Integrand size = 32, antiderivative size = 25 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (-a+b x^n+c x^{2 n}\right )^8} \, dx=\frac {1}{7 n \left (a-b x^n-c x^{2 n}\right )^7} \] Output:
1/7/n/(a-b*x^n-c*x^(2*n))^7
Time = 0.09 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (-a+b x^n+c x^{2 n}\right )^8} \, dx=\frac {1}{7 n \left (a-x^n \left (b+c x^n\right )\right )^7} \] Input:
Integrate[(x^(-1 + n)*(b + 2*c*x^n))/(-a + b*x^n + c*x^(2*n))^8,x]
Output:
1/(7*n*(a - x^n*(b + c*x^n))^7)
Time = 0.18 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {1798, 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.
\(\displaystyle \int \frac {x^{n-1} \left (b+2 c x^n\right )}{\left (-a+b x^n+c x^{2 n}\right )^8} \, dx\) |
\(\Big \downarrow \) 1798 |
\(\displaystyle \frac {\int \frac {2 c x^n+b}{\left (-b x^n-c x^{2 n}+a\right )^8}dx^n}{n}\) |
\(\Big \downarrow \) 1104 |
\(\displaystyle \frac {1}{7 n \left (a-b x^n-c x^{2 n}\right )^7}\) |
Input:
Int[(x^(-1 + n)*(b + 2*c*x^n))/(-a + b*x^n + c*x^(2*n))^8,x]
Output:
1/(7*n*(a - b*x^n - c*x^(2*n))^7)
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]
Time = 224.96 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.96
Input:
int(x^(-1+n)*(2*c*x^n+b)/(-a+b*x^n+c*x^(2*n))^8,x,method=_RETURNVERBOSE)
Output:
1/7/n/(-c*(x^n)^2-b*x^n+a)^7
Leaf count of result is larger than twice the leaf count of optimal. 397 vs. \(2 (23) = 46\).
Time = 0.10 (sec) , antiderivative size = 397, normalized size of antiderivative = 15.88 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (-a+b x^n+c x^{2 n}\right )^8} \, dx=-\frac {1}{7 \, {\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 7 \, a^{6} b n x^{n} - a^{7} n + 7 \, {\left (3 \, b^{2} c^{5} - a c^{6}\right )} n x^{12 \, n} + 7 \, {\left (5 \, b^{3} c^{4} - 6 \, a b c^{5}\right )} n x^{11 \, n} + 7 \, {\left (5 \, b^{4} c^{3} - 15 \, a b^{2} c^{4} + 3 \, a^{2} c^{5}\right )} n x^{10 \, n} + 7 \, {\left (3 \, b^{5} c^{2} - 20 \, a b^{3} c^{3} + 15 \, a^{2} b c^{4}\right )} n x^{9 \, n} + 7 \, {\left (b^{6} c - 15 \, a b^{4} c^{2} + 30 \, a^{2} b^{2} c^{3} - 5 \, a^{3} c^{4}\right )} n x^{8 \, n} + {\left (b^{7} - 42 \, a b^{5} c + 210 \, a^{2} b^{3} c^{2} - 140 \, a^{3} b c^{3}\right )} n x^{7 \, n} - 7 \, {\left (a b^{6} - 15 \, a^{2} b^{4} c + 30 \, a^{3} b^{2} c^{2} - 5 \, a^{4} c^{3}\right )} n x^{6 \, n} + 7 \, {\left (3 \, a^{2} b^{5} - 20 \, a^{3} b^{3} c + 15 \, a^{4} b c^{2}\right )} n x^{5 \, n} - 7 \, {\left (5 \, a^{3} b^{4} - 15 \, a^{4} b^{2} c + 3 \, a^{5} c^{2}\right )} n x^{4 \, n} + 7 \, {\left (5 \, a^{4} b^{3} - 6 \, a^{5} b c\right )} n x^{3 \, n} - 7 \, {\left (3 \, a^{5} b^{2} - a^{6} c\right )} n x^{2 \, n}\right )}} \] Input:
integrate(x^(-1+n)*(b+2*c*x^n)/(-a+b*x^n+c*x^(2*n))^8,x, algorithm="fricas ")
Output:
-1/7/(c^7*n*x^(14*n) + 7*b*c^6*n*x^(13*n) + 7*a^6*b*n*x^n - a^7*n + 7*(3*b ^2*c^5 - a*c^6)*n*x^(12*n) + 7*(5*b^3*c^4 - 6*a*b*c^5)*n*x^(11*n) + 7*(5*b ^4*c^3 - 15*a*b^2*c^4 + 3*a^2*c^5)*n*x^(10*n) + 7*(3*b^5*c^2 - 20*a*b^3*c^ 3 + 15*a^2*b*c^4)*n*x^(9*n) + 7*(b^6*c - 15*a*b^4*c^2 + 30*a^2*b^2*c^3 - 5 *a^3*c^4)*n*x^(8*n) + (b^7 - 42*a*b^5*c + 210*a^2*b^3*c^2 - 140*a^3*b*c^3) *n*x^(7*n) - 7*(a*b^6 - 15*a^2*b^4*c + 30*a^3*b^2*c^2 - 5*a^4*c^3)*n*x^(6* n) + 7*(3*a^2*b^5 - 20*a^3*b^3*c + 15*a^4*b*c^2)*n*x^(5*n) - 7*(5*a^3*b^4 - 15*a^4*b^2*c + 3*a^5*c^2)*n*x^(4*n) + 7*(5*a^4*b^3 - 6*a^5*b*c)*n*x^(3*n ) - 7*(3*a^5*b^2 - a^6*c)*n*x^(2*n))
Timed out. \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (-a+b x^n+c x^{2 n}\right )^8} \, dx=\text {Timed out} \] Input:
integrate(x**(-1+n)*(b+2*c*x**n)/(-a+b*x**n+c*x**(2*n))**8,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (23) = 46\).
Time = 0.38 (sec) , antiderivative size = 419, normalized size of antiderivative = 16.76 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (-a+b x^n+c x^{2 n}\right )^8} \, dx=-\frac {1}{7 \, {\left (c^{7} n x^{14 \, n} + 7 \, b c^{6} n x^{13 \, n} + 7 \, a^{6} b n x^{n} - a^{7} n + 7 \, {\left (3 \, b^{2} c^{5} n - a c^{6} n\right )} x^{12 \, n} + 7 \, {\left (5 \, b^{3} c^{4} n - 6 \, a b c^{5} n\right )} x^{11 \, n} + 7 \, {\left (5 \, b^{4} c^{3} n - 15 \, a b^{2} c^{4} n + 3 \, a^{2} c^{5} n\right )} x^{10 \, n} + 7 \, {\left (3 \, b^{5} c^{2} n - 20 \, a b^{3} c^{3} n + 15 \, a^{2} b c^{4} n\right )} x^{9 \, n} + 7 \, {\left (b^{6} c n - 15 \, a b^{4} c^{2} n + 30 \, a^{2} b^{2} c^{3} n - 5 \, a^{3} c^{4} n\right )} x^{8 \, n} + {\left (b^{7} n - 42 \, a b^{5} c n + 210 \, a^{2} b^{3} c^{2} n - 140 \, a^{3} b c^{3} n\right )} x^{7 \, n} - 7 \, {\left (a b^{6} n - 15 \, a^{2} b^{4} c n + 30 \, a^{3} b^{2} c^{2} n - 5 \, a^{4} c^{3} n\right )} x^{6 \, n} + 7 \, {\left (3 \, a^{2} b^{5} n - 20 \, a^{3} b^{3} c n + 15 \, a^{4} b c^{2} n\right )} x^{5 \, n} - 7 \, {\left (5 \, a^{3} b^{4} n - 15 \, a^{4} b^{2} c n + 3 \, a^{5} c^{2} n\right )} x^{4 \, n} + 7 \, {\left (5 \, a^{4} b^{3} n - 6 \, a^{5} b c n\right )} x^{3 \, n} - 7 \, {\left (3 \, a^{5} b^{2} n - a^{6} c n\right )} x^{2 \, n}\right )}} \] Input:
integrate(x^(-1+n)*(b+2*c*x^n)/(-a+b*x^n+c*x^(2*n))^8,x, algorithm="maxima ")
Output:
-1/7/(c^7*n*x^(14*n) + 7*b*c^6*n*x^(13*n) + 7*a^6*b*n*x^n - a^7*n + 7*(3*b ^2*c^5*n - a*c^6*n)*x^(12*n) + 7*(5*b^3*c^4*n - 6*a*b*c^5*n)*x^(11*n) + 7* (5*b^4*c^3*n - 15*a*b^2*c^4*n + 3*a^2*c^5*n)*x^(10*n) + 7*(3*b^5*c^2*n - 2 0*a*b^3*c^3*n + 15*a^2*b*c^4*n)*x^(9*n) + 7*(b^6*c*n - 15*a*b^4*c^2*n + 30 *a^2*b^2*c^3*n - 5*a^3*c^4*n)*x^(8*n) + (b^7*n - 42*a*b^5*c*n + 210*a^2*b^ 3*c^2*n - 140*a^3*b*c^3*n)*x^(7*n) - 7*(a*b^6*n - 15*a^2*b^4*c*n + 30*a^3* b^2*c^2*n - 5*a^4*c^3*n)*x^(6*n) + 7*(3*a^2*b^5*n - 20*a^3*b^3*c*n + 15*a^ 4*b*c^2*n)*x^(5*n) - 7*(5*a^3*b^4*n - 15*a^4*b^2*c*n + 3*a^5*c^2*n)*x^(4*n ) + 7*(5*a^4*b^3*n - 6*a^5*b*c*n)*x^(3*n) - 7*(3*a^5*b^2*n - a^6*c*n)*x^(2 *n))
Time = 0.14 (sec) , antiderivative size = 23, normalized size of antiderivative = 0.92 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (-a+b x^n+c x^{2 n}\right )^8} \, dx=-\frac {1}{7 \, {\left (c x^{2 \, n} + b x^{n} - a\right )}^{7} n} \] Input:
integrate(x^(-1+n)*(b+2*c*x^n)/(-a+b*x^n+c*x^(2*n))^8,x, algorithm="giac")
Output:
-1/7/((c*x^(2*n) + b*x^n - a)^7*n)
Time = 52.26 (sec) , antiderivative size = 496, normalized size of antiderivative = 19.84 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (-a+b x^n+c x^{2 n}\right )^8} \, dx=-\frac {1}{7\,b^7\,n\,x^{7\,n}-7\,a^7\,n+7\,c^7\,n\,x^{14\,n}+49\,a^6\,b\,n\,x^n-49\,a\,b^6\,n\,x^{6\,n}+49\,a^6\,c\,n\,x^{2\,n}-49\,a\,c^6\,n\,x^{12\,n}+49\,b^6\,c\,n\,x^{8\,n}+49\,b\,c^6\,n\,x^{13\,n}-147\,a^5\,b^2\,n\,x^{2\,n}+245\,a^4\,b^3\,n\,x^{3\,n}-245\,a^3\,b^4\,n\,x^{4\,n}+147\,a^2\,b^5\,n\,x^{5\,n}-147\,a^5\,c^2\,n\,x^{4\,n}+245\,a^4\,c^3\,n\,x^{6\,n}-245\,a^3\,c^4\,n\,x^{8\,n}+147\,a^2\,c^5\,n\,x^{10\,n}+147\,b^5\,c^2\,n\,x^{9\,n}+245\,b^4\,c^3\,n\,x^{10\,n}+245\,b^3\,c^4\,n\,x^{11\,n}+147\,b^2\,c^5\,n\,x^{12\,n}+735\,a^4\,b^2\,c\,n\,x^{4\,n}-980\,a^3\,b^3\,c\,n\,x^{5\,n}+735\,a^4\,b\,c^2\,n\,x^{5\,n}+735\,a^2\,b^4\,c\,n\,x^{6\,n}-980\,a^3\,b\,c^3\,n\,x^{7\,n}-735\,a\,b^4\,c^2\,n\,x^{8\,n}-980\,a\,b^3\,c^3\,n\,x^{9\,n}+735\,a^2\,b\,c^4\,n\,x^{9\,n}-735\,a\,b^2\,c^4\,n\,x^{10\,n}-1470\,a^3\,b^2\,c^2\,n\,x^{6\,n}+1470\,a^2\,b^3\,c^2\,n\,x^{7\,n}+1470\,a^2\,b^2\,c^3\,n\,x^{8\,n}-294\,a^5\,b\,c\,n\,x^{3\,n}-294\,a\,b^5\,c\,n\,x^{7\,n}-294\,a\,b\,c^5\,n\,x^{11\,n}} \] Input:
int((x^(n - 1)*(b + 2*c*x^n))/(b*x^n - a + c*x^(2*n))^8,x)
Output:
-1/(7*b^7*n*x^(7*n) - 7*a^7*n + 7*c^7*n*x^(14*n) + 49*a^6*b*n*x^n - 49*a*b ^6*n*x^(6*n) + 49*a^6*c*n*x^(2*n) - 49*a*c^6*n*x^(12*n) + 49*b^6*c*n*x^(8* n) + 49*b*c^6*n*x^(13*n) - 147*a^5*b^2*n*x^(2*n) + 245*a^4*b^3*n*x^(3*n) - 245*a^3*b^4*n*x^(4*n) + 147*a^2*b^5*n*x^(5*n) - 147*a^5*c^2*n*x^(4*n) + 2 45*a^4*c^3*n*x^(6*n) - 245*a^3*c^4*n*x^(8*n) + 147*a^2*c^5*n*x^(10*n) + 14 7*b^5*c^2*n*x^(9*n) + 245*b^4*c^3*n*x^(10*n) + 245*b^3*c^4*n*x^(11*n) + 14 7*b^2*c^5*n*x^(12*n) + 735*a^4*b^2*c*n*x^(4*n) - 980*a^3*b^3*c*n*x^(5*n) + 735*a^4*b*c^2*n*x^(5*n) + 735*a^2*b^4*c*n*x^(6*n) - 980*a^3*b*c^3*n*x^(7* n) - 735*a*b^4*c^2*n*x^(8*n) - 980*a*b^3*c^3*n*x^(9*n) + 735*a^2*b*c^4*n*x ^(9*n) - 735*a*b^2*c^4*n*x^(10*n) - 1470*a^3*b^2*c^2*n*x^(6*n) + 1470*a^2* b^3*c^2*n*x^(7*n) + 1470*a^2*b^2*c^3*n*x^(8*n) - 294*a^5*b*c*n*x^(3*n) - 2 94*a*b^5*c*n*x^(7*n) - 294*a*b*c^5*n*x^(11*n))
Time = 0.18 (sec) , antiderivative size = 461, normalized size of antiderivative = 18.44 \[ \int \frac {x^{-1+n} \left (b+2 c x^n\right )}{\left (-a+b x^n+c x^{2 n}\right )^8} \, dx=-\frac {1}{7 n \left (-a^{7}+7 x^{13 n} b \,c^{6}-7 x^{12 n} a \,c^{6}+21 x^{12 n} b^{2} c^{5}+35 x^{11 n} b^{3} c^{4}+21 x^{10 n} a^{2} c^{5}+35 x^{10 n} b^{4} c^{3}+21 x^{9 n} b^{5} c^{2}-35 x^{8 n} a^{3} c^{4}+7 x^{8 n} b^{6} c +35 x^{6 n} a^{4} c^{3}-7 x^{6 n} a \,b^{6}+21 x^{5 n} a^{2} b^{5}-21 x^{4 n} a^{5} c^{2}-35 x^{4 n} a^{3} b^{4}+35 x^{3 n} a^{4} b^{3}+7 x^{2 n} a^{6} c -21 x^{2 n} a^{5} b^{2}+7 x^{n} a^{6} b +x^{14 n} c^{7}+x^{7 n} b^{7}-42 x^{11 n} a b \,c^{5}-105 x^{10 n} a \,b^{2} c^{4}+105 x^{9 n} a^{2} b \,c^{4}-140 x^{9 n} a \,b^{3} c^{3}+210 x^{8 n} a^{2} b^{2} c^{3}-105 x^{8 n} a \,b^{4} c^{2}-140 x^{7 n} a^{3} b \,c^{3}+210 x^{7 n} a^{2} b^{3} c^{2}-42 x^{7 n} a \,b^{5} c -210 x^{6 n} a^{3} b^{2} c^{2}+105 x^{6 n} a^{2} b^{4} c +105 x^{5 n} a^{4} b \,c^{2}-140 x^{5 n} a^{3} b^{3} c +105 x^{4 n} a^{4} b^{2} c -42 x^{3 n} a^{5} b c \right )} \] Input:
int(x^(-1+n)*(b+2*c*x^n)/(-a+b*x^n+c*x^(2*n))^8,x)
Output:
( - 1)/(7*n*(x**(14*n)*c**7 + 7*x**(13*n)*b*c**6 - 7*x**(12*n)*a*c**6 + 21 *x**(12*n)*b**2*c**5 - 42*x**(11*n)*a*b*c**5 + 35*x**(11*n)*b**3*c**4 + 21 *x**(10*n)*a**2*c**5 - 105*x**(10*n)*a*b**2*c**4 + 35*x**(10*n)*b**4*c**3 + 105*x**(9*n)*a**2*b*c**4 - 140*x**(9*n)*a*b**3*c**3 + 21*x**(9*n)*b**5*c **2 - 35*x**(8*n)*a**3*c**4 + 210*x**(8*n)*a**2*b**2*c**3 - 105*x**(8*n)*a *b**4*c**2 + 7*x**(8*n)*b**6*c - 140*x**(7*n)*a**3*b*c**3 + 210*x**(7*n)*a **2*b**3*c**2 - 42*x**(7*n)*a*b**5*c + x**(7*n)*b**7 + 35*x**(6*n)*a**4*c* *3 - 210*x**(6*n)*a**3*b**2*c**2 + 105*x**(6*n)*a**2*b**4*c - 7*x**(6*n)*a *b**6 + 105*x**(5*n)*a**4*b*c**2 - 140*x**(5*n)*a**3*b**3*c + 21*x**(5*n)* a**2*b**5 - 21*x**(4*n)*a**5*c**2 + 105*x**(4*n)*a**4*b**2*c - 35*x**(4*n) *a**3*b**4 - 42*x**(3*n)*a**5*b*c + 35*x**(3*n)*a**4*b**3 + 7*x**(2*n)*a** 6*c - 21*x**(2*n)*a**5*b**2 + 7*x**n*a**6*b - a**7))