Integrand size = 32, antiderivative size = 25 \[ \int x^{-1+n} \left (b+2 c x^n\right ) \left (-a+b x^n+c x^{2 n}\right )^{13} \, dx=\frac {\left (a-b x^n-c x^{2 n}\right )^{14}}{14 n} \] Output:
1/14*(a-b*x^n-c*x^(2*n))^14/n
Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96 \[ \int x^{-1+n} \left (b+2 c x^n\right ) \left (-a+b x^n+c x^{2 n}\right )^{13} \, dx=\frac {\left (-a+x^n \left (b+c x^n\right )\right )^{14}}{14 n} \] Input:
Integrate[x^(-1 + n)*(b + 2*c*x^n)*(-a + b*x^n + c*x^(2*n))^13,x]
Output:
(-a + x^n*(b + c*x^n))^14/(14*n)
Time = 0.19 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.094, Rules used = {1798, 25, 1104}
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.
\(\displaystyle \int x^{n-1} \left (b+2 c x^n\right ) \left (-a+b x^n+c x^{2 n}\right )^{13} \, dx\) |
\(\Big \downarrow \) 1798 |
\(\displaystyle \frac {\int -\left (\left (2 c x^n+b\right ) \left (-b x^n-c x^{2 n}+a\right )^{13}\right )dx^n}{n}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \left (2 c x^n+b\right ) \left (-b x^n-c x^{2 n}+a\right )^{13}dx^n}{n}\) |
\(\Big \downarrow \) 1104 |
\(\displaystyle \frac {\left (a-b x^n-c x^{2 n}\right )^{14}}{14 n}\) |
Input:
Int[x^(-1 + n)*(b + 2*c*x^n)*(-a + b*x^n + c*x^(2*n))^13,x]
Output:
(a - b*x^n - c*x^(2*n))^14/(14*n)
Int[((d_) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol ] :> Simp[d*((a + b*x + c*x^2)^(p + 1)/(b*(p + 1))), x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[2*c*d - b*e, 0]
Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.)*((d_) + ( e_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[1/n Subst[Int[(d + e*x)^q*(a + b *x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && EqQ[n2, 2*n] && EqQ[Simplify[m - n + 1], 0]
Leaf count of result is larger than twice the leaf count of optimal. \(2045\) vs. \(2(23)=46\).
Time = 0.01 (sec) , antiderivative size = 2046, normalized size of antiderivative = 81.84
\[\text {Expression too large to display}\]
Input:
int(x^(-1+n)*(2*c*x^n+b)*(-a+b*x^n+c*x^(2*n))^13,x)
Output:
429/2*a^8/n*(x^n)^12*c^6+13/2*a^2/n*(x^n)^12*b^12+13/2*a^12/n*(x^n)^4*c^2+ 143/2*a^10/n*(x^n)^4*b^4-a^13/n*(x^n)^2*c+13/2*a^12/n*(x^n)^2*b^2+143/2*c^ 10/n*(x^n)^20*a^4+429/2*c^6/n*(x^n)^20*b^8-143*c^9/n*(x^n)^18*a^5+143/2*c^ 4/n*(x^n)^18*b^10+13/2*c^12/n*(x^n)^24*a^2+143/2*c^10/n*(x^n)^24*b^4-26*c^ 11/n*(x^n)^22*a^3+429/2*c^8/n*(x^n)^22*b^6-143*a^9/n*(x^n)^10*c^5+143/2*a^ 4/n*(x^n)^10*b^10+143/2*a^10/n*(x^n)^8*c^4+429/2*a^6/n*(x^n)^8*b^8-26*a^11 /n*(x^n)^6*c^3+429/2*a^8/n*(x^n)^6*b^6-1716/7/n*(x^n)^14*c^7*a^7+6435/2*c^ 8/n*(x^n)^18*a^4*b^2-8580*c^7/n*(x^n)^18*a^3*b^4+6006*c^6/n*(x^n)^18*a^2*b ^6-1287*c^5/n*(x^n)^18*a*b^8-5148*c^7/n*(x^n)^16*a^5*b^2+15015*c^6/n*(x^n) ^16*a^4*b^4+78*b*c^11/n*(x^n)^23*a^2-286*b^3*c^10/n*(x^n)^23*a-78*c^11/n*( x^n)^24*a*b^2+13*a^12*b/n*(x^n)^3*c+715*b*c^9/n*(x^n)^19*a^4-4290*b^3*c^8/ n*(x^n)^19*a^3+5148*b^5*c^7/n*(x^n)^19*a^2-1716*b^7*c^6/n*(x^n)^19*a+1716* b*c^7/n*(x^n)^15*a^6-12012*b^3*c^6/n*(x^n)^15*a^5+18018*b^5*c^5/n*(x^n)^15 *a^4-8580*b^7*c^4/n*(x^n)^15*a^3+1430*b^9*c^3/n*(x^n)^15*a^2-78*b^11*c^2/n *(x^n)^15*a-13*b*c^12/n*(x^n)^25*a-1430*c^9/n*(x^n)^20*a^3*b^2-1716*a^7*b/ n*(x^n)^13*c^6+12012*a^6*b^3/n*(x^n)^13*c^5-18018*a^5*b^5/n*(x^n)^13*c^4+8 580*a^4*b^7/n*(x^n)^13*c^3-1430*a^3*b^9/n*(x^n)^13*c^2+78*a^2*b^11/n*(x^n) ^13*c+286*a^10*b/n*(x^n)^7*c^3-1430*a^9*b^3/n*(x^n)^7*c^2+1287*a^8*b^5/n*( x^n)^7*c-78*a^11*b/n*(x^n)^5*c^2+286*a^10*b^3/n*(x^n)^5*c+429*c^10/n*(x^n) ^22*a^2*b^2-1287*b*c^8/n*(x^n)^17*a^5+8580*b^3*c^7/n*(x^n)^17*a^4-12012...
Leaf count of result is larger than twice the leaf count of optimal. 1299 vs. \(2 (23) = 46\).
Time = 0.10 (sec) , antiderivative size = 1299, normalized size of antiderivative = 51.96 \[ \int x^{-1+n} \left (b+2 c x^n\right ) \left (-a+b x^n+c x^{2 n}\right )^{13} \, dx=\text {Too large to display} \] Input:
integrate(x^(-1+n)*(b+2*c*x^n)*(-a+b*x^n+c*x^(2*n))^13,x, algorithm="frica s")
Output:
1/14*(c^14*x^(28*n) + 14*b*c^13*x^(27*n) - 14*a^13*b*x^n + 7*(13*b^2*c^12 - 2*a*c^13)*x^(26*n) + 182*(2*b^3*c^11 - a*b*c^12)*x^(25*n) + 91*(11*b^4*c ^10 - 12*a*b^2*c^11 + a^2*c^12)*x^(24*n) + 182*(11*b^5*c^9 - 22*a*b^3*c^10 + 6*a^2*b*c^11)*x^(23*n) + 91*(33*b^6*c^8 - 110*a*b^4*c^9 + 66*a^2*b^2*c^ 10 - 4*a^3*c^11)*x^(22*n) + 286*(12*b^7*c^7 - 63*a*b^5*c^8 + 70*a^2*b^3*c^ 9 - 14*a^3*b*c^10)*x^(21*n) + 1001*(3*b^8*c^6 - 24*a*b^6*c^7 + 45*a^2*b^4* c^8 - 20*a^3*b^2*c^9 + a^4*c^10)*x^(20*n) + 2002*(b^9*c^5 - 12*a*b^7*c^6 + 36*a^2*b^5*c^7 - 30*a^3*b^3*c^8 + 5*a^4*b*c^9)*x^(19*n) + 1001*(b^10*c^4 - 18*a*b^8*c^5 + 84*a^2*b^6*c^6 - 120*a^3*b^4*c^7 + 45*a^4*b^2*c^8 - 2*a^5 *c^9)*x^(18*n) + 182*(2*b^11*c^3 - 55*a*b^9*c^4 + 396*a^2*b^7*c^5 - 924*a^ 3*b^5*c^6 + 660*a^4*b^3*c^7 - 99*a^5*b*c^8)*x^(17*n) + 91*(b^12*c^2 - 44*a *b^10*c^3 + 495*a^2*b^8*c^4 - 1848*a^3*b^6*c^5 + 2310*a^4*b^4*c^6 - 792*a^ 5*b^2*c^7 + 33*a^6*c^8)*x^(16*n) + 14*(b^13*c - 78*a*b^11*c^2 + 1430*a^2*b ^9*c^3 - 8580*a^3*b^7*c^4 + 18018*a^4*b^5*c^5 - 12012*a^5*b^3*c^6 + 1716*a ^6*b*c^7)*x^(15*n) + (b^14 - 182*a*b^12*c + 6006*a^2*b^10*c^2 - 60060*a^3* b^8*c^3 + 210210*a^4*b^6*c^4 - 252252*a^5*b^4*c^5 + 84084*a^6*b^2*c^6 - 34 32*a^7*c^7)*x^(14*n) - 14*(a*b^13 - 78*a^2*b^11*c + 1430*a^3*b^9*c^2 - 858 0*a^4*b^7*c^3 + 18018*a^5*b^5*c^4 - 12012*a^6*b^3*c^5 + 1716*a^7*b*c^6)*x^ (13*n) + 91*(a^2*b^12 - 44*a^3*b^10*c + 495*a^4*b^8*c^2 - 1848*a^5*b^6*c^3 + 2310*a^6*b^4*c^4 - 792*a^7*b^2*c^5 + 33*a^8*c^6)*x^(12*n) - 182*(2*a...
Timed out. \[ \int x^{-1+n} \left (b+2 c x^n\right ) \left (-a+b x^n+c x^{2 n}\right )^{13} \, dx=\text {Timed out} \] Input:
integrate(x**(-1+n)*(b+2*c*x**n)*(-a+b*x**n+c*x**(2*n))**13,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 2045 vs. \(2 (23) = 46\).
Time = 0.10 (sec) , antiderivative size = 2045, normalized size of antiderivative = 81.80 \[ \int x^{-1+n} \left (b+2 c x^n\right ) \left (-a+b x^n+c x^{2 n}\right )^{13} \, dx=\text {Too large to display} \] Input:
integrate(x^(-1+n)*(b+2*c*x^n)*(-a+b*x^n+c*x^(2*n))^13,x, algorithm="maxim a")
Output:
1/14*c^14*x^(28*n)/n + b*c^13*x^(27*n)/n + 13/2*b^2*c^12*x^(26*n)/n - a*c^ 13*x^(26*n)/n + 26*b^3*c^11*x^(25*n)/n - 13*a*b*c^12*x^(25*n)/n + 143/2*b^ 4*c^10*x^(24*n)/n - 78*a*b^2*c^11*x^(24*n)/n + 13/2*a^2*c^12*x^(24*n)/n + 143*b^5*c^9*x^(23*n)/n - 286*a*b^3*c^10*x^(23*n)/n + 78*a^2*b*c^11*x^(23*n )/n + 429/2*b^6*c^8*x^(22*n)/n - 715*a*b^4*c^9*x^(22*n)/n + 429*a^2*b^2*c^ 10*x^(22*n)/n - 26*a^3*c^11*x^(22*n)/n + 1716/7*b^7*c^7*x^(21*n)/n - 1287* a*b^5*c^8*x^(21*n)/n + 1430*a^2*b^3*c^9*x^(21*n)/n - 286*a^3*b*c^10*x^(21* n)/n + 429/2*b^8*c^6*x^(20*n)/n - 1716*a*b^6*c^7*x^(20*n)/n + 6435/2*a^2*b ^4*c^8*x^(20*n)/n - 1430*a^3*b^2*c^9*x^(20*n)/n + 143/2*a^4*c^10*x^(20*n)/ n + 143*b^9*c^5*x^(19*n)/n - 1716*a*b^7*c^6*x^(19*n)/n + 5148*a^2*b^5*c^7* x^(19*n)/n - 4290*a^3*b^3*c^8*x^(19*n)/n + 715*a^4*b*c^9*x^(19*n)/n + 143/ 2*b^10*c^4*x^(18*n)/n - 1287*a*b^8*c^5*x^(18*n)/n + 6006*a^2*b^6*c^6*x^(18 *n)/n - 8580*a^3*b^4*c^7*x^(18*n)/n + 6435/2*a^4*b^2*c^8*x^(18*n)/n - 143* a^5*c^9*x^(18*n)/n + 26*b^11*c^3*x^(17*n)/n - 715*a*b^9*c^4*x^(17*n)/n + 5 148*a^2*b^7*c^5*x^(17*n)/n - 12012*a^3*b^5*c^6*x^(17*n)/n + 8580*a^4*b^3*c ^7*x^(17*n)/n - 1287*a^5*b*c^8*x^(17*n)/n + 13/2*b^12*c^2*x^(16*n)/n - 286 *a*b^10*c^3*x^(16*n)/n + 6435/2*a^2*b^8*c^4*x^(16*n)/n - 12012*a^3*b^6*c^5 *x^(16*n)/n + 15015*a^4*b^4*c^6*x^(16*n)/n - 5148*a^5*b^2*c^7*x^(16*n)/n + 429/2*a^6*c^8*x^(16*n)/n + b^13*c*x^(15*n)/n - 78*a*b^11*c^2*x^(15*n)/n + 1430*a^2*b^9*c^3*x^(15*n)/n - 8580*a^3*b^7*c^4*x^(15*n)/n + 18018*a^4*...
Leaf count of result is larger than twice the leaf count of optimal. 1693 vs. \(2 (23) = 46\).
Time = 0.19 (sec) , antiderivative size = 1693, normalized size of antiderivative = 67.72 \[ \int x^{-1+n} \left (b+2 c x^n\right ) \left (-a+b x^n+c x^{2 n}\right )^{13} \, dx=\text {Too large to display} \] Input:
integrate(x^(-1+n)*(b+2*c*x^n)*(-a+b*x^n+c*x^(2*n))^13,x, algorithm="giac" )
Output:
1/14*(c^14*x^(28*n) + 14*b*c^13*x^(27*n) + 91*b^2*c^12*x^(26*n) - 14*a*c^1 3*x^(26*n) + 364*b^3*c^11*x^(25*n) - 182*a*b*c^12*x^(25*n) + 1001*b^4*c^10 *x^(24*n) - 1092*a*b^2*c^11*x^(24*n) + 91*a^2*c^12*x^(24*n) + 2002*b^5*c^9 *x^(23*n) - 4004*a*b^3*c^10*x^(23*n) + 1092*a^2*b*c^11*x^(23*n) + 3003*b^6 *c^8*x^(22*n) - 10010*a*b^4*c^9*x^(22*n) + 6006*a^2*b^2*c^10*x^(22*n) - 36 4*a^3*c^11*x^(22*n) + 3432*b^7*c^7*x^(21*n) - 18018*a*b^5*c^8*x^(21*n) + 2 0020*a^2*b^3*c^9*x^(21*n) - 4004*a^3*b*c^10*x^(21*n) + 3003*b^8*c^6*x^(20* n) - 24024*a*b^6*c^7*x^(20*n) + 45045*a^2*b^4*c^8*x^(20*n) - 20020*a^3*b^2 *c^9*x^(20*n) + 1001*a^4*c^10*x^(20*n) + 2002*b^9*c^5*x^(19*n) - 24024*a*b ^7*c^6*x^(19*n) + 72072*a^2*b^5*c^7*x^(19*n) - 60060*a^3*b^3*c^8*x^(19*n) + 10010*a^4*b*c^9*x^(19*n) + 1001*b^10*c^4*x^(18*n) - 18018*a*b^8*c^5*x^(1 8*n) + 84084*a^2*b^6*c^6*x^(18*n) - 120120*a^3*b^4*c^7*x^(18*n) + 45045*a^ 4*b^2*c^8*x^(18*n) - 2002*a^5*c^9*x^(18*n) + 364*b^11*c^3*x^(17*n) - 10010 *a*b^9*c^4*x^(17*n) + 72072*a^2*b^7*c^5*x^(17*n) - 168168*a^3*b^5*c^6*x^(1 7*n) + 120120*a^4*b^3*c^7*x^(17*n) - 18018*a^5*b*c^8*x^(17*n) + 91*b^12*c^ 2*x^(16*n) - 4004*a*b^10*c^3*x^(16*n) + 45045*a^2*b^8*c^4*x^(16*n) - 16816 8*a^3*b^6*c^5*x^(16*n) + 210210*a^4*b^4*c^6*x^(16*n) - 72072*a^5*b^2*c^7*x ^(16*n) + 3003*a^6*c^8*x^(16*n) + 14*b^13*c*x^(15*n) - 1092*a*b^11*c^2*x^( 15*n) + 20020*a^2*b^9*c^3*x^(15*n) - 120120*a^3*b^7*c^4*x^(15*n) + 252252* a^4*b^5*c^5*x^(15*n) - 168168*a^5*b^3*c^6*x^(15*n) + 24024*a^6*b*c^7*x^...
Time = 26.20 (sec) , antiderivative size = 1401, normalized size of antiderivative = 56.04 \[ \int x^{-1+n} \left (b+2 c x^n\right ) \left (-a+b x^n+c x^{2 n}\right )^{13} \, dx=\text {Too large to display} \] Input:
int(x^(n - 1)*(b + 2*c*x^n)*(b*x^n - a + c*x^(2*n))^13,x)
Output:
x^(n - 1)*((x^(11*n + 1)*((13*a^2*b^12)/2 + (429*a^8*c^6)/2 - 286*a^3*b^10 *c + (6435*a^4*b^8*c^2)/2 - 12012*a^5*b^6*c^3 + 15015*a^6*b^4*c^4 - 5148*a ^7*b^2*c^5))/n + (x^(15*n + 1)*((429*a^6*c^8)/2 + (13*b^12*c^2)/2 - 286*a* b^10*c^3 + (6435*a^2*b^8*c^4)/2 - 12012*a^3*b^6*c^5 + 15015*a^4*b^4*c^6 - 5148*a^5*b^2*c^7))/n - (x^(12*n + 1)*(a*b^13 - 78*a^2*b^11*c + 1716*a^7*b* c^6 + 1430*a^3*b^9*c^2 - 8580*a^4*b^7*c^3 + 18018*a^5*b^5*c^4 - 12012*a^6* b^3*c^5))/n + (x^(14*n + 1)*(b^13*c - 78*a*b^11*c^2 + 1716*a^6*b*c^7 + 143 0*a^2*b^9*c^3 - 8580*a^3*b^7*c^4 + 18018*a^4*b^5*c^5 - 12012*a^5*b^3*c^6)) /n + (x^(5*n + 1)*((429*a^8*b^6)/2 - 26*a^11*c^3 - 715*a^9*b^4*c + 429*a^1 0*b^2*c^2))/n - (x^(21*n + 1)*(26*a^3*c^11 - (429*b^6*c^8)/2 + 715*a*b^4*c ^9 - 429*a^2*b^2*c^10))/n + (x^(9*n + 1)*((143*a^4*b^10)/2 - 143*a^9*c^5 - 1287*a^5*b^8*c + 6006*a^6*b^6*c^2 - 8580*a^7*b^4*c^3 + (6435*a^8*b^2*c^4) /2))/n - (x^(17*n + 1)*(143*a^5*c^9 - (143*b^10*c^4)/2 + 1287*a*b^8*c^5 - 6006*a^2*b^6*c^6 + 8580*a^3*b^4*c^7 - (6435*a^4*b^2*c^8)/2))/n + (x^(13*n + 1)*(b^14/14 - (1716*a^7*c^7)/7 + 429*a^2*b^10*c^2 - 4290*a^3*b^8*c^3 + 1 5015*a^4*b^6*c^4 - 18018*a^5*b^4*c^5 + 6006*a^6*b^2*c^6 - 13*a*b^12*c))/n + (x^(7*n + 1)*((429*a^6*b^8)/2 + (143*a^10*c^4)/2 - 1716*a^7*b^6*c + (643 5*a^8*b^4*c^2)/2 - 1430*a^9*b^2*c^3))/n + (x^(19*n + 1)*((143*a^4*c^10)/2 + (429*b^8*c^6)/2 - 1716*a*b^6*c^7 + (6435*a^2*b^4*c^8)/2 - 1430*a^3*b^2*c ^9))/n + (c^14*x^(27*n + 1))/(14*n) - (a^12*x^(n + 1)*(a*c - (13*b^2)/2...
Time = 0.16 (sec) , antiderivative size = 1689, normalized size of antiderivative = 67.56 \[ \int x^{-1+n} \left (b+2 c x^n\right ) \left (-a+b x^n+c x^{2 n}\right )^{13} \, dx =\text {Too large to display} \] Input:
int(x^(-1+n)*(b+2*c*x^n)*(-a+b*x^n+c*x^(2*n))^13,x)
Output:
(x**n*(x**(27*n)*c**14 + 14*x**(26*n)*b*c**13 - 14*x**(25*n)*a*c**13 + 91* x**(25*n)*b**2*c**12 - 182*x**(24*n)*a*b*c**12 + 364*x**(24*n)*b**3*c**11 + 91*x**(23*n)*a**2*c**12 - 1092*x**(23*n)*a*b**2*c**11 + 1001*x**(23*n)*b **4*c**10 + 1092*x**(22*n)*a**2*b*c**11 - 4004*x**(22*n)*a*b**3*c**10 + 20 02*x**(22*n)*b**5*c**9 - 364*x**(21*n)*a**3*c**11 + 6006*x**(21*n)*a**2*b* *2*c**10 - 10010*x**(21*n)*a*b**4*c**9 + 3003*x**(21*n)*b**6*c**8 - 4004*x **(20*n)*a**3*b*c**10 + 20020*x**(20*n)*a**2*b**3*c**9 - 18018*x**(20*n)*a *b**5*c**8 + 3432*x**(20*n)*b**7*c**7 + 1001*x**(19*n)*a**4*c**10 - 20020* x**(19*n)*a**3*b**2*c**9 + 45045*x**(19*n)*a**2*b**4*c**8 - 24024*x**(19*n )*a*b**6*c**7 + 3003*x**(19*n)*b**8*c**6 + 10010*x**(18*n)*a**4*b*c**9 - 6 0060*x**(18*n)*a**3*b**3*c**8 + 72072*x**(18*n)*a**2*b**5*c**7 - 24024*x** (18*n)*a*b**7*c**6 + 2002*x**(18*n)*b**9*c**5 - 2002*x**(17*n)*a**5*c**9 + 45045*x**(17*n)*a**4*b**2*c**8 - 120120*x**(17*n)*a**3*b**4*c**7 + 84084* x**(17*n)*a**2*b**6*c**6 - 18018*x**(17*n)*a*b**8*c**5 + 1001*x**(17*n)*b* *10*c**4 - 18018*x**(16*n)*a**5*b*c**8 + 120120*x**(16*n)*a**4*b**3*c**7 - 168168*x**(16*n)*a**3*b**5*c**6 + 72072*x**(16*n)*a**2*b**7*c**5 - 10010* x**(16*n)*a*b**9*c**4 + 364*x**(16*n)*b**11*c**3 + 3003*x**(15*n)*a**6*c** 8 - 72072*x**(15*n)*a**5*b**2*c**7 + 210210*x**(15*n)*a**4*b**4*c**6 - 168 168*x**(15*n)*a**3*b**6*c**5 + 45045*x**(15*n)*a**2*b**8*c**4 - 4004*x**(1 5*n)*a*b**10*c**3 + 91*x**(15*n)*b**12*c**2 + 24024*x**(14*n)*a**6*b*c*...