\(\int \frac {(a+b x)^{10}}{x^{14}} \, dx\) [148]

Optimal result

Integrand size = 11, antiderivative size = 56 \[ \int \frac {(a+b x)^{10}}{x^{14}} \, dx=-\frac {(a+b x)^{11}}{13 a x^{13}}+\frac {b (a+b x)^{11}}{78 a^2 x^{12}}-\frac {b^2 (a+b x)^{11}}{858 a^3 x^{11}} \]

[Out]

-1/13*(b*x+a)^11/a/x^13+1/78*b*(b*x+a)^11/a^2/x^12-1/858*b^2*(b*x+a)^11/a^3/x^11

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 56, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {47, 37} \[ \int \frac {(a+b x)^{10}}{x^{14}} \, dx=-\frac {b^2 (a+b x)^{11}}{858 a^3 x^{11}}+\frac {b (a+b x)^{11}}{78 a^2 x^{12}}-\frac {(a+b x)^{11}}{13 a x^{13}} \]

[In]

Int[(a + b*x)^10/x^14,x]

[Out]

-1/13*(a + b*x)^11/(a*x^13) + (b*(a + b*x)^11)/(78*a^2*x^12) - (b^2*(a + b*x)^11)/(858*a^3*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}}{13 a x^{13}}-\frac {(2 b) \int \frac {(a+b x)^{10}}{x^{13}} \, dx}{13 a} \\ & = -\frac {(a+b x)^{11}}{13 a x^{13}}+\frac {b (a+b x)^{11}}{78 a^2 x^{12}}+\frac {b^2 \int \frac {(a+b x)^{10}}{x^{12}} \, dx}{78 a^2} \\ & = -\frac {(a+b x)^{11}}{13 a x^{13}}+\frac {b (a+b x)^{11}}{78 a^2 x^{12}}-\frac {b^2 (a+b x)^{11}}{858 a^3 x^{11}} \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(126\) vs. \(2(56)=112\).

Time = 0.01 (sec) , antiderivative size = 126, normalized size of antiderivative = 2.25 \[ \int \frac {(a+b x)^{10}}{x^{14}} \, dx=-\frac {a^{10}}{13 x^{13}}-\frac {5 a^9 b}{6 x^{12}}-\frac {45 a^8 b^2}{11 x^{11}}-\frac {12 a^7 b^3}{x^{10}}-\frac {70 a^6 b^4}{3 x^9}-\frac {63 a^5 b^5}{2 x^8}-\frac {30 a^4 b^6}{x^7}-\frac {20 a^3 b^7}{x^6}-\frac {9 a^2 b^8}{x^5}-\frac {5 a b^9}{2 x^4}-\frac {b^{10}}{3 x^3} \]

[In]

Integrate[(a + b*x)^10/x^14,x]

[Out]

-1/13*a^10/x^13 - (5*a^9*b)/(6*x^12) - (45*a^8*b^2)/(11*x^11) - (12*a^7*b^3)/x^10 - (70*a^6*b^4)/(3*x^9) - (63
*a^5*b^5)/(2*x^8) - (30*a^4*b^6)/x^7 - (20*a^3*b^7)/x^6 - (9*a^2*b^8)/x^5 - (5*a*b^9)/(2*x^4) - b^10/(3*x^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(111\) vs. \(2(50)=100\).

Time = 0.17 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.00

method	result	size
norman	\(\frac {-\frac {1}{3} b^{10} x^{10}-\frac {5}{2} a \,b^{9} x^{9}-9 a^{2} b^{8} x^{8}-20 a^{3} b^{7} x^{7}-30 a^{4} b^{6} x^{6}-\frac {63}{2} a^{5} b^{5} x^{5}-\frac {70}{3} a^{6} b^{4} x^{4}-12 a^{7} b^{3} x^{3}-\frac {45}{11} a^{8} b^{2} x^{2}-\frac {5}{6} a^{9} b x -\frac {1}{13} a^{10}}{x^{13}}\)	\(112\)
risch	\(\frac {-\frac {1}{3} b^{10} x^{10}-\frac {5}{2} a \,b^{9} x^{9}-9 a^{2} b^{8} x^{8}-20 a^{3} b^{7} x^{7}-30 a^{4} b^{6} x^{6}-\frac {63}{2} a^{5} b^{5} x^{5}-\frac {70}{3} a^{6} b^{4} x^{4}-12 a^{7} b^{3} x^{3}-\frac {45}{11} a^{8} b^{2} x^{2}-\frac {5}{6} a^{9} b x -\frac {1}{13} a^{10}}{x^{13}}\)	\(112\)
gosper	\(-\frac {286 b^{10} x^{10}+2145 a \,b^{9} x^{9}+7722 a^{2} b^{8} x^{8}+17160 a^{3} b^{7} x^{7}+25740 a^{4} b^{6} x^{6}+27027 a^{5} b^{5} x^{5}+20020 a^{6} b^{4} x^{4}+10296 a^{7} b^{3} x^{3}+3510 a^{8} b^{2} x^{2}+715 a^{9} b x +66 a^{10}}{858 x^{13}}\)	\(113\)
default	\(-\frac {12 a^{7} b^{3}}{x^{10}}-\frac {20 a^{3} b^{7}}{x^{6}}-\frac {30 a^{4} b^{6}}{x^{7}}-\frac {a^{10}}{13 x^{13}}-\frac {70 a^{6} b^{4}}{3 x^{9}}-\frac {5 a^{9} b}{6 x^{12}}-\frac {b^{10}}{3 x^{3}}-\frac {45 a^{8} b^{2}}{11 x^{11}}-\frac {5 a \,b^{9}}{2 x^{4}}-\frac {9 a^{2} b^{8}}{x^{5}}-\frac {63 a^{5} b^{5}}{2 x^{8}}\)	\(113\)
parallelrisch	\(\frac {-286 b^{10} x^{10}-2145 a \,b^{9} x^{9}-7722 a^{2} b^{8} x^{8}-17160 a^{3} b^{7} x^{7}-25740 a^{4} b^{6} x^{6}-27027 a^{5} b^{5} x^{5}-20020 a^{6} b^{4} x^{4}-10296 a^{7} b^{3} x^{3}-3510 a^{8} b^{2} x^{2}-715 a^{9} b x -66 a^{10}}{858 x^{13}}\)	\(113\)

[In]

int((b*x+a)^10/x^14,x,method=_RETURNVERBOSE)

[Out]

1/x^13*(-1/3*b^10*x^10-5/2*a*b^9*x^9-9*a^2*b^8*x^8-20*a^3*b^7*x^7-30*a^4*b^6*x^6-63/2*a^5*b^5*x^5-70/3*a^6*b^4
*x^4-12*a^7*b^3*x^3-45/11*a^8*b^2*x^2-5/6*a^9*b*x-1/13*a^10)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (50) = 100\).

Time = 0.22 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.00 \[ \int \frac {(a+b x)^{10}}{x^{14}} \, dx=-\frac {286 \, b^{10} x^{10} + 2145 \, a b^{9} x^{9} + 7722 \, a^{2} b^{8} x^{8} + 17160 \, a^{3} b^{7} x^{7} + 25740 \, a^{4} b^{6} x^{6} + 27027 \, a^{5} b^{5} x^{5} + 20020 \, a^{6} b^{4} x^{4} + 10296 \, a^{7} b^{3} x^{3} + 3510 \, a^{8} b^{2} x^{2} + 715 \, a^{9} b x + 66 \, a^{10}}{858 \, x^{13}} \]

[In]

integrate((b*x+a)^10/x^14,x, algorithm="fricas")

[Out]

-1/858*(286*b^10*x^10 + 2145*a*b^9*x^9 + 7722*a^2*b^8*x^8 + 17160*a^3*b^7*x^7 + 25740*a^4*b^6*x^6 + 27027*a^5*
b^5*x^5 + 20020*a^6*b^4*x^4 + 10296*a^7*b^3*x^3 + 3510*a^8*b^2*x^2 + 715*a^9*b*x + 66*a^10)/x^13

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 121 vs. \(2 (48) = 96\).

Time = 0.57 (sec) , antiderivative size = 121, normalized size of antiderivative = 2.16 \[ \int \frac {(a+b x)^{10}}{x^{14}} \, dx=\frac {- 66 a^{10} - 715 a^{9} b x - 3510 a^{8} b^{2} x^{2} - 10296 a^{7} b^{3} x^{3} - 20020 a^{6} b^{4} x^{4} - 27027 a^{5} b^{5} x^{5} - 25740 a^{4} b^{6} x^{6} - 17160 a^{3} b^{7} x^{7} - 7722 a^{2} b^{8} x^{8} - 2145 a b^{9} x^{9} - 286 b^{10} x^{10}}{858 x^{13}} \]

[In]

integrate((b*x+a)**10/x**14,x)

[Out]

(-66*a**10 - 715*a**9*b*x - 3510*a**8*b**2*x**2 - 10296*a**7*b**3*x**3 - 20020*a**6*b**4*x**4 - 27027*a**5*b**
5*x**5 - 25740*a**4*b**6*x**6 - 17160*a**3*b**7*x**7 - 7722*a**2*b**8*x**8 - 2145*a*b**9*x**9 - 286*b**10*x**1
0)/(858*x**13)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (50) = 100\).

Time = 0.20 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.00 \[ \int \frac {(a+b x)^{10}}{x^{14}} \, dx=-\frac {286 \, b^{10} x^{10} + 2145 \, a b^{9} x^{9} + 7722 \, a^{2} b^{8} x^{8} + 17160 \, a^{3} b^{7} x^{7} + 25740 \, a^{4} b^{6} x^{6} + 27027 \, a^{5} b^{5} x^{5} + 20020 \, a^{6} b^{4} x^{4} + 10296 \, a^{7} b^{3} x^{3} + 3510 \, a^{8} b^{2} x^{2} + 715 \, a^{9} b x + 66 \, a^{10}}{858 \, x^{13}} \]

[In]

integrate((b*x+a)^10/x^14,x, algorithm="maxima")

[Out]

-1/858*(286*b^10*x^10 + 2145*a*b^9*x^9 + 7722*a^2*b^8*x^8 + 17160*a^3*b^7*x^7 + 25740*a^4*b^6*x^6 + 27027*a^5*
b^5*x^5 + 20020*a^6*b^4*x^4 + 10296*a^7*b^3*x^3 + 3510*a^8*b^2*x^2 + 715*a^9*b*x + 66*a^10)/x^13

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 112 vs. \(2 (50) = 100\).

Time = 0.29 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.00 \[ \int \frac {(a+b x)^{10}}{x^{14}} \, dx=-\frac {286 \, b^{10} x^{10} + 2145 \, a b^{9} x^{9} + 7722 \, a^{2} b^{8} x^{8} + 17160 \, a^{3} b^{7} x^{7} + 25740 \, a^{4} b^{6} x^{6} + 27027 \, a^{5} b^{5} x^{5} + 20020 \, a^{6} b^{4} x^{4} + 10296 \, a^{7} b^{3} x^{3} + 3510 \, a^{8} b^{2} x^{2} + 715 \, a^{9} b x + 66 \, a^{10}}{858 \, x^{13}} \]

[In]

integrate((b*x+a)^10/x^14,x, algorithm="giac")

[Out]

-1/858*(286*b^10*x^10 + 2145*a*b^9*x^9 + 7722*a^2*b^8*x^8 + 17160*a^3*b^7*x^7 + 25740*a^4*b^6*x^6 + 27027*a^5*
b^5*x^5 + 20020*a^6*b^4*x^4 + 10296*a^7*b^3*x^3 + 3510*a^8*b^2*x^2 + 715*a^9*b*x + 66*a^10)/x^13

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 112, normalized size of antiderivative = 2.00 \[ \int \frac {(a+b x)^{10}}{x^{14}} \, dx=-\frac {\frac {a^{10}}{13}+\frac {5\,a^9\,b\,x}{6}+\frac {45\,a^8\,b^2\,x^2}{11}+12\,a^7\,b^3\,x^3+\frac {70\,a^6\,b^4\,x^4}{3}+\frac {63\,a^5\,b^5\,x^5}{2}+30\,a^4\,b^6\,x^6+20\,a^3\,b^7\,x^7+9\,a^2\,b^8\,x^8+\frac {5\,a\,b^9\,x^9}{2}+\frac {b^{10}\,x^{10}}{3}}{x^{13}} \]

[In]

int((a + b*x)^10/x^14,x)

[Out]

-(a^10/13 + (b^10*x^10)/3 + (5*a*b^9*x^9)/2 + (45*a^8*b^2*x^2)/11 + 12*a^7*b^3*x^3 + (70*a^6*b^4*x^4)/3 + (63*
a^5*b^5*x^5)/2 + 30*a^4*b^6*x^6 + 20*a^3*b^7*x^7 + 9*a^2*b^8*x^8 + (5*a^9*b*x)/6)/x^13