3.225 \(\int \frac{\sqrt{x} \sqrt{1-a^2 x^2}}{\sqrt{1+a x}} \, dx\)

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 = 0.0329066, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {848, 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^2*x^2])/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 848

Int[((d_) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[(d + e*x)
^(m + p)*(f + g*x)^n*(a/d + (c*x)/e)^p, x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && NeQ[e*f - d*g, 0] && EqQ[c
*d^2 + a*e^2, 0] && (IntegerQ[p] || (GtQ[a, 0] && GtQ[d, 0] && EqQ[m + p, 0]))

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 \frac{\sqrt{x} \sqrt{1-a^2 x^2}}{\sqrt{1+a x}} \, dx &=\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 = 0.0077062, 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^2*x^2])/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 [B]  time = 0.135, size = 92, normalized size = 1.5 \begin{align*}{\frac{1}{8}\sqrt{x}\sqrt{-{a}^{2}{x}^{2}+1} \left ( 4\,x{a}^{3/2}\sqrt{-x \left ( ax-1 \right ) }-2\,\sqrt{a}\sqrt{-x \left ( ax-1 \right ) }+\arctan \left ({\frac{2\,ax-1}{2}{\frac{1}{\sqrt{a}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) }}}} \right ) \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{ax+1}}}{\frac{1}{\sqrt{-x \left ( ax-1 \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{-a^{2} x^{2} + 1} \sqrt{x}}{\sqrt{a x + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [B]  time = 1.75568, size = 525, normalized size = 8.33 \begin{align*} \left [\frac{4 \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a^{2} x - a\right )} \sqrt{a x + 1} \sqrt{x} -{\left (a x + 1\right )} \sqrt{-a} \log \left (-\frac{8 \, a^{3} x^{3} - 4 \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a x - 1\right )} \sqrt{a x + 1} \sqrt{-a} \sqrt{x} - 7 \, a x + 1}{a x + 1}\right )}{16 \,{\left (a^{3} x + a^{2}\right )}}, \frac{2 \, \sqrt{-a^{2} x^{2} + 1}{\left (2 \, a^{2} x - a\right )} \sqrt{a x + 1} \sqrt{x} -{\left (a x + 1\right )} \sqrt{a} \arctan \left (\frac{2 \, \sqrt{-a^{2} x^{2} + 1} \sqrt{a x + 1} \sqrt{a} \sqrt{x}}{2 \, a^{2} x^{2} + a x - 1}\right )}{8 \,{\left (a^{3} x + a^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/16*(4*sqrt(-a^2*x^2 + 1)*(2*a^2*x - a)*sqrt(a*x + 1)*sqrt(x) - (a*x + 1)*sqrt(-a)*log(-(8*a^3*x^3 - 4*sqrt(
-a^2*x^2 + 1)*(2*a*x - 1)*sqrt(a*x + 1)*sqrt(-a)*sqrt(x) - 7*a*x + 1)/(a*x + 1)))/(a^3*x + a^2), 1/8*(2*sqrt(-
a^2*x^2 + 1)*(2*a^2*x - a)*sqrt(a*x + 1)*sqrt(x) - (a*x + 1)*sqrt(a)*arctan(2*sqrt(-a^2*x^2 + 1)*sqrt(a*x + 1)
*sqrt(a)*sqrt(x)/(2*a^2*x^2 + a*x - 1)))/(a^3*x + a^2)]

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{x} \sqrt{- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt{a x + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(x)*sqrt(-(a*x - 1)*(a*x + 1))/sqrt(a*x + 1), x)

________________________________________________________________________________________

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

[Out]

Exception raised: NotImplementedError