Integrand size = 11, antiderivative size = 96 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {(a+b x)^{11}}{15 a x^{15}}+\frac {2 b (a+b x)^{11}}{105 a^2 x^{14}}-\frac {2 b^2 (a+b x)^{11}}{455 a^3 x^{13}}+\frac {b^3 (a+b x)^{11}}{1365 a^4 x^{12}}-\frac {b^4 (a+b x)^{11}}{15015 a^5 x^{11}} \] Output:
-1/15*(b*x+a)^11/a/x^15+2/105*b*(b*x+a)^11/a^2/x^14-2/455*b^2*(b*x+a)^11/a ^3/x^13+1/1365*b^3*(b*x+a)^11/a^4/x^12-1/15015*b^4*(b*x+a)^11/a^5/x^11
Time = 0.01 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {a^{10}}{15 x^{15}}-\frac {5 a^9 b}{7 x^{14}}-\frac {45 a^8 b^2}{13 x^{13}}-\frac {10 a^7 b^3}{x^{12}}-\frac {210 a^6 b^4}{11 x^{11}}-\frac {126 a^5 b^5}{5 x^{10}}-\frac {70 a^4 b^6}{3 x^9}-\frac {15 a^3 b^7}{x^8}-\frac {45 a^2 b^8}{7 x^7}-\frac {5 a b^9}{3 x^6}-\frac {b^{10}}{5 x^5} \] Input:
Integrate[(a + b*x)^10/x^16,x]
Output:
-1/15*a^10/x^15 - (5*a^9*b)/(7*x^14) - (45*a^8*b^2)/(13*x^13) - (10*a^7*b^ 3)/x^12 - (210*a^6*b^4)/(11*x^11) - (126*a^5*b^5)/(5*x^10) - (70*a^4*b^6)/ (3*x^9) - (15*a^3*b^7)/x^8 - (45*a^2*b^8)/(7*x^7) - (5*a*b^9)/(3*x^6) - b^ 10/(5*x^5)
Time = 0.17 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.19, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.455, Rules used = {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.
\(\displaystyle \int \frac {(a+b x)^{10}}{x^{16}} \, dx\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {4 b \int \frac {(a+b x)^{10}}{x^{15}}dx}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {4 b \left (-\frac {3 b \int \frac {(a+b x)^{10}}{x^{14}}dx}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \int \frac {(a+b x)^{10}}{x^{13}}dx}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle -\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \left (-\frac {b \int \frac {(a+b x)^{10}}{x^{12}}dx}{12 a}-\frac {(a+b x)^{11}}{12 a x^{12}}\right )}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle -\frac {4 b \left (-\frac {3 b \left (-\frac {2 b \left (\frac {b (a+b x)^{11}}{132 a^2 x^{11}}-\frac {(a+b x)^{11}}{12 a x^{12}}\right )}{13 a}-\frac {(a+b x)^{11}}{13 a x^{13}}\right )}{14 a}-\frac {(a+b x)^{11}}{14 a x^{14}}\right )}{15 a}-\frac {(a+b x)^{11}}{15 a x^{15}}\) |
Input:
Int[(a + b*x)^10/x^16,x]
Output:
-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*(-1/12*(a + b*x)^11/(a*x^12) + (b*(a + b* x)^11)/(132*a^2*x^11)))/(13*a)))/(14*a)))/(15*a)
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 = 0.05 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17
method | result | size |
norman | \(\frac {-\frac {1}{15} a^{10}-\frac {5}{7} a^{9} b x -\frac {45}{13} a^{8} b^{2} x^{2}-10 a^{7} b^{3} x^{3}-\frac {210}{11} a^{6} b^{4} x^{4}-\frac {126}{5} a^{5} b^{5} x^{5}-\frac {70}{3} a^{4} b^{6} x^{6}-15 a^{3} b^{7} x^{7}-\frac {45}{7} a^{2} b^{8} x^{8}-\frac {5}{3} a \,b^{9} x^{9}-\frac {1}{5} b^{10} x^{10}}{x^{15}}\) | \(112\) |
risch | \(\frac {-\frac {1}{15} a^{10}-\frac {5}{7} a^{9} b x -\frac {45}{13} a^{8} b^{2} x^{2}-10 a^{7} b^{3} x^{3}-\frac {210}{11} a^{6} b^{4} x^{4}-\frac {126}{5} a^{5} b^{5} x^{5}-\frac {70}{3} a^{4} b^{6} x^{6}-15 a^{3} b^{7} x^{7}-\frac {45}{7} a^{2} b^{8} x^{8}-\frac {5}{3} a \,b^{9} x^{9}-\frac {1}{5} b^{10} x^{10}}{x^{15}}\) | \(112\) |
gosper | \(-\frac {3003 b^{10} x^{10}+25025 a \,b^{9} x^{9}+96525 a^{2} b^{8} x^{8}+225225 a^{3} b^{7} x^{7}+350350 a^{4} b^{6} x^{6}+378378 a^{5} b^{5} x^{5}+286650 a^{6} b^{4} x^{4}+150150 a^{7} b^{3} x^{3}+51975 a^{8} b^{2} x^{2}+10725 a^{9} b x +1001 a^{10}}{15015 x^{15}}\) | \(113\) |
default | \(-\frac {b^{10}}{5 x^{5}}-\frac {210 a^{6} b^{4}}{11 x^{11}}-\frac {a^{10}}{15 x^{15}}-\frac {45 a^{2} b^{8}}{7 x^{7}}-\frac {15 a^{3} b^{7}}{x^{8}}-\frac {45 a^{8} b^{2}}{13 x^{13}}-\frac {126 a^{5} b^{5}}{5 x^{10}}-\frac {5 a^{9} b}{7 x^{14}}-\frac {5 a \,b^{9}}{3 x^{6}}-\frac {70 a^{4} b^{6}}{3 x^{9}}-\frac {10 a^{7} b^{3}}{x^{12}}\) | \(113\) |
parallelrisch | \(\frac {-3003 b^{10} x^{10}-25025 a \,b^{9} x^{9}-96525 a^{2} b^{8} x^{8}-225225 a^{3} b^{7} x^{7}-350350 a^{4} b^{6} x^{6}-378378 a^{5} b^{5} x^{5}-286650 a^{6} b^{4} x^{4}-150150 a^{7} b^{3} x^{3}-51975 a^{8} b^{2} x^{2}-10725 a^{9} b x -1001 a^{10}}{15015 x^{15}}\) | \(113\) |
orering | \(-\frac {3003 b^{10} x^{10}+25025 a \,b^{9} x^{9}+96525 a^{2} b^{8} x^{8}+225225 a^{3} b^{7} x^{7}+350350 a^{4} b^{6} x^{6}+378378 a^{5} b^{5} x^{5}+286650 a^{6} b^{4} x^{4}+150150 a^{7} b^{3} x^{3}+51975 a^{8} b^{2} x^{2}+10725 a^{9} b x +1001 a^{10}}{15015 x^{15}}\) | \(113\) |
Input:
int((b*x+a)^10/x^16,x,method=_RETURNVERBOSE)
Output:
1/x^15*(-1/15*a^10-5/7*a^9*b*x-45/13*a^8*b^2*x^2-10*a^7*b^3*x^3-210/11*a^6 *b^4*x^4-126/5*a^5*b^5*x^5-70/3*a^4*b^6*x^6-15*a^3*b^7*x^7-45/7*a^2*b^8*x^ 8-5/3*a*b^9*x^9-1/5*b^10*x^10)
Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {3003 \, b^{10} x^{10} + 25025 \, a b^{9} x^{9} + 96525 \, a^{2} b^{8} x^{8} + 225225 \, a^{3} b^{7} x^{7} + 350350 \, a^{4} b^{6} x^{6} + 378378 \, a^{5} b^{5} x^{5} + 286650 \, a^{6} b^{4} x^{4} + 150150 \, a^{7} b^{3} x^{3} + 51975 \, a^{8} b^{2} x^{2} + 10725 \, a^{9} b x + 1001 \, a^{10}}{15015 \, x^{15}} \] Input:
integrate((b*x+a)^10/x^16,x, algorithm="fricas")
Output:
-1/15015*(3003*b^10*x^10 + 25025*a*b^9*x^9 + 96525*a^2*b^8*x^8 + 225225*a^ 3*b^7*x^7 + 350350*a^4*b^6*x^6 + 378378*a^5*b^5*x^5 + 286650*a^6*b^4*x^4 + 150150*a^7*b^3*x^3 + 51975*a^8*b^2*x^2 + 10725*a^9*b*x + 1001*a^10)/x^15
Time = 0.50 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=\frac {- 1001 a^{10} - 10725 a^{9} b x - 51975 a^{8} b^{2} x^{2} - 150150 a^{7} b^{3} x^{3} - 286650 a^{6} b^{4} x^{4} - 378378 a^{5} b^{5} x^{5} - 350350 a^{4} b^{6} x^{6} - 225225 a^{3} b^{7} x^{7} - 96525 a^{2} b^{8} x^{8} - 25025 a b^{9} x^{9} - 3003 b^{10} x^{10}}{15015 x^{15}} \] Input:
integrate((b*x+a)**10/x**16,x)
Output:
(-1001*a**10 - 10725*a**9*b*x - 51975*a**8*b**2*x**2 - 150150*a**7*b**3*x* *3 - 286650*a**6*b**4*x**4 - 378378*a**5*b**5*x**5 - 350350*a**4*b**6*x**6 - 225225*a**3*b**7*x**7 - 96525*a**2*b**8*x**8 - 25025*a*b**9*x**9 - 3003 *b**10*x**10)/(15015*x**15)
Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {3003 \, b^{10} x^{10} + 25025 \, a b^{9} x^{9} + 96525 \, a^{2} b^{8} x^{8} + 225225 \, a^{3} b^{7} x^{7} + 350350 \, a^{4} b^{6} x^{6} + 378378 \, a^{5} b^{5} x^{5} + 286650 \, a^{6} b^{4} x^{4} + 150150 \, a^{7} b^{3} x^{3} + 51975 \, a^{8} b^{2} x^{2} + 10725 \, a^{9} b x + 1001 \, a^{10}}{15015 \, x^{15}} \] Input:
integrate((b*x+a)^10/x^16,x, algorithm="maxima")
Output:
-1/15015*(3003*b^10*x^10 + 25025*a*b^9*x^9 + 96525*a^2*b^8*x^8 + 225225*a^ 3*b^7*x^7 + 350350*a^4*b^6*x^6 + 378378*a^5*b^5*x^5 + 286650*a^6*b^4*x^4 + 150150*a^7*b^3*x^3 + 51975*a^8*b^2*x^2 + 10725*a^9*b*x + 1001*a^10)/x^15
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {3003 \, b^{10} x^{10} + 25025 \, a b^{9} x^{9} + 96525 \, a^{2} b^{8} x^{8} + 225225 \, a^{3} b^{7} x^{7} + 350350 \, a^{4} b^{6} x^{6} + 378378 \, a^{5} b^{5} x^{5} + 286650 \, a^{6} b^{4} x^{4} + 150150 \, a^{7} b^{3} x^{3} + 51975 \, a^{8} b^{2} x^{2} + 10725 \, a^{9} b x + 1001 \, a^{10}}{15015 \, x^{15}} \] Input:
integrate((b*x+a)^10/x^16,x, algorithm="giac")
Output:
-1/15015*(3003*b^10*x^10 + 25025*a*b^9*x^9 + 96525*a^2*b^8*x^8 + 225225*a^ 3*b^7*x^7 + 350350*a^4*b^6*x^6 + 378378*a^5*b^5*x^5 + 286650*a^6*b^4*x^4 + 150150*a^7*b^3*x^3 + 51975*a^8*b^2*x^2 + 10725*a^9*b*x + 1001*a^10)/x^15
Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {\frac {a^{10}}{15}+\frac {5\,a^9\,b\,x}{7}+\frac {45\,a^8\,b^2\,x^2}{13}+10\,a^7\,b^3\,x^3+\frac {210\,a^6\,b^4\,x^4}{11}+\frac {126\,a^5\,b^5\,x^5}{5}+\frac {70\,a^4\,b^6\,x^6}{3}+15\,a^3\,b^7\,x^7+\frac {45\,a^2\,b^8\,x^8}{7}+\frac {5\,a\,b^9\,x^9}{3}+\frac {b^{10}\,x^{10}}{5}}{x^{15}} \] Input:
int((a + b*x)^10/x^16,x)
Output:
-(a^10/15 + (b^10*x^10)/5 + (5*a*b^9*x^9)/3 + (45*a^8*b^2*x^2)/13 + 10*a^7 *b^3*x^3 + (210*a^6*b^4*x^4)/11 + (126*a^5*b^5*x^5)/5 + (70*a^4*b^6*x^6)/3 + 15*a^3*b^7*x^7 + (45*a^2*b^8*x^8)/7 + (5*a^9*b*x)/7)/x^15
Time = 0.15 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=\frac {-3003 b^{10} x^{10}-25025 a \,b^{9} x^{9}-96525 a^{2} b^{8} x^{8}-225225 a^{3} b^{7} x^{7}-350350 a^{4} b^{6} x^{6}-378378 a^{5} b^{5} x^{5}-286650 a^{6} b^{4} x^{4}-150150 a^{7} b^{3} x^{3}-51975 a^{8} b^{2} x^{2}-10725 a^{9} b x -1001 a^{10}}{15015 x^{15}} \] Input:
int((b*x+a)^10/x^16,x)
Output:
( - 1001*a**10 - 10725*a**9*b*x - 51975*a**8*b**2*x**2 - 150150*a**7*b**3* x**3 - 286650*a**6*b**4*x**4 - 378378*a**5*b**5*x**5 - 350350*a**4*b**6*x* *6 - 225225*a**3*b**7*x**7 - 96525*a**2*b**8*x**8 - 25025*a*b**9*x**9 - 30 03*b**10*x**10)/(15015*x**15)