\(\int \frac {(a+b x)^{10}}{x^{18}} \, dx\) [152]

Optimal result

Integrand size = 11, antiderivative size = 136 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {(a+b x)^{11}}{17 a x^{17}}+\frac {3 b (a+b x)^{11}}{136 a^2 x^{16}}-\frac {b^2 (a+b x)^{11}}{136 a^3 x^{15}}+\frac {b^3 (a+b x)^{11}}{476 a^4 x^{14}}-\frac {3 b^4 (a+b x)^{11}}{6188 a^5 x^{13}}+\frac {b^5 (a+b x)^{11}}{12376 a^6 x^{12}}-\frac {b^6 (a+b x)^{11}}{136136 a^7 x^{11}} \]

[Out]

-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^1
4-3/6188*b^4*(b*x+a)^11/a^5/x^13+1/12376*b^5*(b*x+a)^11/a^6/x^12-1/136136*b^6*(b*x+a)^11/a^7/x^11

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {b^6 (a+b x)^{11}}{136136 a^7 x^{11}}+\frac {b^5 (a+b x)^{11}}{12376 a^6 x^{12}}-\frac {3 b^4 (a+b x)^{11}}{6188 a^5 x^{13}}+\frac {b^3 (a+b x)^{11}}{476 a^4 x^{14}}-\frac {b^2 (a+b x)^{11}}{136 a^3 x^{15}}+\frac {3 b (a+b x)^{11}}{136 a^2 x^{16}}-\frac {(a+b x)^{11}}{17 a x^{17}} \]

[In]

Int[(a + b*x)^10/x^18,x]

[Out]

-1/17*(a + b*x)^11/(a*x^17) + (3*b*(a + b*x)^11)/(136*a^2*x^16) - (b^2*(a + b*x)^11)/(136*a^3*x^15) + (b^3*(a
+ b*x)^11)/(476*a^4*x^14) - (3*b^4*(a + b*x)^11)/(6188*a^5*x^13) + (b^5*(a + b*x)^11)/(12376*a^6*x^12) - (b^6*
(a + b*x)^11)/(136136*a^7*x^11)

Rule 37

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] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

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] - Dist[d*(Simplify[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] && NeQ[b*c - a*d, 0] && 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] ||  !SumSimplerQ[n, 1])

Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b x)^{11}}{17 a x^{17}}-\frac {(6 b) \int \frac {(a+b x)^{10}}{x^{17}} \, dx}{17 a} \\ & = -\frac {(a+b x)^{11}}{17 a x^{17}}+\frac {3 b (a+b x)^{11}}{136 a^2 x^{16}}+\frac {\left (15 b^2\right ) \int \frac {(a+b x)^{10}}{x^{16}} \, dx}{136 a^2} \\ & = -\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 \int \frac {(a+b x)^{10}}{x^{15}} \, dx}{34 a^3} \\ & = -\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 {\left (3 b^4\right ) \int \frac {(a+b x)^{10}}{x^{14}} \, dx}{476 a^4} \\ & = -\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 {\left (3 b^5\right ) \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{3094 a^5} \\ & = -\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 \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{12376 a^6} \\ & = -\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.93 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {a^{10}}{17 x^{17}}-\frac {5 a^9 b}{8 x^{16}}-\frac {3 a^8 b^2}{x^{15}}-\frac {60 a^7 b^3}{7 x^{14}}-\frac {210 a^6 b^4}{13 x^{13}}-\frac {21 a^5 b^5}{x^{12}}-\frac {210 a^4 b^6}{11 x^{11}}-\frac {12 a^3 b^7}{x^{10}}-\frac {5 a^2 b^8}{x^9}-\frac {5 a b^9}{4 x^8}-\frac {b^{10}}{7 x^7} \]

[In]

Integrate[(a + b*x)^10/x^18,x]

[Out]

-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) - (2
1*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)

Maple [A] (verified)

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	\(\frac {-19448 b^{10} x^{10}-170170 a \,b^{9} x^{9}-680680 a^{2} b^{8} x^{8}-1633632 a^{3} b^{7} x^{7}-2598960 a^{4} b^{6} x^{6}-2858856 a^{5} b^{5} x^{5}-2199120 a^{6} b^{4} x^{4}-1166880 a^{7} b^{3} x^{3}-408408 a^{8} b^{2} x^{2}-85085 a^{9} b x -8008 a^{10}}{136136 x^{17}}\)	\(113\)

[In]

int((b*x+a)^10/x^18,x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {19448 \, b^{10} x^{10} + 170170 \, a b^{9} x^{9} + 680680 \, a^{2} b^{8} x^{8} + 1633632 \, a^{3} b^{7} x^{7} + 2598960 \, a^{4} b^{6} x^{6} + 2858856 \, a^{5} b^{5} x^{5} + 2199120 \, a^{6} b^{4} x^{4} + 1166880 \, a^{7} b^{3} x^{3} + 408408 \, a^{8} b^{2} x^{2} + 85085 \, a^{9} b x + 8008 \, a^{10}}{136136 \, x^{17}} \]

[In]

integrate((b*x+a)^10/x^18,x, algorithm="fricas")

[Out]

-1/136136*(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)/x^17

Sympy [A] (verification not implemented)

Time = 0.66 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=\frac {- 8008 a^{10} - 85085 a^{9} b x - 408408 a^{8} b^{2} x^{2} - 1166880 a^{7} b^{3} x^{3} - 2199120 a^{6} b^{4} x^{4} - 2858856 a^{5} b^{5} x^{5} - 2598960 a^{4} b^{6} x^{6} - 1633632 a^{3} b^{7} x^{7} - 680680 a^{2} b^{8} x^{8} - 170170 a b^{9} x^{9} - 19448 b^{10} x^{10}}{136136 x^{17}} \]

[In]

integrate((b*x+a)**10/x**18,x)

[Out]

(-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 - 2858
856*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)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {19448 \, b^{10} x^{10} + 170170 \, a b^{9} x^{9} + 680680 \, a^{2} b^{8} x^{8} + 1633632 \, a^{3} b^{7} x^{7} + 2598960 \, a^{4} b^{6} x^{6} + 2858856 \, a^{5} b^{5} x^{5} + 2199120 \, a^{6} b^{4} x^{4} + 1166880 \, a^{7} b^{3} x^{3} + 408408 \, a^{8} b^{2} x^{2} + 85085 \, a^{9} b x + 8008 \, a^{10}}{136136 \, x^{17}} \]

[In]

integrate((b*x+a)^10/x^18,x, algorithm="maxima")

[Out]

-1/136136*(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)/x^17

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {19448 \, b^{10} x^{10} + 170170 \, a b^{9} x^{9} + 680680 \, a^{2} b^{8} x^{8} + 1633632 \, a^{3} b^{7} x^{7} + 2598960 \, a^{4} b^{6} x^{6} + 2858856 \, a^{5} b^{5} x^{5} + 2199120 \, a^{6} b^{4} x^{4} + 1166880 \, a^{7} b^{3} x^{3} + 408408 \, a^{8} b^{2} x^{2} + 85085 \, a^{9} b x + 8008 \, a^{10}}{136136 \, x^{17}} \]

[In]

integrate((b*x+a)^10/x^18,x, algorithm="giac")

[Out]

-1/136136*(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)/x^17

Mupad [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{10}}{x^{18}} \, dx=-\frac {\frac {a^{10}}{17}+\frac {5\,a^9\,b\,x}{8}+3\,a^8\,b^2\,x^2+\frac {60\,a^7\,b^3\,x^3}{7}+\frac {210\,a^6\,b^4\,x^4}{13}+21\,a^5\,b^5\,x^5+\frac {210\,a^4\,b^6\,x^6}{11}+12\,a^3\,b^7\,x^7+5\,a^2\,b^8\,x^8+\frac {5\,a\,b^9\,x^9}{4}+\frac {b^{10}\,x^{10}}{7}}{x^{17}} \]

[In]

int((a + b*x)^10/x^18,x)

[Out]

-(a^10/17 + (b^10*x^10)/7 + (5*a*b^9*x^9)/4 + 3*a^8*b^2*x^2 + (60*a^7*b^3*x^3)/7 + (210*a^6*b^4*x^4)/13 + 21*a
^5*b^5*x^5 + (210*a^4*b^6*x^6)/11 + 12*a^3*b^7*x^7 + 5*a^2*b^8*x^8 + (5*a^9*b*x)/8)/x^17