Integrand size = 39, antiderivative size = 193 \[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{a-2 b^2} \sqrt [4]{-1+b x+a x^2}}{\sqrt {a-2 b^2}-\sqrt {a} \sqrt {-1+b x+a x^2}}\right )}{a^{3/4} \sqrt [4]{a-2 b^2}}-\frac {\sqrt {2} \text {arctanh}\left (\frac {\frac {\sqrt [4]{a-2 b^2}}{\sqrt {2} \sqrt [4]{a}}+\frac {\sqrt [4]{a} \sqrt {-1+b x+a x^2}}{\sqrt {2} \sqrt [4]{a-2 b^2}}}{\sqrt [4]{-1+b x+a x^2}}\right )}{a^{3/4} \sqrt [4]{a-2 b^2}} \]
2^(1/2)*arctan(2^(1/2)*a^(1/4)*(-2*b^2+a)^(1/4)*(a*x^2+b*x-1)^(1/4)/((-2*b ^2+a)^(1/2)-a^(1/2)*(a*x^2+b*x-1)^(1/2)))/a^(3/4)/(-2*b^2+a)^(1/4)-2^(1/2) *arctanh((1/2*(-2*b^2+a)^(1/4)*2^(1/2)/a^(1/4)+1/2*a^(1/4)*(a*x^2+b*x-1)^( 1/2)*2^(1/2)/(-2*b^2+a)^(1/4))/(a*x^2+b*x-1)^(1/4))/a^(3/4)/(-2*b^2+a)^(1/ 4)
Time = 0.43 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.86 \[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=\frac {\sqrt {2} \left (\arctan \left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{a-2 b^2} \sqrt [4]{-1+b x+a x^2}}{\sqrt {a-2 b^2}-\sqrt {a} \sqrt {-1+b x+a x^2}}\right )-\text {arctanh}\left (\frac {\sqrt {a-2 b^2}+\sqrt {a} \sqrt {-1+b x+a x^2}}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{a-2 b^2} \sqrt [4]{-1+b x+a x^2}}\right )\right )}{a^{3/4} \sqrt [4]{a-2 b^2}} \]
(Sqrt[2]*(ArcTan[(Sqrt[2]*a^(1/4)*(a - 2*b^2)^(1/4)*(-1 + b*x + a*x^2)^(1/ 4))/(Sqrt[a - 2*b^2] - Sqrt[a]*Sqrt[-1 + b*x + a*x^2])] - ArcTanh[(Sqrt[a - 2*b^2] + Sqrt[a]*Sqrt[-1 + b*x + a*x^2])/(Sqrt[2]*a^(1/4)*(a - 2*b^2)^(1 /4)*(-1 + b*x + a*x^2)^(1/4))]))/(a^(3/4)*(a - 2*b^2)^(1/4))
Time = 2.22 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.27, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {7293, 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.
\(\displaystyle \int \frac {2 a x+b}{(a x-b) (a x+2 b) \sqrt [4]{a x^2+b x-1}} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (\frac {1}{(a x-b) \sqrt [4]{a x^2+b x-1}}+\frac {1}{(a x+2 b) \sqrt [4]{a x^2+b x-1}}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2 \sqrt [4]{4 a+b^2} \sqrt [4]{\frac {a \left (-a x^2-b x+1\right )}{4 a+b^2}} \arctan \left (\frac {\sqrt [4]{4 a+b^2} \sqrt [4]{1-\frac {(2 a x+b)^2}{4 a+b^2}}}{\sqrt {2} \sqrt [4]{a-2 b^2}}\right )}{a \sqrt [4]{a-2 b^2} \sqrt [4]{a x^2+b x-1}}-\frac {2 \sqrt [4]{4 a+b^2} \sqrt [4]{\frac {a \left (-a x^2-b x+1\right )}{4 a+b^2}} \text {arctanh}\left (\frac {\sqrt [4]{4 a+b^2} \sqrt [4]{1-\frac {(2 a x+b)^2}{4 a+b^2}}}{\sqrt {2} \sqrt [4]{a-2 b^2}}\right )}{a \sqrt [4]{a-2 b^2} \sqrt [4]{a x^2+b x-1}}\) |
(2*(4*a + b^2)^(1/4)*((a*(1 - b*x - a*x^2))/(4*a + b^2))^(1/4)*ArcTan[((4* a + b^2)^(1/4)*(1 - (b + 2*a*x)^2/(4*a + b^2))^(1/4))/(Sqrt[2]*(a - 2*b^2) ^(1/4))])/(a*(a - 2*b^2)^(1/4)*(-1 + b*x + a*x^2)^(1/4)) - (2*(4*a + b^2)^ (1/4)*((a*(1 - b*x - a*x^2))/(4*a + b^2))^(1/4)*ArcTanh[((4*a + b^2)^(1/4) *(1 - (b + 2*a*x)^2/(4*a + b^2))^(1/4))/(Sqrt[2]*(a - 2*b^2)^(1/4))])/(a*( a - 2*b^2)^(1/4)*(-1 + b*x + a*x^2)^(1/4))
3.24.97.3.1 Defintions of rubi rules used
Time = 0.66 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.60
method | result | size |
pseudoelliptic | \(\frac {2 \arctan \left (\frac {\left (a \,x^{2}+b x -1\right )^{\frac {1}{4}}}{\left (\frac {2 b^{2}-a}{a}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {\left (a \,x^{2}+b x -1\right )^{\frac {1}{4}}+\left (\frac {2 b^{2}-a}{a}\right )^{\frac {1}{4}}}{\left (a \,x^{2}+b x -1\right )^{\frac {1}{4}}-\left (\frac {2 b^{2}-a}{a}\right )^{\frac {1}{4}}}\right )}{\left (\frac {2 b^{2}-a}{a}\right )^{\frac {1}{4}} a}\) | \(116\) |
1/((2*b^2-a)/a)^(1/4)*(2*arctan((a*x^2+b*x-1)^(1/4)/((2*b^2-a)/a)^(1/4))-l n(((a*x^2+b*x-1)^(1/4)+((2*b^2-a)/a)^(1/4))/((a*x^2+b*x-1)^(1/4)-((2*b^2-a )/a)^(1/4))))/a
Result contains complex when optimal does not.
Time = 0.30 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.31 \[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=-\frac {\log \left (\frac {2 \, a^{2} b^{2} - a^{3}}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {3}{4}}} + {\left (a x^{2} + b x - 1\right )}^{\frac {1}{4}}\right )}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {1}{4}}} + \frac {\log \left (-\frac {2 \, a^{2} b^{2} - a^{3}}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {3}{4}}} + {\left (a x^{2} + b x - 1\right )}^{\frac {1}{4}}\right )}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {1}{4}}} + \frac {i \, \log \left (\frac {2 i \, a^{2} b^{2} - i \, a^{3}}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {3}{4}}} + {\left (a x^{2} + b x - 1\right )}^{\frac {1}{4}}\right )}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {1}{4}}} - \frac {i \, \log \left (\frac {-2 i \, a^{2} b^{2} + i \, a^{3}}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {3}{4}}} + {\left (a x^{2} + b x - 1\right )}^{\frac {1}{4}}\right )}{{\left (2 \, a^{3} b^{2} - a^{4}\right )}^{\frac {1}{4}}} \]
-log((2*a^2*b^2 - a^3)/(2*a^3*b^2 - a^4)^(3/4) + (a*x^2 + b*x - 1)^(1/4))/ (2*a^3*b^2 - a^4)^(1/4) + log(-(2*a^2*b^2 - a^3)/(2*a^3*b^2 - a^4)^(3/4) + (a*x^2 + b*x - 1)^(1/4))/(2*a^3*b^2 - a^4)^(1/4) + I*log((2*I*a^2*b^2 - I *a^3)/(2*a^3*b^2 - a^4)^(3/4) + (a*x^2 + b*x - 1)^(1/4))/(2*a^3*b^2 - a^4) ^(1/4) - I*log((-2*I*a^2*b^2 + I*a^3)/(2*a^3*b^2 - a^4)^(3/4) + (a*x^2 + b *x - 1)^(1/4))/(2*a^3*b^2 - a^4)^(1/4)
\[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=\int \frac {2 a x + b}{\left (a x - b\right ) \left (a x + 2 b\right ) \sqrt [4]{a x^{2} + b x - 1}}\, dx \]
\[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=\int { \frac {2 \, a x + b}{{\left (a x^{2} + b x - 1\right )}^{\frac {1}{4}} {\left (a x + 2 \, b\right )} {\left (a x - b\right )}} \,d x } \]
\[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=\int { \frac {2 \, a x + b}{{\left (a x^{2} + b x - 1\right )}^{\frac {1}{4}} {\left (a x + 2 \, b\right )} {\left (a x - b\right )}} \,d x } \]
Timed out. \[ \int \frac {b+2 a x}{(-b+a x) (2 b+a x) \sqrt [4]{-1+b x+a x^2}} \, dx=-\int \frac {b+2\,a\,x}{\left (2\,b+a\,x\right )\,\left (b-a\,x\right )\,{\left (a\,x^2+b\,x-1\right )}^{1/4}} \,d x \]