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]

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

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, 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 = 0.01, size = 22, normalized size = 1.00 \[ \frac {2 x^{3/2}}{3 a (a-b x)^{3/2}} \]

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))

________________________________________________________________________________________

fricas [B]  time = 0.42, size = 34, normalized size = 1.55 \[ \frac {2 \, \sqrt {-b x + a} x^{\frac {3}{2}}}{3 \, {\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )}} \]

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)

________________________________________________________________________________________

giac [B]  time = 1.63, size = 102, normalized size = 4.64 \[ \frac {4 \, {\left (3 \, {\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{4} \sqrt {-b} + a^{2} \sqrt {-b} b^{2}\right )} {\left | b \right |}}{3 \, {\left ({\left (\sqrt {-b x + a} \sqrt {-b} - \sqrt {{\left (b x - a\right )} b + a b}\right )}^{2} - a b\right )}^{3} b^{2}} \]

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)

________________________________________________________________________________________

maple [A]  time = 0.00, size = 17, normalized size = 0.77 \[ \frac {2 x^{\frac {3}{2}}}{3 \left (-b x +a \right )^{\frac {3}{2}} a} \]

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)

________________________________________________________________________________________

maxima [A]  time = 1.36, size = 16, normalized size = 0.73 \[ \frac {2 \, x^{\frac {3}{2}}}{3 \, {\left (-b x + a\right )}^{\frac {3}{2}} a} \]

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)

________________________________________________________________________________________

mupad [B]  time = 0.25, size = 37, normalized size = 1.68 \[ \frac {2\,x^{3/2}\,\sqrt {a-b\,x}}{3\,\left (a^3-2\,a^2\,b\,x+a\,b^2\,x^2\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(a - b*x)^(5/2),x)

[Out]

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

________________________________________________________________________________________

sympy [B]  time = 1.51, size = 95, normalized size = 4.32 \[ \begin {cases} \frac {2 i x^{\frac {3}{2}}}{- 3 a^{\frac {5}{2}} \sqrt {-1 + \frac {b x}{a}} + 3 a^{\frac {3}{2}} b x \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {2 x^{\frac {3}{2}}}{- 3 a^{\frac {5}{2}} \sqrt {1 - \frac {b x}{a}} + 3 a^{\frac {3}{2}} b x \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]

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/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))

________________________________________________________________________________________