3.644 \(\int \frac{1}{\sqrt{x} (2-b x)^{5/2}} \, dx\)

Optimal. Leaf size=39 \[ \frac{\sqrt{x}}{3 \sqrt{2-b x}}+\frac{\sqrt{x}}{3 (2-b x)^{3/2}} \]

[Out]

Sqrt[x]/(3*(2 - b*x)^(3/2)) + Sqrt[x]/(3*Sqrt[2 - b*x])

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Rubi [A]  time = 0.003746, antiderivative size = 39, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {45, 37} \[ \frac{\sqrt{x}}{3 \sqrt{2-b x}}+\frac{\sqrt{x}}{3 (2-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(2 - b*x)^(5/2)),x]

[Out]

Sqrt[x]/(3*(2 - b*x)^(3/2)) + Sqrt[x]/(3*Sqrt[2 - b*x])

Rule 45

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] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

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{1}{\sqrt{x} (2-b x)^{5/2}} \, dx &=\frac{\sqrt{x}}{3 (2-b x)^{3/2}}+\frac{1}{3} \int \frac{1}{\sqrt{x} (2-b x)^{3/2}} \, dx\\ &=\frac{\sqrt{x}}{3 (2-b x)^{3/2}}+\frac{\sqrt{x}}{3 \sqrt{2-b x}}\\ \end{align*}

Mathematica [A]  time = 0.0074441, size = 24, normalized size = 0.62 \[ -\frac{\sqrt{x} (b x-3)}{3 (2-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(2 - b*x)^(5/2)),x]

[Out]

-(Sqrt[x]*(-3 + b*x))/(3*(2 - b*x)^(3/2))

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Maple [A]  time = 0.004, size = 19, normalized size = 0.5 \begin{align*} -{\frac{bx-3}{3}\sqrt{x} \left ( -bx+2 \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05575, size = 34, normalized size = 0.87 \begin{align*} \frac{{\left (b - \frac{3 \,{\left (b x - 2\right )}}{x}\right )} x^{\frac{3}{2}}}{6 \,{\left (-b x + 2\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(b - 3*(b*x - 2)/x)*x^(3/2)/(-b*x + 2)^(3/2)

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Fricas [A]  time = 1.62095, size = 82, normalized size = 2.1 \begin{align*} -\frac{{\left (b x - 3\right )} \sqrt{-b x + 2} \sqrt{x}}{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(1/(-b*x+2)^(5/2)/x^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 4.08189, size = 163, normalized size = 4.18 \begin{align*} \begin{cases} \frac{b x}{3 b^{\frac{3}{2}} x \sqrt{-1 + \frac{2}{b x}} - 6 \sqrt{b} \sqrt{-1 + \frac{2}{b x}}} - \frac{3}{3 b^{\frac{3}{2}} x \sqrt{-1 + \frac{2}{b x}} - 6 \sqrt{b} \sqrt{-1 + \frac{2}{b x}}} & \text{for}\: \frac{2}{\left |{b x}\right |} > 1 \\- \frac{i b^{2} x}{3 b^{\frac{5}{2}} x \sqrt{1 - \frac{2}{b x}} - 6 b^{\frac{3}{2}} \sqrt{1 - \frac{2}{b x}}} + \frac{3 i b}{3 b^{\frac{5}{2}} x \sqrt{1 - \frac{2}{b x}} - 6 b^{\frac{3}{2}} \sqrt{1 - \frac{2}{b x}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((b*x/(3*b**(3/2)*x*sqrt(-1 + 2/(b*x)) - 6*sqrt(b)*sqrt(-1 + 2/(b*x))) - 3/(3*b**(3/2)*x*sqrt(-1 + 2/
(b*x)) - 6*sqrt(b)*sqrt(-1 + 2/(b*x))), 2/Abs(b*x) > 1), (-I*b**2*x/(3*b**(5/2)*x*sqrt(1 - 2/(b*x)) - 6*b**(3/
2)*sqrt(1 - 2/(b*x))) + 3*I*b/(3*b**(5/2)*x*sqrt(1 - 2/(b*x)) - 6*b**(3/2)*sqrt(1 - 2/(b*x))), True))

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Giac [B]  time = 1.08853, size = 122, normalized size = 3.13 \begin{align*} \frac{8 \,{\left (3 \,{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} \sqrt{-b} b^{2}}{3 \,{\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}^{3}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3*(3*(sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)*sqrt(-b)*b^2/(((sqrt(-b*x + 2)*sqrt(-b) -
sqrt((b*x - 2)*b + 2*b))^2 - 2*b)^3*abs(b))