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

Optimal. Leaf size=58 \[ -\frac{2 \sqrt{2-b x}}{3 \sqrt{x}}+\frac{2}{3 \sqrt{x} \sqrt{2-b x}}+\frac{1}{3 \sqrt{x} (2-b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0063941, antiderivative size = 58, 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} \[ -\frac{2 \sqrt{2-b x}}{3 \sqrt{x}}+\frac{2}{3 \sqrt{x} \sqrt{2-b x}}+\frac{1}{3 \sqrt{x} (2-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(2 - b*x)^(5/2)),x]

[Out]

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

Mathematica [A]  time = 0.0107665, size = 33, normalized size = 0.57 \[ -\frac{2 b^2 x^2-6 b x+3}{3 \sqrt{x} (2-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(3 - 6*b*x + 2*b^2*x^2)/(3*Sqrt[x]*(2 - b*x)^(3/2))

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Maple [A]  time = 0.003, size = 28, normalized size = 0.5 \begin{align*} -{\frac{2\,{b}^{2}{x}^{2}-6\,bx+3}{3} \left ( -bx+2 \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02109, size = 57, normalized size = 0.98 \begin{align*} \frac{{\left (b^{2} - \frac{6 \,{\left (b x - 2\right )} b}{x}\right )} x^{\frac{3}{2}}}{12 \,{\left (-b x + 2\right )}^{\frac{3}{2}}} - \frac{\sqrt{-b x + 2}}{4 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(b^2 - 6*(b*x - 2)*b/x)*x^(3/2)/(-b*x + 2)^(3/2) - 1/4*sqrt(-b*x + 2)/sqrt(x)

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Fricas [A]  time = 1.58853, size = 107, normalized size = 1.84 \begin{align*} -\frac{{\left (2 \, b^{2} x^{2} - 6 \, b x + 3\right )} \sqrt{-b x + 2} \sqrt{x}}{3 \,{\left (b^{2} x^{3} - 4 \, b x^{2} + 4 \, x\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(2*b^2*x^2 - 6*b*x + 3)*sqrt(-b*x + 2)*sqrt(x)/(b^2*x^3 - 4*b*x^2 + 4*x)

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Sympy [B]  time = 13.9576, size = 243, normalized size = 4.19 \begin{align*} \begin{cases} - \frac{2 b^{\frac{13}{2}} x^{2} \sqrt{-1 + \frac{2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} + \frac{6 b^{\frac{11}{2}} x \sqrt{-1 + \frac{2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} - \frac{3 b^{\frac{9}{2}} \sqrt{-1 + \frac{2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} & \text{for}\: \frac{2}{\left |{b x}\right |} > 1 \\- \frac{2 i b^{\frac{13}{2}} x^{2} \sqrt{1 - \frac{2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} + \frac{6 i b^{\frac{11}{2}} x \sqrt{1 - \frac{2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} - \frac{3 i b^{\frac{9}{2}} \sqrt{1 - \frac{2}{b x}}}{3 b^{6} x^{2} - 12 b^{5} x + 12 b^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.11142, size = 230, normalized size = 3.97 \begin{align*} -\frac{\sqrt{-b x + 2} b^{2}}{4 \, \sqrt{{\left (b x - 2\right )} b + 2 \, b}{\left | b \right |}} - \frac{3 \,{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{4} \sqrt{-b} b^{2} - 24 \,{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} \sqrt{-b} b^{3} + 20 \, \sqrt{-b} b^{4}}{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/x^(3/2)/(-b*x+2)^(5/2),x, algorithm="giac")

[Out]

-1/4*sqrt(-b*x + 2)*b^2/(sqrt((b*x - 2)*b + 2*b)*abs(b)) - 1/3*(3*(sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b
+ 2*b))^4*sqrt(-b)*b^2 - 24*(sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2*sqrt(-b)*b^3 + 20*sqrt(-b)*b
^4)/(((sqrt(-b*x + 2)*sqrt(-b) - sqrt((b*x - 2)*b + 2*b))^2 - 2*b)^3*abs(b))