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

Optimal. Leaf size=19 \[ \frac{2}{3} b x^{3/2}-\frac{2 a}{\sqrt{x}} \]

[Out]

(-2*a)/Sqrt[x] + (2*b*x^(3/2))/3

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Rubi [A]  time = 0.0042045, antiderivative size = 19, 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} \[ \frac{2}{3} b x^{3/2}-\frac{2 a}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/x^(3/2),x]

[Out]

(-2*a)/Sqrt[x] + (2*b*x^(3/2))/3

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

Mathematica [A]  time = 0.0053236, size = 19, normalized size = 1. \[ \frac{2}{3} b x^{3/2}-\frac{2 a}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x^(3/2),x]

[Out]

(-2*a)/Sqrt[x] + (2*b*x^(3/2))/3

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Maple [A]  time = 0.003, size = 16, normalized size = 0.8 \begin{align*} -{\frac{-2\,b{x}^{2}+6\,a}{3}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.74622, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{3} \, b x^{\frac{3}{2}} - \frac{2 \, a}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b*x^(3/2) - 2*a/sqrt(x)

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Fricas [A]  time = 1.56427, size = 36, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (b x^{2} - 3 \, a\right )}}{3 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(b*x^2 - 3*a)/sqrt(x)

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Sympy [A]  time = 0.507086, size = 17, normalized size = 0.89 \begin{align*} - \frac{2 a}{\sqrt{x}} + \frac{2 b x^{\frac{3}{2}}}{3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/sqrt(x) + 2*b*x**(3/2)/3

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Giac [A]  time = 2.49301, size = 18, normalized size = 0.95 \begin{align*} \frac{2}{3} \, b x^{\frac{3}{2}} - \frac{2 \, a}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*b*x^(3/2) - 2*a/sqrt(x)