\(\int \frac {(a+b x)^{10}}{x^{20}} \, dx\) [154]

Optimal result

Integrand size = 11, antiderivative size = 126 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {a^{10}}{19 x^{19}}-\frac {5 a^9 b}{9 x^{18}}-\frac {45 a^8 b^2}{17 x^{17}}-\frac {15 a^7 b^3}{2 x^{16}}-\frac {14 a^6 b^4}{x^{15}}-\frac {18 a^5 b^5}{x^{14}}-\frac {210 a^4 b^6}{13 x^{13}}-\frac {10 a^3 b^7}{x^{12}}-\frac {45 a^2 b^8}{11 x^{11}}-\frac {a b^9}{x^{10}}-\frac {b^{10}}{9 x^9} \]

[Out]

-1/19*a^10/x^19-5/9*a^9*b/x^18-45/17*a^8*b^2/x^17-15/2*a^7*b^3/x^16-14*a^6*b^4/x^15-18*a^5*b^5/x^14-210/13*a^4
*b^6/x^13-10*a^3*b^7/x^12-45/11*a^2*b^8/x^11-a*b^9/x^10-1/9*b^10/x^9

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {a^{10}}{19 x^{19}}-\frac {5 a^9 b}{9 x^{18}}-\frac {45 a^8 b^2}{17 x^{17}}-\frac {15 a^7 b^3}{2 x^{16}}-\frac {14 a^6 b^4}{x^{15}}-\frac {18 a^5 b^5}{x^{14}}-\frac {210 a^4 b^6}{13 x^{13}}-\frac {10 a^3 b^7}{x^{12}}-\frac {45 a^2 b^8}{11 x^{11}}-\frac {a b^9}{x^{10}}-\frac {b^{10}}{9 x^9} \]

[In]

Int[(a + b*x)^10/x^20,x]

[Out]

-1/19*a^10/x^19 - (5*a^9*b)/(9*x^18) - (45*a^8*b^2)/(17*x^17) - (15*a^7*b^3)/(2*x^16) - (14*a^6*b^4)/x^15 - (1
8*a^5*b^5)/x^14 - (210*a^4*b^6)/(13*x^13) - (10*a^3*b^7)/x^12 - (45*a^2*b^8)/(11*x^11) - (a*b^9)/x^10 - b^10/(
9*x^9)

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {a^{10}}{x^{20}}+\frac {10 a^9 b}{x^{19}}+\frac {45 a^8 b^2}{x^{18}}+\frac {120 a^7 b^3}{x^{17}}+\frac {210 a^6 b^4}{x^{16}}+\frac {252 a^5 b^5}{x^{15}}+\frac {210 a^4 b^6}{x^{14}}+\frac {120 a^3 b^7}{x^{13}}+\frac {45 a^2 b^8}{x^{12}}+\frac {10 a b^9}{x^{11}}+\frac {b^{10}}{x^{10}}\right ) \, dx \\ & = -\frac {a^{10}}{19 x^{19}}-\frac {5 a^9 b}{9 x^{18}}-\frac {45 a^8 b^2}{17 x^{17}}-\frac {15 a^7 b^3}{2 x^{16}}-\frac {14 a^6 b^4}{x^{15}}-\frac {18 a^5 b^5}{x^{14}}-\frac {210 a^4 b^6}{13 x^{13}}-\frac {10 a^3 b^7}{x^{12}}-\frac {45 a^2 b^8}{11 x^{11}}-\frac {a b^9}{x^{10}}-\frac {b^{10}}{9 x^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {a^{10}}{19 x^{19}}-\frac {5 a^9 b}{9 x^{18}}-\frac {45 a^8 b^2}{17 x^{17}}-\frac {15 a^7 b^3}{2 x^{16}}-\frac {14 a^6 b^4}{x^{15}}-\frac {18 a^5 b^5}{x^{14}}-\frac {210 a^4 b^6}{13 x^{13}}-\frac {10 a^3 b^7}{x^{12}}-\frac {45 a^2 b^8}{11 x^{11}}-\frac {a b^9}{x^{10}}-\frac {b^{10}}{9 x^9} \]

[In]

Integrate[(a + b*x)^10/x^20,x]

[Out]

-1/19*a^10/x^19 - (5*a^9*b)/(9*x^18) - (45*a^8*b^2)/(17*x^17) - (15*a^7*b^3)/(2*x^16) - (14*a^6*b^4)/x^15 - (1
8*a^5*b^5)/x^14 - (210*a^4*b^6)/(13*x^13) - (10*a^3*b^7)/x^12 - (45*a^2*b^8)/(11*x^11) - (a*b^9)/x^10 - b^10/(
9*x^9)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89

method	result	size
norman	\(\frac {-\frac {1}{19} a^{10}-\frac {5}{9} a^{9} b x -\frac {45}{17} a^{8} b^{2} x^{2}-\frac {15}{2} a^{7} b^{3} x^{3}-14 a^{6} b^{4} x^{4}-18 a^{5} b^{5} x^{5}-\frac {210}{13} a^{4} b^{6} x^{6}-10 a^{3} b^{7} x^{7}-\frac {45}{11} a^{2} b^{8} x^{8}-a \,b^{9} x^{9}-\frac {1}{9} b^{10} x^{10}}{x^{19}}\)	\(112\)
risch	\(\frac {-\frac {1}{19} a^{10}-\frac {5}{9} a^{9} b x -\frac {45}{17} a^{8} b^{2} x^{2}-\frac {15}{2} a^{7} b^{3} x^{3}-14 a^{6} b^{4} x^{4}-18 a^{5} b^{5} x^{5}-\frac {210}{13} a^{4} b^{6} x^{6}-10 a^{3} b^{7} x^{7}-\frac {45}{11} a^{2} b^{8} x^{8}-a \,b^{9} x^{9}-\frac {1}{9} b^{10} x^{10}}{x^{19}}\)	\(112\)
gosper	\(-\frac {92378 b^{10} x^{10}+831402 a \,b^{9} x^{9}+3401190 a^{2} b^{8} x^{8}+8314020 a^{3} b^{7} x^{7}[phone] a^{4} b^{6} x^{6}[phone] a^{5} b^{5} x^{5}[phone] a^{6} b^{4} x^{4}+6235515 a^{7} b^{3} x^{3}+2200770 a^{8} b^{2} x^{2}+461890 a^{9} b x +43758 a^{10}}{831402 x^{19}}\)	\(113\)
default	\(-\frac {a^{10}}{19 x^{19}}-\frac {5 a^{9} b}{9 x^{18}}-\frac {45 a^{8} b^{2}}{17 x^{17}}-\frac {15 a^{7} b^{3}}{2 x^{16}}-\frac {14 a^{6} b^{4}}{x^{15}}-\frac {18 a^{5} b^{5}}{x^{14}}-\frac {210 a^{4} b^{6}}{13 x^{13}}-\frac {10 a^{3} b^{7}}{x^{12}}-\frac {45 a^{2} b^{8}}{11 x^{11}}-\frac {a \,b^{9}}{x^{10}}-\frac {b^{10}}{9 x^{9}}\)	\(113\)
parallelrisch	\(\frac {-92378 b^{10} x^{10}-831402 a \,b^{9} x^{9}-3401190 a^{2} b^{8} x^{8}-8314020 a^{3} b^{7} x^{7}-13430340 a^{4} b^{6} x^{6}-14965236 a^{5} b^{5} x^{5}-11639628 a^{6} b^{4} x^{4}-6235515 a^{7} b^{3} x^{3}-2200770 a^{8} b^{2} x^{2}-461890 a^{9} b x -43758 a^{10}}{831402 x^{19}}\)	\(113\)

[In]

int((b*x+a)^10/x^20,x,method=_RETURNVERBOSE)

[Out]

1/x^19*(-1/19*a^10-5/9*a^9*b*x-45/17*a^8*b^2*x^2-15/2*a^7*b^3*x^3-14*a^6*b^4*x^4-18*a^5*b^5*x^5-210/13*a^4*b^6
*x^6-10*a^3*b^7*x^7-45/11*a^2*b^8*x^8-a*b^9*x^9-1/9*b^10*x^10)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {92378 \, b^{10} x^{10} + 831402 \, a b^{9} x^{9} + 3401190 \, a^{2} b^{8} x^{8} + 8314020 \, a^{3} b^{7} x^{7} + 13430340 \, a^{4} b^{6} x^{6} + 14965236 \, a^{5} b^{5} x^{5} + 11639628 \, a^{6} b^{4} x^{4} + 6235515 \, a^{7} b^{3} x^{3} + 2200770 \, a^{8} b^{2} x^{2} + 461890 \, a^{9} b x + 43758 \, a^{10}}{831402 \, x^{19}} \]

[In]

integrate((b*x+a)^10/x^20,x, algorithm="fricas")

[Out]

-1/831402*(92378*b^10*x^10 + 831402*a*b^9*x^9 + 3401190*a^2*b^8*x^8 + 8314020*a^3*b^7*x^7 + 13430340*a^4*b^6*x
^6 + 14965236*a^5*b^5*x^5 + 11639628*a^6*b^4*x^4 + 6235515*a^7*b^3*x^3 + 2200770*a^8*b^2*x^2 + 461890*a^9*b*x
+ 43758*a^10)/x^19

Sympy [A] (verification not implemented)

Time = 0.70 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.96 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=\frac {- 43758 a^{10} - 461890 a^{9} b x - 2200770 a^{8} b^{2} x^{2} - 6235515 a^{7} b^{3} x^{3} - 11639628 a^{6} b^{4} x^{4} - 14965236 a^{5} b^{5} x^{5} - 13430340 a^{4} b^{6} x^{6} - 8314020 a^{3} b^{7} x^{7} - 3401190 a^{2} b^{8} x^{8} - 831402 a b^{9} x^{9} - 92378 b^{10} x^{10}}{831402 x^{19}} \]

[In]

integrate((b*x+a)**10/x**20,x)

[Out]

(-43758*a**10 - 461890*a**9*b*x - 2200770*a**8*b**2*x**2 - 6235515*a**7*b**3*x**3 - 11639628*a**6*b**4*x**4 -
14965236*a**5*b**5*x**5 - 13430340*a**4*b**6*x**6 - 8314020*a**3*b**7*x**7 - 3401190*a**2*b**8*x**8 - 831402*a
*b**9*x**9 - 92378*b**10*x**10)/(831402*x**19)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {92378 \, b^{10} x^{10} + 831402 \, a b^{9} x^{9} + 3401190 \, a^{2} b^{8} x^{8} + 8314020 \, a^{3} b^{7} x^{7} + 13430340 \, a^{4} b^{6} x^{6} + 14965236 \, a^{5} b^{5} x^{5} + 11639628 \, a^{6} b^{4} x^{4} + 6235515 \, a^{7} b^{3} x^{3} + 2200770 \, a^{8} b^{2} x^{2} + 461890 \, a^{9} b x + 43758 \, a^{10}}{831402 \, x^{19}} \]

[In]

integrate((b*x+a)^10/x^20,x, algorithm="maxima")

[Out]

-1/831402*(92378*b^10*x^10 + 831402*a*b^9*x^9 + 3401190*a^2*b^8*x^8 + 8314020*a^3*b^7*x^7 + 13430340*a^4*b^6*x
^6 + 14965236*a^5*b^5*x^5 + 11639628*a^6*b^4*x^4 + 6235515*a^7*b^3*x^3 + 2200770*a^8*b^2*x^2 + 461890*a^9*b*x
+ 43758*a^10)/x^19

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.89 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {92378 \, b^{10} x^{10} + 831402 \, a b^{9} x^{9} + 3401190 \, a^{2} b^{8} x^{8} + 8314020 \, a^{3} b^{7} x^{7} + 13430340 \, a^{4} b^{6} x^{6} + 14965236 \, a^{5} b^{5} x^{5} + 11639628 \, a^{6} b^{4} x^{4} + 6235515 \, a^{7} b^{3} x^{3} + 2200770 \, a^{8} b^{2} x^{2} + 461890 \, a^{9} b x + 43758 \, a^{10}}{831402 \, x^{19}} \]

[In]

integrate((b*x+a)^10/x^20,x, algorithm="giac")

[Out]

-1/831402*(92378*b^10*x^10 + 831402*a*b^9*x^9 + 3401190*a^2*b^8*x^8 + 8314020*a^3*b^7*x^7 + 13430340*a^4*b^6*x
^6 + 14965236*a^5*b^5*x^5 + 11639628*a^6*b^4*x^4 + 6235515*a^7*b^3*x^3 + 2200770*a^8*b^2*x^2 + 461890*a^9*b*x
+ 43758*a^10)/x^19

Mupad [B] (verification not implemented)

Time = 0.15 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x)^{10}}{x^{20}} \, dx=-\frac {\frac {a^{10}}{19}+\frac {5\,a^9\,b\,x}{9}+\frac {45\,a^8\,b^2\,x^2}{17}+\frac {15\,a^7\,b^3\,x^3}{2}+14\,a^6\,b^4\,x^4+18\,a^5\,b^5\,x^5+\frac {210\,a^4\,b^6\,x^6}{13}+10\,a^3\,b^7\,x^7+\frac {45\,a^2\,b^8\,x^8}{11}+a\,b^9\,x^9+\frac {b^{10}\,x^{10}}{9}}{x^{19}} \]

[In]

int((a + b*x)^10/x^20,x)

[Out]

-(a^10/19 + (b^10*x^10)/9 + a*b^9*x^9 + (45*a^8*b^2*x^2)/17 + (15*a^7*b^3*x^3)/2 + 14*a^6*b^4*x^4 + 18*a^5*b^5
*x^5 + (210*a^4*b^6*x^6)/13 + 10*a^3*b^7*x^7 + (45*a^2*b^8*x^8)/11 + (5*a^9*b*x)/9)/x^19