3.270 \(\int \frac{a+b x^2}{x^{5/2}} \, dx\)

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

[Out]

(-2*a)/(3*x^(3/2)) + 2*b*Sqrt[x]

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Rubi [A]  time = 0.0039198, 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} \[ 2 b \sqrt{x}-\frac{2 a}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2)/x^(5/2),x]

[Out]

(-2*a)/(3*x^(3/2)) + 2*b*Sqrt[x]

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

Mathematica [A]  time = 0.0060218, size = 19, normalized size = 1. \[ 2 b \sqrt{x}-\frac{2 a}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)/x^(5/2),x]

[Out]

(-2*a)/(3*x^(3/2)) + 2*b*Sqrt[x]

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Maple [A]  time = 0.002, size = 14, normalized size = 0.7 \begin{align*} -{\frac{-6\,b{x}^{2}+2\,a}{3}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)/x^(5/2),x)

[Out]

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

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Maxima [A]  time = 2.0212, size = 18, normalized size = 0.95 \begin{align*} 2 \, b \sqrt{x} - \frac{2 \, a}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^(5/2),x, algorithm="maxima")

[Out]

2*b*sqrt(x) - 2/3*a/x^(3/2)

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Fricas [A]  time = 1.44335, size = 36, normalized size = 1.89 \begin{align*} \frac{2 \,{\left (3 \, b x^{2} - a\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^(5/2),x, algorithm="fricas")

[Out]

2/3*(3*b*x^2 - a)/x^(3/2)

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Sympy [A]  time = 0.639299, size = 17, normalized size = 0.89 \begin{align*} - \frac{2 a}{3 x^{\frac{3}{2}}} + 2 b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)/x**(5/2),x)

[Out]

-2*a/(3*x**(3/2)) + 2*b*sqrt(x)

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Giac [A]  time = 2.63459, size = 18, normalized size = 0.95 \begin{align*} 2 \, b \sqrt{x} - \frac{2 \, a}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)/x^(5/2),x, algorithm="giac")

[Out]

2*b*sqrt(x) - 2/3*a/x^(3/2)