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

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

[Out]

x^(3/2)/(3*(2 - b*x)^(3/2))

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Rubi [A]  time = 0.001343, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {37} \[ \frac{x^{3/2}}{3 (2-b x)^{3/2}} \]

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 = 0.0044959, size = 19, normalized size = 1. \[ \frac{x^{3/2}}{3 (2-b x)^{3/2}} \]

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 = 0.002, size = 14, 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 = 1.07818, size = 18, normalized size = 0.95 \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 = 1.6524, size = 68, normalized size = 3.58 \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 [B]  time = 2.26752, size = 65, normalized size = 3.42 \begin{align*} \begin{cases} \frac{i x^{\frac{3}{2}}}{3 b x \sqrt{b x - 2} - 6 \sqrt{b x - 2}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\- \frac{x^{\frac{3}{2}}}{3 b x \sqrt{- b x + 2} - 6 \sqrt{- b x + 2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((I*x**(3/2)/(3*b*x*sqrt(b*x - 2) - 6*sqrt(b*x - 2)), Abs(b*x)/2 > 1), (-x**(3/2)/(3*b*x*sqrt(-b*x +
2) - 6*sqrt(-b*x + 2)), True))

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Giac [B]  time = 1.0942, size = 128, normalized size = 6.74 \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 \, \sqrt{-b} b^{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*sqrt(-b)*b^2)*abs(b)/(((sqrt(-b*x +
2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)^3*b^2)