3.1560 \(\int (a+\frac{b}{x})^2 x^4 \, dx\)

Optimal. Leaf size=30 \[ \frac{a^2 x^5}{5}+\frac{1}{2} a b x^4+\frac{b^2 x^3}{3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.013179, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {263, 43} \[ \frac{a^2 x^5}{5}+\frac{1}{2} a b x^4+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^2*x^4,x]

[Out]

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

Rule 263

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]

Rule 43

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*} \int \left (a+\frac{b}{x}\right )^2 x^4 \, dx &=\int x^2 (b+a x)^2 \, dx\\ &=\int \left (b^2 x^2+2 a b x^3+a^2 x^4\right ) \, dx\\ &=\frac{b^2 x^3}{3}+\frac{1}{2} a b x^4+\frac{a^2 x^5}{5}\\ \end{align*}

Mathematica [A]  time = 0.0016516, size = 30, normalized size = 1. \[ \frac{a^2 x^5}{5}+\frac{1}{2} a b x^4+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^2*x^4,x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.002, size = 25, normalized size = 0.8 \begin{align*}{\frac{{b}^{2}{x}^{3}}{3}}+{\frac{ab{x}^{4}}{2}}+{\frac{{x}^{5}{a}^{2}}{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^2*x^4,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 0.959792, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{5} \, a^{2} x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, b^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^2*x^4,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [A]  time = 1.38612, size = 55, normalized size = 1.83 \begin{align*} \frac{1}{5} \, a^{2} x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, b^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^2*x^4,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

Sympy [A]  time = 0.059444, size = 24, normalized size = 0.8 \begin{align*} \frac{a^{2} x^{5}}{5} + \frac{a b x^{4}}{2} + \frac{b^{2} x^{3}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**2*x**4,x)

[Out]

a**2*x**5/5 + a*b*x**4/2 + b**2*x**3/3

________________________________________________________________________________________

Giac [A]  time = 1.14425, size = 32, normalized size = 1.07 \begin{align*} \frac{1}{5} \, a^{2} x^{5} + \frac{1}{2} \, a b x^{4} + \frac{1}{3} \, b^{2} x^{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^2*x^4,x, algorithm="giac")

[Out]

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