Integrand size = 15, antiderivative size = 162 \[ \int \left (a+b \sqrt {x}\right )^{10} x^3 \, dx=-\frac {2 a^7 \left (a+b \sqrt {x}\right )^{11}}{11 b^8}+\frac {7 a^6 \left (a+b \sqrt {x}\right )^{12}}{6 b^8}-\frac {42 a^5 \left (a+b \sqrt {x}\right )^{13}}{13 b^8}+\frac {5 a^4 \left (a+b \sqrt {x}\right )^{14}}{b^8}-\frac {14 a^3 \left (a+b \sqrt {x}\right )^{15}}{3 b^8}+\frac {21 a^2 \left (a+b \sqrt {x}\right )^{16}}{8 b^8}-\frac {14 a \left (a+b \sqrt {x}\right )^{17}}{17 b^8}+\frac {\left (a+b \sqrt {x}\right )^{18}}{9 b^8} \] Output:
-2/11*a^7*(a+b*x^(1/2))^11/b^8+7/6*a^6*(a+b*x^(1/2))^12/b^8-42/13*a^5*(a+b *x^(1/2))^13/b^8+5*a^4*(a+b*x^(1/2))^14/b^8-14/3*a^3*(a+b*x^(1/2))^15/b^8+ 21/8*a^2*(a+b*x^(1/2))^16/b^8-14/17*a*(a+b*x^(1/2))^17/b^8+1/9*(a+b*x^(1/2 ))^18/b^8
Time = 0.03 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.78 \[ \int \left (a+b \sqrt {x}\right )^{10} x^3 \, dx=\frac {43758 a^{10} x^4+388960 a^9 b x^{9/2}+1575288 a^8 b^2 x^5+3818880 a^7 b^3 x^{11/2}+6126120 a^6 b^4 x^6+6785856 a^5 b^5 x^{13/2}+5250960 a^4 b^6 x^7+2800512 a^3 b^7 x^{15/2}+984555 a^2 b^8 x^8+205920 a b^9 x^{17/2}+19448 b^{10} x^9}{175032} \] Input:
Integrate[(a + b*Sqrt[x])^10*x^3,x]
Output:
(43758*a^10*x^4 + 388960*a^9*b*x^(9/2) + 1575288*a^8*b^2*x^5 + 3818880*a^7 *b^3*x^(11/2) + 6126120*a^6*b^4*x^6 + 6785856*a^5*b^5*x^(13/2) + 5250960*a ^4*b^6*x^7 + 2800512*a^3*b^7*x^(15/2) + 984555*a^2*b^8*x^8 + 205920*a*b^9* x^(17/2) + 19448*b^10*x^9)/175032
Time = 0.47 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, 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.
\(\displaystyle \int x^3 \left (a+b \sqrt {x}\right )^{10} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^{10} x^{7/2}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (\frac {\left (a+b \sqrt {x}\right )^{17}}{b^7}-\frac {7 a \left (a+b \sqrt {x}\right )^{16}}{b^7}+\frac {21 a^2 \left (a+b \sqrt {x}\right )^{15}}{b^7}-\frac {35 a^3 \left (a+b \sqrt {x}\right )^{14}}{b^7}+\frac {35 a^4 \left (a+b \sqrt {x}\right )^{13}}{b^7}-\frac {21 a^5 \left (a+b \sqrt {x}\right )^{12}}{b^7}+\frac {7 a^6 \left (a+b \sqrt {x}\right )^{11}}{b^7}-\frac {a^7 \left (a+b \sqrt {x}\right )^{10}}{b^7}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {a^7 \left (a+b \sqrt {x}\right )^{11}}{11 b^8}+\frac {7 a^6 \left (a+b \sqrt {x}\right )^{12}}{12 b^8}-\frac {21 a^5 \left (a+b \sqrt {x}\right )^{13}}{13 b^8}+\frac {5 a^4 \left (a+b \sqrt {x}\right )^{14}}{2 b^8}-\frac {7 a^3 \left (a+b \sqrt {x}\right )^{15}}{3 b^8}+\frac {21 a^2 \left (a+b \sqrt {x}\right )^{16}}{16 b^8}+\frac {\left (a+b \sqrt {x}\right )^{18}}{18 b^8}-\frac {7 a \left (a+b \sqrt {x}\right )^{17}}{17 b^8}\right )\) |
Input:
Int[(a + b*Sqrt[x])^10*x^3,x]
Output:
2*(-1/11*(a^7*(a + b*Sqrt[x])^11)/b^8 + (7*a^6*(a + b*Sqrt[x])^12)/(12*b^8 ) - (21*a^5*(a + b*Sqrt[x])^13)/(13*b^8) + (5*a^4*(a + b*Sqrt[x])^14)/(2*b ^8) - (7*a^3*(a + b*Sqrt[x])^15)/(3*b^8) + (21*a^2*(a + b*Sqrt[x])^16)/(16 *b^8) - (7*a*(a + b*Sqrt[x])^17)/(17*b^8) + (a + b*Sqrt[x])^18/(18*b^8))
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 = 3.76 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(\frac {b^{10} x^{9}}{9}+\frac {20 a \,b^{9} x^{\frac {17}{2}}}{17}+\frac {45 a^{2} b^{8} x^{8}}{8}+16 a^{3} b^{7} x^{\frac {15}{2}}+30 a^{4} b^{6} x^{7}+\frac {504 a^{5} b^{5} x^{\frac {13}{2}}}{13}+35 a^{6} b^{4} x^{6}+\frac {240 a^{7} b^{3} x^{\frac {11}{2}}}{11}+9 a^{8} b^{2} x^{5}+\frac {20 a^{9} b \,x^{\frac {9}{2}}}{9}+\frac {a^{10} x^{4}}{4}\) | \(113\) |
default | \(\frac {b^{10} x^{9}}{9}+\frac {20 a \,b^{9} x^{\frac {17}{2}}}{17}+\frac {45 a^{2} b^{8} x^{8}}{8}+16 a^{3} b^{7} x^{\frac {15}{2}}+30 a^{4} b^{6} x^{7}+\frac {504 a^{5} b^{5} x^{\frac {13}{2}}}{13}+35 a^{6} b^{4} x^{6}+\frac {240 a^{7} b^{3} x^{\frac {11}{2}}}{11}+9 a^{8} b^{2} x^{5}+\frac {20 a^{9} b \,x^{\frac {9}{2}}}{9}+\frac {a^{10} x^{4}}{4}\) | \(113\) |
orering | \(-\frac {\left (188760 b^{26} x^{13}-1742169 a^{2} b^{24} x^{12}+7177500 a^{4} b^{22} x^{11}-17346150 a^{6} b^{20} x^{10}+27149500 a^{8} b^{18} x^{9}-28616715 a^{10} b^{16} x^{8}+20400408 a^{12} b^{14} x^{7}-9554340 a^{14} b^{12} x^{6}+2705040 a^{16} b^{10} x^{5}+3792360 a^{20} x^{3} b^{6}+6738732 a^{22} x^{2} b^{4}+2625480 a^{24} x \,b^{2}+170170 a^{26}\right ) \left (a +b \sqrt {x}\right )^{10}}{875160 b^{8} \left (-b^{2} x +a^{2}\right )^{9}}+\frac {\left (5720 b^{26} x^{13}-56199 a^{2} b^{24} x^{12}+247500 a^{4} b^{22} x^{11}-642450 a^{6} b^{20} x^{10}+1085980 a^{8} b^{18} x^{9}-1244205 a^{10} b^{16} x^{8}+971448 a^{12} b^{14} x^{7}-502860 a^{14} b^{12} x^{6}+159120 a^{16} b^{10} x^{5}+291720 a^{20} x^{3} b^{6}+612612 a^{22} x^{2} b^{4}+291720 a^{24} x \,b^{2}+24310 a^{26}\right ) \left (5 \left (a +b \sqrt {x}\right )^{9} x^{\frac {5}{2}} b +3 \left (a +b \sqrt {x}\right )^{10} x^{2}\right )}{437580 b^{8} \left (-b^{2} x +a^{2}\right )^{9} x^{2}}\) | \(344\) |
trager | \(\frac {\left (8 b^{10} x^{8}+405 a^{2} b^{8} x^{7}+8 b^{10} x^{7}+2160 a^{4} b^{6} x^{6}+405 a^{2} b^{8} x^{6}+8 b^{10} x^{6}+2520 a^{6} b^{4} x^{5}+2160 a^{4} b^{6} x^{5}+405 a^{2} b^{8} x^{5}+8 b^{10} x^{5}+648 a^{8} b^{2} x^{4}+2520 a^{6} b^{4} x^{4}+2160 a^{4} b^{6} x^{4}+405 a^{2} b^{8} x^{4}+8 b^{10} x^{4}+18 a^{10} x^{3}+648 a^{8} b^{2} x^{3}+2520 a^{6} b^{4} x^{3}+2160 a^{4} b^{6} x^{3}+405 a^{2} b^{8} x^{3}+8 b^{10} x^{3}+18 a^{10} x^{2}+648 a^{8} b^{2} x^{2}+2520 a^{6} b^{4} x^{2}+2160 a^{4} b^{6} x^{2}+405 a^{2} b^{8} x^{2}+8 b^{10} x^{2}+18 a^{10} x +648 a^{8} b^{2} x +2520 a^{6} b^{4} x +2160 a^{4} b^{6} x +405 a^{2} b^{8} x +8 b^{10} x +18 a^{10}+648 a^{8} b^{2}+2520 a^{6} b^{4}+2160 a^{4} b^{6}+405 a^{2} b^{8}+8 b^{10}\right ) \left (-1+x \right )}{72}+\frac {4 a b \,x^{\frac {9}{2}} \left (6435 b^{8} x^{4}+87516 a^{2} b^{6} x^{3}+212058 a^{4} b^{4} x^{2}+119340 a^{6} b^{2} x +12155 a^{8}\right )}{21879}\) | \(420\) |
Input:
int((a+b*x^(1/2))^10*x^3,x,method=_RETURNVERBOSE)
Output:
1/9*b^10*x^9+20/17*a*b^9*x^(17/2)+45/8*a^2*b^8*x^8+16*a^3*b^7*x^(15/2)+30* a^4*b^6*x^7+504/13*a^5*b^5*x^(13/2)+35*a^6*b^4*x^6+240/11*a^7*b^3*x^(11/2) +9*a^8*b^2*x^5+20/9*a^9*b*x^(9/2)+1/4*a^10*x^4
Time = 0.07 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.73 \[ \int \left (a+b \sqrt {x}\right )^{10} x^3 \, dx=\frac {1}{9} \, b^{10} x^{9} + \frac {45}{8} \, a^{2} b^{8} x^{8} + 30 \, a^{4} b^{6} x^{7} + 35 \, a^{6} b^{4} x^{6} + 9 \, a^{8} b^{2} x^{5} + \frac {1}{4} \, a^{10} x^{4} + \frac {4}{21879} \, {\left (6435 \, a b^{9} x^{8} + 87516 \, a^{3} b^{7} x^{7} + 212058 \, a^{5} b^{5} x^{6} + 119340 \, a^{7} b^{3} x^{5} + 12155 \, a^{9} b x^{4}\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2))^10*x^3,x, algorithm="fricas")
Output:
1/9*b^10*x^9 + 45/8*a^2*b^8*x^8 + 30*a^4*b^6*x^7 + 35*a^6*b^4*x^6 + 9*a^8* b^2*x^5 + 1/4*a^10*x^4 + 4/21879*(6435*a*b^9*x^8 + 87516*a^3*b^7*x^7 + 212 058*a^5*b^5*x^6 + 119340*a^7*b^3*x^5 + 12155*a^9*b*x^4)*sqrt(x)
Time = 0.47 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.84 \[ \int \left (a+b \sqrt {x}\right )^{10} x^3 \, dx=\frac {a^{10} x^{4}}{4} + \frac {20 a^{9} b x^{\frac {9}{2}}}{9} + 9 a^{8} b^{2} x^{5} + \frac {240 a^{7} b^{3} x^{\frac {11}{2}}}{11} + 35 a^{6} b^{4} x^{6} + \frac {504 a^{5} b^{5} x^{\frac {13}{2}}}{13} + 30 a^{4} b^{6} x^{7} + 16 a^{3} b^{7} x^{\frac {15}{2}} + \frac {45 a^{2} b^{8} x^{8}}{8} + \frac {20 a b^{9} x^{\frac {17}{2}}}{17} + \frac {b^{10} x^{9}}{9} \] Input:
integrate((a+b*x**(1/2))**10*x**3,x)
Output:
a**10*x**4/4 + 20*a**9*b*x**(9/2)/9 + 9*a**8*b**2*x**5 + 240*a**7*b**3*x** (11/2)/11 + 35*a**6*b**4*x**6 + 504*a**5*b**5*x**(13/2)/13 + 30*a**4*b**6* x**7 + 16*a**3*b**7*x**(15/2) + 45*a**2*b**8*x**8/8 + 20*a*b**9*x**(17/2)/ 17 + b**10*x**9/9
Time = 0.04 (sec) , antiderivative size = 132, normalized size of antiderivative = 0.81 \[ \int \left (a+b \sqrt {x}\right )^{10} x^3 \, dx=\frac {{\left (b \sqrt {x} + a\right )}^{18}}{9 \, b^{8}} - \frac {14 \, {\left (b \sqrt {x} + a\right )}^{17} a}{17 \, b^{8}} + \frac {21 \, {\left (b \sqrt {x} + a\right )}^{16} a^{2}}{8 \, b^{8}} - \frac {14 \, {\left (b \sqrt {x} + a\right )}^{15} a^{3}}{3 \, b^{8}} + \frac {5 \, {\left (b \sqrt {x} + a\right )}^{14} a^{4}}{b^{8}} - \frac {42 \, {\left (b \sqrt {x} + a\right )}^{13} a^{5}}{13 \, b^{8}} + \frac {7 \, {\left (b \sqrt {x} + a\right )}^{12} a^{6}}{6 \, b^{8}} - \frac {2 \, {\left (b \sqrt {x} + a\right )}^{11} a^{7}}{11 \, b^{8}} \] Input:
integrate((a+b*x^(1/2))^10*x^3,x, algorithm="maxima")
Output:
1/9*(b*sqrt(x) + a)^18/b^8 - 14/17*(b*sqrt(x) + a)^17*a/b^8 + 21/8*(b*sqrt (x) + a)^16*a^2/b^8 - 14/3*(b*sqrt(x) + a)^15*a^3/b^8 + 5*(b*sqrt(x) + a)^ 14*a^4/b^8 - 42/13*(b*sqrt(x) + a)^13*a^5/b^8 + 7/6*(b*sqrt(x) + a)^12*a^6 /b^8 - 2/11*(b*sqrt(x) + a)^11*a^7/b^8
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.69 \[ \int \left (a+b \sqrt {x}\right )^{10} x^3 \, dx=\frac {1}{9} \, b^{10} x^{9} + \frac {20}{17} \, a b^{9} x^{\frac {17}{2}} + \frac {45}{8} \, a^{2} b^{8} x^{8} + 16 \, a^{3} b^{7} x^{\frac {15}{2}} + 30 \, a^{4} b^{6} x^{7} + \frac {504}{13} \, a^{5} b^{5} x^{\frac {13}{2}} + 35 \, a^{6} b^{4} x^{6} + \frac {240}{11} \, a^{7} b^{3} x^{\frac {11}{2}} + 9 \, a^{8} b^{2} x^{5} + \frac {20}{9} \, a^{9} b x^{\frac {9}{2}} + \frac {1}{4} \, a^{10} x^{4} \] Input:
integrate((a+b*x^(1/2))^10*x^3,x, algorithm="giac")
Output:
1/9*b^10*x^9 + 20/17*a*b^9*x^(17/2) + 45/8*a^2*b^8*x^8 + 16*a^3*b^7*x^(15/ 2) + 30*a^4*b^6*x^7 + 504/13*a^5*b^5*x^(13/2) + 35*a^6*b^4*x^6 + 240/11*a^ 7*b^3*x^(11/2) + 9*a^8*b^2*x^5 + 20/9*a^9*b*x^(9/2) + 1/4*a^10*x^4
Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.69 \[ \int \left (a+b \sqrt {x}\right )^{10} x^3 \, dx=\frac {a^{10}\,x^4}{4}+\frac {b^{10}\,x^9}{9}+\frac {20\,a^9\,b\,x^{9/2}}{9}+\frac {20\,a\,b^9\,x^{17/2}}{17}+9\,a^8\,b^2\,x^5+35\,a^6\,b^4\,x^6+30\,a^4\,b^6\,x^7+\frac {45\,a^2\,b^8\,x^8}{8}+\frac {240\,a^7\,b^3\,x^{11/2}}{11}+\frac {504\,a^5\,b^5\,x^{13/2}}{13}+16\,a^3\,b^7\,x^{15/2} \] Input:
int(x^3*(a + b*x^(1/2))^10,x)
Output:
(a^10*x^4)/4 + (b^10*x^9)/9 + (20*a^9*b*x^(9/2))/9 + (20*a*b^9*x^(17/2))/1 7 + 9*a^8*b^2*x^5 + 35*a^6*b^4*x^6 + 30*a^4*b^6*x^7 + (45*a^2*b^8*x^8)/8 + (240*a^7*b^3*x^(11/2))/11 + (504*a^5*b^5*x^(13/2))/13 + 16*a^3*b^7*x^(15/ 2)
Time = 0.20 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.72 \[ \int \left (a+b \sqrt {x}\right )^{10} x^3 \, dx=\frac {x^{4} \left (388960 \sqrt {x}\, a^{9} b +3818880 \sqrt {x}\, a^{7} b^{3} x +6785856 \sqrt {x}\, a^{5} b^{5} x^{2}+2800512 \sqrt {x}\, a^{3} b^{7} x^{3}+205920 \sqrt {x}\, a \,b^{9} x^{4}+43758 a^{10}+1575288 a^{8} b^{2} x +6126120 a^{6} b^{4} x^{2}+5250960 a^{4} b^{6} x^{3}+984555 a^{2} b^{8} x^{4}+19448 b^{10} x^{5}\right )}{175032} \] Input:
int((a+b*x^(1/2))^10*x^3,x)
Output:
(x**4*(388960*sqrt(x)*a**9*b + 3818880*sqrt(x)*a**7*b**3*x + 6785856*sqrt( x)*a**5*b**5*x**2 + 2800512*sqrt(x)*a**3*b**7*x**3 + 205920*sqrt(x)*a*b**9 *x**4 + 43758*a**10 + 1575288*a**8*b**2*x + 6126120*a**6*b**4*x**2 + 52509 60*a**4*b**6*x**3 + 984555*a**2*b**8*x**4 + 19448*b**10*x**5))/175032