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

Optimal. Leaf size=46 \[ -\frac{4 b \sqrt{a-b x}}{3 a^2 \sqrt{x}}-\frac{2 \sqrt{a-b x}}{3 a x^{3/2}} \]

[Out]

(-2*Sqrt[a - b*x])/(3*a*x^(3/2)) - (4*b*Sqrt[a - b*x])/(3*a^2*Sqrt[x])

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Rubi [A]  time = 0.005001, antiderivative size = 46, 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} \[ -\frac{4 b \sqrt{a-b x}}{3 a^2 \sqrt{x}}-\frac{2 \sqrt{a-b x}}{3 a x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^(5/2)*Sqrt[a - b*x]),x]

[Out]

(-2*Sqrt[a - b*x])/(3*a*x^(3/2)) - (4*b*Sqrt[a - b*x])/(3*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^{5/2} \sqrt{a-b x}} \, dx &=-\frac{2 \sqrt{a-b x}}{3 a x^{3/2}}+\frac{(2 b) \int \frac{1}{x^{3/2} \sqrt{a-b x}} \, dx}{3 a}\\ &=-\frac{2 \sqrt{a-b x}}{3 a x^{3/2}}-\frac{4 b \sqrt{a-b x}}{3 a^2 \sqrt{x}}\\ \end{align*}

Mathematica [A]  time = 0.0091297, size = 28, normalized size = 0.61 \[ -\frac{2 \sqrt{a-b x} (a+2 b x)}{3 a^2 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.004, size = 23, normalized size = 0.5 \begin{align*} -{\frac{4\,bx+2\,a}{3\,{a}^{2}}\sqrt{-bx+a}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.988458, size = 43, normalized size = 0.93 \begin{align*} -\frac{2 \,{\left (\frac{3 \, \sqrt{-b x + a} b}{\sqrt{x}} + \frac{{\left (-b x + a\right )}^{\frac{3}{2}}}{x^{\frac{3}{2}}}\right )}}{3 \, a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.92803, size = 63, normalized size = 1.37 \begin{align*} -\frac{2 \,{\left (2 \, b x + a\right )} \sqrt{-b x + a}}{3 \, a^{2} x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 3.81764, size = 180, normalized size = 3.91 \begin{align*} \begin{cases} - \frac{2 \sqrt{b} \sqrt{\frac{a}{b x} - 1}}{3 a x} - \frac{4 b^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}}{3 a^{2}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\\frac{2 i a^{2} b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}}{- 3 a^{3} b x + 3 a^{2} b^{2} x^{2}} + \frac{2 i a b^{\frac{5}{2}} x \sqrt{- \frac{a}{b x} + 1}}{- 3 a^{3} b x + 3 a^{2} b^{2} x^{2}} - \frac{4 i b^{\frac{7}{2}} x^{2} \sqrt{- \frac{a}{b x} + 1}}{- 3 a^{3} b x + 3 a^{2} b^{2} x^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-2*sqrt(b)*sqrt(a/(b*x) - 1)/(3*a*x) - 4*b**(3/2)*sqrt(a/(b*x) - 1)/(3*a**2), Abs(a)/(Abs(b)*Abs(x)
) > 1), (2*I*a**2*b**(3/2)*sqrt(-a/(b*x) + 1)/(-3*a**3*b*x + 3*a**2*b**2*x**2) + 2*I*a*b**(5/2)*x*sqrt(-a/(b*x
) + 1)/(-3*a**3*b*x + 3*a**2*b**2*x**2) - 4*I*b**(7/2)*x**2*sqrt(-a/(b*x) + 1)/(-3*a**3*b*x + 3*a**2*b**2*x**2
), True))

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Giac [A]  time = 1.09614, size = 73, normalized size = 1.59 \begin{align*} -\frac{\sqrt{-b x + a} b{\left (\frac{2 \,{\left (b x - a\right )}}{a^{2} b^{3}} + \frac{3}{a b^{3}}\right )}}{24 \,{\left ({\left (b x - a\right )} b + a 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+a)^(1/2),x, algorithm="giac")

[Out]

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