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

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

[Out]

(-2*a)/(3*x^(3/2)) - (2*b)/Sqrt[x]

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Rubi [A]  time = 0.0036581, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ -\frac{2 a}{3 x^{3/2}}-\frac{2 b}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^(5/2),x]

[Out]

(-2*a)/(3*x^(3/2)) - (2*b)/Sqrt[x]

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

Mathematica [A]  time = 0.0049646, size = 15, normalized size = 0.79 \[ -\frac{2 (a+3 b x)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^(5/2),x]

[Out]

(-2*(a + 3*b*x))/(3*x^(3/2))

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Maple [A]  time = 0.002, size = 12, normalized size = 0.6 \begin{align*} -{\frac{6\,bx+2\,a}{3}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.992358, size = 15, normalized size = 0.79 \begin{align*} -\frac{2 \,{\left (3 \, b x + a\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*b*x + a)/x^(3/2)

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Fricas [A]  time = 1.55319, size = 35, normalized size = 1.84 \begin{align*} -\frac{2 \,{\left (3 \, b x + a\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*b*x + a)/x^(3/2)

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Sympy [A]  time = 0.957543, size = 19, normalized size = 1. \begin{align*} - \frac{2 a}{3 x^{\frac{3}{2}}} - \frac{2 b}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/(3*x**(3/2)) - 2*b/sqrt(x)

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Giac [A]  time = 1.23794, size = 15, normalized size = 0.79 \begin{align*} -\frac{2 \,{\left (3 \, b x + a\right )}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/3*(3*b*x + a)/x^(3/2)