\(\int \frac {\sqrt {x}}{(2-b x)^{5/2}} \, dx\) [643]

Optimal result

Integrand size = 16, antiderivative size = 19 \[ \int \frac {\sqrt {x}}{(2-b x)^{5/2}} \, dx=\frac {x^{3/2}}{3 (2-b x)^{3/2}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {37} \[ \int \frac {\sqrt {x}}{(2-b x)^{5/2}} \, dx=\frac {x^{3/2}}{3 (2-b x)^{3/2}} \]

[In]

Int[Sqrt[x]/(2 - b*x)^(5/2),x]

[Out]

x^(3/2)/(3*(2 - 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*} \text {integral}& = \frac {x^{3/2}}{3 (2-b x)^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 19, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {x}}{(2-b x)^{5/2}} \, dx=\frac {x^{3/2}}{3 (2-b x)^{3/2}} \]

[In]

Integrate[Sqrt[x]/(2 - b*x)^(5/2),x]

[Out]

x^(3/2)/(3*(2 - b*x)^(3/2))

Maple [A] (verified)

Time = 0.25 (sec) , antiderivative size = 14, normalized size of antiderivative = 0.74

method	result	size
gosper	\(\frac {x^{\frac {3}{2}}}{3 \left (-b x +2\right )^{\frac {3}{2}}}\)	\(14\)
meijerg	\(\frac {x^{\frac {3}{2}} \sqrt {2}}{12 \left (-\frac {b x}{2}+1\right )^{\frac {3}{2}}}\)	\(17\)
default	\(\frac {\sqrt {x}}{b \left (-b x +2\right )^{\frac {3}{2}}}-\frac {\frac {\sqrt {x}}{3 \left (-b x +2\right )^{\frac {3}{2}}}+\frac {\sqrt {x}}{3 \sqrt {-b x +2}}}{b}\)	\(49\)

[In]

int(x^(1/2)/(-b*x+2)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (13) = 26\).

Time = 0.23 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.47 \[ \int \frac {\sqrt {x}}{(2-b x)^{5/2}} \, dx=\frac {\sqrt {-b x + 2} x^{\frac {3}{2}}}{3 \, {\left (b^{2} x^{2} - 4 \, b x + 4\right )}} \]

[In]

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

[Out]

1/3*sqrt(-b*x + 2)*x^(3/2)/(b^2*x^2 - 4*b*x + 4)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.02 (sec) , antiderivative size = 63, normalized size of antiderivative = 3.32 \[ \int \frac {\sqrt {x}}{(2-b x)^{5/2}} \, dx=\begin {cases} \frac {i x^{\frac {3}{2}}}{3 b x \sqrt {b x - 2} - 6 \sqrt {b x - 2}} & \text {for}\: \left |{b x}\right | > 2 \\- \frac {x^{\frac {3}{2}}}{3 b x \sqrt {- b x + 2} - 6 \sqrt {- b x + 2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((I*x**(3/2)/(3*b*x*sqrt(b*x - 2) - 6*sqrt(b*x - 2)), Abs(b*x) > 2), (-x**(3/2)/(3*b*x*sqrt(-b*x + 2)
- 6*sqrt(-b*x + 2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {x}}{(2-b x)^{5/2}} \, dx=\frac {x^{\frac {3}{2}}}{3 \, {\left (-b x + 2\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

1/3*x^(3/2)/(-b*x + 2)^(3/2)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (13) = 26\).

Time = 0.30 (sec) , antiderivative size = 95, normalized size of antiderivative = 5.00 \[ \int \frac {\sqrt {x}}{(2-b x)^{5/2}} \, dx=\frac {4 \, {\left (3 \, {\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{4} \sqrt {-b} + 4 \, \sqrt {-b} b^{2}\right )} {\left | b \right |}}{3 \, {\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}^{3} b^{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {x}}{(2-b x)^{5/2}} \, dx=\frac {x^{3/2}}{3\,{\left (2-b\,x\right )}^{3/2}} \]

[In]

int(x^(1/2)/(2 - b*x)^(5/2),x)

[Out]

x^(3/2)/(3*(2 - b*x)^(3/2))