Integrand size = 22, antiderivative size = 29 \[ \int x^p \left (a x^n+b x^{1+13 n+p}\right )^{12} \, dx=\frac {\left (a+b x^{1+12 n+p}\right )^{13}}{13 b (1+12 n+p)} \] Output:
1/13*(a+b*x^(1+12*n+p))^13/b/(1+12*n+p)
Leaf count is larger than twice the leaf count of optimal. \(232\) vs. \(2(29)=58\).
Time = 0.13 (sec) , antiderivative size = 232, normalized size of antiderivative = 8.00 \[ \int x^p \left (a x^n+b x^{1+13 n+p}\right )^{12} \, dx=\frac {x^{1+12 n+p} \left (13 a^{12}+78 a^{11} b x^{1+12 n+p}+286 a^{10} b^2 x^{2+24 n+2 p}+715 a^9 b^3 x^{3+36 n+3 p}+1287 a^8 b^4 x^{4+48 n+4 p}+1716 a^7 b^5 x^{5+60 n+5 p}+1716 a^6 b^6 x^{6+72 n+6 p}+1287 a^5 b^7 x^{7+84 n+7 p}+715 a^4 b^8 x^{8+96 n+8 p}+286 a^3 b^9 x^{9+108 n+9 p}+78 a^2 b^{10} x^{10+120 n+10 p}+13 a b^{11} x^{11+132 n+11 p}+b^{12} x^{12+144 n+12 p}\right )}{13 (1+12 n+p)} \] Input:
Integrate[x^p*(a*x^n + b*x^(1 + 13*n + p))^12,x]
Output:
(x^(1 + 12*n + p)*(13*a^12 + 78*a^11*b*x^(1 + 12*n + p) + 286*a^10*b^2*x^( 2 + 24*n + 2*p) + 715*a^9*b^3*x^(3 + 36*n + 3*p) + 1287*a^8*b^4*x^(4 + 48* n + 4*p) + 1716*a^7*b^5*x^(5 + 60*n + 5*p) + 1716*a^6*b^6*x^(6 + 72*n + 6* p) + 1287*a^5*b^7*x^(7 + 84*n + 7*p) + 715*a^4*b^8*x^(8 + 96*n + 8*p) + 28 6*a^3*b^9*x^(9 + 108*n + 9*p) + 78*a^2*b^10*x^(10 + 120*n + 10*p) + 13*a*b ^11*x^(11 + 132*n + 11*p) + b^12*x^(12 + 144*n + 12*p)))/(13*(1 + 12*n + p ))
Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {10, 793}
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.
\(\displaystyle \int x^p \left (a x^n+b x^{13 n+p+1}\right )^{12} \, dx\) |
\(\Big \downarrow \) 10 |
\(\displaystyle \int x^{12 n+p} \left (a+b x^{12 n+p+1}\right )^{12}dx\) |
\(\Big \downarrow \) 793 |
\(\displaystyle \frac {\left (a+b x^{12 n+p+1}\right )^{13}}{13 b (12 n+p+1)}\) |
Input:
Int[x^p*(a*x^n + b*x^(1 + 13*n + p))^12,x]
Output:
(a + b*x^(1 + 12*n + p))^13/(13*b*(1 + 12*n + p))
Int[(u_.)*((e_.)*(x_))^(m_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x _Symbol] :> Simp[1/e^(p*r) Int[u*(e*x)^(m + p*r)*(a + b*x^(s - r))^p, x], x] /; FreeQ[{a, b, e, m, r, s}, x] && IntegerQ[p] && (IntegerQ[p*r] || GtQ [e, 0]) && PosQ[s - r]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n) ^(p + 1)/(b*n*(p + 1)), x] /; FreeQ[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(362\) vs. \(2(27)=54\).
Time = 0.04 (sec) , antiderivative size = 363, normalized size of antiderivative = 12.52
\[\frac {b^{12} x^{13} x^{156 n} x^{13 p}}{13+156 n +13 p}+\frac {a \,b^{11} x^{12} x^{144 n} x^{12 p}}{1+12 n +p}+\frac {6 a^{2} b^{10} x^{11} x^{132 n} x^{11 p}}{1+12 n +p}+\frac {22 a^{3} b^{9} x^{10} x^{120 n} x^{10 p}}{1+12 n +p}+\frac {55 a^{4} b^{8} x^{9} x^{108 n} x^{9 p}}{1+12 n +p}+\frac {99 a^{5} b^{7} x^{8} x^{96 n} x^{8 p}}{1+12 n +p}+\frac {132 a^{6} b^{6} x^{7} x^{84 n} x^{7 p}}{1+12 n +p}+\frac {132 a^{7} b^{5} x^{6} x^{72 n} x^{6 p}}{1+12 n +p}+\frac {99 a^{8} b^{4} x^{5} x^{60 n} x^{5 p}}{1+12 n +p}+\frac {55 a^{9} b^{3} x^{4} x^{48 n} x^{4 p}}{1+12 n +p}+\frac {22 a^{10} b^{2} x^{3} x^{36 n} x^{3 p}}{1+12 n +p}+\frac {6 a^{11} b \,x^{2} x^{24 n} x^{2 p}}{1+12 n +p}+\frac {a^{12} x \,x^{12 n} x^{p}}{1+12 n +p}\]
Input:
int(x^p*(a*x^n+b*x^(1+13*n+p))^12,x)
Output:
1/13*b^12*x^13*(x^n)^156/(1+12*n+p)*(x^p)^13+a*b^11*x^12*(x^n)^144/(1+12*n +p)*(x^p)^12+6*a^2*b^10*x^11*(x^n)^132/(1+12*n+p)*(x^p)^11+22*a^3*b^9*x^10 *(x^n)^120/(1+12*n+p)*(x^p)^10+55*a^4*b^8*x^9*(x^n)^108/(1+12*n+p)*(x^p)^9 +99*a^5*b^7*x^8*(x^n)^96/(1+12*n+p)*(x^p)^8+132*a^6*b^6*x^7*(x^n)^84/(1+12 *n+p)*(x^p)^7+132*a^7*b^5*x^6*(x^n)^72/(1+12*n+p)*(x^p)^6+99*a^8*b^4*x^5*( x^n)^60/(1+12*n+p)*(x^p)^5+55*a^9*b^3*x^4*(x^n)^48/(1+12*n+p)*(x^p)^4+22*a ^10*b^2*x^3*(x^n)^36/(1+12*n+p)*(x^p)^3+6*a^11*b*x^2*(x^n)^24/(1+12*n+p)*( x^p)^2+a^12/(1+12*n+p)*x*(x^n)^12*x^p
Leaf count of result is larger than twice the leaf count of optimal. 297 vs. \(2 (27) = 54\).
Time = 0.10 (sec) , antiderivative size = 297, normalized size of antiderivative = 10.24 \[ \int x^p \left (a x^n+b x^{1+13 n+p}\right )^{12} \, dx=\frac {78 \, a^{2} b^{10} x^{2 \, n} x^{143 \, n + 11 \, p + 11} + 286 \, a^{3} b^{9} x^{3 \, n} x^{130 \, n + 10 \, p + 10} + 715 \, a^{4} b^{8} x^{4 \, n} x^{117 \, n + 9 \, p + 9} + 1287 \, a^{5} b^{7} x^{5 \, n} x^{104 \, n + 8 \, p + 8} + 1716 \, a^{6} b^{6} x^{6 \, n} x^{91 \, n + 7 \, p + 7} + 1716 \, a^{7} b^{5} x^{7 \, n} x^{78 \, n + 6 \, p + 6} + 1287 \, a^{8} b^{4} x^{8 \, n} x^{65 \, n + 5 \, p + 5} + 715 \, a^{9} b^{3} x^{9 \, n} x^{52 \, n + 4 \, p + 4} + 286 \, a^{10} b^{2} x^{10 \, n} x^{39 \, n + 3 \, p + 3} + 78 \, a^{11} b x^{11 \, n} x^{26 \, n + 2 \, p + 2} + 13 \, a^{12} x^{12 \, n} x^{13 \, n + p + 1} + 13 \, a b^{11} x^{156 \, n + 12 \, p + 12} x^{n} + b^{12} x^{169 \, n + 13 \, p + 13}}{13 \, {\left (12 \, n + p + 1\right )} x^{13 \, n}} \] Input:
integrate(x^p*(a*x^n+b*x^(1+13*n+p))^12,x, algorithm="fricas")
Output:
1/13*(78*a^2*b^10*x^(2*n)*x^(143*n + 11*p + 11) + 286*a^3*b^9*x^(3*n)*x^(1 30*n + 10*p + 10) + 715*a^4*b^8*x^(4*n)*x^(117*n + 9*p + 9) + 1287*a^5*b^7 *x^(5*n)*x^(104*n + 8*p + 8) + 1716*a^6*b^6*x^(6*n)*x^(91*n + 7*p + 7) + 1 716*a^7*b^5*x^(7*n)*x^(78*n + 6*p + 6) + 1287*a^8*b^4*x^(8*n)*x^(65*n + 5* p + 5) + 715*a^9*b^3*x^(9*n)*x^(52*n + 4*p + 4) + 286*a^10*b^2*x^(10*n)*x^ (39*n + 3*p + 3) + 78*a^11*b*x^(11*n)*x^(26*n + 2*p + 2) + 13*a^12*x^(12*n )*x^(13*n + p + 1) + 13*a*b^11*x^(156*n + 12*p + 12)*x^n + b^12*x^(169*n + 13*p + 13))/((12*n + p + 1)*x^(13*n))
Leaf count of result is larger than twice the leaf count of optimal. 690 vs. \(2 (22) = 44\).
Time = 147.77 (sec) , antiderivative size = 690, normalized size of antiderivative = 23.79 \[ \int x^p \left (a x^n+b x^{1+13 n+p}\right )^{12} \, dx=\text {Too large to display} \] Input:
integrate(x**p*(a*x**n+b*x**(1+13*n+p))**12,x)
Output:
Piecewise((13*a**12*x*x**(12*n)*x**p/(156*n + 13*p + 13) + 78*a**11*b*x*x* *(11*n)*x**p*x**(13*n + p + 1)/(156*n + 13*p + 13) + 286*a**10*b**2*x*x**( 10*n)*x**p*x**(26*n + 2*p + 2)/(156*n + 13*p + 13) + 715*a**9*b**3*x*x**(9 *n)*x**p*x**(39*n + 3*p + 3)/(156*n + 13*p + 13) + 1287*a**8*b**4*x*x**(8* n)*x**p*x**(52*n + 4*p + 4)/(156*n + 13*p + 13) + 1716*a**7*b**5*x*x**(7*n )*x**p*x**(65*n + 5*p + 5)/(156*n + 13*p + 13) + 1716*a**6*b**6*x*x**(6*n) *x**p*x**(78*n + 6*p + 6)/(156*n + 13*p + 13) + 1287*a**5*b**7*x*x**(5*n)* x**p*x**(91*n + 7*p + 7)/(156*n + 13*p + 13) + 715*a**4*b**8*x*x**(4*n)*x* *p*x**(104*n + 8*p + 8)/(156*n + 13*p + 13) + 286*a**3*b**9*x*x**(3*n)*x** p*x**(117*n + 9*p + 9)/(156*n + 13*p + 13) + 78*a**2*b**10*x*x**(2*n)*x**p *x**(130*n + 10*p + 10)/(156*n + 13*p + 13) + 13*a*b**11*x*x**n*x**p*x**(1 43*n + 11*p + 11)/(156*n + 13*p + 13) + b**12*x*x**p*x**(156*n + 12*p + 12 )/(156*n + 13*p + 13), Ne(n, -p/12 - 1/12)), (a**12*Piecewise((log(x), Eq( p, 0)), (log(x**p)/p, True)) + 12*a**11*b*Piecewise((log(x), Eq(p, 0)), (l og(x**p)/p, True)) + 66*a**10*b**2*Piecewise((log(x), Eq(p, 0)), (log(x**p )/p, True)) + 220*a**9*b**3*Piecewise((log(x), Eq(p, 0)), (log(x**p)/p, Tr ue)) + 495*a**8*b**4*Piecewise((log(x), Eq(p, 0)), (log(x**p)/p, True)) + 792*a**7*b**5*Piecewise((log(x), Eq(p, 0)), (log(x**p)/p, True)) + 924*a** 6*b**6*Piecewise((log(x), Eq(p, 0)), (log(x**p)/p, True)) + 792*a**5*b**7* Piecewise((log(x), Eq(p, 0)), (log(x**p)/p, True)) + 495*a**4*b**8*Piec...
Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (27) = 54\).
Time = 0.04 (sec) , antiderivative size = 325, normalized size of antiderivative = 11.21 \[ \int x^p \left (a x^n+b x^{1+13 n+p}\right )^{12} \, dx=\frac {b^{12} x^{156 \, n + 13 \, p + 13}}{13 \, {\left (12 \, n + p + 1\right )}} + \frac {a b^{11} x^{144 \, n + 12 \, p + 12}}{12 \, n + p + 1} + \frac {6 \, a^{2} b^{10} x^{132 \, n + 11 \, p + 11}}{12 \, n + p + 1} + \frac {22 \, a^{3} b^{9} x^{120 \, n + 10 \, p + 10}}{12 \, n + p + 1} + \frac {55 \, a^{4} b^{8} x^{108 \, n + 9 \, p + 9}}{12 \, n + p + 1} + \frac {99 \, a^{5} b^{7} x^{96 \, n + 8 \, p + 8}}{12 \, n + p + 1} + \frac {132 \, a^{6} b^{6} x^{84 \, n + 7 \, p + 7}}{12 \, n + p + 1} + \frac {132 \, a^{7} b^{5} x^{72 \, n + 6 \, p + 6}}{12 \, n + p + 1} + \frac {99 \, a^{8} b^{4} x^{60 \, n + 5 \, p + 5}}{12 \, n + p + 1} + \frac {55 \, a^{9} b^{3} x^{48 \, n + 4 \, p + 4}}{12 \, n + p + 1} + \frac {22 \, a^{10} b^{2} x^{36 \, n + 3 \, p + 3}}{12 \, n + p + 1} + \frac {6 \, a^{11} b x^{24 \, n + 2 \, p + 2}}{12 \, n + p + 1} + \frac {a^{12} x^{12 \, n + p + 1}}{12 \, n + p + 1} \] Input:
integrate(x^p*(a*x^n+b*x^(1+13*n+p))^12,x, algorithm="maxima")
Output:
1/13*b^12*x^(156*n + 13*p + 13)/(12*n + p + 1) + a*b^11*x^(144*n + 12*p + 12)/(12*n + p + 1) + 6*a^2*b^10*x^(132*n + 11*p + 11)/(12*n + p + 1) + 22* a^3*b^9*x^(120*n + 10*p + 10)/(12*n + p + 1) + 55*a^4*b^8*x^(108*n + 9*p + 9)/(12*n + p + 1) + 99*a^5*b^7*x^(96*n + 8*p + 8)/(12*n + p + 1) + 132*a^ 6*b^6*x^(84*n + 7*p + 7)/(12*n + p + 1) + 132*a^7*b^5*x^(72*n + 6*p + 6)/( 12*n + p + 1) + 99*a^8*b^4*x^(60*n + 5*p + 5)/(12*n + p + 1) + 55*a^9*b^3* x^(48*n + 4*p + 4)/(12*n + p + 1) + 22*a^10*b^2*x^(36*n + 3*p + 3)/(12*n + p + 1) + 6*a^11*b*x^(24*n + 2*p + 2)/(12*n + p + 1) + a^12*x^(12*n + p + 1)/(12*n + p + 1)
Leaf count of result is larger than twice the leaf count of optimal. 50971 vs. \(2 (27) = 54\).
Time = 1.45 (sec) , antiderivative size = 50971, normalized size of antiderivative = 1757.62 \[ \int x^p \left (a x^n+b x^{1+13 n+p}\right )^{12} \, dx=\text {Too large to display} \] Input:
integrate(x^p*(a*x^n+b*x^(1+13*n+p))^12,x, algorithm="giac")
Output:
(31408819200*a^2*b^10*n^10*p*x*x^(2*n)*x^p*e^(455*n*log(x) + 35*p*log(x) + 35*log(x)) + 331405966080*a^2*b^10*n^9*p^2*x*x^(2*n)*x^p*e^(455*n*log(x) + 35*p*log(x) + 35*log(x)) + 1230778965888*a^2*b^10*n^8*p^3*x*x^(2*n)*x^p* e^(455*n*log(x) + 35*p*log(x) + 35*log(x)) + 2139674600448*a^2*b^10*n^7*p^ 4*x*x^(2*n)*x^p*e^(455*n*log(x) + 35*p*log(x) + 35*log(x)) + 1890383812992 *a^2*b^10*n^6*p^5*x*x^(2*n)*x^p*e^(455*n*log(x) + 35*p*log(x) + 35*log(x)) + 874552702464*a^2*b^10*n^5*p^6*x*x^(2*n)*x^p*e^(455*n*log(x) + 35*p*log( x) + 35*log(x)) + 222844093056*a^2*b^10*n^4*p^7*x*x^(2*n)*x^p*e^(455*n*log (x) + 35*p*log(x) + 35*log(x)) + 32330382336*a^2*b^10*n^3*p^8*x*x^(2*n)*x^ p*e^(455*n*log(x) + 35*p*log(x) + 35*log(x)) + 2661766272*a^2*b^10*n^2*p^9 *x*x^(2*n)*x^p*e^(455*n*log(x) + 35*p*log(x) + 35*log(x)) + 115879680*a^2* b^10*n*p^10*x*x^(2*n)*x^p*e^(455*n*log(x) + 35*p*log(x) + 35*log(x)) + 207 3600*a^2*b^10*p^11*x*x^(2*n)*x^p*e^(455*n*log(x) + 35*p*log(x) + 35*log(x) ) + 3156586329600*a^5*b^7*n^10*p*x*x^(5*n)*x^p*e^(442*n*log(x) + 34*p*log( x) + 34*log(x)) + 33302016570240*a^5*b^7*n^9*p^2*x*x^(5*n)*x^p*e^(442*n*lo g(x) + 34*p*log(x) + 34*log(x)) + 123648483714624*a^5*b^7*n^8*p^3*x*x^(5*n )*x^p*e^(442*n*log(x) + 34*p*log(x) + 34*log(x)) + 214873536791232*a^5*b^7 *n^7*p^4*x*x^(5*n)*x^p*e^(442*n*log(x) + 34*p*log(x) + 34*log(x)) + 189706 686719616*a^5*b^7*n^6*p^5*x*x^(5*n)*x^p*e^(442*n*log(x) + 34*p*log(x) + 34 *log(x)) + 87659938485504*a^5*b^7*n^5*p^6*x*x^(5*n)*x^p*e^(442*n*log(x)...
Time = 10.86 (sec) , antiderivative size = 363, normalized size of antiderivative = 12.52 \[ \int x^p \left (a x^n+b x^{1+13 n+p}\right )^{12} \, dx=\frac {a^{12}\,x\,x^p\,x^{12\,n}}{12\,n+p+1}+\frac {b^{12}\,x^{156\,n}\,x^{13\,p}\,x^{13}}{156\,n+13\,p+13}+\frac {22\,a^{10}\,b^2\,x^{36\,n}\,x^{3\,p}\,x^3}{12\,n+p+1}+\frac {55\,a^9\,b^3\,x^{48\,n}\,x^{4\,p}\,x^4}{12\,n+p+1}+\frac {99\,a^8\,b^4\,x^{60\,n}\,x^{5\,p}\,x^5}{12\,n+p+1}+\frac {132\,a^7\,b^5\,x^{72\,n}\,x^{6\,p}\,x^6}{12\,n+p+1}+\frac {132\,a^6\,b^6\,x^{84\,n}\,x^{7\,p}\,x^7}{12\,n+p+1}+\frac {99\,a^5\,b^7\,x^{96\,n}\,x^{8\,p}\,x^8}{12\,n+p+1}+\frac {55\,a^4\,b^8\,x^{108\,n}\,x^{9\,p}\,x^9}{12\,n+p+1}+\frac {22\,a^3\,b^9\,x^{120\,n}\,x^{10\,p}\,x^{10}}{12\,n+p+1}+\frac {6\,a^2\,b^{10}\,x^{132\,n}\,x^{11\,p}\,x^{11}}{12\,n+p+1}+\frac {6\,a^{11}\,b\,x^{24\,n}\,x^{2\,p}\,x^2}{12\,n+p+1}+\frac {a\,b^{11}\,x^{144\,n}\,x^{12\,p}\,x^{12}}{12\,n+p+1} \] Input:
int(x^p*(a*x^n + b*x^(13*n + p + 1))^12,x)
Output:
(a^12*x*x^p*x^(12*n))/(12*n + p + 1) + (b^12*x^(156*n)*x^(13*p)*x^13)/(156 *n + 13*p + 13) + (22*a^10*b^2*x^(36*n)*x^(3*p)*x^3)/(12*n + p + 1) + (55* a^9*b^3*x^(48*n)*x^(4*p)*x^4)/(12*n + p + 1) + (99*a^8*b^4*x^(60*n)*x^(5*p )*x^5)/(12*n + p + 1) + (132*a^7*b^5*x^(72*n)*x^(6*p)*x^6)/(12*n + p + 1) + (132*a^6*b^6*x^(84*n)*x^(7*p)*x^7)/(12*n + p + 1) + (99*a^5*b^7*x^(96*n) *x^(8*p)*x^8)/(12*n + p + 1) + (55*a^4*b^8*x^(108*n)*x^(9*p)*x^9)/(12*n + p + 1) + (22*a^3*b^9*x^(120*n)*x^(10*p)*x^10)/(12*n + p + 1) + (6*a^2*b^10 *x^(132*n)*x^(11*p)*x^11)/(12*n + p + 1) + (6*a^11*b*x^(24*n)*x^(2*p)*x^2) /(12*n + p + 1) + (a*b^11*x^(144*n)*x^(12*p)*x^12)/(12*n + p + 1)
Time = 0.20 (sec) , antiderivative size = 253, normalized size of antiderivative = 8.72 \[ \int x^p \left (a x^n+b x^{1+13 n+p}\right )^{12} \, dx=\frac {x^{12 n +p} x \left (x^{144 n +12 p} b^{12} x^{12}+13 x^{132 n +11 p} a \,b^{11} x^{11}+78 x^{120 n +10 p} a^{2} b^{10} x^{10}+286 x^{108 n +9 p} a^{3} b^{9} x^{9}+715 x^{96 n +8 p} a^{4} b^{8} x^{8}+1287 x^{84 n +7 p} a^{5} b^{7} x^{7}+1716 x^{72 n +6 p} a^{6} b^{6} x^{6}+1716 x^{60 n +5 p} a^{7} b^{5} x^{5}+1287 x^{48 n +4 p} a^{8} b^{4} x^{4}+715 x^{36 n +3 p} a^{9} b^{3} x^{3}+286 x^{24 n +2 p} a^{10} b^{2} x^{2}+78 x^{12 n +p} a^{11} b x +13 a^{12}\right )}{156 n +13 p +13} \] Input:
int(x^p*(a*x^n+b*x^(1+13*n+p))^12,x)
Output:
(x**(12*n + p)*x*(x**(144*n + 12*p)*b**12*x**12 + 13*x**(132*n + 11*p)*a*b **11*x**11 + 78*x**(120*n + 10*p)*a**2*b**10*x**10 + 286*x**(108*n + 9*p)* a**3*b**9*x**9 + 715*x**(96*n + 8*p)*a**4*b**8*x**8 + 1287*x**(84*n + 7*p) *a**5*b**7*x**7 + 1716*x**(72*n + 6*p)*a**6*b**6*x**6 + 1716*x**(60*n + 5* p)*a**7*b**5*x**5 + 1287*x**(48*n + 4*p)*a**8*b**4*x**4 + 715*x**(36*n + 3 *p)*a**9*b**3*x**3 + 286*x**(24*n + 2*p)*a**10*b**2*x**2 + 78*x**(12*n + p )*a**11*b*x + 13*a**12))/(13*(12*n + p + 1))