\(\int \frac {(a+\frac {b}{x})^3}{x^5} \, dx\) [1581]

Optimal result

Integrand size = 13, antiderivative size = 43 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^5} \, dx=-\frac {b^3}{7 x^7}-\frac {a b^2}{2 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {a^3}{4 x^4} \]

[Out]

-1/7*b^3/x^7-1/2*a*b^2/x^6-3/5*a^2*b/x^5-1/4*a^3/x^4

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {269, 45} \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^5} \, dx=-\frac {a^3}{4 x^4}-\frac {3 a^2 b}{5 x^5}-\frac {a b^2}{2 x^6}-\frac {b^3}{7 x^7} \]

[In]

Int[(a + b/x)^3/x^5,x]

[Out]

-1/7*b^3/x^7 - (a*b^2)/(2*x^6) - (3*a^2*b)/(5*x^5) - a^3/(4*x^4)

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])

Rule 269

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Int[x^(m + n*p)*(b + a/x^n)^p, x] /; FreeQ[{a, b, m
, n}, x] && IntegerQ[p] && NegQ[n]

Rubi steps \begin{align*} \text {integral}& = \int \frac {(b+a x)^3}{x^8} \, dx \\ & = \int \left (\frac {b^3}{x^8}+\frac {3 a b^2}{x^7}+\frac {3 a^2 b}{x^6}+\frac {a^3}{x^5}\right ) \, dx \\ & = -\frac {b^3}{7 x^7}-\frac {a b^2}{2 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {a^3}{4 x^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 43, normalized size of antiderivative = 1.00 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^5} \, dx=-\frac {b^3}{7 x^7}-\frac {a b^2}{2 x^6}-\frac {3 a^2 b}{5 x^5}-\frac {a^3}{4 x^4} \]

[In]

Integrate[(a + b/x)^3/x^5,x]

[Out]

-1/7*b^3/x^7 - (a*b^2)/(2*x^6) - (3*a^2*b)/(5*x^5) - a^3/(4*x^4)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81

method	result	size
norman	\(\frac {-\frac {1}{4} a^{3} x^{3}-\frac {3}{5} a^{2} b \,x^{2}-\frac {1}{2} a \,b^{2} x -\frac {1}{7} b^{3}}{x^{7}}\)	\(35\)
risch	\(\frac {-\frac {1}{4} a^{3} x^{3}-\frac {3}{5} a^{2} b \,x^{2}-\frac {1}{2} a \,b^{2} x -\frac {1}{7} b^{3}}{x^{7}}\)	\(35\)
gosper	\(-\frac {35 a^{3} x^{3}+84 a^{2} b \,x^{2}+70 a \,b^{2} x +20 b^{3}}{140 x^{7}}\)	\(36\)
default	\(-\frac {b^{3}}{7 x^{7}}-\frac {a \,b^{2}}{2 x^{6}}-\frac {3 a^{2} b}{5 x^{5}}-\frac {a^{3}}{4 x^{4}}\)	\(36\)
parallelrisch	\(\frac {-35 a^{3} x^{3}-84 a^{2} b \,x^{2}-70 a \,b^{2} x -20 b^{3}}{140 x^{7}}\)	\(36\)

[In]

int((a+b/x)^3/x^5,x,method=_RETURNVERBOSE)

[Out]

(-1/4*a^3*x^3-3/5*a^2*b*x^2-1/2*a*b^2*x-1/7*b^3)/x^7

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^5} \, dx=-\frac {35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \]

[In]

integrate((a+b/x)^3/x^5,x, algorithm="fricas")

[Out]

-1/140*(35*a^3*x^3 + 84*a^2*b*x^2 + 70*a*b^2*x + 20*b^3)/x^7

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^5} \, dx=\frac {- 35 a^{3} x^{3} - 84 a^{2} b x^{2} - 70 a b^{2} x - 20 b^{3}}{140 x^{7}} \]

[In]

integrate((a+b/x)**3/x**5,x)

[Out]

(-35*a**3*x**3 - 84*a**2*b*x**2 - 70*a*b**2*x - 20*b**3)/(140*x**7)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^5} \, dx=-\frac {35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \]

[In]

integrate((a+b/x)^3/x^5,x, algorithm="maxima")

[Out]

-1/140*(35*a^3*x^3 + 84*a^2*b*x^2 + 70*a*b^2*x + 20*b^3)/x^7

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^5} \, dx=-\frac {35 \, a^{3} x^{3} + 84 \, a^{2} b x^{2} + 70 \, a b^{2} x + 20 \, b^{3}}{140 \, x^{7}} \]

[In]

integrate((a+b/x)^3/x^5,x, algorithm="giac")

[Out]

-1/140*(35*a^3*x^3 + 84*a^2*b*x^2 + 70*a*b^2*x + 20*b^3)/x^7

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.81 \[ \int \frac {\left (a+\frac {b}{x}\right )^3}{x^5} \, dx=-\frac {\frac {a^3\,x^3}{4}+\frac {3\,a^2\,b\,x^2}{5}+\frac {a\,b^2\,x}{2}+\frac {b^3}{7}}{x^7} \]

[In]

int((a + b/x)^3/x^5,x)

[Out]

-(b^3/7 + (a^3*x^3)/4 + (3*a^2*b*x^2)/5 + (a*b^2*x)/2)/x^7