Integrand size = 37, antiderivative size = 141 \[ \int \frac {-b+2 a x^2}{\left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {2 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{\sqrt [4]{2} \sqrt [4]{a}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{\sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^2+a x^4}}\right )}{\sqrt [4]{2} \sqrt [4]{a}} \]
2*arctan(a^(1/4)*x/(a*x^4+b*x^2)^(1/4))/a^(1/4)-1/2*arctan(2^(1/4)*a^(1/4) *x/(a*x^4+b*x^2)^(1/4))*2^(3/4)/a^(1/4)+2*arctanh(a^(1/4)*x/(a*x^4+b*x^2)^ (1/4))/a^(1/4)-1/2*arctanh(2^(1/4)*a^(1/4)*x/(a*x^4+b*x^2)^(1/4))*2^(3/4)/ a^(1/4)
Time = 0.48 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.14 \[ \int \frac {-b+2 a x^2}{\left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {\sqrt {x} \sqrt [4]{b+a x^2} \left (4 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )-2^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )+4 \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )-2^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{b+a x^2}}\right )\right )}{2 \sqrt [4]{a} \sqrt [4]{x^2 \left (b+a x^2\right )}} \]
(Sqrt[x]*(b + a*x^2)^(1/4)*(4*ArcTan[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)] - 2^(3/4)*ArcTan[(2^(1/4)*a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)] + 4*ArcTanh[ (a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)] - 2^(3/4)*ArcTanh[(2^(1/4)*a^(1/4)*Sq rt[x])/(b + a*x^2)^(1/4)]))/(2*a^(1/4)*(x^2*(b + a*x^2))^(1/4))
Time = 0.51 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.23, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.081, Rules used = {2467, 446, 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.
| \(\displaystyle \int \frac {2 a x^2-b}{\left (a x^2-b\right ) \sqrt [4]{a x^4+b x^2}} \, dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^2+b} \int \frac {b-2 a x^2}{\sqrt {x} \left (b-a x^2\right ) \sqrt [4]{a x^2+b}}dx}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 446 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^2+b} \int \left (\frac {2}{\sqrt {x} \sqrt [4]{a x^2+b}}-\frac {b}{\sqrt {x} \left (b-a x^2\right ) \sqrt [4]{a x^2+b}}\right )dx}{\sqrt [4]{a x^4+b x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {x} \sqrt [4]{a x^2+b} \left (\frac {2 \arctan \left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{2} \sqrt [4]{a}}+\frac {2 \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{a}}-\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt {x}}{\sqrt [4]{a x^2+b}}\right )}{\sqrt [4]{2} \sqrt [4]{a}}\right )}{\sqrt [4]{a x^4+b x^2}}\) |
(Sqrt[x]*(b + a*x^2)^(1/4)*((2*ArcTan[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)] )/a^(1/4) - ArcTan[(2^(1/4)*a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)]/(2^(1/4)*a ^(1/4)) + (2*ArcTanh[(a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)])/a^(1/4) - ArcTa nh[(2^(1/4)*a^(1/4)*Sqrt[x])/(b + a*x^2)^(1/4)]/(2^(1/4)*a^(1/4))))/(b*x^2 + a*x^4)^(1/4)
3.20.99.3.1 Defintions of rubi rules used
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_)*((e_) + (f_.)*(x_)^2))/( (c_) + (d_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^2)^ p*((e + f*x^2)/(c + d*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Time = 0.47 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.13
| method | result | size |
| pseudoelliptic | \(-\frac {\left (4 \,2^{\frac {1}{4}} \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )-2 \,2^{\frac {1}{4}} \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right )-2 \arctan \left (\frac {\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right )+\ln \left (\frac {x 2^{\frac {1}{4}} a^{\frac {1}{4}}+\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}{-x 2^{\frac {1}{4}} a^{\frac {1}{4}}+\left (x^{2} \left (a \,x^{2}+b \right )\right )^{\frac {1}{4}}}\right )\right ) 2^{\frac {3}{4}}}{4 a^{\frac {1}{4}}}\) | \(160\) |
-1/4/a^(1/4)*(4*2^(1/4)*arctan(1/a^(1/4)*(x^2*(a*x^2+b))^(1/4)/x)-2*2^(1/4 )*ln((a^(1/4)*x+(x^2*(a*x^2+b))^(1/4))/(-a^(1/4)*x+(x^2*(a*x^2+b))^(1/4))) -2*arctan(1/2*(x^2*(a*x^2+b))^(1/4)/x*2^(3/4)/a^(1/4))+ln((x*2^(1/4)*a^(1/ 4)+(x^2*(a*x^2+b))^(1/4))/(-x*2^(1/4)*a^(1/4)+(x^2*(a*x^2+b))^(1/4))))*2^( 3/4)
Timed out. \[ \int \frac {-b+2 a x^2}{\left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\text {Timed out} \]
\[ \int \frac {-b+2 a x^2}{\left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {2 a x^{2} - b}{\sqrt [4]{x^{2} \left (a x^{2} + b\right )} \left (a x^{2} - b\right )}\, dx \]
\[ \int \frac {-b+2 a x^2}{\left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int { \frac {2 \, a x^{2} - b}{{\left (a x^{4} + b x^{2}\right )}^{\frac {1}{4}} {\left (a x^{2} - b\right )}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 377 vs. \(2 (109) = 218\).
Time = 0.30 (sec) , antiderivative size = 377, normalized size of antiderivative = 2.67 \[ \int \frac {-b+2 a x^2}{\left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{a} - \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{2 \, a} + \frac {\sqrt {2} \left (-a\right )^{\frac {3}{4}} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{2 \, a} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{2 \, a} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{2 \, a} + \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{4 \, a} - \frac {2^{\frac {1}{4}} \left (-a\right )^{\frac {3}{4}} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x^{2}}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x^{2}}}\right )}{4 \, a} \]
sqrt(2)*(-a)^(3/4)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x^2)^ (1/4))/(-a)^(1/4))/a + sqrt(2)*(-a)^(3/4)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a )^(1/4) - 2*(a + b/x^2)^(1/4))/(-a)^(1/4))/a - 1/2*sqrt(2)*(-a)^(3/4)*log( sqrt(2)*(-a)^(1/4)*(a + b/x^2)^(1/4) + sqrt(-a) + sqrt(a + b/x^2))/a + 1/2 *sqrt(2)*(-a)^(3/4)*log(-sqrt(2)*(-a)^(1/4)*(a + b/x^2)^(1/4) + sqrt(-a) + sqrt(a + b/x^2))/a - 1/2*2^(1/4)*(-a)^(3/4)*arctan(1/2*2^(1/4)*(2^(3/4)*( -a)^(1/4) + 2*(a + b/x^2)^(1/4))/(-a)^(1/4))/a - 1/2*2^(1/4)*(-a)^(3/4)*ar ctan(-1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) - 2*(a + b/x^2)^(1/4))/(-a)^(1/4))/a + 1/4*2^(1/4)*(-a)^(3/4)*log(2^(3/4)*(-a)^(1/4)*(a + b/x^2)^(1/4) + sqrt( 2)*sqrt(-a) + sqrt(a + b/x^2))/a - 1/4*2^(1/4)*(-a)^(3/4)*log(-2^(3/4)*(-a )^(1/4)*(a + b/x^2)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x^2))/a
Timed out. \[ \int \frac {-b+2 a x^2}{\left (-b+a x^2\right ) \sqrt [4]{b x^2+a x^4}} \, dx=\int \frac {b-2\,a\,x^2}{\left (b-a\,x^2\right )\,{\left (a\,x^4+b\,x^2\right )}^{1/4}} \,d x \]