\(\int \frac {1}{(a+\frac {b}{x})^3 x} \, dx\) [1638]

Optimal result

Integrand size = 13, antiderivative size = 41 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x} \, dx=-\frac {b^2}{2 a^3 (b+a x)^2}+\frac {2 b}{a^3 (b+a x)}+\frac {\log (b+a x)}{a^3} \]

[Out]

-1/2*b^2/a^3/(a*x+b)^2+2*b/a^3/(a*x+b)+ln(a*x+b)/a^3

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 41, 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} \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x} \, dx=-\frac {b^2}{2 a^3 (a x+b)^2}+\frac {2 b}{a^3 (a x+b)}+\frac {\log (a x+b)}{a^3} \]

[In]

Int[1/((a + b/x)^3*x),x]

[Out]

-1/2*b^2/(a^3*(b + a*x)^2) + (2*b)/(a^3*(b + a*x)) + Log[b + a*x]/a^3

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 {x^2}{(b+a x)^3} \, dx \\ & = \int \left (\frac {b^2}{a^2 (b+a x)^3}-\frac {2 b}{a^2 (b+a x)^2}+\frac {1}{a^2 (b+a x)}\right ) \, dx \\ & = -\frac {b^2}{2 a^3 (b+a x)^2}+\frac {2 b}{a^3 (b+a x)}+\frac {\log (b+a x)}{a^3} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.80 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x} \, dx=\frac {\frac {b (3 b+4 a x)}{(b+a x)^2}+2 \log (b+a x)}{2 a^3} \]

[In]

Integrate[1/((a + b/x)^3*x),x]

[Out]

((b*(3*b + 4*a*x))/(b + a*x)^2 + 2*Log[b + a*x])/(2*a^3)

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.88

method	result	size
norman	\(\frac {\frac {3 b^{2}}{2 a^{3}}+\frac {2 b x}{a^{2}}}{\left (a x +b \right )^{2}}+\frac {\ln \left (a x +b \right )}{a^{3}}\)	\(36\)
risch	\(\frac {\frac {3 b^{2}}{2 a^{3}}+\frac {2 b x}{a^{2}}}{\left (a x +b \right )^{2}}+\frac {\ln \left (a x +b \right )}{a^{3}}\)	\(36\)
default	\(-\frac {b^{2}}{2 a^{3} \left (a x +b \right )^{2}}+\frac {2 b}{a^{3} \left (a x +b \right )}+\frac {\ln \left (a x +b \right )}{a^{3}}\)	\(40\)
parallelrisch	\(\frac {2 a^{2} \ln \left (a x +b \right ) x^{2}+4 \ln \left (a x +b \right ) x a b +2 b^{2} \ln \left (a x +b \right )+4 a b x +3 b^{2}}{2 a^{3} \left (a x +b \right )^{2}}\)	\(60\)

[In]

int(1/(a+b/x)^3/x,x,method=_RETURNVERBOSE)

[Out]

(3/2*b^2/a^3+2*b/a^2*x)/(a*x+b)^2+ln(a*x+b)/a^3

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 61, normalized size of antiderivative = 1.49 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x} \, dx=\frac {4 \, a b x + 3 \, b^{2} + 2 \, {\left (a^{2} x^{2} + 2 \, a b x + b^{2}\right )} \log \left (a x + b\right )}{2 \, {\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}} \]

[In]

integrate(1/(a+b/x)^3/x,x, algorithm="fricas")

[Out]

1/2*(4*a*b*x + 3*b^2 + 2*(a^2*x^2 + 2*a*b*x + b^2)*log(a*x + b))/(a^5*x^2 + 2*a^4*b*x + a^3*b^2)

Sympy [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x} \, dx=\frac {4 a b x + 3 b^{2}}{2 a^{5} x^{2} + 4 a^{4} b x + 2 a^{3} b^{2}} + \frac {\log {\left (a x + b \right )}}{a^{3}} \]

[In]

integrate(1/(a+b/x)**3/x,x)

[Out]

(4*a*b*x + 3*b**2)/(2*a**5*x**2 + 4*a**4*b*x + 2*a**3*b**2) + log(a*x + b)/a**3

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.17 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x} \, dx=\frac {4 \, a b x + 3 \, b^{2}}{2 \, {\left (a^{5} x^{2} + 2 \, a^{4} b x + a^{3} b^{2}\right )}} + \frac {\log \left (a x + b\right )}{a^{3}} \]

[In]

integrate(1/(a+b/x)^3/x,x, algorithm="maxima")

[Out]

1/2*(4*a*b*x + 3*b^2)/(a^5*x^2 + 2*a^4*b*x + a^3*b^2) + log(a*x + b)/a^3

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.90 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x} \, dx=\frac {\log \left ({\left | a x + b \right |}\right )}{a^{3}} + \frac {4 \, b x + \frac {3 \, b^{2}}{a}}{2 \, {\left (a x + b\right )}^{2} a^{2}} \]

[In]

integrate(1/(a+b/x)^3/x,x, algorithm="giac")

[Out]

log(abs(a*x + b))/a^3 + 1/2*(4*b*x + 3*b^2/a)/((a*x + b)^2*a^2)

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 46, normalized size of antiderivative = 1.12 \[ \int \frac {1}{\left (a+\frac {b}{x}\right )^3 x} \, dx=\frac {\ln \left (b+a\,x\right )}{a^3}+\frac {\frac {3\,b^2}{2\,a^3}+\frac {2\,b\,x}{a^2}}{a^2\,x^2+2\,a\,b\,x+b^2} \]

[In]

int(1/(x*(a + b/x)^3),x)

[Out]

log(b + a*x)/a^3 + ((3*b^2)/(2*a^3) + (2*b*x)/a^2)/(b^2 + a^2*x^2 + 2*a*b*x)