\(\int \frac {(a+b x)^5}{x^{14}} \, dx\) [97]

Optimal result

Integrand size = 11, antiderivative size = 67 \[ \int \frac {(a+b x)^5}{x^{14}} \, dx=-\frac {a^5}{13 x^{13}}-\frac {5 a^4 b}{12 x^{12}}-\frac {10 a^3 b^2}{11 x^{11}}-\frac {a^2 b^3}{x^{10}}-\frac {5 a b^4}{9 x^9}-\frac {b^5}{8 x^8} \]

[Out]

-1/13*a^5/x^13-5/12*a^4*b/x^12-10/11*a^3*b^2/x^11-a^2*b^3/x^10-5/9*a*b^4/x^9-1/8*b^5/x^8

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 67, 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} \[ \int \frac {(a+b x)^5}{x^{14}} \, dx=-\frac {a^5}{13 x^{13}}-\frac {5 a^4 b}{12 x^{12}}-\frac {10 a^3 b^2}{11 x^{11}}-\frac {a^2 b^3}{x^{10}}-\frac {5 a b^4}{9 x^9}-\frac {b^5}{8 x^8} \]

[In]

Int[(a + b*x)^5/x^14,x]

[Out]

-1/13*a^5/x^13 - (5*a^4*b)/(12*x^12) - (10*a^3*b^2)/(11*x^11) - (a^2*b^3)/x^10 - (5*a*b^4)/(9*x^9) - b^5/(8*x^
8)

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^5}{x^{14}}+\frac {5 a^4 b}{x^{13}}+\frac {10 a^3 b^2}{x^{12}}+\frac {10 a^2 b^3}{x^{11}}+\frac {5 a b^4}{x^{10}}+\frac {b^5}{x^9}\right ) \, dx \\ & = -\frac {a^5}{13 x^{13}}-\frac {5 a^4 b}{12 x^{12}}-\frac {10 a^3 b^2}{11 x^{11}}-\frac {a^2 b^3}{x^{10}}-\frac {5 a b^4}{9 x^9}-\frac {b^5}{8 x^8} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{x^{14}} \, dx=-\frac {a^5}{13 x^{13}}-\frac {5 a^4 b}{12 x^{12}}-\frac {10 a^3 b^2}{11 x^{11}}-\frac {a^2 b^3}{x^{10}}-\frac {5 a b^4}{9 x^9}-\frac {b^5}{8 x^8} \]

[In]

Integrate[(a + b*x)^5/x^14,x]

[Out]

-1/13*a^5/x^13 - (5*a^4*b)/(12*x^12) - (10*a^3*b^2)/(11*x^11) - (a^2*b^3)/x^10 - (5*a*b^4)/(9*x^9) - b^5/(8*x^
8)

Maple [A] (verified)

Time = 0.17 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85

method	result	size
norman	\(\frac {-\frac {1}{8} b^{5} x^{5}-\frac {5}{9} a \,b^{4} x^{4}-a^{2} b^{3} x^{3}-\frac {10}{11} a^{3} b^{2} x^{2}-\frac {5}{12} a^{4} b x -\frac {1}{13} a^{5}}{x^{13}}\)	\(57\)
risch	\(\frac {-\frac {1}{8} b^{5} x^{5}-\frac {5}{9} a \,b^{4} x^{4}-a^{2} b^{3} x^{3}-\frac {10}{11} a^{3} b^{2} x^{2}-\frac {5}{12} a^{4} b x -\frac {1}{13} a^{5}}{x^{13}}\)	\(57\)
gosper	\(-\frac {1287 b^{5} x^{5}+5720 a \,b^{4} x^{4}+10296 a^{2} b^{3} x^{3}+9360 a^{3} b^{2} x^{2}+4290 a^{4} b x +792 a^{5}}{10296 x^{13}}\)	\(58\)
default	\(-\frac {a^{5}}{13 x^{13}}-\frac {5 a^{4} b}{12 x^{12}}-\frac {10 a^{3} b^{2}}{11 x^{11}}-\frac {a^{2} b^{3}}{x^{10}}-\frac {5 a \,b^{4}}{9 x^{9}}-\frac {b^{5}}{8 x^{8}}\)	\(58\)
parallelrisch	\(\frac {-1287 b^{5} x^{5}-5720 a \,b^{4} x^{4}-10296 a^{2} b^{3} x^{3}-9360 a^{3} b^{2} x^{2}-4290 a^{4} b x -792 a^{5}}{10296 x^{13}}\)	\(58\)

[In]

int((b*x+a)^5/x^14,x,method=_RETURNVERBOSE)

[Out]

1/x^13*(-1/8*b^5*x^5-5/9*a*b^4*x^4-a^2*b^3*x^3-10/11*a^3*b^2*x^2-5/12*a^4*b*x-1/13*a^5)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^5}{x^{14}} \, dx=-\frac {1287 \, b^{5} x^{5} + 5720 \, a b^{4} x^{4} + 10296 \, a^{2} b^{3} x^{3} + 9360 \, a^{3} b^{2} x^{2} + 4290 \, a^{4} b x + 792 \, a^{5}}{10296 \, x^{13}} \]

[In]

integrate((b*x+a)^5/x^14,x, algorithm="fricas")

[Out]

-1/10296*(1287*b^5*x^5 + 5720*a*b^4*x^4 + 10296*a^2*b^3*x^3 + 9360*a^3*b^2*x^2 + 4290*a^4*b*x + 792*a^5)/x^13

Sympy [A] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^5}{x^{14}} \, dx=\frac {- 792 a^{5} - 4290 a^{4} b x - 9360 a^{3} b^{2} x^{2} - 10296 a^{2} b^{3} x^{3} - 5720 a b^{4} x^{4} - 1287 b^{5} x^{5}}{10296 x^{13}} \]

[In]

integrate((b*x+a)**5/x**14,x)

[Out]

(-792*a**5 - 4290*a**4*b*x - 9360*a**3*b**2*x**2 - 10296*a**2*b**3*x**3 - 5720*a*b**4*x**4 - 1287*b**5*x**5)/(
10296*x**13)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^5}{x^{14}} \, dx=-\frac {1287 \, b^{5} x^{5} + 5720 \, a b^{4} x^{4} + 10296 \, a^{2} b^{3} x^{3} + 9360 \, a^{3} b^{2} x^{2} + 4290 \, a^{4} b x + 792 \, a^{5}}{10296 \, x^{13}} \]

[In]

integrate((b*x+a)^5/x^14,x, algorithm="maxima")

[Out]

-1/10296*(1287*b^5*x^5 + 5720*a*b^4*x^4 + 10296*a^2*b^3*x^3 + 9360*a^3*b^2*x^2 + 4290*a^4*b*x + 792*a^5)/x^13

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^5}{x^{14}} \, dx=-\frac {1287 \, b^{5} x^{5} + 5720 \, a b^{4} x^{4} + 10296 \, a^{2} b^{3} x^{3} + 9360 \, a^{3} b^{2} x^{2} + 4290 \, a^{4} b x + 792 \, a^{5}}{10296 \, x^{13}} \]

[In]

integrate((b*x+a)^5/x^14,x, algorithm="giac")

[Out]

-1/10296*(1287*b^5*x^5 + 5720*a*b^4*x^4 + 10296*a^2*b^3*x^3 + 9360*a^3*b^2*x^2 + 4290*a^4*b*x + 792*a^5)/x^13

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x)^5}{x^{14}} \, dx=-\frac {\frac {a^5}{13}+\frac {5\,a^4\,b\,x}{12}+\frac {10\,a^3\,b^2\,x^2}{11}+a^2\,b^3\,x^3+\frac {5\,a\,b^4\,x^4}{9}+\frac {b^5\,x^5}{8}}{x^{13}} \]

[In]

int((a + b*x)^5/x^14,x)

[Out]

-(a^5/13 + (b^5*x^5)/8 + (5*a*b^4*x^4)/9 + (10*a^3*b^2*x^2)/11 + a^2*b^3*x^3 + (5*a^4*b*x)/12)/x^13