\(\int \frac {(a+b x)^{10}}{x^{17}} \, dx\) [151]

Optimal result

Integrand size = 11, antiderivative size = 116 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=-\frac {(a+b x)^{11}}{16 a x^{16}}+\frac {b (a+b x)^{11}}{48 a^2 x^{15}}-\frac {b^2 (a+b x)^{11}}{168 a^3 x^{14}}+\frac {b^3 (a+b x)^{11}}{728 a^4 x^{13}}-\frac {b^4 (a+b x)^{11}}{4368 a^5 x^{12}}+\frac {b^5 (a+b x)^{11}}{48048 a^6 x^{11}} \]

[Out]

-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+1/48048*b^5*(b*x+a)^11/a^6/x^11

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=\frac {b^5 (a+b x)^{11}}{48048 a^6 x^{11}}-\frac {b^4 (a+b x)^{11}}{4368 a^5 x^{12}}+\frac {b^3 (a+b x)^{11}}{728 a^4 x^{13}}-\frac {b^2 (a+b x)^{11}}{168 a^3 x^{14}}+\frac {b (a+b x)^{11}}{48 a^2 x^{15}}-\frac {(a+b x)^{11}}{16 a x^{16}} \]

[In]

Int[(a + b*x)^10/x^17,x]

[Out]

-1/16*(a + b*x)^11/(a*x^16) + (b*(a + b*x)^11)/(48*a^2*x^15) - (b^2*(a + b*x)^11)/(168*a^3*x^14) + (b^3*(a + b
*x)^11)/(728*a^4*x^13) - (b^4*(a + b*x)^11)/(4368*a^5*x^12) + (b^5*(a + b*x)^11)/(48048*a^6*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}}{16 a x^{16}}-\frac {(5 b) \int \frac {(a+b x)^{10}}{x^{16}} \, dx}{16 a} \\ & = -\frac {(a+b x)^{11}}{16 a x^{16}}+\frac {b (a+b x)^{11}}{48 a^2 x^{15}}+\frac {b^2 \int \frac {(a+b x)^{10}}{x^{15}} \, dx}{12 a^2} \\ & = -\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 \int \frac {(a+b x)^{10}}{x^{14}} \, dx}{56 a^3} \\ & = -\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 \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{364 a^4} \\ & = -\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 \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{4368 a^5} \\ & = -\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 132, normalized size of antiderivative = 1.14 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=-\frac {a^{10}}{16 x^{16}}-\frac {2 a^9 b}{3 x^{15}}-\frac {45 a^8 b^2}{14 x^{14}}-\frac {120 a^7 b^3}{13 x^{13}}-\frac {35 a^6 b^4}{2 x^{12}}-\frac {252 a^5 b^5}{11 x^{11}}-\frac {21 a^4 b^6}{x^{10}}-\frac {40 a^3 b^7}{3 x^9}-\frac {45 a^2 b^8}{8 x^8}-\frac {10 a b^9}{7 x^7}-\frac {b^{10}}{6 x^6} \]

[In]

Integrate[(a + b*x)^10/x^17,x]

[Out]

-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)

Maple [A] (verified)

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	\(\frac {-8008 b^{10} x^{10}-68640 a \,b^{9} x^{9}-270270 a^{2} b^{8} x^{8}-640640 a^{3} b^{7} x^{7}-1009008 a^{4} b^{6} x^{6}-1100736 a^{5} b^{5} x^{5}-840840 a^{6} b^{4} x^{4}-443520 a^{7} b^{3} x^{3}-154440 a^{8} b^{2} x^{2}-32032 a^{9} b x -3003 a^{10}}{48048 x^{16}}\)	\(113\)

[In]

int((b*x+a)^10/x^17,x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=-\frac {8008 \, b^{10} x^{10} + 68640 \, a b^{9} x^{9} + 270270 \, a^{2} b^{8} x^{8} + 640640 \, a^{3} b^{7} x^{7} + 1009008 \, a^{4} b^{6} x^{6} + 1100736 \, a^{5} b^{5} x^{5} + 840840 \, a^{6} b^{4} x^{4} + 443520 \, a^{7} b^{3} x^{3} + 154440 \, a^{8} b^{2} x^{2} + 32032 \, a^{9} b x + 3003 \, a^{10}}{48048 \, x^{16}} \]

[In]

integrate((b*x+a)^10/x^17,x, algorithm="fricas")

[Out]

-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 + 1
100736*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

Sympy [A] (verification not implemented)

Time = 0.65 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.04 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=\frac {- 3003 a^{10} - 32032 a^{9} b x - 154440 a^{8} b^{2} x^{2} - 443520 a^{7} b^{3} x^{3} - 840840 a^{6} b^{4} x^{4} - 1100736 a^{5} b^{5} x^{5} - 1009008 a^{4} b^{6} x^{6} - 640640 a^{3} b^{7} x^{7} - 270270 a^{2} b^{8} x^{8} - 68640 a b^{9} x^{9} - 8008 b^{10} x^{10}}{48048 x^{16}} \]

[In]

integrate((b*x+a)**10/x**17,x)

[Out]

(-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 - 110073
6*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)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=-\frac {8008 \, b^{10} x^{10} + 68640 \, a b^{9} x^{9} + 270270 \, a^{2} b^{8} x^{8} + 640640 \, a^{3} b^{7} x^{7} + 1009008 \, a^{4} b^{6} x^{6} + 1100736 \, a^{5} b^{5} x^{5} + 840840 \, a^{6} b^{4} x^{4} + 443520 \, a^{7} b^{3} x^{3} + 154440 \, a^{8} b^{2} x^{2} + 32032 \, a^{9} b x + 3003 \, a^{10}}{48048 \, x^{16}} \]

[In]

integrate((b*x+a)^10/x^17,x, algorithm="maxima")

[Out]

-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 + 1
100736*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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=-\frac {8008 \, b^{10} x^{10} + 68640 \, a b^{9} x^{9} + 270270 \, a^{2} b^{8} x^{8} + 640640 \, a^{3} b^{7} x^{7} + 1009008 \, a^{4} b^{6} x^{6} + 1100736 \, a^{5} b^{5} x^{5} + 840840 \, a^{6} b^{4} x^{4} + 443520 \, a^{7} b^{3} x^{3} + 154440 \, a^{8} b^{2} x^{2} + 32032 \, a^{9} b x + 3003 \, a^{10}}{48048 \, x^{16}} \]

[In]

integrate((b*x+a)^10/x^17,x, algorithm="giac")

[Out]

-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 + 1
100736*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

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.97 \[ \int \frac {(a+b x)^{10}}{x^{17}} \, dx=-\frac {\frac {a^{10}}{16}+\frac {2\,a^9\,b\,x}{3}+\frac {45\,a^8\,b^2\,x^2}{14}+\frac {120\,a^7\,b^3\,x^3}{13}+\frac {35\,a^6\,b^4\,x^4}{2}+\frac {252\,a^5\,b^5\,x^5}{11}+21\,a^4\,b^6\,x^6+\frac {40\,a^3\,b^7\,x^7}{3}+\frac {45\,a^2\,b^8\,x^8}{8}+\frac {10\,a\,b^9\,x^9}{7}+\frac {b^{10}\,x^{10}}{6}}{x^{16}} \]

[In]

int((a + b*x)^10/x^17,x)

[Out]

-(a^10/16 + (b^10*x^10)/6 + (10*a*b^9*x^9)/7 + (45*a^8*b^2*x^2)/14 + (120*a^7*b^3*x^3)/13 + (35*a^6*b^4*x^4)/2
+ (252*a^5*b^5*x^5)/11 + 21*a^4*b^6*x^6 + (40*a^3*b^7*x^7)/3 + (45*a^2*b^8*x^8)/8 + (2*a^9*b*x)/3)/x^16