Integrand size = 11, antiderivative size = 38 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=-\frac {a \left (a+b \sqrt {x}\right )^{16}}{8 b^2}+\frac {2 \left (a+b \sqrt {x}\right )^{17}}{17 b^2} \] Output:
-1/8*a*(a+b*x^(1/2))^16/b^2+2/17*(a+b*x^(1/2))^17/b^2
Leaf count is larger than twice the leaf count of optimal. \(185\) vs. \(2(38)=76\).
Time = 0.03 (sec) , antiderivative size = 185, normalized size of antiderivative = 4.87 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=\frac {1}{136} \left (136 a^{15} x+1360 a^{14} b x^{3/2}+7140 a^{13} b^2 x^2+24752 a^{12} b^3 x^{5/2}+61880 a^{11} b^4 x^3+116688 a^{10} b^5 x^{7/2}+170170 a^9 b^6 x^4+194480 a^8 b^7 x^{9/2}+175032 a^7 b^8 x^5+123760 a^6 b^9 x^{11/2}+68068 a^5 b^{10} x^6+28560 a^4 b^{11} x^{13/2}+8840 a^3 b^{12} x^7+1904 a^2 b^{13} x^{15/2}+255 a b^{14} x^8+16 b^{15} x^{17/2}\right ) \] Input:
Integrate[(a + b*Sqrt[x])^15,x]
Output:
(136*a^15*x + 1360*a^14*b*x^(3/2) + 7140*a^13*b^2*x^2 + 24752*a^12*b^3*x^( 5/2) + 61880*a^11*b^4*x^3 + 116688*a^10*b^5*x^(7/2) + 170170*a^9*b^6*x^4 + 194480*a^8*b^7*x^(9/2) + 175032*a^7*b^8*x^5 + 123760*a^6*b^9*x^(11/2) + 6 8068*a^5*b^10*x^6 + 28560*a^4*b^11*x^(13/2) + 8840*a^3*b^12*x^7 + 1904*a^2 *b^13*x^(15/2) + 255*a*b^14*x^8 + 16*b^15*x^(17/2))/136
Time = 0.28 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {774, 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.
\(\displaystyle \int \left (a+b \sqrt {x}\right )^{15} \, dx\) |
\(\Big \downarrow \) 774 |
\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^{15} \sqrt {x}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (\frac {\left (a+b \sqrt {x}\right )^{16}}{b}-\frac {a \left (a+b \sqrt {x}\right )^{15}}{b}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (\frac {\left (a+b \sqrt {x}\right )^{17}}{17 b^2}-\frac {a \left (a+b \sqrt {x}\right )^{16}}{16 b^2}\right )\) |
Input:
Int[(a + b*Sqrt[x])^15,x]
Output:
2*(-1/16*(a*(a + b*Sqrt[x])^16)/b^2 + (a + b*Sqrt[x])^17/(17*b^2))
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[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = Denominator[n]}, Simp[k Subst[Int[x^(k - 1)*(a + b*x^(k*n))^p, x], x, x^(1/k)], x]] /; Fre eQ[{a, b, p}, x] && FractionQ[n]
Leaf count of result is larger than twice the leaf count of optimal. \(164\) vs. \(2(30)=60\).
Time = 23.05 (sec) , antiderivative size = 165, normalized size of antiderivative = 4.34
method | result | size |
derivativedivides | \(\frac {2 b^{15} x^{\frac {17}{2}}}{17}+\frac {15 a \,b^{14} x^{8}}{8}+14 a^{2} b^{13} x^{\frac {15}{2}}+65 a^{3} b^{12} x^{7}+210 a^{4} b^{11} x^{\frac {13}{2}}+\frac {1001 a^{5} b^{10} x^{6}}{2}+910 a^{6} b^{9} x^{\frac {11}{2}}+1287 a^{7} b^{8} x^{5}+1430 a^{8} b^{7} x^{\frac {9}{2}}+\frac {5005 a^{9} b^{6} x^{4}}{4}+858 a^{10} b^{5} x^{\frac {7}{2}}+455 a^{11} b^{4} x^{3}+182 a^{12} b^{3} x^{\frac {5}{2}}+\frac {105 a^{13} b^{2} x^{2}}{2}+10 a^{14} b \,x^{\frac {3}{2}}+a^{15} x\) | \(165\) |
default | \(\frac {2 b^{15} x^{\frac {17}{2}}}{17}+\frac {15 a \,b^{14} x^{8}}{8}+14 a^{2} b^{13} x^{\frac {15}{2}}+65 a^{3} b^{12} x^{7}+210 a^{4} b^{11} x^{\frac {13}{2}}+\frac {1001 a^{5} b^{10} x^{6}}{2}+910 a^{6} b^{9} x^{\frac {11}{2}}+1287 a^{7} b^{8} x^{5}+1430 a^{8} b^{7} x^{\frac {9}{2}}+\frac {5005 a^{9} b^{6} x^{4}}{4}+858 a^{10} b^{5} x^{\frac {7}{2}}+455 a^{11} b^{4} x^{3}+182 a^{12} b^{3} x^{\frac {5}{2}}+\frac {105 a^{13} b^{2} x^{2}}{2}+10 a^{14} b \,x^{\frac {3}{2}}+a^{15} x\) | \(165\) |
orering | \(-\frac {\left (-31 b^{30} x^{15}+435 a^{2} b^{28} x^{14}-2835 a^{4} b^{26} x^{13}+11375 a^{6} b^{24} x^{12}-31395 a^{8} b^{22} x^{11}+63063 a^{10} b^{20} x^{10}-95095 a^{12} b^{18} x^{9}+109395 a^{14} b^{16} x^{8}+8848840 a^{18} b^{12} x^{6}+96588492 a^{20} b^{10} x^{5}+289876860 a^{22} b^{8} x^{4}+289861390 a^{24} b^{6} x^{3}+96622050 a^{26} b^{4} x^{2}+8783730 a^{28} b^{2} x +96526 a^{30}\right ) \left (a +b \sqrt {x}\right )^{15}}{136 b^{2} \left (-b^{2} x +a^{2}\right )^{14}}+\frac {15 \left (-b^{30} x^{15}+15 a^{2} b^{28} x^{14}-105 a^{4} b^{26} x^{13}+455 a^{6} b^{24} x^{12}-1365 a^{8} b^{22} x^{11}+3003 a^{10} b^{20} x^{10}-5005 a^{12} b^{18} x^{9}+6435 a^{14} b^{16} x^{8}+680680 a^{18} b^{12} x^{6}+8780772 a^{20} b^{10} x^{5}+32208540 a^{22} b^{8} x^{4}+41408770 a^{24} b^{6} x^{3}+19324410 a^{26} b^{4} x^{2}+2927910 a^{28} b^{2} x +96526 a^{30}\right ) \sqrt {x}\, \left (a +b \sqrt {x}\right )^{14}}{136 b \left (-b^{2} x +a^{2}\right )^{14}}\) | \(367\) |
trager | \(\frac {a \left (15 x^{7} b^{14}+520 a^{2} b^{12} x^{6}+15 b^{14} x^{6}+4004 a^{4} b^{10} x^{5}+520 a^{2} b^{12} x^{5}+15 b^{14} x^{5}+10296 a^{6} b^{8} x^{4}+4004 a^{4} b^{10} x^{4}+520 b^{12} x^{4} a^{2}+15 b^{14} x^{4}+10010 a^{8} b^{6} x^{3}+10296 a^{6} b^{8} x^{3}+4004 b^{10} x^{3} a^{4}+520 b^{12} x^{3} a^{2}+15 b^{14} x^{3}+3640 a^{10} b^{4} x^{2}+10010 a^{8} b^{6} x^{2}+10296 b^{8} x^{2} a^{6}+4004 b^{10} x^{2} a^{4}+520 b^{12} x^{2} a^{2}+15 b^{14} x^{2}+420 a^{12} b^{2} x +3640 a^{10} b^{4} x +10010 a^{8} b^{6} x +10296 a^{6} b^{8} x +4004 b^{10} x \,a^{4}+520 b^{12} x \,a^{2}+15 b^{14} x +8 a^{14}+420 a^{12} b^{2}+3640 a^{10} b^{4}+10010 a^{8} b^{6}+10296 a^{6} b^{8}+4004 a^{4} b^{10}+520 a^{2} b^{12}+15 b^{14}\right ) \left (-1+x \right )}{8}+\frac {2 b \,x^{\frac {3}{2}} \left (x^{7} b^{14}+119 a^{2} b^{12} x^{6}+1785 a^{4} b^{10} x^{5}+7735 a^{6} b^{8} x^{4}+12155 a^{8} b^{6} x^{3}+7293 a^{10} b^{4} x^{2}+1547 a^{12} b^{2} x +85 a^{14}\right )}{17}\) | \(423\) |
Input:
int((a+b*x^(1/2))^15,x,method=_RETURNVERBOSE)
Output:
2/17*b^15*x^(17/2)+15/8*a*b^14*x^8+14*a^2*b^13*x^(15/2)+65*a^3*b^12*x^7+21 0*a^4*b^11*x^(13/2)+1001/2*a^5*b^10*x^6+910*a^6*b^9*x^(11/2)+1287*a^7*b^8* x^5+1430*a^8*b^7*x^(9/2)+5005/4*a^9*b^6*x^4+858*a^10*b^5*x^(7/2)+455*a^11* b^4*x^3+182*a^12*b^3*x^(5/2)+105/2*a^13*b^2*x^2+10*a^14*b*x^(3/2)+a^15*x
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (30) = 60\).
Time = 0.08 (sec) , antiderivative size = 167, normalized size of antiderivative = 4.39 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=\frac {15}{8} \, a b^{14} x^{8} + 65 \, a^{3} b^{12} x^{7} + \frac {1001}{2} \, a^{5} b^{10} x^{6} + 1287 \, a^{7} b^{8} x^{5} + \frac {5005}{4} \, a^{9} b^{6} x^{4} + 455 \, a^{11} b^{4} x^{3} + \frac {105}{2} \, a^{13} b^{2} x^{2} + a^{15} x + \frac {2}{17} \, {\left (b^{15} x^{8} + 119 \, a^{2} b^{13} x^{7} + 1785 \, a^{4} b^{11} x^{6} + 7735 \, a^{6} b^{9} x^{5} + 12155 \, a^{8} b^{7} x^{4} + 7293 \, a^{10} b^{5} x^{3} + 1547 \, a^{12} b^{3} x^{2} + 85 \, a^{14} b x\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2))^15,x, algorithm="fricas")
Output:
15/8*a*b^14*x^8 + 65*a^3*b^12*x^7 + 1001/2*a^5*b^10*x^6 + 1287*a^7*b^8*x^5 + 5005/4*a^9*b^6*x^4 + 455*a^11*b^4*x^3 + 105/2*a^13*b^2*x^2 + a^15*x + 2 /17*(b^15*x^8 + 119*a^2*b^13*x^7 + 1785*a^4*b^11*x^6 + 7735*a^6*b^9*x^5 + 12155*a^8*b^7*x^4 + 7293*a^10*b^5*x^3 + 1547*a^12*b^3*x^2 + 85*a^14*b*x)*s qrt(x)
Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (32) = 64\).
Time = 0.99 (sec) , antiderivative size = 197, normalized size of antiderivative = 5.18 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=a^{15} x + 10 a^{14} b x^{\frac {3}{2}} + \frac {105 a^{13} b^{2} x^{2}}{2} + 182 a^{12} b^{3} x^{\frac {5}{2}} + 455 a^{11} b^{4} x^{3} + 858 a^{10} b^{5} x^{\frac {7}{2}} + \frac {5005 a^{9} b^{6} x^{4}}{4} + 1430 a^{8} b^{7} x^{\frac {9}{2}} + 1287 a^{7} b^{8} x^{5} + 910 a^{6} b^{9} x^{\frac {11}{2}} + \frac {1001 a^{5} b^{10} x^{6}}{2} + 210 a^{4} b^{11} x^{\frac {13}{2}} + 65 a^{3} b^{12} x^{7} + 14 a^{2} b^{13} x^{\frac {15}{2}} + \frac {15 a b^{14} x^{8}}{8} + \frac {2 b^{15} x^{\frac {17}{2}}}{17} \] Input:
integrate((a+b*x**(1/2))**15,x)
Output:
a**15*x + 10*a**14*b*x**(3/2) + 105*a**13*b**2*x**2/2 + 182*a**12*b**3*x** (5/2) + 455*a**11*b**4*x**3 + 858*a**10*b**5*x**(7/2) + 5005*a**9*b**6*x** 4/4 + 1430*a**8*b**7*x**(9/2) + 1287*a**7*b**8*x**5 + 910*a**6*b**9*x**(11 /2) + 1001*a**5*b**10*x**6/2 + 210*a**4*b**11*x**(13/2) + 65*a**3*b**12*x* *7 + 14*a**2*b**13*x**(15/2) + 15*a*b**14*x**8/8 + 2*b**15*x**(17/2)/17
Time = 0.03 (sec) , antiderivative size = 30, normalized size of antiderivative = 0.79 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{17}}{17 \, b^{2}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a}{8 \, b^{2}} \] Input:
integrate((a+b*x^(1/2))^15,x, algorithm="maxima")
Output:
2/17*(b*sqrt(x) + a)^17/b^2 - 1/8*(b*sqrt(x) + a)^16*a/b^2
Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (30) = 60\).
Time = 0.13 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.32 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=\frac {2}{17} \, b^{15} x^{\frac {17}{2}} + \frac {15}{8} \, a b^{14} x^{8} + 14 \, a^{2} b^{13} x^{\frac {15}{2}} + 65 \, a^{3} b^{12} x^{7} + 210 \, a^{4} b^{11} x^{\frac {13}{2}} + \frac {1001}{2} \, a^{5} b^{10} x^{6} + 910 \, a^{6} b^{9} x^{\frac {11}{2}} + 1287 \, a^{7} b^{8} x^{5} + 1430 \, a^{8} b^{7} x^{\frac {9}{2}} + \frac {5005}{4} \, a^{9} b^{6} x^{4} + 858 \, a^{10} b^{5} x^{\frac {7}{2}} + 455 \, a^{11} b^{4} x^{3} + 182 \, a^{12} b^{3} x^{\frac {5}{2}} + \frac {105}{2} \, a^{13} b^{2} x^{2} + 10 \, a^{14} b x^{\frac {3}{2}} + a^{15} x \] Input:
integrate((a+b*x^(1/2))^15,x, algorithm="giac")
Output:
2/17*b^15*x^(17/2) + 15/8*a*b^14*x^8 + 14*a^2*b^13*x^(15/2) + 65*a^3*b^12* x^7 + 210*a^4*b^11*x^(13/2) + 1001/2*a^5*b^10*x^6 + 910*a^6*b^9*x^(11/2) + 1287*a^7*b^8*x^5 + 1430*a^8*b^7*x^(9/2) + 5005/4*a^9*b^6*x^4 + 858*a^10*b ^5*x^(7/2) + 455*a^11*b^4*x^3 + 182*a^12*b^3*x^(5/2) + 105/2*a^13*b^2*x^2 + 10*a^14*b*x^(3/2) + a^15*x
Time = 0.16 (sec) , antiderivative size = 164, normalized size of antiderivative = 4.32 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=a^{15}\,x+\frac {2\,b^{15}\,x^{17/2}}{17}+10\,a^{14}\,b\,x^{3/2}+\frac {15\,a\,b^{14}\,x^8}{8}+\frac {105\,a^{13}\,b^2\,x^2}{2}+455\,a^{11}\,b^4\,x^3+\frac {5005\,a^9\,b^6\,x^4}{4}+1287\,a^7\,b^8\,x^5+\frac {1001\,a^5\,b^{10}\,x^6}{2}+65\,a^3\,b^{12}\,x^7+182\,a^{12}\,b^3\,x^{5/2}+858\,a^{10}\,b^5\,x^{7/2}+1430\,a^8\,b^7\,x^{9/2}+910\,a^6\,b^9\,x^{11/2}+210\,a^4\,b^{11}\,x^{13/2}+14\,a^2\,b^{13}\,x^{15/2} \] Input:
int((a + b*x^(1/2))^15,x)
Output:
a^15*x + (2*b^15*x^(17/2))/17 + 10*a^14*b*x^(3/2) + (15*a*b^14*x^8)/8 + (1 05*a^13*b^2*x^2)/2 + 455*a^11*b^4*x^3 + (5005*a^9*b^6*x^4)/4 + 1287*a^7*b^ 8*x^5 + (1001*a^5*b^10*x^6)/2 + 65*a^3*b^12*x^7 + 182*a^12*b^3*x^(5/2) + 8 58*a^10*b^5*x^(7/2) + 1430*a^8*b^7*x^(9/2) + 910*a^6*b^9*x^(11/2) + 210*a^ 4*b^11*x^(13/2) + 14*a^2*b^13*x^(15/2)
Time = 0.21 (sec) , antiderivative size = 176, normalized size of antiderivative = 4.63 \[ \int \left (a+b \sqrt {x}\right )^{15} \, dx=\frac {x \left (1360 \sqrt {x}\, a^{14} b +24752 \sqrt {x}\, a^{12} b^{3} x +116688 \sqrt {x}\, a^{10} b^{5} x^{2}+194480 \sqrt {x}\, a^{8} b^{7} x^{3}+123760 \sqrt {x}\, a^{6} b^{9} x^{4}+28560 \sqrt {x}\, a^{4} b^{11} x^{5}+1904 \sqrt {x}\, a^{2} b^{13} x^{6}+16 \sqrt {x}\, b^{15} x^{7}+136 a^{15}+7140 a^{13} b^{2} x +61880 a^{11} b^{4} x^{2}+170170 a^{9} b^{6} x^{3}+175032 a^{7} b^{8} x^{4}+68068 a^{5} b^{10} x^{5}+8840 a^{3} b^{12} x^{6}+255 a \,b^{14} x^{7}\right )}{136} \] Input:
int((a+b*x^(1/2))^15,x)
Output:
(x*(1360*sqrt(x)*a**14*b + 24752*sqrt(x)*a**12*b**3*x + 116688*sqrt(x)*a** 10*b**5*x**2 + 194480*sqrt(x)*a**8*b**7*x**3 + 123760*sqrt(x)*a**6*b**9*x* *4 + 28560*sqrt(x)*a**4*b**11*x**5 + 1904*sqrt(x)*a**2*b**13*x**6 + 16*sqr t(x)*b**15*x**7 + 136*a**15 + 7140*a**13*b**2*x + 61880*a**11*b**4*x**2 + 170170*a**9*b**6*x**3 + 175032*a**7*b**8*x**4 + 68068*a**5*b**10*x**5 + 88 40*a**3*b**12*x**6 + 255*a*b**14*x**7))/136