Integrand size = 38, antiderivative size = 827 \[ \int \frac {\left (b+a x^4\right ) \sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2} \, dx=\frac {x \left (b^{3/2}-i \sqrt {a} b x^2+\sqrt {b} c x^2-a \sqrt {b} x^4-i \sqrt {a} c x^4+i a^{3/2} x^6-i b \sqrt {-b-c x^2+a x^4}-\sqrt {a} \sqrt {b} x^2 \sqrt {-b-c x^2+a x^4}-\frac {1}{2} i c x^2 \sqrt {-b-c x^2+a x^4}+i a x^4 \sqrt {-b-c x^2+a x^4}\right )}{\left (-b+a x^4\right ) \left (2 i b+2 \sqrt {a} \sqrt {b} x^2+i c x^2-2 i a x^4+2 \sqrt {b} \sqrt {-b-c x^2+a x^4}-2 i \sqrt {a} x^2 \sqrt {-b-c x^2+a x^4}\right )}-\frac {\arctan \left (\frac {\sqrt {-2 i \sqrt {a} \sqrt {b}-(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}+c} x}{i \sqrt {b}+\sqrt {a} x^2+\sqrt {-b-c x^2+a x^4}}\right )}{2 \sqrt {-2 i \sqrt {a} \sqrt {b}-(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}+c}}+\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{a} \sqrt [4]{b} \arctan \left (\frac {\sqrt {-2 i \sqrt {a} \sqrt {b}-(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}+c} x}{i \sqrt {b}+\sqrt {a} x^2+\sqrt {-b-c x^2+a x^4}}\right )}{\sqrt {c} \sqrt {-2 i \sqrt {a} \sqrt {b}-(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}+c}}-\frac {\arctan \left (\frac {\sqrt {-2 i \sqrt {a} \sqrt {b}+(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}+c} x}{i \sqrt {b}+\sqrt {a} x^2+\sqrt {-b-c x^2+a x^4}}\right )}{2 \sqrt {-2 i \sqrt {a} \sqrt {b}+(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}+c}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \sqrt [4]{a} \sqrt [4]{b} \arctan \left (\frac {\sqrt {-2 i \sqrt {a} \sqrt {b}+(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}+c} x}{i \sqrt {b}+\sqrt {a} x^2+\sqrt {-b-c x^2+a x^4}}\right )}{\sqrt {c} \sqrt {-2 i \sqrt {a} \sqrt {b}+(2-2 i) \sqrt [4]{a} \sqrt [4]{b} \sqrt {c}+c}} \]
x*(b^(3/2)-I*a^(1/2)*b*x^2+b^(1/2)*c*x^2-a*b^(1/2)*x^4-I*a^(1/2)*c*x^4+I*a ^(3/2)*x^6-I*b*(a*x^4-c*x^2-b)^(1/2)-a^(1/2)*b^(1/2)*x^2*(a*x^4-c*x^2-b)^( 1/2)-1/2*I*c*x^2*(a*x^4-c*x^2-b)^(1/2)+I*a*x^4*(a*x^4-c*x^2-b)^(1/2))/(a*x ^4-b)/(2*I*b+2*a^(1/2)*b^(1/2)*x^2+I*c*x^2-2*I*a*x^4+2*b^(1/2)*(a*x^4-c*x^ 2-b)^(1/2)-2*I*a^(1/2)*x^2*(a*x^4-c*x^2-b)^(1/2))-1/2*arctan((-2*I*a^(1/2) *b^(1/2)+(-2+2*I)*a^(1/4)*b^(1/4)*c^(1/2)+c)^(1/2)*x/(I*b^(1/2)+a^(1/2)*x^ 2+(a*x^4-c*x^2-b)^(1/2)))/(-2*I*a^(1/2)*b^(1/2)+(-2+2*I)*a^(1/4)*b^(1/4)*c ^(1/2)+c)^(1/2)+(1/2-1/2*I)*a^(1/4)*b^(1/4)*arctan((-2*I*a^(1/2)*b^(1/2)+( -2+2*I)*a^(1/4)*b^(1/4)*c^(1/2)+c)^(1/2)*x/(I*b^(1/2)+a^(1/2)*x^2+(a*x^4-c *x^2-b)^(1/2)))/c^(1/2)/(-2*I*a^(1/2)*b^(1/2)+(-2+2*I)*a^(1/4)*b^(1/4)*c^( 1/2)+c)^(1/2)-1/2*arctan((-2*I*a^(1/2)*b^(1/2)+(2-2*I)*a^(1/4)*b^(1/4)*c^( 1/2)+c)^(1/2)*x/(I*b^(1/2)+a^(1/2)*x^2+(a*x^4-c*x^2-b)^(1/2)))/(-2*I*a^(1/ 2)*b^(1/2)+(2-2*I)*a^(1/4)*b^(1/4)*c^(1/2)+c)^(1/2)+(-1/2+1/2*I)*a^(1/4)*b ^(1/4)*arctan((-2*I*a^(1/2)*b^(1/2)+(2-2*I)*a^(1/4)*b^(1/4)*c^(1/2)+c)^(1/ 2)*x/(I*b^(1/2)+a^(1/2)*x^2+(a*x^4-c*x^2-b)^(1/2)))/c^(1/2)/(-2*I*a^(1/2)* b^(1/2)+(2-2*I)*a^(1/4)*b^(1/4)*c^(1/2)+c)^(1/2)
Time = 0.64 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.09 \[ \int \frac {\left (b+a x^4\right ) \sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2} \, dx=-\frac {x \sqrt {-b-c x^2+a x^4}}{2 \left (-b+a x^4\right )}-\frac {\arctan \left (\frac {\sqrt {c} x}{\sqrt {-b-c x^2+a x^4}}\right )}{2 \sqrt {c}} \]
-1/2*(x*Sqrt[-b - c*x^2 + a*x^4])/(-b + a*x^4) - ArcTan[(Sqrt[c]*x)/Sqrt[- b - c*x^2 + a*x^4]]/(2*Sqrt[c])
Result contains higher order function than in optimal. Order 4 vs. order 3 in optimal.
Time = 4.15 (sec) , antiderivative size = 908, normalized size of antiderivative = 1.10, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, 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.
| \(\displaystyle \int \frac {\left (a x^4+b\right ) \sqrt {a x^4-b-c x^2}}{\left (a x^4-b\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {2 b \sqrt {a x^4-b-c x^2}}{\left (a x^4-b\right )^2}+\frac {\sqrt {a x^4-b-c x^2}}{a x^4-b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {a x^4-c x^2-b} x}{4 \sqrt {b} \left (\sqrt {b}-\sqrt {a} x^2\right )}+\frac {\sqrt {a x^4-c x^2-b} x}{4 \sqrt {b} \left (\sqrt {a} x^2+\sqrt {b}\right )}+\frac {\left (c+2 \sqrt {a} \sqrt {b}-\sqrt {c^2+4 a b}\right ) \sqrt {c+\sqrt {c^2+4 a b}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {c^2+4 a b}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {c^2+4 a b}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {c^2+4 a b}}}\right ),\frac {c+\sqrt {c^2+4 a b}}{c-\sqrt {c^2+4 a b}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {a x^4-c x^2-b}}+\frac {\left (-c+2 \sqrt {a} \sqrt {b}+\sqrt {c^2+4 a b}\right ) \sqrt {c+\sqrt {c^2+4 a b}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {c^2+4 a b}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {c^2+4 a b}}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {c^2+4 a b}}}\right ),\frac {c+\sqrt {c^2+4 a b}}{c-\sqrt {c^2+4 a b}}\right )}{8 \sqrt {2} a \sqrt {b} \sqrt {a x^4-c x^2-b}}-\frac {\sqrt {c+\sqrt {c^2+4 a b}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {c^2+4 a b}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {c^2+4 a b}}} \operatorname {EllipticPi}\left (-\frac {c+\sqrt {c^2+4 a b}}{2 \sqrt {a} \sqrt {b}},\arcsin \left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {c^2+4 a b}}}\right ),\frac {c+\sqrt {c^2+4 a b}}{c-\sqrt {c^2+4 a b}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a x^4-c x^2-b}}-\frac {\sqrt {c+\sqrt {c^2+4 a b}} \sqrt {1-\frac {2 a x^2}{c-\sqrt {c^2+4 a b}}} \sqrt {1-\frac {2 a x^2}{c+\sqrt {c^2+4 a b}}} \operatorname {EllipticPi}\left (\frac {c+\sqrt {c^2+4 a b}}{2 \sqrt {a} \sqrt {b}},\arcsin \left (\frac {\sqrt {2} \sqrt {a} x}{\sqrt {c+\sqrt {c^2+4 a b}}}\right ),\frac {c+\sqrt {c^2+4 a b}}{c-\sqrt {c^2+4 a b}}\right )}{2 \sqrt {2} \sqrt {a} \sqrt {a x^4-c x^2-b}}\) |
(x*Sqrt[-b - c*x^2 + a*x^4])/(4*Sqrt[b]*(Sqrt[b] - Sqrt[a]*x^2)) + (x*Sqrt [-b - c*x^2 + a*x^4])/(4*Sqrt[b]*(Sqrt[b] + Sqrt[a]*x^2)) + ((2*Sqrt[a]*Sq rt[b] + c - Sqrt[4*a*b + c^2])*Sqrt[c + Sqrt[4*a*b + c^2]]*Sqrt[1 - (2*a*x ^2)/(c - Sqrt[4*a*b + c^2])]*Sqrt[1 - (2*a*x^2)/(c + Sqrt[4*a*b + c^2])]*E llipticF[ArcSin[(Sqrt[2]*Sqrt[a]*x)/Sqrt[c + Sqrt[4*a*b + c^2]]], (c + Sqr t[4*a*b + c^2])/(c - Sqrt[4*a*b + c^2])])/(8*Sqrt[2]*a*Sqrt[b]*Sqrt[-b - c *x^2 + a*x^4]) + ((2*Sqrt[a]*Sqrt[b] - c + Sqrt[4*a*b + c^2])*Sqrt[c + Sqr t[4*a*b + c^2]]*Sqrt[1 - (2*a*x^2)/(c - Sqrt[4*a*b + c^2])]*Sqrt[1 - (2*a* x^2)/(c + Sqrt[4*a*b + c^2])]*EllipticF[ArcSin[(Sqrt[2]*Sqrt[a]*x)/Sqrt[c + Sqrt[4*a*b + c^2]]], (c + Sqrt[4*a*b + c^2])/(c - Sqrt[4*a*b + c^2])])/( 8*Sqrt[2]*a*Sqrt[b]*Sqrt[-b - c*x^2 + a*x^4]) - (Sqrt[c + Sqrt[4*a*b + c^2 ]]*Sqrt[1 - (2*a*x^2)/(c - Sqrt[4*a*b + c^2])]*Sqrt[1 - (2*a*x^2)/(c + Sqr t[4*a*b + c^2])]*EllipticPi[-1/2*(c + Sqrt[4*a*b + c^2])/(Sqrt[a]*Sqrt[b]) , ArcSin[(Sqrt[2]*Sqrt[a]*x)/Sqrt[c + Sqrt[4*a*b + c^2]]], (c + Sqrt[4*a*b + c^2])/(c - Sqrt[4*a*b + c^2])])/(2*Sqrt[2]*Sqrt[a]*Sqrt[-b - c*x^2 + a* x^4]) - (Sqrt[c + Sqrt[4*a*b + c^2]]*Sqrt[1 - (2*a*x^2)/(c - Sqrt[4*a*b + c^2])]*Sqrt[1 - (2*a*x^2)/(c + Sqrt[4*a*b + c^2])]*EllipticPi[(c + Sqrt[4* a*b + c^2])/(2*Sqrt[a]*Sqrt[b]), ArcSin[(Sqrt[2]*Sqrt[a]*x)/Sqrt[c + Sqrt[ 4*a*b + c^2]]], (c + Sqrt[4*a*b + c^2])/(c - Sqrt[4*a*b + c^2])])/(2*Sqrt[ 2]*Sqrt[a]*Sqrt[-b - c*x^2 + a*x^4])
3.32.33.3.1 Defintions of rubi rules used
Time = 2.22 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.08
| method | result | size |
| default | \(-\frac {\sqrt {a \,x^{4}-c \,x^{2}-b}\, x}{2 a \,x^{4}-2 b}+\frac {\arctan \left (\frac {\sqrt {a \,x^{4}-c \,x^{2}-b}}{x \sqrt {c}}\right )}{2 \sqrt {c}}\) | \(64\) |
| pseudoelliptic | \(-\frac {\sqrt {a \,x^{4}-c \,x^{2}-b}\, x}{2 a \,x^{4}-2 b}+\frac {\arctan \left (\frac {\sqrt {a \,x^{4}-c \,x^{2}-b}}{x \sqrt {c}}\right )}{2 \sqrt {c}}\) | \(64\) |
| elliptic | \(\frac {\left (-\frac {\sqrt {a \,x^{4}-c \,x^{2}-b}\, \sqrt {2}}{4 x \left (\frac {a \,x^{4}-c \,x^{2}-b}{2 x^{2}}+\frac {c}{2}\right )}+\frac {\sqrt {2}\, \arctan \left (\frac {\sqrt {a \,x^{4}-c \,x^{2}-b}}{x \sqrt {c}}\right )}{2 \sqrt {c}}\right ) \sqrt {2}}{2}\) | \(91\) |
Time = 0.30 (sec) , antiderivative size = 246, normalized size of antiderivative = 0.30 \[ \int \frac {\left (b+a x^4\right ) \sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2} \, dx=\left [-\frac {4 \, \sqrt {a x^{4} - c x^{2} - b} c x + {\left (a x^{4} - b\right )} \sqrt {-c} \log \left (-\frac {a^{2} x^{8} - 8 \, a c x^{6} - 2 \, {\left (a b - 4 \, c^{2}\right )} x^{4} + 8 \, b c x^{2} + b^{2} - 4 \, {\left (a x^{5} - 2 \, c x^{3} - b x\right )} \sqrt {a x^{4} - c x^{2} - b} \sqrt {-c}}{a^{2} x^{8} - 2 \, a b x^{4} + b^{2}}\right )}{8 \, {\left (a c x^{4} - b c\right )}}, -\frac {2 \, \sqrt {a x^{4} - c x^{2} - b} c x + {\left (a x^{4} - b\right )} \sqrt {c} \arctan \left (\frac {2 \, \sqrt {a x^{4} - c x^{2} - b} \sqrt {c} x}{a x^{4} - 2 \, c x^{2} - b}\right )}{4 \, {\left (a c x^{4} - b c\right )}}\right ] \]
[-1/8*(4*sqrt(a*x^4 - c*x^2 - b)*c*x + (a*x^4 - b)*sqrt(-c)*log(-(a^2*x^8 - 8*a*c*x^6 - 2*(a*b - 4*c^2)*x^4 + 8*b*c*x^2 + b^2 - 4*(a*x^5 - 2*c*x^3 - b*x)*sqrt(a*x^4 - c*x^2 - b)*sqrt(-c))/(a^2*x^8 - 2*a*b*x^4 + b^2)))/(a*c *x^4 - b*c), -1/4*(2*sqrt(a*x^4 - c*x^2 - b)*c*x + (a*x^4 - b)*sqrt(c)*arc tan(2*sqrt(a*x^4 - c*x^2 - b)*sqrt(c)*x/(a*x^4 - 2*c*x^2 - b)))/(a*c*x^4 - b*c)]
\[ \int \frac {\left (b+a x^4\right ) \sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2} \, dx=\int \frac {\left (a x^{4} + b\right ) \sqrt {a x^{4} - b - c x^{2}}}{\left (a x^{4} - b\right )^{2}}\, dx \]
\[ \int \frac {\left (b+a x^4\right ) \sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2} \, dx=\int { \frac {\sqrt {a x^{4} - c x^{2} - b} {\left (a x^{4} + b\right )}}{{\left (a x^{4} - b\right )}^{2}} \,d x } \]
\[ \int \frac {\left (b+a x^4\right ) \sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2} \, dx=\int { \frac {\sqrt {a x^{4} - c x^{2} - b} {\left (a x^{4} + b\right )}}{{\left (a x^{4} - b\right )}^{2}} \,d x } \]
Timed out. \[ \int \frac {\left (b+a x^4\right ) \sqrt {-b-c x^2+a x^4}}{\left (-b+a x^4\right )^2} \, dx=\int \frac {\left (a\,x^4+b\right )\,\sqrt {a\,x^4-c\,x^2-b}}{{\left (b-a\,x^4\right )}^2} \,d x \]