a b^9 x^{19/2}+92378 b^{10} x^{10}}{923780} \] Input:
Integrate[(a + b*Sqrt[x])^10*x^4,x]
Output:
(184756*a^10*x^5 + 1679600*a^9*b*x^(11/2) + 6928350*a^8*b^2*x^6 + 17054400 *a^7*b^3*x^(13/2) + 27713400*a^6*b^4*x^7 + 31039008*a^5*b^5*x^(15/2) + 242 49225*a^4*b^6*x^8 + 13041600*a^3*b^7*x^(17/2) + 4618900*a^2*b^8*x^9 + 9724 00*a*b^9*x^(19/2) + 92378*b^10*x^10)/923780
Time = 0.47 (sec) , antiderivative size = 144, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {798, 49, 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.
\(\displaystyle \int x^4 \left (a+b \sqrt {x}\right )^{10} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^{10} x^{9/2}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (x^{9/2} a^{10}+10 b x^5 a^9+45 b^2 x^{11/2} a^8+120 b^3 x^6 a^7+210 b^4 x^{13/2} a^6+252 b^5 x^7 a^5+210 b^6 x^{15/2} a^4+120 b^7 x^8 a^3+45 b^8 x^{17/2} a^2+10 b^9 x^9 a+b^{10} x^{19/2}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {a^{10} x^5}{10}+\frac {10}{11} a^9 b x^{11/2}+\frac {15}{4} a^8 b^2 x^6+\frac {120}{13} a^7 b^3 x^{13/2}+15 a^6 b^4 x^7+\frac {84}{5} a^5 b^5 x^{15/2}+\frac {105}{8} a^4 b^6 x^8+\frac {120}{17} a^3 b^7 x^{17/2}+\frac {5}{2} a^2 b^8 x^9+\frac {10}{19} a b^9 x^{19/2}+\frac {b^{10} x^{10}}{20}\right )\) |
Input:
Int[(a + b*Sqrt[x])^10*x^4,x]
Output:
2*((a^10*x^5)/10 + (10*a^9*b*x^(11/2))/11 + (15*a^8*b^2*x^6)/4 + (120*a^7* b^3*x^(13/2))/13 + 15*a^6*b^4*x^7 + (84*a^5*b^5*x^(15/2))/5 + (105*a^4*b^6 *x^8)/8 + (120*a^3*b^7*x^(17/2))/17 + (5*a^2*b^8*x^9)/2 + (10*a*b^9*x^(19/ 2))/19 + (b^10*x^10)/20)
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}, x] && IGtQ[m, 0] && IGtQ[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 = 3.74 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.81
method | result | size |
derivativedivides | \(\frac {a^{10} x^{5}}{5}+\frac {20 a^{9} b \,x^{\frac {11}{2}}}{11}+\frac {15 a^{8} b^{2} x^{6}}{2}+\frac {240 a^{7} b^{3} x^{\frac {13}{2}}}{13}+30 a^{6} b^{4} x^{7}+\frac {168 a^{5} b^{5} x^{\frac {15}{2}}}{5}+\frac {105 a^{4} b^{6} x^{8}}{4}+\frac {240 a^{3} b^{7} x^{\frac {17}{2}}}{17}+5 a^{2} b^{8} x^{9}+\frac {20 a \,b^{9} x^{\frac {19}{2}}}{19}+\frac {b^{10} x^{10}}{10}\) | \(113\) |
default | \(\frac {a^{10} x^{5}}{5}+\frac {20 a^{9} b \,x^{\frac {11}{2}}}{11}+\frac {15 a^{8} b^{2} x^{6}}{2}+\frac {240 a^{7} b^{3} x^{\frac {13}{2}}}{13}+30 a^{6} b^{4} x^{7}+\frac {168 a^{5} b^{5} x^{\frac {15}{2}}}{5}+\frac {105 a^{4} b^{6} x^{8}}{4}+\frac {240 a^{3} b^{7} x^{\frac {17}{2}}}{17}+5 a^{2} b^{8} x^{9}+\frac {20 a \,b^{9} x^{\frac {19}{2}}}{19}+\frac {b^{10} x^{10}}{10}\) | \(113\) |
orering | \(\frac {x^{5} \left (-899470 b^{18} x^{9}+8308300 a^{2} b^{16} x^{8}-34245783 a^{4} b^{14} x^{7}+82760700 a^{6} b^{12} x^{6}-129416850 a^{8} b^{10} x^{5}+136083780 a^{10} b^{8} x^{4}-96522375 a^{12} b^{6} x^{3}+44752296 a^{14} b^{4} x^{2}-12412890 a^{16} b^{2} x +1595620 a^{18}\right ) \left (a +b \sqrt {x}\right )^{10}}{4618900 \left (-b^{2} x +a^{2}\right )^{9}}-\frac {x^{2} \left (-24310 b^{18} x^{9}+237380 a^{2} b^{16} x^{8}-1037751 a^{4} b^{14} x^{7}+2669700 a^{6} b^{12} x^{6}-4462650 a^{8} b^{10} x^{5}+5040140 a^{10} b^{8} x^{4}-3860895 a^{12} b^{6} x^{3}+1945752 a^{14} b^{4} x^{2}-591090 a^{16} b^{2} x +83980 a^{18}\right ) \left (5 \left (a +b \sqrt {x}\right )^{9} x^{\frac {7}{2}} b +4 \left (a +b \sqrt {x}\right )^{10} x^{3}\right )}{2309450 \left (-b^{2} x +a^{2}\right )^{9}}\) | \(275\) |
trager | \(\frac {\left (2 b^{10} x^{9}+100 a^{2} b^{8} x^{8}+2 b^{10} x^{8}+525 a^{4} b^{6} x^{7}+100 a^{2} b^{8} x^{7}+2 b^{10} x^{7}+600 a^{6} b^{4} x^{6}+525 a^{4} b^{6} x^{6}+100 a^{2} b^{8} x^{6}+2 b^{10} x^{6}+150 a^{8} b^{2} x^{5}+600 a^{6} b^{4} x^{5}+525 a^{4} b^{6} x^{5}+100 a^{2} b^{8} x^{5}+2 b^{10} x^{5}+4 a^{10} x^{4}+150 a^{8} b^{2} x^{4}+600 a^{6} b^{4} x^{4}+525 a^{4} b^{6} x^{4}+100 a^{2} b^{8} x^{4}+2 b^{10} x^{4}+4 a^{10} x^{3}+150 a^{8} b^{2} x^{3}+600 a^{6} b^{4} x^{3}+525 a^{4} b^{6} x^{3}+100 a^{2} b^{8} x^{3}+2 b^{10} x^{3}+4 a^{10} x^{2}+150 a^{8} b^{2} x^{2}+600 a^{6} b^{4} x^{2}+525 a^{4} b^{6} x^{2}+100 a^{2} b^{8} x^{2}+2 b^{10} x^{2}+4 a^{10} x +150 a^{8} b^{2} x +600 a^{6} b^{4} x +525 a^{4} b^{6} x +100 a^{2} b^{8} x +2 b^{10} x +4 a^{10}+150 a^{8} b^{2}+600 a^{6} b^{4}+525 a^{4} b^{6}+100 a^{2} b^{8}+2 b^{10}\right ) \left (-1+x \right )}{20}+\frac {4 a b \,x^{\frac {11}{2}} \left (60775 b^{8} x^{4}+815100 a^{2} b^{6} x^{3}+1939938 a^{4} b^{4} x^{2}+1065900 a^{6} b^{2} x +104975 a^{8}\right )}{230945}\) | \(480\) |
Input:
int((a+b*x^(1/2))^10*x^4,x,method=_RETURNVERBOSE)
Output:
1/5*a^10*x^5+20/11*a^9*b*x^(11/2)+15/2*a^8*b^2*x^6+240/13*a^7*b^3*x^(13/2) +30*a^6*b^4*x^7+168/5*a^5*b^5*x^(15/2)+105/4*a^4*b^6*x^8+240/17*a^3*b^7*x^ (17/2)+5*a^2*b^8*x^9+20/19*a*b^9*x^(19/2)+1/10*b^10*x^10
Time = 0.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.84 \[ \int \left (a+b \sqrt {x}\right )^{10} x^4 \, dx=\frac {1}{10} \, b^{10} x^{10} + 5 \, a^{2} b^{8} x^{9} + \frac {105}{4} \, a^{4} b^{6} x^{8} + 30 \, a^{6} b^{4} x^{7} + \frac {15}{2} \, a^{8} b^{2} x^{6} + \frac {1}{5} \, a^{10} x^{5} + \frac {4}{230945} \, {\left (60775 \, a b^{9} x^{9} + 815100 \, a^{3} b^{7} x^{8} + 1939938 \, a^{5} b^{5} x^{7} + 1065900 \, a^{7} b^{3} x^{6} + 104975 \, a^{9} b x^{5}\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2))^10*x^4,x, algorithm="fricas")
Output:
1/10*b^10*x^10 + 5*a^2*b^8*x^9 + 105/4*a^4*b^6*x^8 + 30*a^6*b^4*x^7 + 15/2 *a^8*b^2*x^6 + 1/5*a^10*x^5 + 4/230945*(60775*a*b^9*x^9 + 815100*a^3*b^7*x ^8 + 1939938*a^5*b^5*x^7 + 1065900*a^7*b^3*x^6 + 104975*a^9*b*x^5)*sqrt(x)
Time = 0.61 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.99 \[ \int \left (a+b \sqrt {x}\right )^{10} x^4 \, dx=\frac {a^{10} x^{5}}{5} + \frac {20 a^{9} b x^{\frac {11}{2}}}{11} + \frac {15 a^{8} b^{2} x^{6}}{2} + \frac {240 a^{7} b^{3} x^{\frac {13}{2}}}{13} + 30 a^{6} b^{4} x^{7} + \frac {168 a^{5} b^{5} x^{\frac {15}{2}}}{5} + \frac {105 a^{4} b^{6} x^{8}}{4} + \frac {240 a^{3} b^{7} x^{\frac {17}{2}}}{17} + 5 a^{2} b^{8} x^{9} + \frac {20 a b^{9} x^{\frac {19}{2}}}{19} + \frac {b^{10} x^{10}}{10} \] Input:
integrate((a+b*x**(1/2))**10*x**4,x)
Output:
a**10*x**5/5 + 20*a**9*b*x**(11/2)/11 + 15*a**8*b**2*x**6/2 + 240*a**7*b** 3*x**(13/2)/13 + 30*a**6*b**4*x**7 + 168*a**5*b**5*x**(15/2)/5 + 105*a**4* b**6*x**8/4 + 240*a**3*b**7*x**(17/2)/17 + 5*a**2*b**8*x**9 + 20*a*b**9*x* *(19/2)/19 + b**10*x**10/10
Time = 0.03 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.19 \[ \int \left (a+b \sqrt {x}\right )^{10} x^4 \, dx=\frac {{\left (b \sqrt {x} + a\right )}^{20}}{10 \, b^{10}} - \frac {18 \, {\left (b \sqrt {x} + a\right )}^{19} a}{19 \, b^{10}} + \frac {4 \, {\left (b \sqrt {x} + a\right )}^{18} a^{2}}{b^{10}} - \frac {168 \, {\left (b \sqrt {x} + a\right )}^{17} a^{3}}{17 \, b^{10}} + \frac {63 \, {\left (b \sqrt {x} + a\right )}^{16} a^{4}}{4 \, b^{10}} - \frac {84 \, {\left (b \sqrt {x} + a\right )}^{15} a^{5}}{5 \, b^{10}} + \frac {12 \, {\left (b \sqrt {x} + a\right )}^{14} a^{6}}{b^{10}} - \frac {72 \, {\left (b \sqrt {x} + a\right )}^{13} a^{7}}{13 \, b^{10}} + \frac {3 \, {\left (b \sqrt {x} + a\right )}^{12} a^{8}}{2 \, b^{10}} - \frac {2 \, {\left (b \sqrt {x} + a\right )}^{11} a^{9}}{11 \, b^{10}} \] Input:
integrate((a+b*x^(1/2))^10*x^4,x, algorithm="maxima")
Output:
1/10*(b*sqrt(x) + a)^20/b^10 - 18/19*(b*sqrt(x) + a)^19*a/b^10 + 4*(b*sqrt (x) + a)^18*a^2/b^10 - 168/17*(b*sqrt(x) + a)^17*a^3/b^10 + 63/4*(b*sqrt(x ) + a)^16*a^4/b^10 - 84/5*(b*sqrt(x) + a)^15*a^5/b^10 + 12*(b*sqrt(x) + a) ^14*a^6/b^10 - 72/13*(b*sqrt(x) + a)^13*a^7/b^10 + 3/2*(b*sqrt(x) + a)^12* a^8/b^10 - 2/11*(b*sqrt(x) + a)^11*a^9/b^10
Time = 0.13 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.80 \[ \int \left (a+b \sqrt {x}\right )^{10} x^4 \, dx=\frac {1}{10} \, b^{10} x^{10} + \frac {20}{19} \, a b^{9} x^{\frac {19}{2}} + 5 \, a^{2} b^{8} x^{9} + \frac {240}{17} \, a^{3} b^{7} x^{\frac {17}{2}} + \frac {105}{4} \, a^{4} b^{6} x^{8} + \frac {168}{5} \, a^{5} b^{5} x^{\frac {15}{2}} + 30 \, a^{6} b^{4} x^{7} + \frac {240}{13} \, a^{7} b^{3} x^{\frac {13}{2}} + \frac {15}{2} \, a^{8} b^{2} x^{6} + \frac {20}{11} \, a^{9} b x^{\frac {11}{2}} + \frac {1}{5} \, a^{10} x^{5} \] Input:
integrate((a+b*x^(1/2))^10*x^4,x, algorithm="giac")
Output:
1/10*b^10*x^10 + 20/19*a*b^9*x^(19/2) + 5*a^2*b^8*x^9 + 240/17*a^3*b^7*x^( 17/2) + 105/4*a^4*b^6*x^8 + 168/5*a^5*b^5*x^(15/2) + 30*a^6*b^4*x^7 + 240/ 13*a^7*b^3*x^(13/2) + 15/2*a^8*b^2*x^6 + 20/11*a^9*b*x^(11/2) + 1/5*a^10*x ^5
Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.80 \[ \int \left (a+b \sqrt {x}\right )^{10} x^4 \, dx=\frac {a^{10}\,x^5}{5}+\frac {b^{10}\,x^{10}}{10}+\frac {20\,a^9\,b\,x^{11/2}}{11}+\frac {20\,a\,b^9\,x^{19/2}}{19}+\frac {15\,a^8\,b^2\,x^6}{2}+30\,a^6\,b^4\,x^7+\frac {105\,a^4\,b^6\,x^8}{4}+5\,a^2\,b^8\,x^9+\frac {240\,a^7\,b^3\,x^{13/2}}{13}+\frac {168\,a^5\,b^5\,x^{15/2}}{5}+\frac {240\,a^3\,b^7\,x^{17/2}}{17} \] Input:
int(x^4*(a + b*x^(1/2))^10,x)
Output:
(a^10*x^5)/5 + (b^10*x^10)/10 + (20*a^9*b*x^(11/2))/11 + (20*a*b^9*x^(19/2 ))/19 + (15*a^8*b^2*x^6)/2 + 30*a^6*b^4*x^7 + (105*a^4*b^6*x^8)/4 + 5*a^2* b^8*x^9 + (240*a^7*b^3*x^(13/2))/13 + (168*a^5*b^5*x^(15/2))/5 + (240*a^3* b^7*x^(17/2))/17
Time = 0.21 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.84 \[ \int \left (a+b \sqrt {x}\right )^{10} x^4 \, dx=\frac {x^{5} \left (1679600 \sqrt {x}\, a^{9} b +17054400 \sqrt {x}\, a^{7} b^{3} x +31039008 \sqrt {x}\, a^{5} b^{5} x^{2}+13041600 \sqrt {x}\, a^{3} b^{7} x^{3}+972400 \sqrt {x}\, a \,b^{9} x^{4}+184756 a^{10}+6928350 a^{8} b^{2} x +27713400 a^{6} b^{4} x^{2}+24249225 a^{4} b^{6} x^{3}+4618900 a^{2} b^{8} x^{4}+92378 b^{10} x^{5}\right )}{923780} \] Input:
int((a+b*x^(1/2))^10*x^4,x)
Output:
(x**5*(1679600*sqrt(x)*a**9*b + 17054400*sqrt(x)*a**7*b**3*x + 31039008*sq rt(x)*a**5*b**5*x**2 + 13041600*sqrt(x)*a**3*b**7*x**3 + 972400*sqrt(x)*a* b**9*x**4 + 184756*a**10 + 6928350*a**8*b**2*x + 27713400*a**6*b**4*x**2 + 24249225*a**4*b**6*x**3 + 4618900*a**2*b**8*x**4 + 92378*b**10*x**5))/923 780