3.1563 \(\int (a+\frac{b}{x})^2 x \, dx\)

Optimal. Leaf size=22 \[ \frac{a^2 x^2}{2}+2 a b x+b^2 \log (x) \]

[Out]

2*a*b*x + (a^2*x^2)/2 + b^2*Log[x]

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Rubi [A]  time = 0.0080695, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {263, 43} \[ \frac{a^2 x^2}{2}+2 a b x+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^2*x,x]

[Out]

2*a*b*x + (a^2*x^2)/2 + b^2*Log[x]

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

Mathematica [A]  time = 0.000959, size = 22, normalized size = 1. \[ \frac{a^2 x^2}{2}+2 a b x+b^2 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^2*x,x]

[Out]

2*a*b*x + (a^2*x^2)/2 + b^2*Log[x]

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Maple [A]  time = 0.002, size = 21, normalized size = 1. \begin{align*} 2\,xab+{\frac{{a}^{2}{x}^{2}}{2}}+{b}^{2}\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.948466, size = 27, normalized size = 1.23 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + 2 \, a b x + b^{2} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*x^2 + 2*a*b*x + b^2*log(x)

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Fricas [A]  time = 1.46167, size = 49, normalized size = 2.23 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + 2 \, a b x + b^{2} \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*a^2*x^2 + 2*a*b*x + b^2*log(x)

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Sympy [A]  time = 0.246043, size = 20, normalized size = 0.91 \begin{align*} \frac{a^{2} x^{2}}{2} + 2 a b x + b^{2} \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a**2*x**2/2 + 2*a*b*x + b**2*log(x)

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Giac [A]  time = 1.13985, size = 28, normalized size = 1.27 \begin{align*} \frac{1}{2} \, a^{2} x^{2} + 2 \, a b x + b^{2} \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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