\(\int \frac {1}{\sqrt {x} (2-b x)^{5/2}} \, dx\) [644]

Optimal result

Integrand size = 16, antiderivative size = 39 \[ \int \frac {1}{\sqrt {x} (2-b x)^{5/2}} \, dx=\frac {\sqrt {x}}{3 (2-b x)^{3/2}}+\frac {\sqrt {x}}{3 \sqrt {2-b x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {47, 37} \[ \int \frac {1}{\sqrt {x} (2-b x)^{5/2}} \, dx=\frac {\sqrt {x}}{3 \sqrt {2-b x}}+\frac {\sqrt {x}}{3 (2-b x)^{3/2}} \]

[In]

Int[1/(Sqrt[x]*(2 - b*x)^(5/2)),x]

[Out]

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

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]

Rule 47

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])

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {x}}{3 (2-b x)^{3/2}}+\frac {1}{3} \int \frac {1}{\sqrt {x} (2-b x)^{3/2}} \, dx \\ & = \frac {\sqrt {x}}{3 (2-b x)^{3/2}}+\frac {\sqrt {x}}{3 \sqrt {2-b x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.12 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.62 \[ \int \frac {1}{\sqrt {x} (2-b x)^{5/2}} \, dx=-\frac {\sqrt {x} (-3+b x)}{3 (2-b x)^{3/2}} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.49

method	result	size
gosper	\(-\frac {\sqrt {x}\, \left (b x -3\right )}{3 \left (-b x +2\right )^{\frac {3}{2}}}\)	\(19\)
meijerg	\(\frac {\sqrt {x}\, \sqrt {2}\, \left (-b x +3\right )}{12 \left (-\frac {b x}{2}+1\right )^{\frac {3}{2}}}\)	\(23\)
default	\(\frac {\sqrt {x}}{3 \left (-b x +2\right )^{\frac {3}{2}}}+\frac {\sqrt {x}}{3 \sqrt {-b x +2}}\)	\(28\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.24 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.85 \[ \int \frac {1}{\sqrt {x} (2-b x)^{5/2}} \, dx=-\frac {{\left (b x - 3\right )} \sqrt {-b x + 2} \sqrt {x}}{3 \, {\left (b^{2} x^{2} - 4 \, b x + 4\right )}} \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.70 (sec) , antiderivative size = 165, normalized size of antiderivative = 4.23 \[ \int \frac {1}{\sqrt {x} (2-b x)^{5/2}} \, dx=\begin {cases} \frac {b^{2} x}{3 b^{\frac {5}{2}} x \sqrt {-1 + \frac {2}{b x}} - 6 b^{\frac {3}{2}} \sqrt {-1 + \frac {2}{b x}}} - \frac {3 b}{3 b^{\frac {5}{2}} x \sqrt {-1 + \frac {2}{b x}} - 6 b^{\frac {3}{2}} \sqrt {-1 + \frac {2}{b x}}} & \text {for}\: \frac {1}{\left |{b x}\right |} > \frac {1}{2} \\- \frac {i b x}{3 b^{\frac {3}{2}} x \sqrt {1 - \frac {2}{b x}} - 6 \sqrt {b} \sqrt {1 - \frac {2}{b x}}} + \frac {3 i}{3 b^{\frac {3}{2}} x \sqrt {1 - \frac {2}{b x}} - 6 \sqrt {b} \sqrt {1 - \frac {2}{b x}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\sqrt {x} (2-b x)^{5/2}} \, dx=\frac {{\left (b - \frac {3 \, {\left (b x - 2\right )}}{x}\right )} x^{\frac {3}{2}}}{6 \, {\left (-b x + 2\right )}^{\frac {3}{2}}} \]

[In]

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

[Out]

1/6*(b - 3*(b*x - 2)/x)*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. 90 vs. \(2 (27) = 54\).

Time = 0.29 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.31 \[ \int \frac {1}{\sqrt {x} (2-b x)^{5/2}} \, dx=\frac {8 \, {\left (3 \, {\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )} \sqrt {-b} b^{2}}{3 \, {\left ({\left (\sqrt {-b x + 2} \sqrt {-b} - \sqrt {{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b\right )}^{3} {\left | b \right |}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.54 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\sqrt {x} (2-b x)^{5/2}} \, dx=\frac {3\,\sqrt {x}\,\sqrt {2-b\,x}-b\,x^{3/2}\,\sqrt {2-b\,x}}{3\,b^2\,x^2-12\,b\,x+12} \]

[In]

int(1/(x^(1/2)*(2 - b*x)^(5/2)),x)

[Out]

(3*x^(1/2)*(2 - b*x)^(1/2) - b*x^(3/2)*(2 - b*x)^(1/2))/(3*b^2*x^2 - 12*b*x + 12)