\(\int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1-a x}} \, dx\) [223]

Optimal result

Integrand size = 30, antiderivative size = 34 \[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1-a x}} \, dx=\sqrt {x} \sqrt {1+a x}+\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \]

[Out]

arcsinh(a^(1/2)*x^(1/2))/a^(1/2)+x^(1/2)*(a*x+1)^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {862, 52, 56, 221} \[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1-a x}} \, dx=\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}}+\sqrt {x} \sqrt {a x+1} \]

[In]

Int[Sqrt[1 - a^2*x^2]/(Sqrt[x]*Sqrt[1 - a*x]),x]

[Out]

Sqrt[x]*Sqrt[1 + a*x] + ArcSinh[Sqrt[a]*Sqrt[x]]/Sqrt[a]

Rule 52

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 56

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 221

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

Rule 862

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/e)*x)^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]))

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sqrt {1+a x}}{\sqrt {x}} \, dx \\ & = \sqrt {x} \sqrt {1+a x}+\frac {1}{2} \int \frac {1}{\sqrt {x} \sqrt {1+a x}} \, dx \\ & = \sqrt {x} \sqrt {1+a x}+\text {Subst}\left (\int \frac {1}{\sqrt {1+a x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \sqrt {x} \sqrt {1+a x}+\frac {\sinh ^{-1}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.50 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1-a x}} \, dx=\sqrt {x} \sqrt {1+a x}+\frac {\text {arcsinh}\left (\sqrt {a} \sqrt {x}\right )}{\sqrt {a}} \]

[In]

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

[Out]

Sqrt[x]*Sqrt[1 + a*x] + ArcSinh[Sqrt[a]*Sqrt[x]]/Sqrt[a]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(24)=48\).

Time = 0.36 (sec) , antiderivative size = 86, normalized size of antiderivative = 2.53

method	result	size
default	\(-\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {x}\, \sqrt {-a x +1}\, \left (2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+\ln \left (\frac {2 \sqrt {\left (a x +1\right ) x}\, \sqrt {a}+2 a x +1}{2 \sqrt {a}}\right )\right )}{2 \left (a x -1\right ) \sqrt {\left (a x +1\right ) x}\, \sqrt {a}}\)	\(86\)
risch	\(-\frac {\left (a x +1\right ) \sqrt {x}\, \sqrt {\frac {\left (-a^{2} x^{2}+1\right ) x \left (-a x +1\right )}{\left (a x -1\right )^{2}}}\, \left (a x -1\right )}{\sqrt {\left (a x +1\right ) x}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {-a x +1}}-\frac {\ln \left (\frac {\frac {1}{2}+a x}{\sqrt {a}}+\sqrt {a \,x^{2}+x}\right ) \sqrt {\frac {\left (-a^{2} x^{2}+1\right ) x \left (-a x +1\right )}{\left (a x -1\right )^{2}}}\, \left (a x -1\right )}{2 \sqrt {a}\, \sqrt {-a^{2} x^{2}+1}\, \sqrt {x}\, \sqrt {-a x +1}}\)	\(153\)

[In]

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

[Out]

-1/2*(-a^2*x^2+1)^(1/2)*x^(1/2)*(-a*x+1)^(1/2)*(2*((a*x+1)*x)^(1/2)*a^(1/2)+ln(1/2*(2*((a*x+1)*x)^(1/2)*a^(1/2
)+2*a*x+1)/a^(1/2)))/(a*x-1)/((a*x+1)*x)^(1/2)/a^(1/2)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 99 vs. \(2 (24) = 48\).

Time = 0.29 (sec) , antiderivative size = 208, normalized size of antiderivative = 6.12 \[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1-a x}} \, dx=\left [-\frac {4 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a x + 1} a \sqrt {x} - {\left (a x - 1\right )} \sqrt {a} \log \left (-\frac {8 \, a^{3} x^{3} - 4 \, \sqrt {-a^{2} x^{2} + 1} {\left (2 \, a x + 1\right )} \sqrt {-a x + 1} \sqrt {a} \sqrt {x} - 7 \, a x - 1}{a x - 1}\right )}{4 \, {\left (a^{2} x - a\right )}}, -\frac {2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a x + 1} a \sqrt {x} - {\left (a x - 1\right )} \sqrt {-a} \arctan \left (\frac {2 \, \sqrt {-a^{2} x^{2} + 1} \sqrt {-a x + 1} \sqrt {-a} \sqrt {x}}{2 \, a^{2} x^{2} - a x - 1}\right )}{2 \, {\left (a^{2} x - a\right )}}\right ] \]

[In]

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

[Out]

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

Sympy [F]

\[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1-a x}} \, dx=\int \frac {\sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}{\sqrt {x} \sqrt {- a x + 1}}\, dx \]

[In]

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

[Out]

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

Maxima [F]

\[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1-a x}} \, dx=\int { \frac {\sqrt {-a^{2} x^{2} + 1}}{\sqrt {-a x + 1} \sqrt {x}} \,d x } \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 89 vs. \(2 (24) = 48\).

Time = 5.45 (sec) , antiderivative size = 89, normalized size of antiderivative = 2.62 \[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1-a x}} \, dx=-\frac {a {\left (\frac {\sqrt {2} - \log \left ({\left | -\sqrt {2} \sqrt {a} + \sqrt {a} \right |}\right )}{\sqrt {a}} + \frac {\log \left ({\left | -\sqrt {a x + 1} \sqrt {a} + \sqrt {{\left (a x + 1\right )} a - a} \right |}\right )}{\sqrt {a}} - \frac {\sqrt {{\left (a x + 1\right )} a - a} \sqrt {a x + 1}}{a}\right )}}{{\left | a \right |}} \]

[In]

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

[Out]

-a*((sqrt(2) - log(abs(-sqrt(2)*sqrt(a) + sqrt(a))))/sqrt(a) + log(abs(-sqrt(a*x + 1)*sqrt(a) + sqrt((a*x + 1)
*a - a)))/sqrt(a) - sqrt((a*x + 1)*a - a)*sqrt(a*x + 1)/a)/abs(a)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {1-a^2 x^2}}{\sqrt {x} \sqrt {1-a x}} \, dx=\int \frac {\sqrt {1-a^2\,x^2}}{\sqrt {x}\,\sqrt {1-a\,x}} \,d x \]

[In]

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

[Out]

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