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

Optimal. Leaf size=22 \[ \frac{2 x^{3/2}}{3 a (a-b x)^{3/2}} \]

[Out]

(2*x^(3/2))/(3*a*(a - b*x)^(3/2))

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Rubi [A]  time = 0.0016898, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {37} \[ \frac{2 x^{3/2}}{3 a (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(a - b*x)^(5/2),x]

[Out]

(2*x^(3/2))/(3*a*(a - b*x)^(3/2))

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{\sqrt{x}}{(a-b x)^{5/2}} \, dx &=\frac{2 x^{3/2}}{3 a (a-b x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0055816, size = 22, normalized size = 1. \[ \frac{2 x^{3/2}}{3 a (a-b x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(a - b*x)^(5/2),x]

[Out]

(2*x^(3/2))/(3*a*(a - b*x)^(3/2))

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Maple [A]  time = 0.003, size = 17, normalized size = 0.8 \begin{align*}{\frac{2}{3\,a}{x}^{{\frac{3}{2}}} \left ( -bx+a \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.05743, size = 22, normalized size = 1. \begin{align*} \frac{2 \, x^{\frac{3}{2}}}{3 \,{\left (-b x + a\right )}^{\frac{3}{2}} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*x^(3/2)/((-b*x + a)^(3/2)*a)

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Fricas [B]  time = 1.77506, size = 78, normalized size = 3.55 \begin{align*} \frac{2 \, \sqrt{-b x + a} x^{\frac{3}{2}}}{3 \,{\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 2.25948, size = 97, normalized size = 4.41 \begin{align*} \begin{cases} \frac{2 i x^{\frac{3}{2}}}{- 3 a^{\frac{5}{2}} \sqrt{-1 + \frac{b x}{a}} + 3 a^{\frac{3}{2}} b x \sqrt{-1 + \frac{b x}{a}}} & \text{for}\: \frac{\left |{b x}\right |}{\left |{a}\right |} > 1 \\- \frac{2 x^{\frac{3}{2}}}{- 3 a^{\frac{5}{2}} \sqrt{1 - \frac{b x}{a}} + 3 a^{\frac{3}{2}} b x \sqrt{1 - \frac{b x}{a}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.1171, size = 138, normalized size = 6.27 \begin{align*} \frac{4 \,{\left (3 \,{\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{4} \sqrt{-b} + a^{2} \sqrt{-b} b^{2}\right )}{\left | b \right |}}{3 \,{\left ({\left (\sqrt{-b x + a} \sqrt{-b} - \sqrt{{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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