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

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

[Out]

x^(3/2)/(3*(2 + b*x)^(3/2))

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Rubi [A]  time = 0.0012821, 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} \[ \frac{x^{3/2}}{3 (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(2 + b*x)^(5/2),x]

[Out]

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

Mathematica [A]  time = 0.0042477, size = 18, normalized size = 1. \[ \frac{x^{3/2}}{3 (b x+2)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(2 + b*x)^(5/2),x]

[Out]

x^(3/2)/(3*(2 + b*x)^(3/2))

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Maple [A]  time = 0.004, size = 13, normalized size = 0.7 \begin{align*}{\frac{1}{3}{x}^{{\frac{3}{2}}} \left ( bx+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01698, size = 16, normalized size = 0.89 \begin{align*} \frac{x^{\frac{3}{2}}}{3 \,{\left (b x + 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*x^(3/2)/(b*x + 2)^(3/2)

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Fricas [B]  time = 1.52731, size = 66, normalized size = 3.67 \begin{align*} \frac{\sqrt{b x + 2} x^{\frac{3}{2}}}{3 \,{\left (b^{2} x^{2} + 4 \, b x + 4\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.27685, size = 27, normalized size = 1.5 \begin{align*} \frac{x^{\frac{3}{2}}}{3 b x \sqrt{b x + 2} + 6 \sqrt{b x + 2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**(3/2)/(3*b*x*sqrt(b*x + 2) + 6*sqrt(b*x + 2))

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Giac [B]  time = 1.10089, size = 111, normalized size = 6.17 \begin{align*} \frac{4 \,{\left (3 \,{\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{4} \sqrt{b} + 4 \, b^{\frac{5}{2}}\right )}{\left | b \right |}}{3 \,{\left ({\left (\sqrt{b x + 2} \sqrt{b} - \sqrt{{\left (b x + 2\right )} b - 2 \, b}\right )}^{2} + 2 \, b\right )}^{3} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/3*(3*(sqrt(b*x + 2)*sqrt(b) - sqrt((b*x + 2)*b - 2*b))^4*sqrt(b) + 4*b^(5/2))*abs(b)/(((sqrt(b*x + 2)*sqrt(b
) - sqrt((b*x + 2)*b - 2*b))^2 + 2*b)^3*b^2)