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

Optimal. Leaf size=18 \[ -\frac{(b x+2)^{3/2}}{3 x^{3/2}} \]

[Out]

-(2 + b*x)^(3/2)/(3*x^(3/2))

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Rubi [A]  time = 0.0014173, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {37} \[ -\frac{(b x+2)^{3/2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[2 + b*x]/x^(5/2),x]

[Out]

-(2 + b*x)^(3/2)/(3*x^(3/2))

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0132774, size = 18, normalized size = 1. \[ -\frac{(b x+2)^{3/2}}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[2 + b*x]/x^(5/2),x]

[Out]

-(2 + b*x)^(3/2)/(3*x^(3/2))

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Maple [A]  time = 0.002, size = 13, normalized size = 0.7 \begin{align*} -{\frac{1}{3} \left ( bx+2 \right ) ^{{\frac{3}{2}}}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+2)^(1/2)/x^(5/2),x)

[Out]

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

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Maxima [A]  time = 1.03068, size = 16, normalized size = 0.89 \begin{align*} -\frac{{\left (b x + 2\right )}^{\frac{3}{2}}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+2)^(1/2)/x^(5/2),x, algorithm="maxima")

[Out]

-1/3*(b*x + 2)^(3/2)/x^(3/2)

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Fricas [A]  time = 1.50226, size = 41, normalized size = 2.28 \begin{align*} -\frac{{\left (b x + 2\right )}^{\frac{3}{2}}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+2)^(1/2)/x^(5/2),x, algorithm="fricas")

[Out]

-1/3*(b*x + 2)^(3/2)/x^(3/2)

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Sympy [B]  time = 2.11646, size = 37, normalized size = 2.06 \begin{align*} - \frac{b^{\frac{3}{2}} \sqrt{1 + \frac{2}{b x}}}{3} - \frac{2 \sqrt{b} \sqrt{1 + \frac{2}{b x}}}{3 x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+2)**(1/2)/x**(5/2),x)

[Out]

-b**(3/2)*sqrt(1 + 2/(b*x))/3 - 2*sqrt(b)*sqrt(1 + 2/(b*x))/(3*x)

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Giac [B]  time = 1.23251, size = 39, normalized size = 2.17 \begin{align*} -\frac{{\left (b x + 2\right )}^{\frac{3}{2}} b^{4}}{3 \,{\left ({\left (b x + 2\right )} b - 2 \, b\right )}^{\frac{3}{2}}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+2)^(1/2)/x^(5/2),x, algorithm="giac")

[Out]

-1/3*(b*x + 2)^(3/2)*b^4/(((b*x + 2)*b - 2*b)^(3/2)*abs(b))