Integrand size = 11, antiderivative size = 136 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {(a+b x)^{11}}{17 a x^{17}}+\frac {3 b (a+b x)^{11}}{136 a^2 x^{16}}-\frac {b^2 (a+b x)^{11}}{136 a^3 x^{15}}+\frac {b^3 (a+b x)^{11}}{476 a^4 x^{14}}-\frac {3 b^4 (a+b x)^{11}}{6188 a^5 x^{13}}+\frac {b^5 (a+b x)^{11}}{12376 a^6 x^{12}}-\frac {b^6 (a+b x)^{11}}{136136 a^7 x^{11}} \]
-1/17*(b*x+a)^11/a/x^17+3/136*b*(b*x+a)^11/a^2/x^16-1/136*b^2*(b*x+a)^11/a ^3/x^15+1/476*b^3*(b*x+a)^11/a^4/x^14-3/6188*b^4*(b*x+a)^11/a^5/x^13+1/123 76*b^5*(b*x+a)^11/a^6/x^12-1/136136*b^6*(b*x+a)^11/a^7/x^11
Time = 0.01 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {a^{10}}{17 x^{17}}-\frac {5 a^9 b}{8 x^{16}}-\frac {3 a^8 b^2}{x^{15}}-\frac {60 a^7 b^3}{7 x^{14}}-\frac {210 a^6 b^4}{13 x^{13}}-\frac {21 a^5 b^5}{x^{12}}-\frac {210 a^4 b^6}{11 x^{11}}-\frac {12 a^3 b^7}{x^{10}}-\frac {5 a^2 b^8}{x^9}-\frac {5 a b^9}{4 x^8}-\frac {b^{10}}{7 x^7} \]
-1/17*a^10/x^17 - (5*a^9*b)/(8*x^16) - (3*a^8*b^2)/x^15 - (60*a^7*b^3)/(7* x^14) - (210*a^6*b^4)/(13*x^13) - (21*a^5*b^5)/x^12 - (210*a^4*b^6)/(11*x^ 11) - (12*a^3*b^7)/x^10 - (5*a^2*b^8)/x^9 - (5*a*b^9)/(4*x^8) - b^10/(7*x^ 7)
Time = 0.20 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.22, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.636, Rules used = {55, 55, 55, 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.
\(\displaystyle \int \frac {(a+b x)^{10}}{x^{18}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {6 b \int \frac {(a+b x)^{10}}{x^{17}}dx}{17 a}-\frac {(a+b x)^{11}}{17 a x^{17}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {6 b \left (-\frac {5 b \int \frac {(a+b x)^{10}}{x^{16}}dx}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\right )}{17 a}-\frac {(a+b x)^{11}}{17 a x^{17}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {6 b \left (-\frac {5 b \left (-\frac {4 b \int \frac {(a+b x)^{10}}{x^{15}}dx}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\right )}{17 a}-\frac {(a+b x)^{11}}{17 a x^{17}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {6 b \left (-\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \int \frac {(a+b x)^{10}}{x^{14}}dx}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\right )}{17 a}-\frac {(a+b x)^{11}}{17 a x^{17}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {6 b \left (-\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \int \frac {(a+b x)^{10}}{x^{13}}dx}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\right )}{17 a}-\frac {(a+b x)^{11}}{17 a x^{17}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {6 b \left (-\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \left (-\frac {b \int \frac {(a+b x)^{10}}{x^{12}}dx}{12 a}-\frac {(a+b x)^{11}}{12 a x^{12}}\right )}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\right )}{17 a}-\frac {(a+b x)^{11}}{17 a x^{17}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {6 b \left (-\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \left (\frac {b (a+b x)^{11}}{132 a^2 x^{11}}-\frac {(a+b x)^{11}}{12 a x^{12}}\right )}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\right )}{17 a}-\frac {(a+b x)^{11}}{17 a x^{17}}\) |
-1/17*(a + b*x)^11/(a*x^17) - (6*b*(-1/16*(a + b*x)^11/(a*x^16) - (5*b*(-1 /15*(a + b*x)^11/(a*x^15) - (4*b*(-1/14*(a + b*x)^11/(a*x^14) - (3*b*(-1/1 3*(a + b*x)^11/(a*x^13) - (2*b*(-1/12*(a + b*x)^11/(a*x^12) + (b*(a + b*x) ^11)/(132*a^2*x^11)))/(13*a)))/(14*a)))/(15*a)))/(16*a)))/(17*a)
3.2.52.3.1 Defintions of rubi rules used
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])
Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82
method | result | size |
norman | \(\frac {-\frac {1}{17} a^{10}-\frac {5}{8} a^{9} b x -3 a^{8} b^{2} x^{2}-\frac {60}{7} a^{7} b^{3} x^{3}-\frac {210}{13} a^{6} b^{4} x^{4}-21 a^{5} b^{5} x^{5}-\frac {210}{11} a^{4} b^{6} x^{6}-12 a^{3} b^{7} x^{7}-5 a^{2} b^{8} x^{8}-\frac {5}{4} a \,b^{9} x^{9}-\frac {1}{7} b^{10} x^{10}}{x^{17}}\) | \(112\) |
risch | \(\frac {-\frac {1}{17} a^{10}-\frac {5}{8} a^{9} b x -3 a^{8} b^{2} x^{2}-\frac {60}{7} a^{7} b^{3} x^{3}-\frac {210}{13} a^{6} b^{4} x^{4}-21 a^{5} b^{5} x^{5}-\frac {210}{11} a^{4} b^{6} x^{6}-12 a^{3} b^{7} x^{7}-5 a^{2} b^{8} x^{8}-\frac {5}{4} a \,b^{9} x^{9}-\frac {1}{7} b^{10} x^{10}}{x^{17}}\) | \(112\) |
gosper | \(-\frac {19448 b^{10} x^{10}+170170 a \,b^{9} x^{9}+680680 a^{2} b^{8} x^{8}+1633632 a^{3} b^{7} x^{7}+2598960 a^{4} b^{6} x^{6}+2858856 a^{5} b^{5} x^{5}+2199120 a^{6} b^{4} x^{4}+1166880 a^{7} b^{3} x^{3}+408408 a^{8} b^{2} x^{2}+85085 a^{9} b x +8008 a^{10}}{136136 x^{17}}\) | \(113\) |
default | \(-\frac {12 a^{3} b^{7}}{x^{10}}-\frac {3 a^{8} b^{2}}{x^{15}}-\frac {60 a^{7} b^{3}}{7 x^{14}}-\frac {b^{10}}{7 x^{7}}-\frac {210 a^{6} b^{4}}{13 x^{13}}-\frac {5 a^{2} b^{8}}{x^{9}}-\frac {21 a^{5} b^{5}}{x^{12}}-\frac {210 a^{4} b^{6}}{11 x^{11}}-\frac {5 a^{9} b}{8 x^{16}}-\frac {a^{10}}{17 x^{17}}-\frac {5 a \,b^{9}}{4 x^{8}}\) | \(113\) |
parallelrisch | \(\frac {-19448 b^{10} x^{10}-170170 a \,b^{9} x^{9}-680680 a^{2} b^{8} x^{8}-1633632 a^{3} b^{7} x^{7}-2598960 a^{4} b^{6} x^{6}-2858856 a^{5} b^{5} x^{5}-2199120 a^{6} b^{4} x^{4}-1166880 a^{7} b^{3} x^{3}-408408 a^{8} b^{2} x^{2}-85085 a^{9} b x -8008 a^{10}}{136136 x^{17}}\) | \(113\) |
1/x^17*(-1/17*a^10-5/8*a^9*b*x-3*a^8*b^2*x^2-60/7*a^7*b^3*x^3-210/13*a^6*b ^4*x^4-21*a^5*b^5*x^5-210/11*a^4*b^6*x^6-12*a^3*b^7*x^7-5*a^2*b^8*x^8-5/4* a*b^9*x^9-1/7*b^10*x^10)
Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {19448 \, b^{10} x^{10} + 170170 \, a b^{9} x^{9} + 680680 \, a^{2} b^{8} x^{8} + 1633632 \, a^{3} b^{7} x^{7} + 2598960 \, a^{4} b^{6} x^{6} + 2858856 \, a^{5} b^{5} x^{5} + 2199120 \, a^{6} b^{4} x^{4} + 1166880 \, a^{7} b^{3} x^{3} + 408408 \, a^{8} b^{2} x^{2} + 85085 \, a^{9} b x + 8008 \, a^{10}}{136136 \, x^{17}} \]
-1/136136*(19448*b^10*x^10 + 170170*a*b^9*x^9 + 680680*a^2*b^8*x^8 + 16336 32*a^3*b^7*x^7 + 2598960*a^4*b^6*x^6 + 2858856*a^5*b^5*x^5 + 2199120*a^6*b ^4*x^4 + 1166880*a^7*b^3*x^3 + 408408*a^8*b^2*x^2 + 85085*a^9*b*x + 8008*a ^10)/x^17
Time = 0.66 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=\frac {- 8008 a^{10} - 85085 a^{9} b x - 408408 a^{8} b^{2} x^{2} - 1166880 a^{7} b^{3} x^{3} - 2199120 a^{6} b^{4} x^{4} - 2858856 a^{5} b^{5} x^{5} - 2598960 a^{4} b^{6} x^{6} - 1633632 a^{3} b^{7} x^{7} - 680680 a^{2} b^{8} x^{8} - 170170 a b^{9} x^{9} - 19448 b^{10} x^{10}}{136136 x^{17}} \]
(-8008*a**10 - 85085*a**9*b*x - 408408*a**8*b**2*x**2 - 1166880*a**7*b**3* x**3 - 2199120*a**6*b**4*x**4 - 2858856*a**5*b**5*x**5 - 2598960*a**4*b**6 *x**6 - 1633632*a**3*b**7*x**7 - 680680*a**2*b**8*x**8 - 170170*a*b**9*x** 9 - 19448*b**10*x**10)/(136136*x**17)
Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {19448 \, b^{10} x^{10} + 170170 \, a b^{9} x^{9} + 680680 \, a^{2} b^{8} x^{8} + 1633632 \, a^{3} b^{7} x^{7} + 2598960 \, a^{4} b^{6} x^{6} + 2858856 \, a^{5} b^{5} x^{5} + 2199120 \, a^{6} b^{4} x^{4} + 1166880 \, a^{7} b^{3} x^{3} + 408408 \, a^{8} b^{2} x^{2} + 85085 \, a^{9} b x + 8008 \, a^{10}}{136136 \, x^{17}} \]
-1/136136*(19448*b^10*x^10 + 170170*a*b^9*x^9 + 680680*a^2*b^8*x^8 + 16336 32*a^3*b^7*x^7 + 2598960*a^4*b^6*x^6 + 2858856*a^5*b^5*x^5 + 2199120*a^6*b ^4*x^4 + 1166880*a^7*b^3*x^3 + 408408*a^8*b^2*x^2 + 85085*a^9*b*x + 8008*a ^10)/x^17
Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {19448 \, b^{10} x^{10} + 170170 \, a b^{9} x^{9} + 680680 \, a^{2} b^{8} x^{8} + 1633632 \, a^{3} b^{7} x^{7} + 2598960 \, a^{4} b^{6} x^{6} + 2858856 \, a^{5} b^{5} x^{5} + 2199120 \, a^{6} b^{4} x^{4} + 1166880 \, a^{7} b^{3} x^{3} + 408408 \, a^{8} b^{2} x^{2} + 85085 \, a^{9} b x + 8008 \, a^{10}}{136136 \, x^{17}} \]
-1/136136*(19448*b^10*x^10 + 170170*a*b^9*x^9 + 680680*a^2*b^8*x^8 + 16336 32*a^3*b^7*x^7 + 2598960*a^4*b^6*x^6 + 2858856*a^5*b^5*x^5 + 2199120*a^6*b ^4*x^4 + 1166880*a^7*b^3*x^3 + 408408*a^8*b^2*x^2 + 85085*a^9*b*x + 8008*a ^10)/x^17
Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {\frac {a^{10}}{17}+\frac {5\,a^9\,b\,x}{8}+3\,a^8\,b^2\,x^2+\frac {60\,a^7\,b^3\,x^3}{7}+\frac {210\,a^6\,b^4\,x^4}{13}+21\,a^5\,b^5\,x^5+\frac {210\,a^4\,b^6\,x^6}{11}+12\,a^3\,b^7\,x^7+5\,a^2\,b^8\,x^8+\frac {5\,a\,b^9\,x^9}{4}+\frac {b^{10}\,x^{10}}{7}}{x^{17}} \]