\(\int \frac {\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx\) [1210]

Optimal result

Integrand size = 59, antiderivative size = 88 \[ \int \frac {\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=-\frac {\sqrt {2} b \log \left (-a x-b \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}+\sqrt {2} \sqrt {a} \sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}\right )}{\sqrt {a}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.62 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.52, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.034, Rules used = {2155, 221} \[ \int \frac {\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\frac {\sqrt {2} b \text {arcsinh}\left (\frac {b \sqrt {\frac {a^2 x^2}{b^2}-\frac {a}{b^2}}+a x}{\sqrt {a}}\right )}{\sqrt {a}} \]

[In]

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

[Out]

(Sqrt[2]*b*ArcSinh[(a*x + b*Sqrt[-(a/b^2) + (a^2*x^2)/b^2])/Sqrt[a]])/Sqrt[a]

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 2155

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

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

Mathematica [A] (verified)

Time = 5.18 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.22 \[ \int \frac {\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=-\frac {\sqrt {2} b \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \sqrt {x \left (a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )} \arctan \left (\sqrt {2} \sqrt {x \left (-a x+b \sqrt {\frac {a \left (-1+a x^2\right )}{b^2}}\right )}\right )}{a x} \]

[In]

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

[Out]

-((Sqrt[2]*b*Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*Sqrt[x*(a*x + b*Sqrt[(a*(-1 + a*x^2))/b^2])]*ArcT
an[Sqrt[2]*Sqrt[x*(-(a*x) + b*Sqrt[(a*(-1 + a*x^2))/b^2])]])/(a*x))

Maple [F]

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

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 5.77 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.83 \[ \int \frac {\sqrt {a x^2+b x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}}}{x \sqrt {-\frac {a}{b^2}+\frac {a^2 x^2}{b^2}}} \, dx=\left [\frac {\sqrt {2} b \log \left (-4 \, a x^{2} - 4 \, b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}} - 2 \, \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} {\left (\sqrt {2} \sqrt {a} x + \frac {\sqrt {2} b \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}}{\sqrt {a}}\right )} + 1\right )}{2 \, \sqrt {a}}, -\sqrt {2} b \sqrt {-\frac {1}{a}} \arctan \left (\frac {\sqrt {2} \sqrt {a x^{2} + b x \sqrt {\frac {a^{2} x^{2} - a}{b^{2}}}} \sqrt {-\frac {1}{a}}}{2 \, x}\right )\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral(sqrt(x*(a*x + b*sqrt(a**2*x**2/b**2 - a/b**2)))/(x*sqrt(a*(a*x**2 - 1)/b**2)), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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