\(\int \frac {(a+b x)^{10}}{x^{15}} \, dx\) [149]

Optimal result

Integrand size = 11, antiderivative size = 76 \[ \int \frac {(a+b x)^{10}}{x^{15}} \, dx=-\frac {(a+b x)^{11}}{14 a x^{14}}+\frac {3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac {b^2 (a+b x)^{11}}{364 a^3 x^{12}}+\frac {b^3 (a+b x)^{11}}{4004 a^4 x^{11}} \]

[Out]

-1/14*(b*x+a)^11/a/x^14+3/182*b*(b*x+a)^11/a^2/x^13-1/364*b^2*(b*x+a)^11/a^3/x^12+1/4004*b^3*(b*x+a)^11/a^4/x^
11

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^{10}}{x^{15}} \, dx=\frac {b^3 (a+b x)^{11}}{4004 a^4 x^{11}}-\frac {b^2 (a+b x)^{11}}{364 a^3 x^{12}}+\frac {3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac {(a+b x)^{11}}{14 a x^{14}} \]

[In]

Int[(a + b*x)^10/x^15,x]

[Out]

-1/14*(a + b*x)^11/(a*x^14) + (3*b*(a + b*x)^11)/(182*a^2*x^13) - (b^2*(a + b*x)^11)/(364*a^3*x^12) + (b^3*(a
+ b*x)^11)/(4004*a^4*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}}{14 a x^{14}}-\frac {(3 b) \int \frac {(a+b x)^{10}}{x^{14}} \, dx}{14 a} \\ & = -\frac {(a+b x)^{11}}{14 a x^{14}}+\frac {3 b (a+b x)^{11}}{182 a^2 x^{13}}+\frac {\left (3 b^2\right ) \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{91 a^2} \\ & = -\frac {(a+b x)^{11}}{14 a x^{14}}+\frac {3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac {b^2 (a+b x)^{11}}{364 a^3 x^{12}}-\frac {b^3 \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{364 a^3} \\ & = -\frac {(a+b x)^{11}}{14 a x^{14}}+\frac {3 b (a+b x)^{11}}{182 a^2 x^{13}}-\frac {b^2 (a+b x)^{11}}{364 a^3 x^{12}}+\frac {b^3 (a+b x)^{11}}{4004 a^4 x^{11}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.68 \[ \int \frac {(a+b x)^{10}}{x^{15}} \, dx=-\frac {a^{10}}{14 x^{14}}-\frac {10 a^9 b}{13 x^{13}}-\frac {15 a^8 b^2}{4 x^{12}}-\frac {120 a^7 b^3}{11 x^{11}}-\frac {21 a^6 b^4}{x^{10}}-\frac {28 a^5 b^5}{x^9}-\frac {105 a^4 b^6}{4 x^8}-\frac {120 a^3 b^7}{7 x^7}-\frac {15 a^2 b^8}{2 x^6}-\frac {2 a b^9}{x^5}-\frac {b^{10}}{4 x^4} \]

[In]

Integrate[(a + b*x)^10/x^15,x]

[Out]

-1/14*a^10/x^14 - (10*a^9*b)/(13*x^13) - (15*a^8*b^2)/(4*x^12) - (120*a^7*b^3)/(11*x^11) - (21*a^6*b^4)/x^10 -
(28*a^5*b^5)/x^9 - (105*a^4*b^6)/(4*x^8) - (120*a^3*b^7)/(7*x^7) - (15*a^2*b^8)/(2*x^6) - (2*a*b^9)/x^5 - b^1
0/(4*x^4)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.47

method	result	size
norman	\(\frac {-\frac {1}{14} a^{10}-\frac {10}{13} a^{9} b x -\frac {15}{4} a^{8} b^{2} x^{2}-\frac {120}{11} a^{7} b^{3} x^{3}-21 a^{6} b^{4} x^{4}-28 a^{5} b^{5} x^{5}-\frac {105}{4} a^{4} b^{6} x^{6}-\frac {120}{7} a^{3} b^{7} x^{7}-\frac {15}{2} a^{2} b^{8} x^{8}-2 a \,b^{9} x^{9}-\frac {1}{4} b^{10} x^{10}}{x^{14}}\)	\(112\)
risch	\(\frac {-\frac {1}{14} a^{10}-\frac {10}{13} a^{9} b x -\frac {15}{4} a^{8} b^{2} x^{2}-\frac {120}{11} a^{7} b^{3} x^{3}-21 a^{6} b^{4} x^{4}-28 a^{5} b^{5} x^{5}-\frac {105}{4} a^{4} b^{6} x^{6}-\frac {120}{7} a^{3} b^{7} x^{7}-\frac {15}{2} a^{2} b^{8} x^{8}-2 a \,b^{9} x^{9}-\frac {1}{4} b^{10} x^{10}}{x^{14}}\)	\(112\)
gosper	\(-\frac {1001 b^{10} x^{10}+8008 a \,b^{9} x^{9}+30030 a^{2} b^{8} x^{8}+68640 a^{3} b^{7} x^{7}+105105 a^{4} b^{6} x^{6}+112112 a^{5} b^{5} x^{5}+84084 a^{6} b^{4} x^{4}+43680 a^{7} b^{3} x^{3}+15015 a^{8} b^{2} x^{2}+3080 a^{9} b x +286 a^{10}}{4004 x^{14}}\)	\(113\)
default	\(-\frac {21 a^{6} b^{4}}{x^{10}}-\frac {15 a^{2} b^{8}}{2 x^{6}}-\frac {a^{10}}{14 x^{14}}-\frac {120 a^{3} b^{7}}{7 x^{7}}-\frac {10 a^{9} b}{13 x^{13}}-\frac {28 a^{5} b^{5}}{x^{9}}-\frac {15 a^{8} b^{2}}{4 x^{12}}-\frac {120 a^{7} b^{3}}{11 x^{11}}-\frac {b^{10}}{4 x^{4}}-\frac {2 a \,b^{9}}{x^{5}}-\frac {105 a^{4} b^{6}}{4 x^{8}}\)	\(113\)
parallelrisch	\(\frac {-1001 b^{10} x^{10}-8008 a \,b^{9} x^{9}-30030 a^{2} b^{8} x^{8}-68640 a^{3} b^{7} x^{7}-105105 a^{4} b^{6} x^{6}-112112 a^{5} b^{5} x^{5}-84084 a^{6} b^{4} x^{4}-43680 a^{7} b^{3} x^{3}-15015 a^{8} b^{2} x^{2}-3080 a^{9} b x -286 a^{10}}{4004 x^{14}}\)	\(113\)

[In]

int((b*x+a)^10/x^15,x,method=_RETURNVERBOSE)

[Out]

1/x^14*(-1/14*a^10-10/13*a^9*b*x-15/4*a^8*b^2*x^2-120/11*a^7*b^3*x^3-21*a^6*b^4*x^4-28*a^5*b^5*x^5-105/4*a^4*b
^6*x^6-120/7*a^3*b^7*x^7-15/2*a^2*b^8*x^8-2*a*b^9*x^9-1/4*b^10*x^10)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b x)^{10}}{x^{15}} \, dx=-\frac {1001 \, b^{10} x^{10} + 8008 \, a b^{9} x^{9} + 30030 \, a^{2} b^{8} x^{8} + 68640 \, a^{3} b^{7} x^{7} + 105105 \, a^{4} b^{6} x^{6} + 112112 \, a^{5} b^{5} x^{5} + 84084 \, a^{6} b^{4} x^{4} + 43680 \, a^{7} b^{3} x^{3} + 15015 \, a^{8} b^{2} x^{2} + 3080 \, a^{9} b x + 286 \, a^{10}}{4004 \, x^{14}} \]

[In]

integrate((b*x+a)^10/x^15,x, algorithm="fricas")

[Out]

-1/4004*(1001*b^10*x^10 + 8008*a*b^9*x^9 + 30030*a^2*b^8*x^8 + 68640*a^3*b^7*x^7 + 105105*a^4*b^6*x^6 + 112112
*a^5*b^5*x^5 + 84084*a^6*b^4*x^4 + 43680*a^7*b^3*x^3 + 15015*a^8*b^2*x^2 + 3080*a^9*b*x + 286*a^10)/x^14

Sympy [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.59 \[ \int \frac {(a+b x)^{10}}{x^{15}} \, dx=\frac {- 286 a^{10} - 3080 a^{9} b x - 15015 a^{8} b^{2} x^{2} - 43680 a^{7} b^{3} x^{3} - 84084 a^{6} b^{4} x^{4} - 112112 a^{5} b^{5} x^{5} - 105105 a^{4} b^{6} x^{6} - 68640 a^{3} b^{7} x^{7} - 30030 a^{2} b^{8} x^{8} - 8008 a b^{9} x^{9} - 1001 b^{10} x^{10}}{4004 x^{14}} \]

[In]

integrate((b*x+a)**10/x**15,x)

[Out]

(-286*a**10 - 3080*a**9*b*x - 15015*a**8*b**2*x**2 - 43680*a**7*b**3*x**3 - 84084*a**6*b**4*x**4 - 112112*a**5
*b**5*x**5 - 105105*a**4*b**6*x**6 - 68640*a**3*b**7*x**7 - 30030*a**2*b**8*x**8 - 8008*a*b**9*x**9 - 1001*b**
10*x**10)/(4004*x**14)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b x)^{10}}{x^{15}} \, dx=-\frac {1001 \, b^{10} x^{10} + 8008 \, a b^{9} x^{9} + 30030 \, a^{2} b^{8} x^{8} + 68640 \, a^{3} b^{7} x^{7} + 105105 \, a^{4} b^{6} x^{6} + 112112 \, a^{5} b^{5} x^{5} + 84084 \, a^{6} b^{4} x^{4} + 43680 \, a^{7} b^{3} x^{3} + 15015 \, a^{8} b^{2} x^{2} + 3080 \, a^{9} b x + 286 \, a^{10}}{4004 \, x^{14}} \]

[In]

integrate((b*x+a)^10/x^15,x, algorithm="maxima")

[Out]

-1/4004*(1001*b^10*x^10 + 8008*a*b^9*x^9 + 30030*a^2*b^8*x^8 + 68640*a^3*b^7*x^7 + 105105*a^4*b^6*x^6 + 112112
*a^5*b^5*x^5 + 84084*a^6*b^4*x^4 + 43680*a^7*b^3*x^3 + 15015*a^8*b^2*x^2 + 3080*a^9*b*x + 286*a^10)/x^14

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b x)^{10}}{x^{15}} \, dx=-\frac {1001 \, b^{10} x^{10} + 8008 \, a b^{9} x^{9} + 30030 \, a^{2} b^{8} x^{8} + 68640 \, a^{3} b^{7} x^{7} + 105105 \, a^{4} b^{6} x^{6} + 112112 \, a^{5} b^{5} x^{5} + 84084 \, a^{6} b^{4} x^{4} + 43680 \, a^{7} b^{3} x^{3} + 15015 \, a^{8} b^{2} x^{2} + 3080 \, a^{9} b x + 286 \, a^{10}}{4004 \, x^{14}} \]

[In]

integrate((b*x+a)^10/x^15,x, algorithm="giac")

[Out]

-1/4004*(1001*b^10*x^10 + 8008*a*b^9*x^9 + 30030*a^2*b^8*x^8 + 68640*a^3*b^7*x^7 + 105105*a^4*b^6*x^6 + 112112
*a^5*b^5*x^5 + 84084*a^6*b^4*x^4 + 43680*a^7*b^3*x^3 + 15015*a^8*b^2*x^2 + 3080*a^9*b*x + 286*a^10)/x^14

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.47 \[ \int \frac {(a+b x)^{10}}{x^{15}} \, dx=-\frac {\frac {a^{10}}{14}+\frac {10\,a^9\,b\,x}{13}+\frac {15\,a^8\,b^2\,x^2}{4}+\frac {120\,a^7\,b^3\,x^3}{11}+21\,a^6\,b^4\,x^4+28\,a^5\,b^5\,x^5+\frac {105\,a^4\,b^6\,x^6}{4}+\frac {120\,a^3\,b^7\,x^7}{7}+\frac {15\,a^2\,b^8\,x^8}{2}+2\,a\,b^9\,x^9+\frac {b^{10}\,x^{10}}{4}}{x^{14}} \]

[In]

int((a + b*x)^10/x^15,x)

[Out]

-(a^10/14 + (b^10*x^10)/4 + 2*a*b^9*x^9 + (15*a^8*b^2*x^2)/4 + (120*a^7*b^3*x^3)/11 + 21*a^6*b^4*x^4 + 28*a^5*
b^5*x^5 + (105*a^4*b^6*x^6)/4 + (120*a^3*b^7*x^7)/7 + (15*a^2*b^8*x^8)/2 + (10*a^9*b*x)/13)/x^14