\(\int \frac {(a+b x)^5}{x^{13}} \, dx\) [96]

Optimal result

Integrand size = 11, antiderivative size = 67 \[ \int \frac {(a+b x)^5}{x^{13}} \, dx=-\frac {a^5}{12 x^{12}}-\frac {5 a^4 b}{11 x^{11}}-\frac {a^3 b^2}{x^{10}}-\frac {10 a^2 b^3}{9 x^9}-\frac {5 a b^4}{8 x^8}-\frac {b^5}{7 x^7} \]

[Out]

-1/12*a^5/x^12-5/11*a^4*b/x^11-a^3*b^2/x^10-10/9*a^2*b^3/x^9-5/8*a*b^4/x^8-1/7*b^5/x^7

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} \[ \int \frac {(a+b x)^5}{x^{13}} \, dx=-\frac {a^5}{12 x^{12}}-\frac {5 a^4 b}{11 x^{11}}-\frac {a^3 b^2}{x^{10}}-\frac {10 a^2 b^3}{9 x^9}-\frac {5 a b^4}{8 x^8}-\frac {b^5}{7 x^7} \]

[In]

Int[(a + b*x)^5/x^13,x]

[Out]

-1/12*a^5/x^12 - (5*a^4*b)/(11*x^11) - (a^3*b^2)/x^10 - (10*a^2*b^3)/(9*x^9) - (5*a*b^4)/(8*x^8) - b^5/(7*x^7)

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^{13}}+\frac {5 a^4 b}{x^{12}}+\frac {10 a^3 b^2}{x^{11}}+\frac {10 a^2 b^3}{x^{10}}+\frac {5 a b^4}{x^9}+\frac {b^5}{x^8}\right ) \, dx \\ & = -\frac {a^5}{12 x^{12}}-\frac {5 a^4 b}{11 x^{11}}-\frac {a^3 b^2}{x^{10}}-\frac {10 a^2 b^3}{9 x^9}-\frac {5 a b^4}{8 x^8}-\frac {b^5}{7 x^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.00 \[ \int \frac {(a+b x)^5}{x^{13}} \, dx=-\frac {a^5}{12 x^{12}}-\frac {5 a^4 b}{11 x^{11}}-\frac {a^3 b^2}{x^{10}}-\frac {10 a^2 b^3}{9 x^9}-\frac {5 a b^4}{8 x^8}-\frac {b^5}{7 x^7} \]

[In]

Integrate[(a + b*x)^5/x^13,x]

[Out]

-1/12*a^5/x^12 - (5*a^4*b)/(11*x^11) - (a^3*b^2)/x^10 - (10*a^2*b^3)/(9*x^9) - (5*a*b^4)/(8*x^8) - b^5/(7*x^7)

Maple [A] (verified)

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

method	result	size
norman	\(\frac {-\frac {1}{7} b^{5} x^{5}-\frac {5}{8} a \,b^{4} x^{4}-\frac {10}{9} a^{2} b^{3} x^{3}-a^{3} b^{2} x^{2}-\frac {5}{11} a^{4} b x -\frac {1}{12} a^{5}}{x^{12}}\)	\(57\)
risch	\(\frac {-\frac {1}{7} b^{5} x^{5}-\frac {5}{8} a \,b^{4} x^{4}-\frac {10}{9} a^{2} b^{3} x^{3}-a^{3} b^{2} x^{2}-\frac {5}{11} a^{4} b x -\frac {1}{12} a^{5}}{x^{12}}\)	\(57\)
gosper	\(-\frac {792 b^{5} x^{5}+3465 a \,b^{4} x^{4}+6160 a^{2} b^{3} x^{3}+5544 a^{3} b^{2} x^{2}+2520 a^{4} b x +462 a^{5}}{5544 x^{12}}\)	\(58\)
default	\(-\frac {a^{5}}{12 x^{12}}-\frac {5 a^{4} b}{11 x^{11}}-\frac {a^{3} b^{2}}{x^{10}}-\frac {10 a^{2} b^{3}}{9 x^{9}}-\frac {5 a \,b^{4}}{8 x^{8}}-\frac {b^{5}}{7 x^{7}}\)	\(58\)
parallelrisch	\(\frac {-792 b^{5} x^{5}-3465 a \,b^{4} x^{4}-6160 a^{2} b^{3} x^{3}-5544 a^{3} b^{2} x^{2}-2520 a^{4} b x -462 a^{5}}{5544 x^{12}}\)	\(58\)

[In]

int((b*x+a)^5/x^13,x,method=_RETURNVERBOSE)

[Out]

1/x^12*(-1/7*b^5*x^5-5/8*a*b^4*x^4-10/9*a^2*b^3*x^3-a^3*b^2*x^2-5/11*a^4*b*x-1/12*a^5)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^5}{x^{13}} \, dx=-\frac {792 \, b^{5} x^{5} + 3465 \, a b^{4} x^{4} + 6160 \, a^{2} b^{3} x^{3} + 5544 \, a^{3} b^{2} x^{2} + 2520 \, a^{4} b x + 462 \, a^{5}}{5544 \, x^{12}} \]

[In]

integrate((b*x+a)^5/x^13,x, algorithm="fricas")

[Out]

-1/5544*(792*b^5*x^5 + 3465*a*b^4*x^4 + 6160*a^2*b^3*x^3 + 5544*a^3*b^2*x^2 + 2520*a^4*b*x + 462*a^5)/x^12

Sympy [A] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.91 \[ \int \frac {(a+b x)^5}{x^{13}} \, dx=\frac {- 462 a^{5} - 2520 a^{4} b x - 5544 a^{3} b^{2} x^{2} - 6160 a^{2} b^{3} x^{3} - 3465 a b^{4} x^{4} - 792 b^{5} x^{5}}{5544 x^{12}} \]

[In]

integrate((b*x+a)**5/x**13,x)

[Out]

(-462*a**5 - 2520*a**4*b*x - 5544*a**3*b**2*x**2 - 6160*a**2*b**3*x**3 - 3465*a*b**4*x**4 - 792*b**5*x**5)/(55
44*x**12)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^5}{x^{13}} \, dx=-\frac {792 \, b^{5} x^{5} + 3465 \, a b^{4} x^{4} + 6160 \, a^{2} b^{3} x^{3} + 5544 \, a^{3} b^{2} x^{2} + 2520 \, a^{4} b x + 462 \, a^{5}}{5544 \, x^{12}} \]

[In]

integrate((b*x+a)^5/x^13,x, algorithm="maxima")

[Out]

-1/5544*(792*b^5*x^5 + 3465*a*b^4*x^4 + 6160*a^2*b^3*x^3 + 5544*a^3*b^2*x^2 + 2520*a^4*b*x + 462*a^5)/x^12

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.85 \[ \int \frac {(a+b x)^5}{x^{13}} \, dx=-\frac {792 \, b^{5} x^{5} + 3465 \, a b^{4} x^{4} + 6160 \, a^{2} b^{3} x^{3} + 5544 \, a^{3} b^{2} x^{2} + 2520 \, a^{4} b x + 462 \, a^{5}}{5544 \, x^{12}} \]

[In]

integrate((b*x+a)^5/x^13,x, algorithm="giac")

[Out]

-1/5544*(792*b^5*x^5 + 3465*a*b^4*x^4 + 6160*a^2*b^3*x^3 + 5544*a^3*b^2*x^2 + 2520*a^4*b*x + 462*a^5)/x^12

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.84 \[ \int \frac {(a+b x)^5}{x^{13}} \, dx=-\frac {\frac {a^5}{12}+\frac {5\,a^4\,b\,x}{11}+a^3\,b^2\,x^2+\frac {10\,a^2\,b^3\,x^3}{9}+\frac {5\,a\,b^4\,x^4}{8}+\frac {b^5\,x^5}{7}}{x^{12}} \]

[In]

int((a + b*x)^5/x^13,x)

[Out]

-(a^5/12 + (b^5*x^5)/7 + (5*a*b^4*x^4)/8 + a^3*b^2*x^2 + (10*a^2*b^3*x^3)/9 + (5*a^4*b*x)/11)/x^12