3.224 \(\int \sqrt{x} \sqrt{1-a x} \, dx\)

Optimal. Leaf size=63 \[ \frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{1-a x}-\frac{\sqrt{x} \sqrt{1-a x}}{4 a} \]

[Out]

-(Sqrt[x]*Sqrt[1 - a*x])/(4*a) + (x^(3/2)*Sqrt[1 - a*x])/2 + ArcSin[Sqrt[a]*Sqrt[x]]/(4*a^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0150052, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.188, Rules used = {50, 54, 216} \[ \frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2}}+\frac{1}{2} x^{3/2} \sqrt{1-a x}-\frac{\sqrt{x} \sqrt{1-a x}}{4 a} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*Sqrt[1 - a*x],x]

[Out]

-(Sqrt[x]*Sqrt[1 - a*x])/(4*a) + (x^(3/2)*Sqrt[1 - a*x])/2 + ArcSin[Sqrt[a]*Sqrt[x]]/(4*a^(3/2))

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 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 \sqrt{x} \sqrt{1-a x} \, dx &=\frac{1}{2} x^{3/2} \sqrt{1-a x}+\frac{1}{4} \int \frac{\sqrt{x}}{\sqrt{1-a x}} \, dx\\ &=-\frac{\sqrt{x} \sqrt{1-a x}}{4 a}+\frac{1}{2} x^{3/2} \sqrt{1-a x}+\frac{\int \frac{1}{\sqrt{x} \sqrt{1-a x}} \, dx}{8 a}\\ &=-\frac{\sqrt{x} \sqrt{1-a x}}{4 a}+\frac{1}{2} x^{3/2} \sqrt{1-a x}+\frac{\operatorname{Subst}\left (\int \frac{1}{\sqrt{1-a x^2}} \, dx,x,\sqrt{x}\right )}{4 a}\\ &=-\frac{\sqrt{x} \sqrt{1-a x}}{4 a}+\frac{1}{2} x^{3/2} \sqrt{1-a x}+\frac{\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.021092, size = 49, normalized size = 0.78 \[ \frac{\sqrt{a} \sqrt{x} \sqrt{1-a x} (2 a x-1)+\sin ^{-1}\left (\sqrt{a} \sqrt{x}\right )}{4 a^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*Sqrt[1 - a*x],x]

[Out]

(Sqrt[a]*Sqrt[x]*Sqrt[1 - a*x]*(-1 + 2*a*x) + ArcSin[Sqrt[a]*Sqrt[x]])/(4*a^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.135, size = 79, normalized size = 1.3 \begin{align*}{\frac{1}{2}{x}^{{\frac{3}{2}}}\sqrt{-ax+1}}-{\frac{1}{4\,a}\sqrt{x}\sqrt{-ax+1}}+{\frac{1}{8}\sqrt{ \left ( -ax+1 \right ) x}\arctan \left ({\sqrt{a} \left ( x-{\frac{1}{2\,a}} \right ){\frac{1}{\sqrt{-a{x}^{2}+x}}}} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{-ax+1}}}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x^(3/2)*(-a*x+1)^(1/2)-1/4*x^(1/2)*(-a*x+1)^(1/2)/a+1/8/a^(3/2)*((-a*x+1)*x)^(1/2)/(-a*x+1)^(1/2)/x^(1/2)*
arctan(a^(1/2)*(x-1/2/a)/(-a*x^2+x)^(1/2))

________________________________________________________________________________________

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)*(-a*x+1)^(1/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.57949, size = 292, normalized size = 4.63 \begin{align*} \left [\frac{2 \,{\left (2 \, a^{2} x - a\right )} \sqrt{-a x + 1} \sqrt{x} - \sqrt{-a} \log \left (-2 \, a x + 2 \, \sqrt{-a x + 1} \sqrt{-a} \sqrt{x} + 1\right )}{8 \, a^{2}}, \frac{{\left (2 \, a^{2} x - a\right )} \sqrt{-a x + 1} \sqrt{x} - \sqrt{a} \arctan \left (\frac{\sqrt{-a x + 1}}{\sqrt{a} \sqrt{x}}\right )}{4 \, a^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(2*(2*a^2*x - a)*sqrt(-a*x + 1)*sqrt(x) - sqrt(-a)*log(-2*a*x + 2*sqrt(-a*x + 1)*sqrt(-a)*sqrt(x) + 1))/a
^2, 1/4*((2*a^2*x - a)*sqrt(-a*x + 1)*sqrt(x) - sqrt(a)*arctan(sqrt(-a*x + 1)/(sqrt(a)*sqrt(x))))/a^2]

________________________________________________________________________________________

Sympy [A]  time = 3.54705, size = 148, normalized size = 2.35 \begin{align*} \begin{cases} \frac{i a x^{\frac{5}{2}}}{2 \sqrt{a x - 1}} - \frac{3 i x^{\frac{3}{2}}}{4 \sqrt{a x - 1}} + \frac{i \sqrt{x}}{4 a \sqrt{a x - 1}} - \frac{i \operatorname{acosh}{\left (\sqrt{a} \sqrt{x} \right )}}{4 a^{\frac{3}{2}}} & \text{for}\: \left |{a x}\right | > 1 \\- \frac{a x^{\frac{5}{2}}}{2 \sqrt{- a x + 1}} + \frac{3 x^{\frac{3}{2}}}{4 \sqrt{- a x + 1}} - \frac{\sqrt{x}}{4 a \sqrt{- a x + 1}} + \frac{\operatorname{asin}{\left (\sqrt{a} \sqrt{x} \right )}}{4 a^{\frac{3}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(1/2)*(-a*x+1)**(1/2),x)

[Out]

Piecewise((I*a*x**(5/2)/(2*sqrt(a*x - 1)) - 3*I*x**(3/2)/(4*sqrt(a*x - 1)) + I*sqrt(x)/(4*a*sqrt(a*x - 1)) - I
*acosh(sqrt(a)*sqrt(x))/(4*a**(3/2)), Abs(a*x) > 1), (-a*x**(5/2)/(2*sqrt(-a*x + 1)) + 3*x**(3/2)/(4*sqrt(-a*x
+ 1)) - sqrt(x)/(4*a*sqrt(-a*x + 1)) + asin(sqrt(a)*sqrt(x))/(4*a**(3/2)), True))

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: NotImplementedError