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

Optimal. Leaf size=40 \[ -\frac{\sqrt{2-b x}}{3 x^{3/2}}-\frac{b \sqrt{2-b x}}{3 \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0039226, antiderivative size = 40, 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} \[ -\frac{\sqrt{2-b x}}{3 x^{3/2}}-\frac{b \sqrt{2-b x}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*Sqrt[2 - b*x]),x]

[Out]

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

Mathematica [A]  time = 0.0112824, size = 24, normalized size = 0.6 \[ -\frac{\sqrt{2-b x} (b x+1)}{3 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.002, size = 19, normalized size = 0.5 \begin{align*} -{\frac{bx+1}{3}\sqrt{-bx+2}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01113, size = 38, normalized size = 0.95 \begin{align*} -\frac{\sqrt{-b x + 2} b}{2 \, \sqrt{x}} - \frac{{\left (-b x + 2\right )}^{\frac{3}{2}}}{6 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.60481, size = 53, normalized size = 1.32 \begin{align*} -\frac{{\left (b x + 1\right )} \sqrt{-b x + 2}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(b*x + 1)*sqrt(-b*x + 2)/x^(3/2)

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Sympy [A]  time = 4.10904, size = 139, normalized size = 3.48 \begin{align*} \begin{cases} - \frac{b^{\frac{3}{2}} \sqrt{-1 + \frac{2}{b x}}}{3} - \frac{\sqrt{b} \sqrt{-1 + \frac{2}{b x}}}{3 x} & \text{for}\: \frac{2}{\left |{b x}\right |} > 1 \\- \frac{i b^{\frac{7}{2}} x^{2} \sqrt{1 - \frac{2}{b x}}}{3 b^{2} x^{2} - 6 b x} + \frac{i b^{\frac{5}{2}} x \sqrt{1 - \frac{2}{b x}}}{3 b^{2} x^{2} - 6 b x} + \frac{2 i b^{\frac{3}{2}} \sqrt{1 - \frac{2}{b x}}}{3 b^{2} x^{2} - 6 b x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-b**(3/2)*sqrt(-1 + 2/(b*x))/3 - sqrt(b)*sqrt(-1 + 2/(b*x))/(3*x), 2/Abs(b*x) > 1), (-I*b**(7/2)*x*
*2*sqrt(1 - 2/(b*x))/(3*b**2*x**2 - 6*b*x) + I*b**(5/2)*x*sqrt(1 - 2/(b*x))/(3*b**2*x**2 - 6*b*x) + 2*I*b**(3/
2)*sqrt(1 - 2/(b*x))/(3*b**2*x**2 - 6*b*x), True))

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Giac [A]  time = 1.06432, size = 58, normalized size = 1.45 \begin{align*} -\frac{{\left ({\left (b x - 2\right )} b^{3} + 3 \, b^{3}\right )} \sqrt{-b x + 2} b}{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(1/x^(5/2)/(-b*x+2)^(1/2),x, algorithm="giac")

[Out]

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