3.1656 \(\int \frac{(a+\frac{b}{x})^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=32 \[ 2 a^2 \sqrt{x}-\frac{4 a b}{\sqrt{x}}-\frac{2 b^2}{3 x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0091209, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {263, 43} \[ 2 a^2 \sqrt{x}-\frac{4 a b}{\sqrt{x}}-\frac{2 b^2}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^2/Sqrt[x],x]

[Out]

(-2*b^2)/(3*x^(3/2)) - (4*a*b)/Sqrt[x] + 2*a^2*Sqrt[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 \frac{\left (a+\frac{b}{x}\right )^2}{\sqrt{x}} \, dx &=\int \frac{(b+a x)^2}{x^{5/2}} \, dx\\ &=\int \left (\frac{b^2}{x^{5/2}}+\frac{2 a b}{x^{3/2}}+\frac{a^2}{\sqrt{x}}\right ) \, dx\\ &=-\frac{2 b^2}{3 x^{3/2}}-\frac{4 a b}{\sqrt{x}}+2 a^2 \sqrt{x}\\ \end{align*}

Mathematica [A]  time = 0.0077322, size = 26, normalized size = 0.81 \[ -\frac{2 \left (-3 a^2 x^2+6 a b x+b^2\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^2/Sqrt[x],x]

[Out]

(-2*(b^2 + 6*a*b*x - 3*a^2*x^2))/(3*x^(3/2))

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Maple [A]  time = 0.003, size = 25, normalized size = 0.8 \begin{align*}{\frac{6\,{a}^{2}{x}^{2}-12\,xab-2\,{b}^{2}}{3}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^2/x^(1/2),x)

[Out]

2/3*(3*a^2*x^2-6*a*b*x-b^2)/x^(3/2)

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Maxima [A]  time = 0.982192, size = 32, normalized size = 1. \begin{align*} 2 \, a^{2} \sqrt{x} - \frac{4 \, a b}{\sqrt{x}} - \frac{2 \, b^{2}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^2/x^(1/2),x, algorithm="maxima")

[Out]

2*a^2*sqrt(x) - 4*a*b/sqrt(x) - 2/3*b^2/x^(3/2)

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Fricas [A]  time = 1.97556, size = 55, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (3 \, a^{2} x^{2} - 6 \, a b x - b^{2}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^2/x^(1/2),x, algorithm="fricas")

[Out]

2/3*(3*a^2*x^2 - 6*a*b*x - b^2)/x^(3/2)

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Sympy [A]  time = 0.487387, size = 31, normalized size = 0.97 \begin{align*} 2 a^{2} \sqrt{x} - \frac{4 a b}{\sqrt{x}} - \frac{2 b^{2}}{3 x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**2/x**(1/2),x)

[Out]

2*a**2*sqrt(x) - 4*a*b/sqrt(x) - 2*b**2/(3*x**(3/2))

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Giac [A]  time = 1.10572, size = 31, normalized size = 0.97 \begin{align*} 2 \, a^{2} \sqrt{x} - \frac{2 \,{\left (6 \, a b x + b^{2}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^2/x^(1/2),x, algorithm="giac")

[Out]

2*a^2*sqrt(x) - 2/3*(6*a*b*x + b^2)/x^(3/2)