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

Optimal. Leaf size=41 \[ \frac{2}{a \sqrt{x} \sqrt{a-b x}}-\frac{4 \sqrt{a-b x}}{a^2 \sqrt{x}} \]

[Out]

2/(a*Sqrt[x]*Sqrt[a - b*x]) - (4*Sqrt[a - b*x])/(a^2*Sqrt[x])

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Rubi [A]  time = 0.0049461, antiderivative size = 41, 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} \[ \frac{2}{a \sqrt{x} \sqrt{a-b x}}-\frac{4 \sqrt{a-b x}}{a^2 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(3/2)*(a - b*x)^(3/2)),x]

[Out]

2/(a*Sqrt[x]*Sqrt[a - b*x]) - (4*Sqrt[a - b*x])/(a^2*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} (a-b x)^{3/2}} \, dx &=\frac{2}{a \sqrt{x} \sqrt{a-b x}}+\frac{2 \int \frac{1}{x^{3/2} \sqrt{a-b x}} \, dx}{a}\\ &=\frac{2}{a \sqrt{x} \sqrt{a-b x}}-\frac{4 \sqrt{a-b x}}{a^2 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0082571, size = 26, normalized size = 0.63 \[ -\frac{2 (a-2 b x)}{a^2 \sqrt{x} \sqrt{a-b x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 23, normalized size = 0.6 \begin{align*} -2\,{\frac{-2\,bx+a}{{a}^{2}\sqrt{x}\sqrt{-bx+a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05453, size = 46, normalized size = 1.12 \begin{align*} \frac{2 \, b \sqrt{x}}{\sqrt{-b x + a} a^{2}} - \frac{2 \, \sqrt{-b x + a}}{a^{2} \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*b*sqrt(x)/(sqrt(-b*x + a)*a^2) - 2*sqrt(-b*x + a)/(a^2*sqrt(x))

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Fricas [A]  time = 1.74941, size = 80, normalized size = 1.95 \begin{align*} -\frac{2 \,{\left (2 \, b x - a\right )} \sqrt{-b x + a} \sqrt{x}}{a^{2} b x^{2} - a^{3} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 2.53384, size = 116, normalized size = 2.83 \begin{align*} \begin{cases} - \frac{2}{a \sqrt{b} x \sqrt{\frac{a}{b x} - 1}} + \frac{4 \sqrt{b}}{a^{2} \sqrt{\frac{a}{b x} - 1}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\\frac{2 i a b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}}{- a^{3} b + a^{2} b^{2} x} - \frac{4 i b^{\frac{5}{2}} x \sqrt{- \frac{a}{b x} + 1}}{- a^{3} b + a^{2} b^{2} x} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2/(a*sqrt(b)*x*sqrt(a/(b*x) - 1)) + 4*sqrt(b)/(a**2*sqrt(a/(b*x) - 1)), Abs(a)/(Abs(b)*Abs(x)) > 1
), (2*I*a*b**(3/2)*sqrt(-a/(b*x) + 1)/(-a**3*b + a**2*b**2*x) - 4*I*b**(5/2)*x*sqrt(-a/(b*x) + 1)/(-a**3*b + a
**2*b**2*x), True))

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Giac [B]  time = 1.07418, size = 127, normalized size = 3.1 \begin{align*} -\frac{4 \, \sqrt{-b} b^{2}}{{\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} a{\left | b \right |}} - \frac{2 \, \sqrt{-b x + a} b^{2}}{\sqrt{{\left (b x - a\right )} b + a b} a^{2}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*sqrt(-b)*b^2/(((sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2 - a*b)*a*abs(b)) - 2*sqrt(-b*x + a)*b^
2/(sqrt((b*x - a)*b + a*b)*a^2*abs(b))