Integrand size = 13, antiderivative size = 72 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^5} \, dx=\frac {a^3 \left (a+\frac {b}{x}\right )^9}{9 b^4}-\frac {3 a^2 \left (a+\frac {b}{x}\right )^{10}}{10 b^4}+\frac {3 a \left (a+\frac {b}{x}\right )^{11}}{11 b^4}-\frac {\left (a+\frac {b}{x}\right )^{12}}{12 b^4} \] Output:
1/9*a^3*(a+b/x)^9/b^4-3/10*a^2*(a+b/x)^10/b^4+3/11*a*(a+b/x)^11/b^4-1/12*( a+b/x)^12/b^4
Time = 0.00 (sec) , antiderivative size = 106, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^5} \, dx=-\frac {b^8}{12 x^{12}}-\frac {8 a b^7}{11 x^{11}}-\frac {14 a^2 b^6}{5 x^{10}}-\frac {56 a^3 b^5}{9 x^9}-\frac {35 a^4 b^4}{4 x^8}-\frac {8 a^5 b^3}{x^7}-\frac {14 a^6 b^2}{3 x^6}-\frac {8 a^7 b}{5 x^5}-\frac {a^8}{4 x^4} \] Input:
Integrate[(a + b/x)^8/x^5,x]
Output:
-1/12*b^8/x^12 - (8*a*b^7)/(11*x^11) - (14*a^2*b^6)/(5*x^10) - (56*a^3*b^5 )/(9*x^9) - (35*a^4*b^4)/(4*x^8) - (8*a^5*b^3)/x^7 - (14*a^6*b^2)/(3*x^6) - (8*a^7*b)/(5*x^5) - a^8/(4*x^4)
Time = 0.29 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.22, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {795, 55, 55, 55, 48}
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.
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^8}{x^5} \, dx\) |
\(\Big \downarrow \) 795 |
\(\displaystyle \int \frac {(a x+b)^8}{x^{13}}dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {a \int \frac {(b+a x)^8}{x^{12}}dx}{4 b}-\frac {(a x+b)^9}{12 b x^{12}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {a \left (-\frac {2 a \int \frac {(b+a x)^8}{x^{11}}dx}{11 b}-\frac {(a x+b)^9}{11 b x^{11}}\right )}{4 b}-\frac {(a x+b)^9}{12 b x^{12}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {a \left (-\frac {2 a \left (-\frac {a \int \frac {(b+a x)^8}{x^{10}}dx}{10 b}-\frac {(a x+b)^9}{10 b x^{10}}\right )}{11 b}-\frac {(a x+b)^9}{11 b x^{11}}\right )}{4 b}-\frac {(a x+b)^9}{12 b x^{12}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {a \left (-\frac {2 a \left (\frac {a (a x+b)^9}{90 b^2 x^9}-\frac {(a x+b)^9}{10 b x^{10}}\right )}{11 b}-\frac {(a x+b)^9}{11 b x^{11}}\right )}{4 b}-\frac {(a x+b)^9}{12 b x^{12}}\) |
Input:
Int[(a + b/x)^8/x^5,x]
Output:
-1/12*(b + a*x)^9/(b*x^12) - (a*(-1/11*(b + a*x)^9/(b*x^11) - (2*a*(-1/10* (b + a*x)^9/(b*x^10) + (a*(b + a*x)^9)/(90*b^2*x^9)))/(11*b)))/(4*b)
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp [(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)* (b + a/x^n)^p, x] /; FreeQ[{a, b, m, n}, x] && IntegerQ[p] && NegQ[n]
Time = 0.05 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.25
method | result | size |
norman | \(\frac {-\frac {1}{4} a^{8} x^{8}-\frac {8}{5} a^{7} b \,x^{7}-\frac {14}{3} a^{6} b^{2} x^{6}-8 a^{5} b^{3} x^{5}-\frac {35}{4} a^{4} x^{4} b^{4}-\frac {56}{9} a^{3} b^{5} x^{3}-\frac {14}{5} a^{2} b^{6} x^{2}-\frac {8}{11} a \,b^{7} x -\frac {1}{12} b^{8}}{x^{12}}\) | \(90\) |
risch | \(\frac {-\frac {1}{4} a^{8} x^{8}-\frac {8}{5} a^{7} b \,x^{7}-\frac {14}{3} a^{6} b^{2} x^{6}-8 a^{5} b^{3} x^{5}-\frac {35}{4} a^{4} x^{4} b^{4}-\frac {56}{9} a^{3} b^{5} x^{3}-\frac {14}{5} a^{2} b^{6} x^{2}-\frac {8}{11} a \,b^{7} x -\frac {1}{12} b^{8}}{x^{12}}\) | \(90\) |
gosper | \(-\frac {495 a^{8} x^{8}+3168 a^{7} b \,x^{7}+9240 a^{6} b^{2} x^{6}+15840 a^{5} b^{3} x^{5}+17325 a^{4} x^{4} b^{4}+12320 a^{3} b^{5} x^{3}+5544 a^{2} b^{6} x^{2}+1440 a \,b^{7} x +165 b^{8}}{1980 x^{12}}\) | \(91\) |
default | \(-\frac {8 a^{7} b}{5 x^{5}}-\frac {8 a \,b^{7}}{11 x^{11}}-\frac {8 a^{5} b^{3}}{x^{7}}-\frac {a^{8}}{4 x^{4}}-\frac {35 a^{4} b^{4}}{4 x^{8}}-\frac {14 a^{2} b^{6}}{5 x^{10}}-\frac {14 a^{6} b^{2}}{3 x^{6}}-\frac {56 a^{3} b^{5}}{9 x^{9}}-\frac {b^{8}}{12 x^{12}}\) | \(91\) |
parallelrisch | \(\frac {-495 a^{8} x^{8}-3168 a^{7} b \,x^{7}-9240 a^{6} b^{2} x^{6}-15840 a^{5} b^{3} x^{5}-17325 a^{4} x^{4} b^{4}-12320 a^{3} b^{5} x^{3}-5544 a^{2} b^{6} x^{2}-1440 a \,b^{7} x -165 b^{8}}{1980 x^{12}}\) | \(91\) |
orering | \(-\frac {\left (495 a^{8} x^{8}+3168 a^{7} b \,x^{7}+9240 a^{6} b^{2} x^{6}+15840 a^{5} b^{3} x^{5}+17325 a^{4} x^{4} b^{4}+12320 a^{3} b^{5} x^{3}+5544 a^{2} b^{6} x^{2}+1440 a \,b^{7} x +165 b^{8}\right ) \left (a +\frac {b}{x}\right )^{8}}{1980 x^{4} \left (a x +b \right )^{8}}\) | \(107\) |
Input:
int((a+b/x)^8/x^5,x,method=_RETURNVERBOSE)
Output:
(-1/4*a^8*x^8-8/5*a^7*b*x^7-14/3*a^6*b^2*x^6-8*a^5*b^3*x^5-35/4*a^4*x^4*b^ 4-56/9*a^3*b^5*x^3-14/5*a^2*b^6*x^2-8/11*a*b^7*x-1/12*b^8)/x^12
Time = 0.06 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^5} \, dx=-\frac {495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \] Input:
integrate((a+b/x)^8/x^5,x, algorithm="fricas")
Output:
-1/1980*(495*a^8*x^8 + 3168*a^7*b*x^7 + 9240*a^6*b^2*x^6 + 15840*a^5*b^3*x ^5 + 17325*a^4*b^4*x^4 + 12320*a^3*b^5*x^3 + 5544*a^2*b^6*x^2 + 1440*a*b^7 *x + 165*b^8)/x^12
Time = 0.40 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^5} \, dx=\frac {- 495 a^{8} x^{8} - 3168 a^{7} b x^{7} - 9240 a^{6} b^{2} x^{6} - 15840 a^{5} b^{3} x^{5} - 17325 a^{4} b^{4} x^{4} - 12320 a^{3} b^{5} x^{3} - 5544 a^{2} b^{6} x^{2} - 1440 a b^{7} x - 165 b^{8}}{1980 x^{12}} \] Input:
integrate((a+b/x)**8/x**5,x)
Output:
(-495*a**8*x**8 - 3168*a**7*b*x**7 - 9240*a**6*b**2*x**6 - 15840*a**5*b**3 *x**5 - 17325*a**4*b**4*x**4 - 12320*a**3*b**5*x**3 - 5544*a**2*b**6*x**2 - 1440*a*b**7*x - 165*b**8)/(1980*x**12)
Time = 0.03 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^5} \, dx=-\frac {495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \] Input:
integrate((a+b/x)^8/x^5,x, algorithm="maxima")
Output:
-1/1980*(495*a^8*x^8 + 3168*a^7*b*x^7 + 9240*a^6*b^2*x^6 + 15840*a^5*b^3*x ^5 + 17325*a^4*b^4*x^4 + 12320*a^3*b^5*x^3 + 5544*a^2*b^6*x^2 + 1440*a*b^7 *x + 165*b^8)/x^12
Time = 0.12 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^5} \, dx=-\frac {495 \, a^{8} x^{8} + 3168 \, a^{7} b x^{7} + 9240 \, a^{6} b^{2} x^{6} + 15840 \, a^{5} b^{3} x^{5} + 17325 \, a^{4} b^{4} x^{4} + 12320 \, a^{3} b^{5} x^{3} + 5544 \, a^{2} b^{6} x^{2} + 1440 \, a b^{7} x + 165 \, b^{8}}{1980 \, x^{12}} \] Input:
integrate((a+b/x)^8/x^5,x, algorithm="giac")
Output:
-1/1980*(495*a^8*x^8 + 3168*a^7*b*x^7 + 9240*a^6*b^2*x^6 + 15840*a^5*b^3*x ^5 + 17325*a^4*b^4*x^4 + 12320*a^3*b^5*x^3 + 5544*a^2*b^6*x^2 + 1440*a*b^7 *x + 165*b^8)/x^12
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^5} \, dx=-\frac {\frac {a^8\,x^8}{4}+\frac {8\,a^7\,b\,x^7}{5}+\frac {14\,a^6\,b^2\,x^6}{3}+8\,a^5\,b^3\,x^5+\frac {35\,a^4\,b^4\,x^4}{4}+\frac {56\,a^3\,b^5\,x^3}{9}+\frac {14\,a^2\,b^6\,x^2}{5}+\frac {8\,a\,b^7\,x}{11}+\frac {b^8}{12}}{x^{12}} \] Input:
int((a + b/x)^8/x^5,x)
Output:
-(b^8/12 + (a^8*x^8)/4 + (8*a^7*b*x^7)/5 + (14*a^2*b^6*x^2)/5 + (56*a^3*b^ 5*x^3)/9 + (35*a^4*b^4*x^4)/4 + 8*a^5*b^3*x^5 + (14*a^6*b^2*x^6)/3 + (8*a* b^7*x)/11)/x^12
Time = 0.21 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.25 \[ \int \frac {\left (a+\frac {b}{x}\right )^8}{x^5} \, dx=\frac {-495 a^{8} x^{8}-3168 a^{7} b \,x^{7}-9240 a^{6} b^{2} x^{6}-15840 a^{5} b^{3} x^{5}-17325 a^{4} b^{4} x^{4}-12320 a^{3} b^{5} x^{3}-5544 a^{2} b^{6} x^{2}-1440 a \,b^{7} x -165 b^{8}}{1980 x^{12}} \] Input:
int((a+b/x)^8/x^5,x)
Output:
( - 495*a**8*x**8 - 3168*a**7*b*x**7 - 9240*a**6*b**2*x**6 - 15840*a**5*b* *3*x**5 - 17325*a**4*b**4*x**4 - 12320*a**3*b**5*x**3 - 5544*a**2*b**6*x** 2 - 1440*a*b**7*x - 165*b**8)/(1980*x**12)