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

Optimal. Leaf size=45 \[ \frac{4 \sqrt{x}}{3 a^2 \sqrt{a-b x}}+\frac{2 \sqrt{x}}{3 a (a-b x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0050084, antiderivative size = 45, 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} \[ \frac{4 \sqrt{x}}{3 a^2 \sqrt{a-b x}}+\frac{2 \sqrt{x}}{3 a (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(a - b*x)^(5/2)),x]

[Out]

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

Mathematica [A]  time = 0.009055, size = 30, normalized size = 0.67 \[ \frac{2 \sqrt{x} (3 a-2 b x)}{3 a^2 (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.003, size = 25, normalized size = 0.6 \begin{align*}{\frac{-4\,bx+6\,a}{3\,{a}^{2}}\sqrt{x} \left ( -bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.02868, size = 41, normalized size = 0.91 \begin{align*} \frac{2 \,{\left (b - \frac{3 \,{\left (b x - a\right )}}{x}\right )} x^{\frac{3}{2}}}{3 \,{\left (-b x + a\right )}^{\frac{3}{2}} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.87127, size = 101, normalized size = 2.24 \begin{align*} -\frac{2 \,{\left (2 \, b x - 3 \, a\right )} \sqrt{-b x + a} \sqrt{x}}{3 \,{\left (a^{2} b^{2} x^{2} - 2 \, a^{3} b x + a^{4}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 3.56506, size = 201, normalized size = 4.47 \begin{align*} \begin{cases} - \frac{6 a}{- 3 a^{3} \sqrt{b} \sqrt{\frac{a}{b x} - 1} + 3 a^{2} b^{\frac{3}{2}} x \sqrt{\frac{a}{b x} - 1}} + \frac{4 b x}{- 3 a^{3} \sqrt{b} \sqrt{\frac{a}{b x} - 1} + 3 a^{2} b^{\frac{3}{2}} x \sqrt{\frac{a}{b x} - 1}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\\frac{6 i a b}{- 3 a^{3} b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1} + 3 a^{2} b^{\frac{5}{2}} x \sqrt{- \frac{a}{b x} + 1}} - \frac{4 i b^{2} x}{- 3 a^{3} b^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1} + 3 a^{2} b^{\frac{5}{2}} x \sqrt{- \frac{a}{b x} + 1}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-6*a/(-3*a**3*sqrt(b)*sqrt(a/(b*x) - 1) + 3*a**2*b**(3/2)*x*sqrt(a/(b*x) - 1)) + 4*b*x/(-3*a**3*sqr
t(b)*sqrt(a/(b*x) - 1) + 3*a**2*b**(3/2)*x*sqrt(a/(b*x) - 1)), Abs(a)/(Abs(b)*Abs(x)) > 1), (6*I*a*b/(-3*a**3*
b**(3/2)*sqrt(-a/(b*x) + 1) + 3*a**2*b**(5/2)*x*sqrt(-a/(b*x) + 1)) - 4*I*b**2*x/(-3*a**3*b**(3/2)*sqrt(-a/(b*
x) + 1) + 3*a**2*b**(5/2)*x*sqrt(-a/(b*x) + 1)), True))

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Giac [B]  time = 1.0963, size = 130, normalized size = 2.89 \begin{align*} \frac{8 \,{\left (3 \,{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )} \sqrt{-b} b^{2}}{3 \,{\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3}{\left | b \right |}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

8/3*(3*(sqrt(-b*x + a)*sqrt(-b) - sqrt((b*x - a)*b + a*b))^2 - a*b)*sqrt(-b)*b^2/(((sqrt(-b*x + a)*sqrt(-b) -
sqrt((b*x - a)*b + a*b))^2 - a*b)^3*abs(b))