\(\int (a+\frac {b}{x})^3 x^2 \, dx\) [1574]

Optimal result

Integrand size = 13, antiderivative size = 35 \[ \int \left (a+\frac {b}{x}\right )^3 x^2 \, dx=3 a b^2 x+\frac {3}{2} a^2 b x^2+\frac {a^3 x^3}{3}+b^3 \log (x) \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 35, 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} \[ \int \left (a+\frac {b}{x}\right )^3 x^2 \, dx=\frac {a^3 x^3}{3}+\frac {3}{2} a^2 b x^2+3 a b^2 x+b^3 \log (x) \]

[In]

Int[(a + b/x)^3*x^2,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right )^3 x^2 \, dx=3 a b^2 x+\frac {3}{2} a^2 b x^2+\frac {a^3 x^3}{3}+b^3 \log (x) \]

[In]

Integrate[(a + b/x)^3*x^2,x]

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91

method	result	size
default	\(3 a \,b^{2} x +\frac {3 a^{2} b \,x^{2}}{2}+\frac {a^{3} x^{3}}{3}+b^{3} \ln \left (x \right )\)	\(32\)
risch	\(3 a \,b^{2} x +\frac {3 a^{2} b \,x^{2}}{2}+\frac {a^{3} x^{3}}{3}+b^{3} \ln \left (x \right )\)	\(32\)
parallelrisch	\(3 a \,b^{2} x +\frac {3 a^{2} b \,x^{2}}{2}+\frac {a^{3} x^{3}}{3}+b^{3} \ln \left (x \right )\)	\(32\)
norman	\(\frac {\frac {1}{3} a^{3} x^{5}+3 a \,b^{2} x^{3}+\frac {3}{2} a^{2} b \,x^{4}}{x^{2}}+b^{3} \ln \left (x \right )\)	\(39\)

[In]

int((a+b/x)^3*x^2,x,method=_RETURNVERBOSE)

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \left (a+\frac {b}{x}\right )^3 x^2 \, dx=\frac {1}{3} \, a^{3} x^{3} + \frac {3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3} \log \left (x\right ) \]

[In]

integrate((a+b/x)^3*x^2,x, algorithm="fricas")

[Out]

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

Sympy [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.97 \[ \int \left (a+\frac {b}{x}\right )^3 x^2 \, dx=\frac {a^{3} x^{3}}{3} + \frac {3 a^{2} b x^{2}}{2} + 3 a b^{2} x + b^{3} \log {\left (x \right )} \]

[In]

integrate((a+b/x)**3*x**2,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \left (a+\frac {b}{x}\right )^3 x^2 \, dx=\frac {1}{3} \, a^{3} x^{3} + \frac {3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3} \log \left (x\right ) \]

[In]

integrate((a+b/x)^3*x^2,x, algorithm="maxima")

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.91 \[ \int \left (a+\frac {b}{x}\right )^3 x^2 \, dx=\frac {1}{3} \, a^{3} x^{3} + \frac {3}{2} \, a^{2} b x^{2} + 3 \, a b^{2} x + b^{3} \log \left ({\left | x \right |}\right ) \]

[In]

integrate((a+b/x)^3*x^2,x, algorithm="giac")

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 31, normalized size of antiderivative = 0.89 \[ \int \left (a+\frac {b}{x}\right )^3 x^2 \, dx=b^3\,\ln \left (x\right )+\frac {a^3\,x^3}{3}+\frac {3\,a^2\,b\,x^2}{2}+3\,a\,b^2\,x \]

[In]

int(x^2*(a + b/x)^3,x)

[Out]

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