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

Optimal. Leaf size=39 \[ \frac {\sqrt {x}}{3 \sqrt {2-b x}}+\frac {\sqrt {x}}{3 (2-b x)^{3/2}} \]

________________________________________________________________________________________

Rubi [A]  time = 0.00, antiderivative size = 39, normalized size of antiderivative = 1.00, 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} \begin {gather*} \frac {\sqrt {x}}{3 \sqrt {2-b x}}+\frac {\sqrt {x}}{3 (2-b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[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 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])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {x} (2-b x)^{5/2}} \, dx &=\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]  time = 0.01, size = 24, normalized size = 0.62 \begin {gather*} -\frac {\sqrt {x} (b x-3)}{3 (2-b x)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.10, size = 31, normalized size = 0.79 \begin {gather*} -\frac {\sqrt {x} \sqrt {2-b x} (b x-3)}{3 (b x-2)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.31, size = 33, normalized size = 0.85 \begin {gather*} -\frac {{\left (b x - 3\right )} \sqrt {-b x + 2} \sqrt {x}}{3 \, {\left (b^{2} x^{2} - 4 \, b x + 4\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

giac [B]  time = 1.10, size = 90, normalized size = 2.31 \begin {gather*} \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 |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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))

________________________________________________________________________________________

maple [A]  time = 0.00, size = 19, normalized size = 0.49 \begin {gather*} -\frac {\left (b x -3\right ) \sqrt {x}}{3 \left (-b x +2\right )^{\frac {3}{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 1.30, size = 25, normalized size = 0.64 \begin {gather*} \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}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [B]  time = 0.36, size = 45, normalized size = 1.15 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

sympy [C]  time = 1.91, size = 177, normalized size = 4.54 \begin {gather*} \begin {cases} \frac {i b x}{3 i b^{\frac {3}{2}} x \sqrt {-1 + \frac {2}{b x}} - 6 i \sqrt {b} \sqrt {-1 + \frac {2}{b x}}} - \frac {3 i}{3 i b^{\frac {3}{2}} x \sqrt {-1 + \frac {2}{b x}} - 6 i \sqrt {b} \sqrt {-1 + \frac {2}{b x}}} & \text {for}\: \frac {2}{\left |{b x}\right |} > 1 \\\frac {b^{2} x}{3 i b^{\frac {5}{2}} x \sqrt {1 - \frac {2}{b x}} - 6 i b^{\frac {3}{2}} \sqrt {1 - \frac {2}{b x}}} - \frac {3 b}{3 i b^{\frac {5}{2}} x \sqrt {1 - \frac {2}{b x}} - 6 i b^{\frac {3}{2}} \sqrt {1 - \frac {2}{b x}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________