3.640 \(\int \frac{1}{x^{5/2} (2-b x)^{3/2}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{2 \sqrt{2-b x}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{2-b x}}-\frac{2 b \sqrt{2-b x}}{3 \sqrt{x}} \]

[Out]

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

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Rubi [A]  time = 0.0076754, antiderivative size = 56, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {45, 37} \[ -\frac{2 \sqrt{2-b x}}{3 x^{3/2}}+\frac{1}{x^{3/2} \sqrt{2-b x}}-\frac{2 b \sqrt{2-b x}}{3 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*(2 - b*x)^(3/2)),x]

[Out]

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

Mathematica [A]  time = 0.0074608, size = 33, normalized size = 0.59 \[ \frac{2 b^2 x^2-2 b x-1}{3 x^{3/2} \sqrt{2-b x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(5/2)*(2 - b*x)^(3/2)),x]

[Out]

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

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Maple [A]  time = 0.004, size = 28, normalized size = 0.5 \begin{align*}{\frac{2\,{b}^{2}{x}^{2}-2\,bx-1}{3}{x}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-bx+2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^(5/2)/(-b*x+2)^(3/2),x)

[Out]

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

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Maxima [A]  time = 1.00596, size = 59, normalized size = 1.05 \begin{align*} \frac{b^{2} \sqrt{x}}{4 \, \sqrt{-b x + 2}} - \frac{\sqrt{-b x + 2} b}{2 \, \sqrt{x}} - \frac{{\left (-b x + 2\right )}^{\frac{3}{2}}}{12 \, 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)^(3/2),x, algorithm="maxima")

[Out]

1/4*b^2*sqrt(x)/sqrt(-b*x + 2) - 1/2*sqrt(-b*x + 2)*b/sqrt(x) - 1/12*(-b*x + 2)^(3/2)/x^(3/2)

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Fricas [A]  time = 1.4673, size = 93, normalized size = 1.66 \begin{align*} -\frac{{\left (2 \, b^{2} x^{2} - 2 \, b x - 1\right )} \sqrt{-b x + 2} \sqrt{x}}{3 \,{\left (b x^{3} - 2 \, x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-b*x+2)^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 13.4028, size = 354, normalized size = 6.32 \begin{align*} \begin{cases} - \frac{2 b^{\frac{15}{2}} x^{3} \sqrt{-1 + \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} + \frac{6 b^{\frac{13}{2}} x^{2} \sqrt{-1 + \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} - \frac{3 b^{\frac{11}{2}} x \sqrt{-1 + \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} - \frac{2 b^{\frac{9}{2}} \sqrt{-1 + \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} & \text{for}\: \frac{2}{\left |{b x}\right |} > 1 \\- \frac{2 i b^{\frac{15}{2}} x^{3} \sqrt{1 - \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} + \frac{6 i b^{\frac{13}{2}} x^{2} \sqrt{1 - \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} - \frac{3 i b^{\frac{11}{2}} x \sqrt{1 - \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} x} - \frac{2 i b^{\frac{9}{2}} \sqrt{1 - \frac{2}{b x}}}{3 b^{6} x^{3} - 12 b^{5} x^{2} + 12 b^{4} 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)**(3/2),x)

[Out]

Piecewise((-2*b**(15/2)*x**3*sqrt(-1 + 2/(b*x))/(3*b**6*x**3 - 12*b**5*x**2 + 12*b**4*x) + 6*b**(13/2)*x**2*sq
rt(-1 + 2/(b*x))/(3*b**6*x**3 - 12*b**5*x**2 + 12*b**4*x) - 3*b**(11/2)*x*sqrt(-1 + 2/(b*x))/(3*b**6*x**3 - 12
*b**5*x**2 + 12*b**4*x) - 2*b**(9/2)*sqrt(-1 + 2/(b*x))/(3*b**6*x**3 - 12*b**5*x**2 + 12*b**4*x), 2/Abs(b*x) >
1), (-2*I*b**(15/2)*x**3*sqrt(1 - 2/(b*x))/(3*b**6*x**3 - 12*b**5*x**2 + 12*b**4*x) + 6*I*b**(13/2)*x**2*sqrt
(1 - 2/(b*x))/(3*b**6*x**3 - 12*b**5*x**2 + 12*b**4*x) - 3*I*b**(11/2)*x*sqrt(1 - 2/(b*x))/(3*b**6*x**3 - 12*b
**5*x**2 + 12*b**4*x) - 2*I*b**(9/2)*sqrt(1 - 2/(b*x))/(3*b**6*x**3 - 12*b**5*x**2 + 12*b**4*x), True))

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Giac [B]  time = 1.06899, size = 130, normalized size = 2.32 \begin{align*} -\frac{\sqrt{-b} b^{3}}{{\left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}{\left | b \right |}} - \frac{{\left (5 \,{\left (b x - 2\right )} b^{2}{\left | b \right |} + 12 \, b^{2}{\left | b \right |}\right )} \sqrt{-b x + 2}}{12 \,{\left ({\left (b x - 2\right )} b + 2 \, b\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^(5/2)/(-b*x+2)^(3/2),x, algorithm="giac")

[Out]

-sqrt(-b)*b^3/(((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)*abs(b)) - 1/12*(5*(b*x - 2)*b^2*a
bs(b) + 12*b^2*abs(b))*sqrt(-b*x + 2)/((b*x - 2)*b + 2*b)^(3/2)