\(\int \frac {(a+b x)^{10}}{x^{16}} \, dx\) [150]

Optimal result

Integrand size = 11, antiderivative size = 96 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {(a+b x)^{11}}{15 a x^{15}}+\frac {2 b (a+b x)^{11}}{105 a^2 x^{14}}-\frac {2 b^2 (a+b x)^{11}}{455 a^3 x^{13}}+\frac {b^3 (a+b x)^{11}}{1365 a^4 x^{12}}-\frac {b^4 (a+b x)^{11}}{15015 a^5 x^{11}} \]

[Out]

-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

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {b^4 (a+b x)^{11}}{15015 a^5 x^{11}}+\frac {b^3 (a+b x)^{11}}{1365 a^4 x^{12}}-\frac {2 b^2 (a+b x)^{11}}{455 a^3 x^{13}}+\frac {2 b (a+b x)^{11}}{105 a^2 x^{14}}-\frac {(a+b x)^{11}}{15 a x^{15}} \]

[In]

Int[(a + b*x)^10/x^16,x]

[Out]

-1/15*(a + b*x)^11/(a*x^15) + (2*b*(a + b*x)^11)/(105*a^2*x^14) - (2*b^2*(a + b*x)^11)/(455*a^3*x^13) + (b^3*(
a + b*x)^11)/(1365*a^4*x^12) - (b^4*(a + b*x)^11)/(15015*a^5*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}}{15 a x^{15}}-\frac {(4 b) \int \frac {(a+b x)^{10}}{x^{15}} \, dx}{15 a} \\ & = -\frac {(a+b x)^{11}}{15 a x^{15}}+\frac {2 b (a+b x)^{11}}{105 a^2 x^{14}}+\frac {\left (2 b^2\right ) \int \frac {(a+b x)^{10}}{x^{14}} \, dx}{35 a^2} \\ & = -\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 {\left (4 b^3\right ) \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{455 a^3} \\ & = -\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 \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{1365 a^4} \\ & = -\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}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.35 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {a^{10}}{15 x^{15}}-\frac {5 a^9 b}{7 x^{14}}-\frac {45 a^8 b^2}{13 x^{13}}-\frac {10 a^7 b^3}{x^{12}}-\frac {210 a^6 b^4}{11 x^{11}}-\frac {126 a^5 b^5}{5 x^{10}}-\frac {70 a^4 b^6}{3 x^9}-\frac {15 a^3 b^7}{x^8}-\frac {45 a^2 b^8}{7 x^7}-\frac {5 a b^9}{3 x^6}-\frac {b^{10}}{5 x^5} \]

[In]

Integrate[(a + b*x)^10/x^16,x]

[Out]

-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)

Maple [A] (verified)

Time = 0.17 (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	\(-\frac {126 a^{5} b^{5}}{5 x^{10}}-\frac {5 a \,b^{9}}{3 x^{6}}-\frac {a^{10}}{15 x^{15}}-\frac {5 a^{9} b}{7 x^{14}}-\frac {45 a^{2} b^{8}}{7 x^{7}}-\frac {45 a^{8} b^{2}}{13 x^{13}}-\frac {70 a^{4} b^{6}}{3 x^{9}}-\frac {10 a^{7} b^{3}}{x^{12}}-\frac {210 a^{6} b^{4}}{11 x^{11}}-\frac {b^{10}}{5 x^{5}}-\frac {15 a^{3} b^{7}}{x^{8}}\)	\(113\)
parallelrisch	\(\frac {-3003 b^{10} x^{10}-25025 a \,b^{9} x^{9}-96525 a^{2} b^{8} x^{8}-225225 a^{3} b^{7} x^{7}-350350 a^{4} b^{6} x^{6}-378378 a^{5} b^{5} x^{5}-286650 a^{6} b^{4} x^{4}-150150 a^{7} b^{3} x^{3}-51975 a^{8} b^{2} x^{2}-10725 a^{9} b x -1001 a^{10}}{15015 x^{15}}\)	\(113\)

[In]

int((b*x+a)^10/x^16,x,method=_RETURNVERBOSE)

[Out]

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)

Fricas [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {3003 \, b^{10} x^{10} + 25025 \, a b^{9} x^{9} + 96525 \, a^{2} b^{8} x^{8} + 225225 \, a^{3} b^{7} x^{7} + 350350 \, a^{4} b^{6} x^{6} + 378378 \, a^{5} b^{5} x^{5} + 286650 \, a^{6} b^{4} x^{4} + 150150 \, a^{7} b^{3} x^{3} + 51975 \, a^{8} b^{2} x^{2} + 10725 \, a^{9} b x + 1001 \, a^{10}}{15015 \, x^{15}} \]

[In]

integrate((b*x+a)^10/x^16,x, algorithm="fricas")

[Out]

-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 + 378
378*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^1
5

Sympy [A] (verification not implemented)

Time = 0.61 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=\frac {- 1001 a^{10} - 10725 a^{9} b x - 51975 a^{8} b^{2} x^{2} - 150150 a^{7} b^{3} x^{3} - 286650 a^{6} b^{4} x^{4} - 378378 a^{5} b^{5} x^{5} - 350350 a^{4} b^{6} x^{6} - 225225 a^{3} b^{7} x^{7} - 96525 a^{2} b^{8} x^{8} - 25025 a b^{9} x^{9} - 3003 b^{10} x^{10}}{15015 x^{15}} \]

[In]

integrate((b*x+a)**10/x**16,x)

[Out]

(-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)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {3003 \, b^{10} x^{10} + 25025 \, a b^{9} x^{9} + 96525 \, a^{2} b^{8} x^{8} + 225225 \, a^{3} b^{7} x^{7} + 350350 \, a^{4} b^{6} x^{6} + 378378 \, a^{5} b^{5} x^{5} + 286650 \, a^{6} b^{4} x^{4} + 150150 \, a^{7} b^{3} x^{3} + 51975 \, a^{8} b^{2} x^{2} + 10725 \, a^{9} b x + 1001 \, a^{10}}{15015 \, x^{15}} \]

[In]

integrate((b*x+a)^10/x^16,x, algorithm="maxima")

[Out]

-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 + 378
378*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^1
5

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {3003 \, b^{10} x^{10} + 25025 \, a b^{9} x^{9} + 96525 \, a^{2} b^{8} x^{8} + 225225 \, a^{3} b^{7} x^{7} + 350350 \, a^{4} b^{6} x^{6} + 378378 \, a^{5} b^{5} x^{5} + 286650 \, a^{6} b^{4} x^{4} + 150150 \, a^{7} b^{3} x^{3} + 51975 \, a^{8} b^{2} x^{2} + 10725 \, a^{9} b x + 1001 \, a^{10}}{15015 \, x^{15}} \]

[In]

integrate((b*x+a)^10/x^16,x, algorithm="giac")

[Out]

-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 + 378
378*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^1
5

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.17 \[ \int \frac {(a+b x)^{10}}{x^{16}} \, dx=-\frac {\frac {a^{10}}{15}+\frac {5\,a^9\,b\,x}{7}+\frac {45\,a^8\,b^2\,x^2}{13}+10\,a^7\,b^3\,x^3+\frac {210\,a^6\,b^4\,x^4}{11}+\frac {126\,a^5\,b^5\,x^5}{5}+\frac {70\,a^4\,b^6\,x^6}{3}+15\,a^3\,b^7\,x^7+\frac {45\,a^2\,b^8\,x^8}{7}+\frac {5\,a\,b^9\,x^9}{3}+\frac {b^{10}\,x^{10}}{5}}{x^{15}} \]

[In]

int((a + b*x)^10/x^16,x)

[Out]

-(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 + (1
26*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