\(\int (a+\frac {b}{x})^2 x^3 \, dx\) [1561]

Optimal result

Integrand size = 13, antiderivative size = 30 \[ \int \left (a+\frac {b}{x}\right )^2 x^3 \, dx=\frac {b^2 x^2}{2}+\frac {2}{3} a b x^3+\frac {a^2 x^4}{4} \]

[Out]

1/2*b^2*x^2+2/3*a*b*x^3+1/4*a^2*x^4

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 30, 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} \[ \int \left (a+\frac {b}{x}\right )^2 x^3 \, dx=\frac {a^2 x^4}{4}+\frac {2}{3} a b x^3+\frac {b^2 x^2}{2} \]

[In]

Int[(a + b/x)^2*x^3,x]

[Out]

(b^2*x^2)/2 + (2*a*b*x^3)/3 + (a^2*x^4)/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 x (b+a x)^2 \, dx \\ & = \int \left (b^2 x+2 a b x^2+a^2 x^3\right ) \, dx \\ & = \frac {b^2 x^2}{2}+\frac {2}{3} a b x^3+\frac {a^2 x^4}{4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \left (a+\frac {b}{x}\right )^2 x^3 \, dx=\frac {b^2 x^2}{2}+\frac {2}{3} a b x^3+\frac {a^2 x^4}{4} \]

[In]

Integrate[(a + b/x)^2*x^3,x]

[Out]

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

Maple [A] (verified)

Time = 0.02 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.83

method	result	size
gosper	\(\frac {x^{2} \left (3 a^{2} x^{2}+8 a b x +6 b^{2}\right )}{12}\)	\(25\)
default	\(\frac {1}{2} b^{2} x^{2}+\frac {2}{3} a b \,x^{3}+\frac {1}{4} a^{2} x^{4}\)	\(25\)
risch	\(\frac {1}{2} b^{2} x^{2}+\frac {2}{3} a b \,x^{3}+\frac {1}{4} a^{2} x^{4}\)	\(25\)
parallelrisch	\(\frac {1}{2} b^{2} x^{2}+\frac {2}{3} a b \,x^{3}+\frac {1}{4} a^{2} x^{4}\)	\(25\)
norman	\(\frac {\frac {1}{2} b^{2} x^{3}+\frac {1}{4} x^{5} a^{2}+\frac {2}{3} a b \,x^{4}}{x}\)	\(29\)

[In]

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

[Out]

1/12*x^2*(3*a^2*x^2+8*a*b*x+6*b^2)

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \left (a+\frac {b}{x}\right )^2 x^3 \, dx=\frac {1}{4} \, a^{2} x^{4} + \frac {2}{3} \, a b x^{3} + \frac {1}{2} \, b^{2} x^{2} \]

[In]

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

[Out]

1/4*a^2*x^4 + 2/3*a*b*x^3 + 1/2*b^2*x^2

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.87 \[ \int \left (a+\frac {b}{x}\right )^2 x^3 \, dx=\frac {a^{2} x^{4}}{4} + \frac {2 a b x^{3}}{3} + \frac {b^{2} x^{2}}{2} \]

[In]

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

[Out]

a**2*x**4/4 + 2*a*b*x**3/3 + b**2*x**2/2

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \left (a+\frac {b}{x}\right )^2 x^3 \, dx=\frac {1}{4} \, a^{2} x^{4} + \frac {2}{3} \, a b x^{3} + \frac {1}{2} \, b^{2} x^{2} \]

[In]

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

[Out]

1/4*a^2*x^4 + 2/3*a*b*x^3 + 1/2*b^2*x^2

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \left (a+\frac {b}{x}\right )^2 x^3 \, dx=\frac {1}{4} \, a^{2} x^{4} + \frac {2}{3} \, a b x^{3} + \frac {1}{2} \, b^{2} x^{2} \]

[In]

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

[Out]

1/4*a^2*x^4 + 2/3*a*b*x^3 + 1/2*b^2*x^2

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.80 \[ \int \left (a+\frac {b}{x}\right )^2 x^3 \, dx=\frac {a^2\,x^4}{4}+\frac {2\,a\,b\,x^3}{3}+\frac {b^2\,x^2}{2} \]

[In]

int(x^3*(a + b/x)^2,x)

[Out]

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