3.1655 \(\int (a+\frac{b}{x})^2 \sqrt{x} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0092571, 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} \[ \frac{2}{3} a^2 x^{3/2}+4 a b \sqrt{x}-\frac{2 b^2}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^2*Sqrt[x],x]

[Out]

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

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

Mathematica [A]  time = 0.0075991, size = 27, normalized size = 0.84 \[ \frac{2 \left (a^2 x^2+6 a b x-3 b^2\right )}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^2*Sqrt[x],x]

[Out]

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

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Maple [A]  time = 0.004, size = 24, normalized size = 0.8 \begin{align*}{\frac{2\,{a}^{2}{x}^{2}+12\,xab-6\,{b}^{2}}{3}{\frac{1}{\sqrt{x}}}} \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*(a^2*x^2+6*a*b*x-3*b^2)/x^(1/2)

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Maxima [A]  time = 0.971106, size = 34, normalized size = 1.06 \begin{align*} \frac{2}{3} \,{\left (a^{2} + \frac{6 \, a b}{x}\right )} x^{\frac{3}{2}} - \frac{2 \, b^{2}}{\sqrt{x}} \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]

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

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Fricas [A]  time = 1.94281, size = 55, normalized size = 1.72 \begin{align*} \frac{2 \,{\left (a^{2} x^{2} + 6 \, a b x - 3 \, b^{2}\right )}}{3 \, \sqrt{x}} \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*(a^2*x^2 + 6*a*b*x - 3*b^2)/sqrt(x)

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Sympy [A]  time = 0.489724, size = 31, normalized size = 0.97 \begin{align*} \frac{2 a^{2} x^{\frac{3}{2}}}{3} + 4 a b \sqrt{x} - \frac{2 b^{2}}{\sqrt{x}} \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*x**(3/2)/3 + 4*a*b*sqrt(x) - 2*b**2/sqrt(x)

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Giac [A]  time = 1.11774, size = 32, normalized size = 1. \begin{align*} \frac{2}{3} \, a^{2} x^{\frac{3}{2}} + 4 \, a b \sqrt{x} - \frac{2 \, b^{2}}{\sqrt{x}} \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]

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