Integrand size = 29, antiderivative size = 21 \[ \int \frac {x^{-7 (-1+n)} \left (b+2 c x^n\right )}{\left (b x+c x^{1+n}\right )^8} \, dx=-\frac {x^{-7 n}}{7 n \left (b+c x^n\right )^7} \] Output:
-1/7/n/(x^(7*n))/(b+c*x^n)^7
Time = 0.01 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {x^{-7 (-1+n)} \left (b+2 c x^n\right )}{\left (b x+c x^{1+n}\right )^8} \, dx=-\frac {x^{-7 n}}{7 n \left (b+c x^n\right )^7} \] Input:
Integrate[(b + 2*c*x^n)/(x^(7*(-1 + n))*(b*x + c*x^(1 + n))^8),x]
Output:
-1/7*1/(n*x^(7*n)*(b + c*x^n)^7)
Time = 0.25 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {10, 948, 83}
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.
\(\displaystyle \int \frac {x^{-7 (n-1)} \left (b+2 c x^n\right )}{\left (b x+c x^{n+1}\right )^8} \, dx\) |
\(\Big \downarrow \) 10 |
\(\displaystyle \int \frac {x^{-7 n-1} \left (b+2 c x^n\right )}{\left (b+c x^n\right )^8}dx\) |
\(\Big \downarrow \) 948 |
\(\displaystyle \frac {\int \frac {x^{-8 n} \left (2 c x^n+b\right )}{\left (c x^n+b\right )^8}dx^n}{n}\) |
\(\Big \downarrow \) 83 |
\(\displaystyle -\frac {x^{-7 n}}{7 n \left (b+c x^n\right )^7}\) |
Input:
Int[(b + 2*c*x^n)/(x^(7*(-1 + n))*(b*x + c*x^(1 + n))^8),x]
Output:
-1/7*1/(n*x^(7*n)*(b + c*x^n)^7)
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[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] && EqQ[a*d*f *(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_. ), x_Symbol] :> Simp[1/n Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^ p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] && NeQ [b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]
Leaf count of result is larger than twice the leaf count of optimal. \(202\) vs. \(2(21)=42\).
Time = 0.01 (sec) , antiderivative size = 203, normalized size of antiderivative = 9.67
\[-\frac {132 c^{6} x^{-n}}{b^{13} n}+\frac {66 c^{5} x^{-2 n}}{b^{12} n}-\frac {30 c^{4} x^{-3 n}}{b^{11} n}+\frac {12 c^{3} x^{-4 n}}{b^{10} n}-\frac {4 c^{2} x^{-5 n}}{b^{9} n}+\frac {c \,x^{-6 n}}{b^{8} n}-\frac {x^{-7 n}}{7 b^{7} n}+\frac {c^{7} \left (924 x^{6 n} c^{6}+6006 b \,c^{5} x^{5 n}+16380 b^{2} c^{4} x^{4 n}+24024 b^{3} c^{3} x^{3 n}+20020 b^{4} c^{2} x^{2 n}+9009 b^{5} c \,x^{n}+1716 b^{6}\right )}{7 b^{13} n \left (b +c \,x^{n}\right )^{7}}\]
Input:
int((2*c*x^n+b)/(x^(-7+7*n))/(b*x+c*x^(1+n))^8,x)
Output:
-132/b^13*c^6/n/(x^n)+66/b^12*c^5/n/(x^n)^2-30/b^11*c^4/n/(x^n)^3+12/b^10* c^3/n/(x^n)^4-4/b^9*c^2/n/(x^n)^5+1/b^8*c/n/(x^n)^6-1/7/b^7/n/(x^n)^7+1/7* c^7*(924*(x^n)^6*c^6+6006*b*c^5*(x^n)^5+16380*b^2*c^4*(x^n)^4+24024*b^3*c^ 3*(x^n)^3+20020*b^4*c^2*(x^n)^2+9009*b^5*c*x^n+1716*b^6)/b^13/n/(b+c*x^n)^ 7
Leaf count of result is larger than twice the leaf count of optimal. 143 vs. \(2 (21) = 42\).
Time = 0.20 (sec) , antiderivative size = 143, normalized size of antiderivative = 6.81 \[ \int \frac {x^{-7 (-1+n)} \left (b+2 c x^n\right )}{\left (b x+c x^{1+n}\right )^8} \, dx=-\frac {x^{14}}{7 \, {\left (b^{7} n x^{7} x^{7 \, n + 7} + 7 \, b^{6} c n x^{6} x^{8 \, n + 8} + 21 \, b^{5} c^{2} n x^{5} x^{9 \, n + 9} + 35 \, b^{4} c^{3} n x^{4} x^{10 \, n + 10} + 35 \, b^{3} c^{4} n x^{3} x^{11 \, n + 11} + 21 \, b^{2} c^{5} n x^{2} x^{12 \, n + 12} + 7 \, b c^{6} n x x^{13 \, n + 13} + c^{7} n x^{14 \, n + 14}\right )}} \] Input:
integrate((b+2*c*x^n)/(x^(-7+7*n))/(b*x+c*x^(1+n))^8,x, algorithm="fricas" )
Output:
-1/7*x^14/(b^7*n*x^7*x^(7*n + 7) + 7*b^6*c*n*x^6*x^(8*n + 8) + 21*b^5*c^2* n*x^5*x^(9*n + 9) + 35*b^4*c^3*n*x^4*x^(10*n + 10) + 35*b^3*c^4*n*x^3*x^(1 1*n + 11) + 21*b^2*c^5*n*x^2*x^(12*n + 12) + 7*b*c^6*n*x*x^(13*n + 13) + c ^7*n*x^(14*n + 14))
Timed out. \[ \int \frac {x^{-7 (-1+n)} \left (b+2 c x^n\right )}{\left (b x+c x^{1+n}\right )^8} \, dx=\text {Timed out} \] Input:
integrate((b+2*c*x**n)/(x**(-7+7*n))/(b*x+c*x**(1+n))**8,x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 612 vs. \(2 (21) = 42\).
Time = 0.08 (sec) , antiderivative size = 612, normalized size of antiderivative = 29.14 \[ \int \frac {x^{-7 (-1+n)} \left (b+2 c x^n\right )}{\left (b x+c x^{1+n}\right )^8} \, dx=-\frac {1}{105} \, b {\left (\frac {360360 \, c^{13} x^{13 \, n} + 2342340 \, b c^{12} x^{12 \, n} + 6426420 \, b^{2} c^{11} x^{11 \, n} + 9579570 \, b^{3} c^{10} x^{10 \, n} + 8270262 \, b^{4} c^{9} x^{9 \, n} + 4018014 \, b^{5} c^{8} x^{8 \, n} + 934362 \, b^{6} c^{7} x^{7 \, n} + 45045 \, b^{7} c^{6} x^{6 \, n} - 5005 \, b^{8} c^{5} x^{5 \, n} + 1001 \, b^{9} c^{4} x^{4 \, n} - 273 \, b^{10} c^{3} x^{3 \, n} + 91 \, b^{11} c^{2} x^{2 \, n} - 35 \, b^{12} c x^{n} + 15 \, b^{13}}{b^{14} c^{7} n x^{14 \, n} + 7 \, b^{15} c^{6} n x^{13 \, n} + 21 \, b^{16} c^{5} n x^{12 \, n} + 35 \, b^{17} c^{4} n x^{11 \, n} + 35 \, b^{18} c^{3} n x^{10 \, n} + 21 \, b^{19} c^{2} n x^{9 \, n} + 7 \, b^{20} c n x^{8 \, n} + b^{21} n x^{7 \, n}} + \frac {360360 \, c^{7} \log \left (x\right )}{b^{15}} - \frac {360360 \, c^{7} \log \left (\frac {c x^{n} + b}{c}\right )}{b^{15} n}\right )} + \frac {1}{105} \, c {\left (\frac {360360 \, c^{12} x^{12 \, n} + 2342340 \, b c^{11} x^{11 \, n} + 6426420 \, b^{2} c^{10} x^{10 \, n} + 9579570 \, b^{3} c^{9} x^{9 \, n} + 8270262 \, b^{4} c^{8} x^{8 \, n} + 4018014 \, b^{5} c^{7} x^{7 \, n} + 934362 \, b^{6} c^{6} x^{6 \, n} + 45045 \, b^{7} c^{5} x^{5 \, n} - 5005 \, b^{8} c^{4} x^{4 \, n} + 1001 \, b^{9} c^{3} x^{3 \, n} - 273 \, b^{10} c^{2} x^{2 \, n} + 91 \, b^{11} c x^{n} - 35 \, b^{12}}{b^{13} c^{7} n x^{13 \, n} + 7 \, b^{14} c^{6} n x^{12 \, n} + 21 \, b^{15} c^{5} n x^{11 \, n} + 35 \, b^{16} c^{4} n x^{10 \, n} + 35 \, b^{17} c^{3} n x^{9 \, n} + 21 \, b^{18} c^{2} n x^{8 \, n} + 7 \, b^{19} c n x^{7 \, n} + b^{20} n x^{6 \, n}} + \frac {360360 \, c^{6} \log \left (x\right )}{b^{14}} - \frac {360360 \, c^{6} \log \left (\frac {c x^{n} + b}{c}\right )}{b^{14} n}\right )} \] Input:
integrate((b+2*c*x^n)/(x^(-7+7*n))/(b*x+c*x^(1+n))^8,x, algorithm="maxima" )
Output:
-1/105*b*((360360*c^13*x^(13*n) + 2342340*b*c^12*x^(12*n) + 6426420*b^2*c^ 11*x^(11*n) + 9579570*b^3*c^10*x^(10*n) + 8270262*b^4*c^9*x^(9*n) + 401801 4*b^5*c^8*x^(8*n) + 934362*b^6*c^7*x^(7*n) + 45045*b^7*c^6*x^(6*n) - 5005* b^8*c^5*x^(5*n) + 1001*b^9*c^4*x^(4*n) - 273*b^10*c^3*x^(3*n) + 91*b^11*c^ 2*x^(2*n) - 35*b^12*c*x^n + 15*b^13)/(b^14*c^7*n*x^(14*n) + 7*b^15*c^6*n*x ^(13*n) + 21*b^16*c^5*n*x^(12*n) + 35*b^17*c^4*n*x^(11*n) + 35*b^18*c^3*n* x^(10*n) + 21*b^19*c^2*n*x^(9*n) + 7*b^20*c*n*x^(8*n) + b^21*n*x^(7*n)) + 360360*c^7*log(x)/b^15 - 360360*c^7*log((c*x^n + b)/c)/(b^15*n)) + 1/105*c *((360360*c^12*x^(12*n) + 2342340*b*c^11*x^(11*n) + 6426420*b^2*c^10*x^(10 *n) + 9579570*b^3*c^9*x^(9*n) + 8270262*b^4*c^8*x^(8*n) + 4018014*b^5*c^7* x^(7*n) + 934362*b^6*c^6*x^(6*n) + 45045*b^7*c^5*x^(5*n) - 5005*b^8*c^4*x^ (4*n) + 1001*b^9*c^3*x^(3*n) - 273*b^10*c^2*x^(2*n) + 91*b^11*c*x^n - 35*b ^12)/(b^13*c^7*n*x^(13*n) + 7*b^14*c^6*n*x^(12*n) + 21*b^15*c^5*n*x^(11*n) + 35*b^16*c^4*n*x^(10*n) + 35*b^17*c^3*n*x^(9*n) + 21*b^18*c^2*n*x^(8*n) + 7*b^19*c*n*x^(7*n) + b^20*n*x^(6*n)) + 360360*c^6*log(x)/b^14 - 360360*c ^6*log((c*x^n + b)/c)/(b^14*n))
\[ \int \frac {x^{-7 (-1+n)} \left (b+2 c x^n\right )}{\left (b x+c x^{1+n}\right )^8} \, dx=\int { \frac {2 \, c x^{n} + b}{{\left (b x + c x^{n + 1}\right )}^{8} x^{7 \, n - 7}} \,d x } \] Input:
integrate((b+2*c*x^n)/(x^(-7+7*n))/(b*x+c*x^(1+n))^8,x, algorithm="giac")
Output:
integrate((2*c*x^n + b)/((b*x + c*x^(n + 1))^8*x^(7*n - 7)), x)
Timed out. \[ \int \frac {x^{-7 (-1+n)} \left (b+2 c x^n\right )}{\left (b x+c x^{1+n}\right )^8} \, dx=\int \frac {x^{7-7\,n}\,\left (b+2\,c\,x^n\right )}{{\left (b\,x+c\,x^{n+1}\right )}^8} \,d x \] Input:
int((x^(7 - 7*n)*(b + 2*c*x^n))/(b*x + c*x^(n + 1))^8,x)
Output:
int((x^(7 - 7*n)*(b + 2*c*x^n))/(b*x + c*x^(n + 1))^8, x)
Time = 0.18 (sec) , antiderivative size = 99, normalized size of antiderivative = 4.71 \[ \int \frac {x^{-7 (-1+n)} \left (b+2 c x^n\right )}{\left (b x+c x^{1+n}\right )^8} \, dx=-\frac {1}{7 x^{7 n} n \left (x^{7 n} c^{7}+7 x^{6 n} b \,c^{6}+21 x^{5 n} b^{2} c^{5}+35 x^{4 n} b^{3} c^{4}+35 x^{3 n} b^{4} c^{3}+21 x^{2 n} b^{5} c^{2}+7 x^{n} b^{6} c +b^{7}\right )} \] Input:
int((b+2*c*x^n)/(x^(-7+7*n))/(b*x+c*x^(1+n))^8,x)
Output:
( - 1)/(7*x**(7*n)*n*(x**(7*n)*c**7 + 7*x**(6*n)*b*c**6 + 21*x**(5*n)*b**2 *c**5 + 35*x**(4*n)*b**3*c**4 + 35*x**(3*n)*b**4*c**3 + 21*x**(2*n)*b**5*c **2 + 7*x**n*b**6*c + b**7))