Integrand size = 37, antiderivative size = 70 \[ \int \frac {\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\frac {\sqrt {b^2+a^2 x^4}}{x}-\sqrt {2} \sqrt {a} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {b^2+a^2 x^4}}\right ) \]
(a^2*x^4+b^2)^(1/2)/x-2^(1/2)*a^(1/2)*b^(1/2)*arctanh(2^(1/2)*a^(1/2)*b^(1 /2)*x/(a^2*x^4+b^2)^(1/2))
Time = 0.43 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00 \[ \int \frac {\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\frac {\sqrt {b^2+a^2 x^4}}{x}-\sqrt {2} \sqrt {a} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {b^2+a^2 x^4}}\right ) \]
Sqrt[b^2 + a^2*x^4]/x - Sqrt[2]*Sqrt[a]*Sqrt[b]*ArcTanh[(Sqrt[2]*Sqrt[a]*S qrt[b]*x)/Sqrt[b^2 + a^2*x^4]]
Time = 0.54 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {2249, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a x^2+b\right ) \sqrt {a^2 x^4+b^2}}{x^2 \left (a x^2-b\right )} \, dx\) |
\(\Big \downarrow \) 2249 |
\(\displaystyle \int \left (\frac {2 a b}{\sqrt {a^2 x^4+b^2}}+\frac {4 a b^2}{\left (a x^2-b\right ) \sqrt {a^2 x^4+b^2}}-\frac {b^2}{x^2 \sqrt {a^2 x^4+b^2}}+\frac {a^2 x^2}{\sqrt {a^2 x^4+b^2}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {a^2 x^4+b^2}}{x}-\sqrt {2} \sqrt {a} \sqrt {b} \text {arctanh}\left (\frac {\sqrt {2} \sqrt {a} \sqrt {b} x}{\sqrt {a^2 x^4+b^2}}\right )\) |
Sqrt[b^2 + a^2*x^4]/x - Sqrt[2]*Sqrt[a]*Sqrt[b]*ArcTanh[(Sqrt[2]*Sqrt[a]*S qrt[b]*x)/Sqrt[b^2 + a^2*x^4]]
3.10.21.3.1 Defintions of rubi rules used
Int[(Px_)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (c_.)*(x_) ^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[1/Sqrt[a + c*x^4], Px*(f*x)^m*(d + e*x^2)^q*(a + c*x^4)^(p + 1/2), x], x] /; FreeQ[{a, c, d, e, f, m}, x] & & PolyQ[Px, x] && IntegerQ[p + 1/2] && IntegerQ[q]
Time = 10.79 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.90
method | result | size |
elliptic | \(\frac {\left (\frac {\sqrt {x^{4} a^{2}+b^{2}}\, \sqrt {2}}{x}-\frac {2 a b \,\operatorname {arctanh}\left (\frac {\sqrt {x^{4} a^{2}+b^{2}}\, \sqrt {2}}{2 x \sqrt {a b}}\right )}{\sqrt {a b}}\right ) \sqrt {2}}{2}\) | \(63\) |
risch | \(\frac {\sqrt {x^{4} a^{2}+b^{2}}}{x}-\frac {b a \sqrt {2}\, \left (2 \ln \left (2\right )+\ln \left (-\frac {2 \left (-\frac {\sqrt {2}\, \sqrt {a b}\, \sqrt {x^{4} a^{2}+b^{2}}}{2}+\left (a \,x^{2}+b \right ) \sqrt {a b}+a b x \right ) a}{a \,x^{2}+2 x \sqrt {a b}+b}\right )+\ln \left (-\frac {2 a \left (-\frac {\sqrt {2}\, \sqrt {a b}\, \sqrt {x^{4} a^{2}+b^{2}}}{2}+\left (-a \,x^{2}-b \right ) \sqrt {a b}+a b x \right )}{a \,x^{2}-2 x \sqrt {a b}+b}\right )\right )}{2 \sqrt {a b}}\) | \(163\) |
default | \(-\frac {2 x b a \sqrt {2}\, \ln \left (2\right )+x b a \sqrt {2}\, \ln \left (-\frac {2 \left (-\frac {\sqrt {2}\, \sqrt {a b}\, \sqrt {x^{4} a^{2}+b^{2}}}{2}+\left (a \,x^{2}+b \right ) \sqrt {a b}+a b x \right ) a}{a \,x^{2}+2 x \sqrt {a b}+b}\right )+x b a \sqrt {2}\, \ln \left (-\frac {2 a \left (-\frac {\sqrt {2}\, \sqrt {a b}\, \sqrt {x^{4} a^{2}+b^{2}}}{2}+\left (-a \,x^{2}-b \right ) \sqrt {a b}+a b x \right )}{a \,x^{2}-2 x \sqrt {a b}+b}\right )-2 \sqrt {x^{4} a^{2}+b^{2}}\, \sqrt {a b}}{2 \sqrt {a b}\, x}\) | \(183\) |
pseudoelliptic | \(-\frac {2 x b a \sqrt {2}\, \ln \left (2\right )+x b a \sqrt {2}\, \ln \left (-\frac {2 \left (-\frac {\sqrt {2}\, \sqrt {a b}\, \sqrt {x^{4} a^{2}+b^{2}}}{2}+\left (a \,x^{2}+b \right ) \sqrt {a b}+a b x \right ) a}{a \,x^{2}+2 x \sqrt {a b}+b}\right )+x b a \sqrt {2}\, \ln \left (-\frac {2 a \left (-\frac {\sqrt {2}\, \sqrt {a b}\, \sqrt {x^{4} a^{2}+b^{2}}}{2}+\left (-a \,x^{2}-b \right ) \sqrt {a b}+a b x \right )}{a \,x^{2}-2 x \sqrt {a b}+b}\right )-2 \sqrt {x^{4} a^{2}+b^{2}}\, \sqrt {a b}}{2 \sqrt {a b}\, x}\) | \(183\) |
1/2*((a^2*x^4+b^2)^(1/2)*2^(1/2)/x-2*a*b/(a*b)^(1/2)*arctanh(1/2*(a^2*x^4+ b^2)^(1/2)*2^(1/2)/x/(a*b)^(1/2)))*2^(1/2)
Time = 0.70 (sec) , antiderivative size = 159, normalized size of antiderivative = 2.27 \[ \int \frac {\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\left [\frac {\sqrt {2} \sqrt {a b} x \log \left (\frac {a^{2} x^{4} + 2 \, a b x^{2} - 2 \, \sqrt {2} \sqrt {a^{2} x^{4} + b^{2}} \sqrt {a b} x + b^{2}}{a^{2} x^{4} - 2 \, a b x^{2} + b^{2}}\right ) + 2 \, \sqrt {a^{2} x^{4} + b^{2}}}{2 \, x}, \frac {\sqrt {2} \sqrt {-a b} x \arctan \left (\frac {\sqrt {2} \sqrt {a^{2} x^{4} + b^{2}} \sqrt {-a b}}{2 \, a b x}\right ) + \sqrt {a^{2} x^{4} + b^{2}}}{x}\right ] \]
[1/2*(sqrt(2)*sqrt(a*b)*x*log((a^2*x^4 + 2*a*b*x^2 - 2*sqrt(2)*sqrt(a^2*x^ 4 + b^2)*sqrt(a*b)*x + b^2)/(a^2*x^4 - 2*a*b*x^2 + b^2)) + 2*sqrt(a^2*x^4 + b^2))/x, (sqrt(2)*sqrt(-a*b)*x*arctan(1/2*sqrt(2)*sqrt(a^2*x^4 + b^2)*sq rt(-a*b)/(a*b*x)) + sqrt(a^2*x^4 + b^2))/x]
\[ \int \frac {\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\int \frac {\left (a x^{2} + b\right ) \sqrt {a^{2} x^{4} + b^{2}}}{x^{2} \left (a x^{2} - b\right )}\, dx \]
\[ \int \frac {\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\int { \frac {\sqrt {a^{2} x^{4} + b^{2}} {\left (a x^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{2}} \,d x } \]
\[ \int \frac {\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (-b+a x^2\right )} \, dx=\int { \frac {\sqrt {a^{2} x^{4} + b^{2}} {\left (a x^{2} + b\right )}}{{\left (a x^{2} - b\right )} x^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (b+a x^2\right ) \sqrt {b^2+a^2 x^4}}{x^2 \left (-b+a x^2\right )} \, dx=-\int \frac {\sqrt {a^2\,x^4+b^2}\,\left (a\,x^2+b\right )}{x^2\,\left (b-a\,x^2\right )} \,d x \]