3.637 \(\int \frac{\sqrt{x}}{(2-b x)^{3/2}} \, dx\)

Optimal. Leaf size=45 \[ \frac{2 \sqrt{x}}{b \sqrt{2-b x}}-\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.0094423, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {47, 54, 216} \[ \frac{2 \sqrt{x}}{b \sqrt{2-b x}}-\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(2 - b*x)^(3/2),x]

[Out]

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

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int \frac{\sqrt{x}}{(2-b x)^{3/2}} \, dx &=\frac{2 \sqrt{x}}{b \sqrt{2-b x}}-\frac{\int \frac{1}{\sqrt{x} \sqrt{2-b x}} \, dx}{b}\\ &=\frac{2 \sqrt{x}}{b \sqrt{2-b x}}-\frac{2 \operatorname{Subst}\left (\int \frac{1}{\sqrt{2-b x^2}} \, dx,x,\sqrt{x}\right )}{b}\\ &=\frac{2 \sqrt{x}}{b \sqrt{2-b x}}-\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0391526, size = 45, normalized size = 1. \[ \frac{2 \sqrt{x}}{b \sqrt{2-b x}}-\frac{2 \sin ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{2}}\right )}{b^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(2 - b*x)^(3/2),x]

[Out]

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

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Maple [A]  time = 0.028, size = 67, normalized size = 1.5 \begin{align*} -2\,{\frac{1}{\sqrt{-b}\sqrt{\pi }b} \left ( 1/2\,{\frac{\sqrt{\pi }\sqrt{x}\sqrt{2} \left ( -b \right ) ^{3/2}}{b\sqrt{-1/2\,bx+1}}}-{\frac{\sqrt{\pi } \left ( -b \right ) ^{3/2}\arcsin \left ( 1/2\,\sqrt{b}\sqrt{x}\sqrt{2} \right ) }{{b}^{3/2}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(-b*x+2)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56244, size = 306, normalized size = 6.8 \begin{align*} \left [-\frac{{\left (b x - 2\right )} \sqrt{-b} \log \left (-b x - \sqrt{-b x + 2} \sqrt{-b} \sqrt{x} + 1\right ) + 2 \, \sqrt{-b x + 2} b \sqrt{x}}{b^{3} x - 2 \, b^{2}}, \frac{2 \,{\left ({\left (b x - 2\right )} \sqrt{b} \arctan \left (\frac{\sqrt{-b x + 2}}{\sqrt{b} \sqrt{x}}\right ) - \sqrt{-b x + 2} b \sqrt{x}\right )}}{b^{3} x - 2 \, b^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(-b*x+2)^(3/2),x, algorithm="fricas")

[Out]

[-((b*x - 2)*sqrt(-b)*log(-b*x - sqrt(-b*x + 2)*sqrt(-b)*sqrt(x) + 1) + 2*sqrt(-b*x + 2)*b*sqrt(x))/(b^3*x - 2
*b^2), 2*((b*x - 2)*sqrt(b)*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))) - sqrt(-b*x + 2)*b*sqrt(x))/(b^3*x - 2*b^
2)]

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Sympy [A]  time = 1.83719, size = 92, normalized size = 2.04 \begin{align*} \begin{cases} - \frac{2 i \sqrt{x}}{b \sqrt{b x - 2}} + \frac{2 i \operatorname{acosh}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} & \text{for}\: \frac{\left |{b x}\right |}{2} > 1 \\\frac{2 \sqrt{x}}{b \sqrt{- b x + 2}} - \frac{2 \operatorname{asin}{\left (\frac{\sqrt{2} \sqrt{b} \sqrt{x}}{2} \right )}}{b^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/2)/(-b*x+2)**(3/2),x)

[Out]

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

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Giac [B]  time = 17.9262, size = 124, normalized size = 2.76 \begin{align*} -\frac{{\left (\frac{\log \left ({\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2}\right )}{\sqrt{-b}} + \frac{8 \, \sqrt{-b}}{{\left (\sqrt{-b x + 2} \sqrt{-b} - \sqrt{{\left (b x - 2\right )} b + 2 \, b}\right )}^{2} - 2 \, b}\right )}{\left | b \right |}}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(1/2)/(-b*x+2)^(3/2),x, algorithm="giac")

[Out]

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