3.1661 \(\int (a+\frac{b}{x})^3 \sqrt{x} \, dx\)

Optimal. Leaf size=47 \[ 6 a^2 b \sqrt{x}+\frac{2}{3} a^3 x^{3/2}-\frac{6 a b^2}{\sqrt{x}}-\frac{2 b^3}{3 x^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0130861, antiderivative size = 47, 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} \[ 6 a^2 b \sqrt{x}+\frac{2}{3} a^3 x^{3/2}-\frac{6 a b^2}{\sqrt{x}}-\frac{2 b^3}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)^3*Sqrt[x],x]

[Out]

(-2*b^3)/(3*x^(3/2)) - (6*a*b^2)/Sqrt[x] + 6*a^2*b*Sqrt[x] + (2*a^3*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 )^3 \sqrt{x} \, dx &=\int \frac{(b+a x)^3}{x^{5/2}} \, dx\\ &=\int \left (\frac{b^3}{x^{5/2}}+\frac{3 a b^2}{x^{3/2}}+\frac{3 a^2 b}{\sqrt{x}}+a^3 \sqrt{x}\right ) \, dx\\ &=-\frac{2 b^3}{3 x^{3/2}}-\frac{6 a b^2}{\sqrt{x}}+6 a^2 b \sqrt{x}+\frac{2}{3} a^3 x^{3/2}\\ \end{align*}

Mathematica [A]  time = 0.0099883, size = 38, normalized size = 0.81 \[ \frac{2 \left (9 a^2 b x^2+a^3 x^3-9 a b^2 x-b^3\right )}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)^3*Sqrt[x],x]

[Out]

(2*(-b^3 - 9*a*b^2*x + 9*a^2*b*x^2 + a^3*x^3))/(3*x^(3/2))

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Maple [A]  time = 0.004, size = 35, normalized size = 0.7 \begin{align*}{\frac{2\,{a}^{3}{x}^{3}+18\,{a}^{2}b{x}^{2}-18\,xa{b}^{2}-2\,{b}^{3}}{3}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b/x)^3*x^(1/2),x)

[Out]

2/3*(a^3*x^3+9*a^2*b*x^2-9*a*b^2*x-b^3)/x^(3/2)

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Maxima [A]  time = 0.971368, size = 49, normalized size = 1.04 \begin{align*} -\frac{6 \, a b^{2}}{\sqrt{x}} + \frac{2}{3} \,{\left (a^{3} + \frac{9 \, a^{2} b}{x}\right )} x^{\frac{3}{2}} - \frac{2 \, b^{3}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3*x^(1/2),x, algorithm="maxima")

[Out]

-6*a*b^2/sqrt(x) + 2/3*(a^3 + 9*a^2*b/x)*x^(3/2) - 2/3*b^3/x^(3/2)

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Fricas [A]  time = 1.69702, size = 74, normalized size = 1.57 \begin{align*} \frac{2 \,{\left (a^{3} x^{3} + 9 \, a^{2} b x^{2} - 9 \, a b^{2} x - b^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3*x^(1/2),x, algorithm="fricas")

[Out]

2/3*(a^3*x^3 + 9*a^2*b*x^2 - 9*a*b^2*x - b^3)/x^(3/2)

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Sympy [A]  time = 0.757705, size = 46, normalized size = 0.98 \begin{align*} \frac{2 a^{3} x^{\frac{3}{2}}}{3} + 6 a^{2} b \sqrt{x} - \frac{6 a b^{2}}{\sqrt{x}} - \frac{2 b^{3}}{3 x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)**3*x**(1/2),x)

[Out]

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

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Giac [A]  time = 1.09305, size = 46, normalized size = 0.98 \begin{align*} \frac{2}{3} \, a^{3} x^{\frac{3}{2}} + 6 \, a^{2} b \sqrt{x} - \frac{2 \,{\left (9 \, a b^{2} x + b^{3}\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b/x)^3*x^(1/2),x, algorithm="giac")

[Out]

2/3*a^3*x^(3/2) + 6*a^2*b*sqrt(x) - 2/3*(9*a*b^2*x + b^3)/x^(3/2)