3.1648 \(\int (a+\frac{b}{x}) x^{3/2} \, dx\)

Optimal. Leaf size=21 \[ \frac{2}{5} a x^{5/2}+\frac{2}{3} b x^{3/2} \]

[Out]

(2*b*x^(3/2))/3 + (2*a*x^(5/2))/5

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Rubi [A]  time = 0.0044522, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used = {14} \[ \frac{2}{5} a x^{5/2}+\frac{2}{3} b x^{3/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b/x)*x^(3/2),x]

[Out]

(2*b*x^(3/2))/3 + (2*a*x^(5/2))/5

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

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

Mathematica [A]  time = 0.0043081, size = 17, normalized size = 0.81 \[ \frac{2}{15} x^{3/2} (3 a x+5 b) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b/x)*x^(3/2),x]

[Out]

(2*x^(3/2)*(5*b + 3*a*x))/15

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Maple [A]  time = 0.003, size = 14, normalized size = 0.7 \begin{align*}{\frac{6\,ax+10\,b}{15}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*a*x+5*b)*x^(3/2)

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Maxima [A]  time = 0.967832, size = 20, normalized size = 0.95 \begin{align*} \frac{2}{15} \,{\left (3 \, a + \frac{5 \, b}{x}\right )} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*a + 5*b/x)*x^(5/2)

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Fricas [A]  time = 1.75132, size = 43, normalized size = 2.05 \begin{align*} \frac{2}{15} \,{\left (3 \, a x^{2} + 5 \, b x\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/15*(3*a*x^2 + 5*b*x)*sqrt(x)

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Sympy [A]  time = 0.601857, size = 19, normalized size = 0.9 \begin{align*} \frac{2 a x^{\frac{5}{2}}}{5} + \frac{2 b x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*x**(5/2)/5 + 2*b*x**(3/2)/3

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Giac [A]  time = 1.09439, size = 18, normalized size = 0.86 \begin{align*} \frac{2}{5} \, a x^{\frac{5}{2}} + \frac{2}{3} \, b x^{\frac{3}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*a*x^(5/2) + 2/3*b*x^(3/2)