Integrand size = 13, antiderivative size = 80 \[ \int \left (a+b \sqrt {x}\right )^{15} x \, dx=-\frac {a^3 \left (a+b \sqrt {x}\right )^{16}}{8 b^4}+\frac {6 a^2 \left (a+b \sqrt {x}\right )^{17}}{17 b^4}-\frac {a \left (a+b \sqrt {x}\right )^{18}}{3 b^4}+\frac {2 \left (a+b \sqrt {x}\right )^{19}}{19 b^4} \] Output:
-1/8*a^3*(a+b*x^(1/2))^16/b^4+6/17*a^2*(a+b*x^(1/2))^17/b^4-1/3*a*(a+b*x^( 1/2))^18/b^4+2/19*(a+b*x^(1/2))^19/b^4
Leaf count is larger than twice the leaf count of optimal. \(187\) vs. \(2(80)=160\).
Time = 0.03 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.34 \[ \int \left (a+b \sqrt {x}\right )^{15} x \, dx=\frac {3876 a^{15} x^2+46512 a^{14} b x^{5/2}+271320 a^{13} b^2 x^3+1007760 a^{12} b^3 x^{7/2}+2645370 a^{11} b^4 x^4+5173168 a^{10} b^5 x^{9/2}+7759752 a^9 b^6 x^5+9069840 a^8 b^7 x^{11/2}+8314020 a^7 b^8 x^6+5969040 a^6 b^9 x^{13/2}+3325608 a^5 b^{10} x^7+1410864 a^4 b^{11} x^{15/2}+440895 a^3 b^{12} x^8+95760 a^2 b^{13} x^{17/2}+12920 a b^{14} x^9+816 b^{15} x^{19/2}}{7752} \] Input:
Integrate[(a + b*Sqrt[x])^15*x,x]
Output:
(3876*a^15*x^2 + 46512*a^14*b*x^(5/2) + 271320*a^13*b^2*x^3 + 1007760*a^12 *b^3*x^(7/2) + 2645370*a^11*b^4*x^4 + 5173168*a^10*b^5*x^(9/2) + 7759752*a ^9*b^6*x^5 + 9069840*a^8*b^7*x^(11/2) + 8314020*a^7*b^8*x^6 + 5969040*a^6* b^9*x^(13/2) + 3325608*a^5*b^10*x^7 + 1410864*a^4*b^11*x^(15/2) + 440895*a ^3*b^12*x^8 + 95760*a^2*b^13*x^(17/2) + 12920*a*b^14*x^9 + 816*b^15*x^(19/ 2))/7752
Time = 0.38 (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.
\(\displaystyle \int x \left (a+b \sqrt {x}\right )^{15} \, dx\) |
\(\Big \downarrow \) 798 |
\(\displaystyle 2 \int \left (a+b \sqrt {x}\right )^{15} x^{3/2}d\sqrt {x}\) |
\(\Big \downarrow \) 49 |
\(\displaystyle 2 \int \left (\frac {\left (a+b \sqrt {x}\right )^{18}}{b^3}-\frac {3 a \left (a+b \sqrt {x}\right )^{17}}{b^3}+\frac {3 a^2 \left (a+b \sqrt {x}\right )^{16}}{b^3}-\frac {a^3 \left (a+b \sqrt {x}\right )^{15}}{b^3}\right )d\sqrt {x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle 2 \left (-\frac {a^3 \left (a+b \sqrt {x}\right )^{16}}{16 b^4}+\frac {3 a^2 \left (a+b \sqrt {x}\right )^{17}}{17 b^4}+\frac {\left (a+b \sqrt {x}\right )^{19}}{19 b^4}-\frac {a \left (a+b \sqrt {x}\right )^{18}}{6 b^4}\right )\) |
Input:
Int[(a + b*Sqrt[x])^15*x,x]
Output:
2*(-1/16*(a^3*(a + b*Sqrt[x])^16)/b^4 + (3*a^2*(a + b*Sqrt[x])^17)/(17*b^4 ) - (a*(a + b*Sqrt[x])^18)/(6*b^4) + (a + b*Sqrt[x])^19/(19*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]]
Leaf count of result is larger than twice the leaf count of optimal. \(167\) vs. \(2(64)=128\).
Time = 23.13 (sec) , antiderivative size = 168, normalized size of antiderivative = 2.10
method | result | size |
derivativedivides | \(\frac {2 b^{15} x^{\frac {19}{2}}}{19}+\frac {5 a \,b^{14} x^{9}}{3}+\frac {210 a^{2} b^{13} x^{\frac {17}{2}}}{17}+\frac {455 a^{3} b^{12} x^{8}}{8}+182 a^{4} b^{11} x^{\frac {15}{2}}+429 a^{5} b^{10} x^{7}+770 a^{6} b^{9} x^{\frac {13}{2}}+\frac {2145 a^{7} b^{8} x^{6}}{2}+1170 a^{8} b^{7} x^{\frac {11}{2}}+1001 a^{9} b^{6} x^{5}+\frac {2002 a^{10} b^{5} x^{\frac {9}{2}}}{3}+\frac {1365 a^{11} b^{4} x^{4}}{4}+130 a^{12} b^{3} x^{\frac {7}{2}}+35 a^{13} b^{2} x^{3}+6 a^{14} b \,x^{\frac {5}{2}}+\frac {a^{15} x^{2}}{2}\) | \(168\) |
default | \(\frac {2 b^{15} x^{\frac {19}{2}}}{19}+\frac {5 a \,b^{14} x^{9}}{3}+\frac {210 a^{2} b^{13} x^{\frac {17}{2}}}{17}+\frac {455 a^{3} b^{12} x^{8}}{8}+182 a^{4} b^{11} x^{\frac {15}{2}}+429 a^{5} b^{10} x^{7}+770 a^{6} b^{9} x^{\frac {13}{2}}+\frac {2145 a^{7} b^{8} x^{6}}{2}+1170 a^{8} b^{7} x^{\frac {11}{2}}+1001 a^{9} b^{6} x^{5}+\frac {2002 a^{10} b^{5} x^{\frac {9}{2}}}{3}+\frac {1365 a^{11} b^{4} x^{4}}{4}+130 a^{12} b^{3} x^{\frac {7}{2}}+35 a^{13} b^{2} x^{3}+6 a^{14} b \,x^{\frac {5}{2}}+\frac {a^{15} x^{2}}{2}\) | \(168\) |
orering | \(\frac {\left (4760 x^{16} b^{32}-67221 x^{15} b^{30} a^{2}+441285 x^{14} b^{28} a^{4}-1785385 x^{13} b^{26} a^{6}+4975425 x^{12} b^{24} a^{8}-10107825 x^{11} b^{22} a^{10}+15448433 x^{10} b^{20} a^{12}-18063045 b^{18} x^{9} a^{14}+16261245 x^{8} b^{16} a^{16}+1047566520 b^{12} x^{6} a^{20}+11733803172 b^{10} x^{5} a^{22}+36412636260 b^{8} x^{4} a^{24}+38303416890 b^{6} x^{3} a^{26}+13902775950 x^{2} a^{28} b^{4}+1504627670 a^{30} x \,b^{2}+29761866 a^{32}\right ) \left (a +b \sqrt {x}\right )^{15}}{23256 b^{4} \left (-b^{2} x +a^{2}\right )^{14}}-\frac {\left (136 x^{16} b^{32}-2037 x^{15} b^{30} a^{2}+14235 x^{14} b^{28} a^{4}-61565 x^{13} b^{26} a^{6}+184275 x^{12} b^{24} a^{8}-404313 x^{11} b^{22} a^{10}+671671 x^{10} b^{20} a^{12}-860145 b^{18} x^{9} a^{14}+855855 x^{8} b^{16} a^{16}+69837768 b^{12} x^{6} a^{20}+902600244 b^{10} x^{5} a^{22}+3310239660 b^{8} x^{4} a^{24}+4255935210 b^{6} x^{3} a^{26}+1986110850 x^{2} a^{28} b^{4}+300925534 a^{30} x \,b^{2}+9920622 a^{32}\right ) \left (\frac {15 \left (a +b \sqrt {x}\right )^{14} \sqrt {x}\, b}{2}+\left (a +b \sqrt {x}\right )^{15}\right )}{11628 b^{4} \left (-b^{2} x +a^{2}\right )^{14}}\) | \(402\) |
trager | \(\frac {a \left (40 b^{14} x^{8}+1365 a^{2} b^{12} x^{7}+40 x^{7} b^{14}+10296 a^{4} b^{10} x^{6}+1365 a^{2} b^{12} x^{6}+40 b^{14} x^{6}+25740 a^{6} b^{8} x^{5}+10296 a^{4} b^{10} x^{5}+1365 a^{2} b^{12} x^{5}+40 b^{14} x^{5}+24024 x^{4} b^{6} a^{8}+25740 a^{6} b^{8} x^{4}+10296 a^{4} b^{10} x^{4}+1365 b^{12} x^{4} a^{2}+40 b^{14} x^{4}+8190 a^{10} b^{4} x^{3}+24024 a^{8} b^{6} x^{3}+25740 a^{6} b^{8} x^{3}+10296 b^{10} x^{3} a^{4}+1365 b^{12} x^{3} a^{2}+40 b^{14} x^{3}+840 a^{12} b^{2} x^{2}+8190 a^{10} b^{4} x^{2}+24024 a^{8} b^{6} x^{2}+25740 b^{8} x^{2} a^{6}+10296 b^{10} x^{2} a^{4}+1365 b^{12} x^{2} a^{2}+40 b^{14} x^{2}+12 a^{14} x +840 a^{12} b^{2} x +8190 a^{10} b^{4} x +24024 a^{8} b^{6} x +25740 a^{6} b^{8} x +10296 b^{10} x \,a^{4}+1365 b^{12} x \,a^{2}+40 b^{14} x +12 a^{14}+840 a^{12} b^{2}+8190 a^{10} b^{4}+24024 a^{8} b^{6}+25740 a^{6} b^{8}+10296 a^{4} b^{10}+1365 a^{2} b^{12}+40 b^{14}\right ) \left (-1+x \right )}{24}+\frac {2 b \,x^{\frac {5}{2}} \left (51 x^{7} b^{14}+5985 a^{2} b^{12} x^{6}+88179 a^{4} b^{10} x^{5}+373065 a^{6} b^{8} x^{4}+566865 a^{8} b^{6} x^{3}+323323 a^{10} b^{4} x^{2}+62985 a^{12} b^{2} x +2907 a^{14}\right )}{969}\) | \(504\) |
Input:
int((a+b*x^(1/2))^15*x,x,method=_RETURNVERBOSE)
Output:
2/19*b^15*x^(19/2)+5/3*a*b^14*x^9+210/17*a^2*b^13*x^(17/2)+455/8*a^3*b^12* x^8+182*a^4*b^11*x^(15/2)+429*a^5*b^10*x^7+770*a^6*b^9*x^(13/2)+2145/2*a^7 *b^8*x^6+1170*a^8*b^7*x^(11/2)+1001*a^9*b^6*x^5+2002/3*a^10*b^5*x^(9/2)+13 65/4*a^11*b^4*x^4+130*a^12*b^3*x^(7/2)+35*a^13*b^2*x^3+6*a^14*b*x^(5/2)+1/ 2*a^15*x^2
Leaf count of result is larger than twice the leaf count of optimal. 173 vs. \(2 (64) = 128\).
Time = 0.08 (sec) , antiderivative size = 173, normalized size of antiderivative = 2.16 \[ \int \left (a+b \sqrt {x}\right )^{15} x \, dx=\frac {5}{3} \, a b^{14} x^{9} + \frac {455}{8} \, a^{3} b^{12} x^{8} + 429 \, a^{5} b^{10} x^{7} + \frac {2145}{2} \, a^{7} b^{8} x^{6} + 1001 \, a^{9} b^{6} x^{5} + \frac {1365}{4} \, a^{11} b^{4} x^{4} + 35 \, a^{13} b^{2} x^{3} + \frac {1}{2} \, a^{15} x^{2} + \frac {2}{969} \, {\left (51 \, b^{15} x^{9} + 5985 \, a^{2} b^{13} x^{8} + 88179 \, a^{4} b^{11} x^{7} + 373065 \, a^{6} b^{9} x^{6} + 566865 \, a^{8} b^{7} x^{5} + 323323 \, a^{10} b^{5} x^{4} + 62985 \, a^{12} b^{3} x^{3} + 2907 \, a^{14} b x^{2}\right )} \sqrt {x} \] Input:
integrate((a+b*x^(1/2))^15*x,x, algorithm="fricas")
Output:
5/3*a*b^14*x^9 + 455/8*a^3*b^12*x^8 + 429*a^5*b^10*x^7 + 2145/2*a^7*b^8*x^ 6 + 1001*a^9*b^6*x^5 + 1365/4*a^11*b^4*x^4 + 35*a^13*b^2*x^3 + 1/2*a^15*x^ 2 + 2/969*(51*b^15*x^9 + 5985*a^2*b^13*x^8 + 88179*a^4*b^11*x^7 + 373065*a ^6*b^9*x^6 + 566865*a^8*b^7*x^5 + 323323*a^10*b^5*x^4 + 62985*a^12*b^3*x^3 + 2907*a^14*b*x^2)*sqrt(x)
Leaf count of result is larger than twice the leaf count of optimal. 204 vs. \(2 (71) = 142\).
Time = 1.22 (sec) , antiderivative size = 204, normalized size of antiderivative = 2.55 \[ \int \left (a+b \sqrt {x}\right )^{15} x \, dx=\frac {a^{15} x^{2}}{2} + 6 a^{14} b x^{\frac {5}{2}} + 35 a^{13} b^{2} x^{3} + 130 a^{12} b^{3} x^{\frac {7}{2}} + \frac {1365 a^{11} b^{4} x^{4}}{4} + \frac {2002 a^{10} b^{5} x^{\frac {9}{2}}}{3} + 1001 a^{9} b^{6} x^{5} + 1170 a^{8} b^{7} x^{\frac {11}{2}} + \frac {2145 a^{7} b^{8} x^{6}}{2} + 770 a^{6} b^{9} x^{\frac {13}{2}} + 429 a^{5} b^{10} x^{7} + 182 a^{4} b^{11} x^{\frac {15}{2}} + \frac {455 a^{3} b^{12} x^{8}}{8} + \frac {210 a^{2} b^{13} x^{\frac {17}{2}}}{17} + \frac {5 a b^{14} x^{9}}{3} + \frac {2 b^{15} x^{\frac {19}{2}}}{19} \] Input:
integrate((a+b*x**(1/2))**15*x,x)
Output:
a**15*x**2/2 + 6*a**14*b*x**(5/2) + 35*a**13*b**2*x**3 + 130*a**12*b**3*x* *(7/2) + 1365*a**11*b**4*x**4/4 + 2002*a**10*b**5*x**(9/2)/3 + 1001*a**9*b **6*x**5 + 1170*a**8*b**7*x**(11/2) + 2145*a**7*b**8*x**6/2 + 770*a**6*b** 9*x**(13/2) + 429*a**5*b**10*x**7 + 182*a**4*b**11*x**(15/2) + 455*a**3*b* *12*x**8/8 + 210*a**2*b**13*x**(17/2)/17 + 5*a*b**14*x**9/3 + 2*b**15*x**( 19/2)/19
Time = 0.02 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.80 \[ \int \left (a+b \sqrt {x}\right )^{15} x \, dx=\frac {2 \, {\left (b \sqrt {x} + a\right )}^{19}}{19 \, b^{4}} - \frac {{\left (b \sqrt {x} + a\right )}^{18} a}{3 \, b^{4}} + \frac {6 \, {\left (b \sqrt {x} + a\right )}^{17} a^{2}}{17 \, b^{4}} - \frac {{\left (b \sqrt {x} + a\right )}^{16} a^{3}}{8 \, b^{4}} \] Input:
integrate((a+b*x^(1/2))^15*x,x, algorithm="maxima")
Output:
2/19*(b*sqrt(x) + a)^19/b^4 - 1/3*(b*sqrt(x) + a)^18*a/b^4 + 6/17*(b*sqrt( x) + a)^17*a^2/b^4 - 1/8*(b*sqrt(x) + a)^16*a^3/b^4
Leaf count of result is larger than twice the leaf count of optimal. 167 vs. \(2 (64) = 128\).
Time = 0.12 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.09 \[ \int \left (a+b \sqrt {x}\right )^{15} x \, dx=\frac {2}{19} \, b^{15} x^{\frac {19}{2}} + \frac {5}{3} \, a b^{14} x^{9} + \frac {210}{17} \, a^{2} b^{13} x^{\frac {17}{2}} + \frac {455}{8} \, a^{3} b^{12} x^{8} + 182 \, a^{4} b^{11} x^{\frac {15}{2}} + 429 \, a^{5} b^{10} x^{7} + 770 \, a^{6} b^{9} x^{\frac {13}{2}} + \frac {2145}{2} \, a^{7} b^{8} x^{6} + 1170 \, a^{8} b^{7} x^{\frac {11}{2}} + 1001 \, a^{9} b^{6} x^{5} + \frac {2002}{3} \, a^{10} b^{5} x^{\frac {9}{2}} + \frac {1365}{4} \, a^{11} b^{4} x^{4} + 130 \, a^{12} b^{3} x^{\frac {7}{2}} + 35 \, a^{13} b^{2} x^{3} + 6 \, a^{14} b x^{\frac {5}{2}} + \frac {1}{2} \, a^{15} x^{2} \] Input:
integrate((a+b*x^(1/2))^15*x,x, algorithm="giac")
Output:
2/19*b^15*x^(19/2) + 5/3*a*b^14*x^9 + 210/17*a^2*b^13*x^(17/2) + 455/8*a^3 *b^12*x^8 + 182*a^4*b^11*x^(15/2) + 429*a^5*b^10*x^7 + 770*a^6*b^9*x^(13/2 ) + 2145/2*a^7*b^8*x^6 + 1170*a^8*b^7*x^(11/2) + 1001*a^9*b^6*x^5 + 2002/3 *a^10*b^5*x^(9/2) + 1365/4*a^11*b^4*x^4 + 130*a^12*b^3*x^(7/2) + 35*a^13*b ^2*x^3 + 6*a^14*b*x^(5/2) + 1/2*a^15*x^2
Time = 0.16 (sec) , antiderivative size = 167, normalized size of antiderivative = 2.09 \[ \int \left (a+b \sqrt {x}\right )^{15} x \, dx=\frac {a^{15}\,x^2}{2}+\frac {2\,b^{15}\,x^{19/2}}{19}+6\,a^{14}\,b\,x^{5/2}+\frac {5\,a\,b^{14}\,x^9}{3}+35\,a^{13}\,b^2\,x^3+\frac {1365\,a^{11}\,b^4\,x^4}{4}+1001\,a^9\,b^6\,x^5+\frac {2145\,a^7\,b^8\,x^6}{2}+429\,a^5\,b^{10}\,x^7+\frac {455\,a^3\,b^{12}\,x^8}{8}+130\,a^{12}\,b^3\,x^{7/2}+\frac {2002\,a^{10}\,b^5\,x^{9/2}}{3}+1170\,a^8\,b^7\,x^{11/2}+770\,a^6\,b^9\,x^{13/2}+182\,a^4\,b^{11}\,x^{15/2}+\frac {210\,a^2\,b^{13}\,x^{17/2}}{17} \] Input:
int(x*(a + b*x^(1/2))^15,x)
Output:
(a^15*x^2)/2 + (2*b^15*x^(19/2))/19 + 6*a^14*b*x^(5/2) + (5*a*b^14*x^9)/3 + 35*a^13*b^2*x^3 + (1365*a^11*b^4*x^4)/4 + 1001*a^9*b^6*x^5 + (2145*a^7*b ^8*x^6)/2 + 429*a^5*b^10*x^7 + (455*a^3*b^12*x^8)/8 + 130*a^12*b^3*x^(7/2) + (2002*a^10*b^5*x^(9/2))/3 + 1170*a^8*b^7*x^(11/2) + 770*a^6*b^9*x^(13/2 ) + 182*a^4*b^11*x^(15/2) + (210*a^2*b^13*x^(17/2))/17
Time = 0.20 (sec) , antiderivative size = 178, normalized size of antiderivative = 2.22 \[ \int \left (a+b \sqrt {x}\right )^{15} x \, dx=\frac {x^{2} \left (46512 \sqrt {x}\, a^{14} b +1007760 \sqrt {x}\, a^{12} b^{3} x +5173168 \sqrt {x}\, a^{10} b^{5} x^{2}+9069840 \sqrt {x}\, a^{8} b^{7} x^{3}+5969040 \sqrt {x}\, a^{6} b^{9} x^{4}+1410864 \sqrt {x}\, a^{4} b^{11} x^{5}+95760 \sqrt {x}\, a^{2} b^{13} x^{6}+816 \sqrt {x}\, b^{15} x^{7}+3876 a^{15}+271320 a^{13} b^{2} x +2645370 a^{11} b^{4} x^{2}+7759752 a^{9} b^{6} x^{3}+8314020 a^{7} b^{8} x^{4}+3325608 a^{5} b^{10} x^{5}+440895 a^{3} b^{12} x^{6}+12920 a \,b^{14} x^{7}\right )}{7752} \] Input:
int((a+b*x^(1/2))^15*x,x)
Output:
(x**2*(46512*sqrt(x)*a**14*b + 1007760*sqrt(x)*a**12*b**3*x + 5173168*sqrt (x)*a**10*b**5*x**2 + 9069840*sqrt(x)*a**8*b**7*x**3 + 5969040*sqrt(x)*a** 6*b**9*x**4 + 1410864*sqrt(x)*a**4*b**11*x**5 + 95760*sqrt(x)*a**2*b**13*x **6 + 816*sqrt(x)*b**15*x**7 + 3876*a**15 + 271320*a**13*b**2*x + 2645370* a**11*b**4*x**2 + 7759752*a**9*b**6*x**3 + 8314020*a**7*b**8*x**4 + 332560 8*a**5*b**10*x**5 + 440895*a**3*b**12*x**6 + 12920*a*b**14*x**7))/7752