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

Optimal. Leaf size=17 \[ 2 b \sqrt{x}-\frac{2 a}{\sqrt{x}} \]

[Out]

(-2*a)/Sqrt[x] + 2*b*Sqrt[x]

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Rubi [A]  time = 0.0040094, antiderivative size = 17, 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} \[ 2 b \sqrt{x}-\frac{2 a}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)/x^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0051799, size = 14, normalized size = 0.82 \[ \frac{2 (b x-a)}{\sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)/x^(3/2),x]

[Out]

(2*(-a + b*x))/Sqrt[x]

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Maple [A]  time = 0.001, size = 12, normalized size = 0.7 \begin{align*} -2\,{\frac{-bx+a}{\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02495, size = 18, normalized size = 1.06 \begin{align*} 2 \, b \sqrt{x} - \frac{2 \, a}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*sqrt(x) - 2*a/sqrt(x)

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Fricas [A]  time = 1.52989, size = 28, normalized size = 1.65 \begin{align*} \frac{2 \,{\left (b x - a\right )}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(b*x - a)/sqrt(x)

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Sympy [A]  time = 0.510208, size = 15, normalized size = 0.88 \begin{align*} - \frac{2 a}{\sqrt{x}} + 2 b \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a/sqrt(x) + 2*b*sqrt(x)

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Giac [A]  time = 1.25551, size = 18, normalized size = 1.06 \begin{align*} 2 \, b \sqrt{x} - \frac{2 \, a}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*sqrt(x) - 2*a/sqrt(x)