Integrand size = 11, antiderivative size = 116 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=-\frac {(a+b x)^{11}}{16 a x^{16}}+\frac {b (a+b x)^{11}}{48 a^2 x^{15}}-\frac {b^2 (a+b x)^{11}}{168 a^3 x^{14}}+\frac {b^3 (a+b x)^{11}}{728 a^4 x^{13}}-\frac {b^4 (a+b x)^{11}}{4368 a^5 x^{12}}+\frac {b^5 (a+b x)^{11}}{48048 a^6 x^{11}} \]
-1/16*(b*x+a)^11/a/x^16+1/48*b*(b*x+a)^11/a^2/x^15-1/168*b^2*(b*x+a)^11/a^ 3/x^14+1/728*b^3*(b*x+a)^11/a^4/x^13-1/4368*b^4*(b*x+a)^11/a^5/x^12+1/4804 8*b^5*(b*x+a)^11/a^6/x^11
Time = 0.00 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=-\frac {a^{10}}{16 x^{16}}-\frac {2 a^9 b}{3 x^{15}}-\frac {45 a^8 b^2}{14 x^{14}}-\frac {120 a^7 b^3}{13 x^{13}}-\frac {35 a^6 b^4}{2 x^{12}}-\frac {252 a^5 b^5}{11 x^{11}}-\frac {21 a^4 b^6}{x^{10}}-\frac {40 a^3 b^7}{3 x^9}-\frac {45 a^2 b^8}{8 x^8}-\frac {10 a b^9}{7 x^7}-\frac {b^{10}}{6 x^6} \]
-1/16*a^10/x^16 - (2*a^9*b)/(3*x^15) - (45*a^8*b^2)/(14*x^14) - (120*a^7*b ^3)/(13*x^13) - (35*a^6*b^4)/(2*x^12) - (252*a^5*b^5)/(11*x^11) - (21*a^4* b^6)/x^10 - (40*a^3*b^7)/(3*x^9) - (45*a^2*b^8)/(8*x^8) - (10*a*b^9)/(7*x^ 7) - b^10/(6*x^6)
Time = 0.19 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.21, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {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.
\(\displaystyle \int \frac {(a+b x)^{10}}{x^{17}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {5 b \int \frac {(a+b x)^{10}}{x^{16}}dx}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {5 b \left (-\frac {4 b \int \frac {(a+b x)^{10}}{x^{15}}dx}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \int \frac {(a+b x)^{10}}{x^{14}}dx}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \int \frac {(a+b x)^{10}}{x^{13}}dx}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \left (-\frac {b \int \frac {(a+b x)^{10}}{x^{12}}dx}{12 a}-\frac {(a+b x)^{11}}{12 a x^{12}}\right )}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {5 b \left (-\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \left (\frac {b (a+b x)^{11}}{132 a^2 x^{11}}-\frac {(a+b x)^{11}}{12 a x^{12}}\right )}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\right )}{16 a}-\frac {(a+b x)^{11}}{16 a x^{16}}\) |
-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*(-1/13*(a + b*x)^11/(a*x^13) - (2*b*(-1/1 2*(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)
3.2.51.3.1 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 = 0.97
method | result | size |
norman | \(\frac {-\frac {1}{16} a^{10}-\frac {2}{3} a^{9} b x -\frac {45}{14} a^{8} b^{2} x^{2}-\frac {120}{13} a^{7} b^{3} x^{3}-\frac {35}{2} a^{6} b^{4} x^{4}-\frac {252}{11} a^{5} b^{5} x^{5}-21 a^{4} b^{6} x^{6}-\frac {40}{3} a^{3} b^{7} x^{7}-\frac {45}{8} a^{2} b^{8} x^{8}-\frac {10}{7} a \,b^{9} x^{9}-\frac {1}{6} b^{10} x^{10}}{x^{16}}\) | \(112\) |
risch | \(\frac {-\frac {1}{16} a^{10}-\frac {2}{3} a^{9} b x -\frac {45}{14} a^{8} b^{2} x^{2}-\frac {120}{13} a^{7} b^{3} x^{3}-\frac {35}{2} a^{6} b^{4} x^{4}-\frac {252}{11} a^{5} b^{5} x^{5}-21 a^{4} b^{6} x^{6}-\frac {40}{3} a^{3} b^{7} x^{7}-\frac {45}{8} a^{2} b^{8} x^{8}-\frac {10}{7} a \,b^{9} x^{9}-\frac {1}{6} b^{10} x^{10}}{x^{16}}\) | \(112\) |
gosper | \(-\frac {8008 b^{10} x^{10}+68640 a \,b^{9} x^{9}+270270 a^{2} b^{8} x^{8}+640640 a^{3} b^{7} x^{7}+1009008 a^{4} b^{6} x^{6}+1100736 a^{5} b^{5} x^{5}+840840 a^{6} b^{4} x^{4}+443520 a^{7} b^{3} x^{3}+154440 a^{8} b^{2} x^{2}+32032 a^{9} b x +3003 a^{10}}{48048 x^{16}}\) | \(113\) |
default | \(-\frac {21 a^{4} b^{6}}{x^{10}}-\frac {b^{10}}{6 x^{6}}-\frac {2 a^{9} b}{3 x^{15}}-\frac {45 a^{8} b^{2}}{14 x^{14}}-\frac {10 a \,b^{9}}{7 x^{7}}-\frac {120 a^{7} b^{3}}{13 x^{13}}-\frac {40 a^{3} b^{7}}{3 x^{9}}-\frac {35 a^{6} b^{4}}{2 x^{12}}-\frac {252 a^{5} b^{5}}{11 x^{11}}-\frac {a^{10}}{16 x^{16}}-\frac {45 a^{2} b^{8}}{8 x^{8}}\) | \(113\) |
parallelrisch | \(\frac {-8008 b^{10} x^{10}-68640 a \,b^{9} x^{9}-270270 a^{2} b^{8} x^{8}-640640 a^{3} b^{7} x^{7}-1009008 a^{4} b^{6} x^{6}-1100736 a^{5} b^{5} x^{5}-840840 a^{6} b^{4} x^{4}-443520 a^{7} b^{3} x^{3}-154440 a^{8} b^{2} x^{2}-32032 a^{9} b x -3003 a^{10}}{48048 x^{16}}\) | \(113\) |
1/x^16*(-1/16*a^10-2/3*a^9*b*x-45/14*a^8*b^2*x^2-120/13*a^7*b^3*x^3-35/2*a ^6*b^4*x^4-252/11*a^5*b^5*x^5-21*a^4*b^6*x^6-40/3*a^3*b^7*x^7-45/8*a^2*b^8 *x^8-10/7*a*b^9*x^9-1/6*b^10*x^10)
Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=-\frac {8008 \, b^{10} x^{10} + 68640 \, a b^{9} x^{9} + 270270 \, a^{2} b^{8} x^{8} + 640640 \, a^{3} b^{7} x^{7} + 1009008 \, a^{4} b^{6} x^{6} + 1100736 \, a^{5} b^{5} x^{5} + 840840 \, a^{6} b^{4} x^{4} + 443520 \, a^{7} b^{3} x^{3} + 154440 \, a^{8} b^{2} x^{2} + 32032 \, a^{9} b x + 3003 \, a^{10}}{48048 \, x^{16}} \]
-1/48048*(8008*b^10*x^10 + 68640*a*b^9*x^9 + 270270*a^2*b^8*x^8 + 640640*a ^3*b^7*x^7 + 1009008*a^4*b^6*x^6 + 1100736*a^5*b^5*x^5 + 840840*a^6*b^4*x^ 4 + 443520*a^7*b^3*x^3 + 154440*a^8*b^2*x^2 + 32032*a^9*b*x + 3003*a^10)/x ^16
Time = 0.65 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=\frac {- 3003 a^{10} - 32032 a^{9} b x - 154440 a^{8} b^{2} x^{2} - 443520 a^{7} b^{3} x^{3} - 840840 a^{6} b^{4} x^{4} - 1100736 a^{5} b^{5} x^{5} - 1009008 a^{4} b^{6} x^{6} - 640640 a^{3} b^{7} x^{7} - 270270 a^{2} b^{8} x^{8} - 68640 a b^{9} x^{9} - 8008 b^{10} x^{10}}{48048 x^{16}} \]
(-3003*a**10 - 32032*a**9*b*x - 154440*a**8*b**2*x**2 - 443520*a**7*b**3*x **3 - 840840*a**6*b**4*x**4 - 1100736*a**5*b**5*x**5 - 1009008*a**4*b**6*x **6 - 640640*a**3*b**7*x**7 - 270270*a**2*b**8*x**8 - 68640*a*b**9*x**9 - 8008*b**10*x**10)/(48048*x**16)
Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=-\frac {8008 \, b^{10} x^{10} + 68640 \, a b^{9} x^{9} + 270270 \, a^{2} b^{8} x^{8} + 640640 \, a^{3} b^{7} x^{7} + 1009008 \, a^{4} b^{6} x^{6} + 1100736 \, a^{5} b^{5} x^{5} + 840840 \, a^{6} b^{4} x^{4} + 443520 \, a^{7} b^{3} x^{3} + 154440 \, a^{8} b^{2} x^{2} + 32032 \, a^{9} b x + 3003 \, a^{10}}{48048 \, x^{16}} \]
-1/48048*(8008*b^10*x^10 + 68640*a*b^9*x^9 + 270270*a^2*b^8*x^8 + 640640*a ^3*b^7*x^7 + 1009008*a^4*b^6*x^6 + 1100736*a^5*b^5*x^5 + 840840*a^6*b^4*x^ 4 + 443520*a^7*b^3*x^3 + 154440*a^8*b^2*x^2 + 32032*a^9*b*x + 3003*a^10)/x ^16
Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=-\frac {8008 \, b^{10} x^{10} + 68640 \, a b^{9} x^{9} + 270270 \, a^{2} b^{8} x^{8} + 640640 \, a^{3} b^{7} x^{7} + 1009008 \, a^{4} b^{6} x^{6} + 1100736 \, a^{5} b^{5} x^{5} + 840840 \, a^{6} b^{4} x^{4} + 443520 \, a^{7} b^{3} x^{3} + 154440 \, a^{8} b^{2} x^{2} + 32032 \, a^{9} b x + 3003 \, a^{10}}{48048 \, x^{16}} \]
-1/48048*(8008*b^10*x^10 + 68640*a*b^9*x^9 + 270270*a^2*b^8*x^8 + 640640*a ^3*b^7*x^7 + 1009008*a^4*b^6*x^6 + 1100736*a^5*b^5*x^5 + 840840*a^6*b^4*x^ 4 + 443520*a^7*b^3*x^3 + 154440*a^8*b^2*x^2 + 32032*a^9*b*x + 3003*a^10)/x ^16
Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=-\frac {\frac {a^{10}}{16}+\frac {2\,a^9\,b\,x}{3}+\frac {45\,a^8\,b^2\,x^2}{14}+\frac {120\,a^7\,b^3\,x^3}{13}+\frac {35\,a^6\,b^4\,x^4}{2}+\frac {252\,a^5\,b^5\,x^5}{11}+21\,a^4\,b^6\,x^6+\frac {40\,a^3\,b^7\,x^7}{3}+\frac {45\,a^2\,b^8\,x^8}{8}+\frac {10\,a\,b^9\,x^9}{7}+\frac {b^{10}\,x^{10}}{6}}{x^{16}} \]