Integrand size = 13, antiderivative size = 80 \[ \int \left (a+b \sqrt {x}\right )^{10} x \, dx=-\frac {2 a^3 \left (a+b \sqrt {x}\right )^{11}}{11 b^4}+\frac {a^2 \left (a+b \sqrt {x}\right )^{12}}{2 b^4}-\frac {6 a \left (a+b \sqrt {x}\right )^{13}}{13 b^4}+\frac {\left (a+b \sqrt {x}\right )^{14}}{7 b^4} \] Output:
-2/11*a^3*(a+b*x^(1/2))^11/b^4+1/2*a^2*(a+b*x^(1/2))^12/b^4-6/13*a*(a+b*x^ (1/2))^13/b^4+1/7*(a+b*x^(1/2))^14/b^4
Time = 0.03 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.58 \[ \int \left (a+b \sqrt {x}\right )^{10} x \, dx=\frac {1001 a^{10} x^2+8008 a^9 b x^{5/2}+30030 a^8 b^2 x^3+68640 a^7 b^3 x^{7/2}+105105 a^6 b^4 x^4+112112 a^5 b^5 x^{9/2}+84084 a^4 b^6 x^5+43680 a^3 b^7 x^{11/2}+15015 a^2 b^8 x^6+3080 a b^9 x^{13/2}+286 b^{10} x^7}{2002} \] Input:
Integrate[(a + b*Sqrt[x])^10*x,x]
Output:
(1001*a^10*x^2 + 8008*a^9*b*x^(5/2) + 30030*a^8*b^2*x^3 + 68640*a^7*b^3*x^ (7/2) + 105105*a^6*b^4*x^4 + 112112*a^5*b^5*x^(9/2) + 84084*a^4*b^6*x^5 + 43680*a^3*b^7*x^(11/2) + 15015*a^2*b^8*x^6 + 3080*a*b^9*x^(13/2) + 286*b^1 0*x^7)/2002
Time = 0.36 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.231, 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.
\(\displaystyle \int x \left (a+b \sqrt {x}\right )^{10} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^{10} x^{3/2}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (\frac {\left (a+b \sqrt {x}\right )^{13}}{b^3}-\frac {3 a \left (a+b \sqrt {x}\right )^{12}}{b^3}+\frac {3 a^2 \left (a+b \sqrt {x}\right )^{11}}{b^3}-\frac {a^3 \left (a+b \sqrt {x}\right )^{10}}{b^3}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {a^3 \left (a+b \sqrt {x}\right )^{11}}{11 b^4}+\frac {a^2 \left (a+b \sqrt {x}\right )^{12}}{4 b^4}+\frac {\left (a+b \sqrt {x}\right )^{14}}{14 b^4}-\frac {3 a \left (a+b \sqrt {x}\right )^{13}}{13 b^4}\right )\) |
Input:
Int[(a + b*Sqrt[x])^10*x,x]
Output:
2*(-1/11*(a^3*(a + b*Sqrt[x])^11)/b^4 + (a^2*(a + b*Sqrt[x])^12)/(4*b^4) - (3*a*(a + b*Sqrt[x])^13)/(13*b^4) + (a + b*Sqrt[x])^14/(14*b^4))
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 = 3.75 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.41
method | result | size |
derivativedivides | \(\frac {b^{10} x^{7}}{7}+\frac {20 a \,b^{9} x^{\frac {13}{2}}}{13}+\frac {15 a^{2} b^{8} x^{6}}{2}+\frac {240 a^{3} b^{7} x^{\frac {11}{2}}}{11}+42 a^{4} b^{6} x^{5}+56 a^{5} b^{5} x^{\frac {9}{2}}+\frac {105 a^{6} b^{4} x^{4}}{2}+\frac {240 a^{7} b^{3} x^{\frac {7}{2}}}{7}+15 a^{8} b^{2} x^{3}+4 a^{9} b \,x^{\frac {5}{2}}+\frac {a^{10} x^{2}}{2}\) | \(113\) |
default | \(\frac {b^{10} x^{7}}{7}+\frac {20 a \,b^{9} x^{\frac {13}{2}}}{13}+\frac {15 a^{2} b^{8} x^{6}}{2}+\frac {240 a^{3} b^{7} x^{\frac {11}{2}}}{11}+42 a^{4} b^{6} x^{5}+56 a^{5} b^{5} x^{\frac {9}{2}}+\frac {105 a^{6} b^{4} x^{4}}{2}+\frac {240 a^{7} b^{3} x^{\frac {7}{2}}}{7}+15 a^{8} b^{2} x^{3}+4 a^{9} b \,x^{\frac {5}{2}}+\frac {a^{10} x^{2}}{2}\) | \(113\) |
orering | \(-\frac {\left (550 b^{22} x^{11}-5037 a^{2} b^{20} x^{10}+20580 a^{4} b^{18} x^{9}-49305 a^{6} b^{16} x^{8}+76500 a^{8} b^{14} x^{7}-80010 a^{10} b^{12} x^{6}+56784 a^{12} b^{10} x^{5}+270270 a^{16} b^{6} x^{3}+427427 a^{18} x^{2} b^{4}+145860 a^{20} x \,b^{2}+7293 a^{22}\right ) \left (a +b \sqrt {x}\right )^{10}}{2002 b^{4} \left (-b^{2} x +a^{2}\right )^{9}}+\frac {\left (22 b^{22} x^{11}-219 a^{2} b^{20} x^{10}+980 a^{4} b^{18} x^{9}-2595 a^{6} b^{16} x^{8}+4500 a^{8} b^{14} x^{7}-5334 a^{10} b^{12} x^{6}+4368 a^{12} b^{10} x^{5}+30030 a^{16} b^{6} x^{3}+61061 a^{18} x^{2} b^{4}+29172 a^{20} x \,b^{2}+2431 a^{22}\right ) \left (5 \left (a +b \sqrt {x}\right )^{9} \sqrt {x}\, b +\left (a +b \sqrt {x}\right )^{10}\right )}{1001 b^{4} \left (-b^{2} x +a^{2}\right )^{9}}\) | \(292\) |
trager | \(\frac {\left (2 b^{10} x^{6}+105 a^{2} b^{8} x^{5}+2 b^{10} x^{5}+588 a^{4} b^{6} x^{4}+105 a^{2} b^{8} x^{4}+2 b^{10} x^{4}+735 a^{6} b^{4} x^{3}+588 a^{4} b^{6} x^{3}+105 a^{2} b^{8} x^{3}+2 b^{10} x^{3}+210 a^{8} b^{2} x^{2}+735 a^{6} b^{4} x^{2}+588 a^{4} b^{6} x^{2}+105 a^{2} b^{8} x^{2}+2 b^{10} x^{2}+7 a^{10} x +210 a^{8} b^{2} x +735 a^{6} b^{4} x +588 a^{4} b^{6} x +105 a^{2} b^{8} x +2 b^{10} x +7 a^{10}+210 a^{8} b^{2}+735 a^{6} b^{4}+588 a^{4} b^{6}+105 a^{2} b^{8}+2 b^{10}\right ) \left (-1+x \right )}{14}+\frac {4 a b \,x^{\frac {5}{2}} \left (385 b^{8} x^{4}+5460 a^{2} b^{6} x^{3}+14014 a^{4} b^{4} x^{2}+8580 a^{6} b^{2} x +1001 a^{8}\right )}{1001}\) | \(300\) |
Input:
int((a+b*x^(1/2))^10*x,x,method=_RETURNVERBOSE)
Output:
1/7*b^10*x^7+20/13*a*b^9*x^(13/2)+15/2*a^2*b^8*x^6+240/11*a^3*b^7*x^(11/2) +42*a^4*b^6*x^5+56*a^5*b^5*x^(9/2)+105/2*a^6*b^4*x^4+240/7*a^7*b^3*x^(7/2) +15*a^8*b^2*x^3+4*a^9*b*x^(5/2)+1/2*a^10*x^2
Time = 0.08 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.48 \[ \int \left (a+b \sqrt {x}\right )^{10} x \, dx=\frac {1}{7} \, b^{10} x^{7} + \frac {15}{2} \, a^{2} b^{8} x^{6} + 42 \, a^{4} b^{6} x^{5} + \frac {105}{2} \, a^{6} b^{4} x^{4} + 15 \, a^{8} b^{2} x^{3} + \frac {1}{2} \, a^{10} x^{2} + \frac {4}{1001} \, {\left (385 \, a b^{9} x^{6} + 5460 \, a^{3} b^{7} x^{5} + 14014 \, a^{5} b^{5} x^{4} + 8580 \, a^{7} b^{3} x^{3} + 1001 \, a^{9} b x^{2}\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2))^10*x,x, algorithm="fricas")
Output:
1/7*b^10*x^7 + 15/2*a^2*b^8*x^6 + 42*a^4*b^6*x^5 + 105/2*a^6*b^4*x^4 + 15* a^8*b^2*x^3 + 1/2*a^10*x^2 + 4/1001*(385*a*b^9*x^6 + 5460*a^3*b^7*x^5 + 14 014*a^5*b^5*x^4 + 8580*a^7*b^3*x^3 + 1001*a^9*b*x^2)*sqrt(x)
Time = 0.31 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.70 \[ \int \left (a+b \sqrt {x}\right )^{10} x \, dx=\frac {a^{10} x^{2}}{2} + 4 a^{9} b x^{\frac {5}{2}} + 15 a^{8} b^{2} x^{3} + \frac {240 a^{7} b^{3} x^{\frac {7}{2}}}{7} + \frac {105 a^{6} b^{4} x^{4}}{2} + 56 a^{5} b^{5} x^{\frac {9}{2}} + 42 a^{4} b^{6} x^{5} + \frac {240 a^{3} b^{7} x^{\frac {11}{2}}}{11} + \frac {15 a^{2} b^{8} x^{6}}{2} + \frac {20 a b^{9} x^{\frac {13}{2}}}{13} + \frac {b^{10} x^{7}}{7} \] Input:
integrate((a+b*x**(1/2))**10*x,x)
Output:
a**10*x**2/2 + 4*a**9*b*x**(5/2) + 15*a**8*b**2*x**3 + 240*a**7*b**3*x**(7 /2)/7 + 105*a**6*b**4*x**4/2 + 56*a**5*b**5*x**(9/2) + 42*a**4*b**6*x**5 + 240*a**3*b**7*x**(11/2)/11 + 15*a**2*b**8*x**6/2 + 20*a*b**9*x**(13/2)/13 + b**10*x**7/7
Time = 0.03 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \left (a+b \sqrt {x}\right )^{10} x \, dx=\frac {{\left (b \sqrt {x} + a\right )}^{14}}{7 \, b^{4}} - \frac {6 \, {\left (b \sqrt {x} + a\right )}^{13} a}{13 \, b^{4}} + \frac {{\left (b \sqrt {x} + a\right )}^{12} a^{2}}{2 \, b^{4}} - \frac {2 \, {\left (b \sqrt {x} + a\right )}^{11} a^{3}}{11 \, b^{4}} \] Input:
integrate((a+b*x^(1/2))^10*x,x, algorithm="maxima")
Output:
1/7*(b*sqrt(x) + a)^14/b^4 - 6/13*(b*sqrt(x) + a)^13*a/b^4 + 1/2*(b*sqrt(x ) + a)^12*a^2/b^4 - 2/11*(b*sqrt(x) + a)^11*a^3/b^4
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.40 \[ \int \left (a+b \sqrt {x}\right )^{10} x \, dx=\frac {1}{7} \, b^{10} x^{7} + \frac {20}{13} \, a b^{9} x^{\frac {13}{2}} + \frac {15}{2} \, a^{2} b^{8} x^{6} + \frac {240}{11} \, a^{3} b^{7} x^{\frac {11}{2}} + 42 \, a^{4} b^{6} x^{5} + 56 \, a^{5} b^{5} x^{\frac {9}{2}} + \frac {105}{2} \, a^{6} b^{4} x^{4} + \frac {240}{7} \, a^{7} b^{3} x^{\frac {7}{2}} + 15 \, a^{8} b^{2} x^{3} + 4 \, a^{9} b x^{\frac {5}{2}} + \frac {1}{2} \, a^{10} x^{2} \] Input:
integrate((a+b*x^(1/2))^10*x,x, algorithm="giac")
Output:
1/7*b^10*x^7 + 20/13*a*b^9*x^(13/2) + 15/2*a^2*b^8*x^6 + 240/11*a^3*b^7*x^ (11/2) + 42*a^4*b^6*x^5 + 56*a^5*b^5*x^(9/2) + 105/2*a^6*b^4*x^4 + 240/7*a ^7*b^3*x^(7/2) + 15*a^8*b^2*x^3 + 4*a^9*b*x^(5/2) + 1/2*a^10*x^2
Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.40 \[ \int \left (a+b \sqrt {x}\right )^{10} x \, dx=\frac {a^{10}\,x^2}{2}+\frac {b^{10}\,x^7}{7}+4\,a^9\,b\,x^{5/2}+\frac {20\,a\,b^9\,x^{13/2}}{13}+15\,a^8\,b^2\,x^3+\frac {105\,a^6\,b^4\,x^4}{2}+42\,a^4\,b^6\,x^5+\frac {15\,a^2\,b^8\,x^6}{2}+\frac {240\,a^7\,b^3\,x^{7/2}}{7}+56\,a^5\,b^5\,x^{9/2}+\frac {240\,a^3\,b^7\,x^{11/2}}{11} \] Input:
int(x*(a + b*x^(1/2))^10,x)
Output:
(a^10*x^2)/2 + (b^10*x^7)/7 + 4*a^9*b*x^(5/2) + (20*a*b^9*x^(13/2))/13 + 1 5*a^8*b^2*x^3 + (105*a^6*b^4*x^4)/2 + 42*a^4*b^6*x^5 + (15*a^2*b^8*x^6)/2 + (240*a^7*b^3*x^(7/2))/7 + 56*a^5*b^5*x^(9/2) + (240*a^3*b^7*x^(11/2))/11
Time = 0.20 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.46 \[ \int \left (a+b \sqrt {x}\right )^{10} x \, dx=\frac {x^{2} \left (8008 \sqrt {x}\, a^{9} b +68640 \sqrt {x}\, a^{7} b^{3} x +112112 \sqrt {x}\, a^{5} b^{5} x^{2}+43680 \sqrt {x}\, a^{3} b^{7} x^{3}+3080 \sqrt {x}\, a \,b^{9} x^{4}+1001 a^{10}+30030 a^{8} b^{2} x +105105 a^{6} b^{4} x^{2}+84084 a^{4} b^{6} x^{3}+15015 a^{2} b^{8} x^{4}+286 b^{10} x^{5}\right )}{2002} \] Input:
int((a+b*x^(1/2))^10*x,x)
Output:
(x**2*(8008*sqrt(x)*a**9*b + 68640*sqrt(x)*a**7*b**3*x + 112112*sqrt(x)*a* *5*b**5*x**2 + 43680*sqrt(x)*a**3*b**7*x**3 + 3080*sqrt(x)*a*b**9*x**4 + 1 001*a**10 + 30030*a**8*b**2*x + 105105*a**6*b**4*x**2 + 84084*a**4*b**6*x* *3 + 15015*a**2*b**8*x**4 + 286*b**10*x**5))/2002