Integrand size = 22, antiderivative size = 1006 \[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^2} \, dx=\frac {e \left (b c d^2+b^2 e^2-2 a c e^2+c \left (2 c d^2+b e^2\right ) x^2\right )}{2 \left (b^2-4 a c\right ) \left (c d^4+b d^2 e^2+a e^4\right ) \left (a+b x^2+c x^4\right )}-\frac {d x \left (a b c e^2-\left (b^2-2 a c\right ) \left (c d^2+b e^2\right )-c \left (b c d^2+b^2 e^2-2 a c e^2\right ) x^2\right )}{2 a \left (b^2-4 a c\right ) \left (c d^4+b d^2 e^2+a e^4\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {c} d e^4 \left (e^2+\frac {2 c d^2+b e^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^4+b d^2 e^2+a e^4\right )^2}+\frac {\sqrt {c} d \left (b^3 e^2+b c \left (\sqrt {b^2-4 a c} d^2-8 a e^2\right )+b^2 \left (c d^2+\sqrt {b^2-4 a c} e^2\right )-2 a c \left (6 c d^2+\sqrt {b^2-4 a c} e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^4+b d^2 e^2+a e^4\right )}+\frac {\sqrt {c} d e^4 \left (e^2-\frac {2 c d^2+b e^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^4+b d^2 e^2+a e^4\right )^2}+\frac {\sqrt {c} d \left (b c d^2+b^2 e^2-2 a c e^2-\frac {b^2 c d^2-12 a c^2 d^2+b^3 e^2-8 a b c e^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^4+b d^2 e^2+a e^4\right )}+\frac {e^5 \left (2 c d^2+b e^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt {b^2-4 a c} \left (c d^4+b d^2 e^2+a e^4\right )^2}-\frac {c e \left (2 c d^2+b e^2\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^4+b d^2 e^2+a e^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+b d^2 e^2+a e^4\right )^2}-\frac {e^7 \log \left (a+b x^2+c x^4\right )}{4 \left (c d^4+b d^2 e^2+a e^4\right )^2} \] Output:
1/2*e*(b*c*d^2+b^2*e^2-2*a*c*e^2+c*(b*e^2+2*c*d^2)*x^2)/(-4*a*c+b^2)/(a*e^ 4+b*d^2*e^2+c*d^4)/(c*x^4+b*x^2+a)-1/2*d*x*(a*b*c*e^2-(-2*a*c+b^2)*(b*e^2+ c*d^2)-c*(-2*a*c*e^2+b^2*e^2+b*c*d^2)*x^2)/a/(-4*a*c+b^2)/(a*e^4+b*d^2*e^2 +c*d^4)/(c*x^4+b*x^2+a)+1/2*c^(1/2)*d*e^4*(e^2+(b*e^2+2*c*d^2)/(-4*a*c+b^2 )^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)/(b -(-4*a*c+b^2)^(1/2))^(1/2)/(a*e^4+b*d^2*e^2+c*d^4)^2+1/4*c^(1/2)*d*(b^3*e^ 2+b*c*((-4*a*c+b^2)^(1/2)*d^2-8*a*e^2)+b^2*(c*d^2+(-4*a*c+b^2)^(1/2)*e^2)- 2*a*c*(6*c*d^2+(-4*a*c+b^2)^(1/2)*e^2))*arctan(2^(1/2)*c^(1/2)*x/(b-(-4*a* c+b^2)^(1/2))^(1/2))*2^(1/2)/a/(-4*a*c+b^2)^(3/2)/(b-(-4*a*c+b^2)^(1/2))^( 1/2)/(a*e^4+b*d^2*e^2+c*d^4)+1/2*c^(1/2)*d*e^4*(e^2-(b*e^2+2*c*d^2)/(-4*a* c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/ 2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)/(a*e^4+b*d^2*e^2+c*d^4)^2+1/4*c^(1/2)*d*(b *c*d^2+b^2*e^2-2*a*c*e^2-(-8*a*b*c*e^2-12*a*c^2*d^2+b^3*e^2+b^2*c*d^2)/(-4 *a*c+b^2)^(1/2))*arctan(2^(1/2)*c^(1/2)*x/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^ (1/2)/a/(-4*a*c+b^2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)/(a*e^4+b*d^2*e^2+c*d^4)+ 1/2*e^5*(b*e^2+2*c*d^2)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/(-4*a*c+b^ 2)^(1/2)/(a*e^4+b*d^2*e^2+c*d^4)^2-c*e*(b*e^2+2*c*d^2)*arctanh((2*c*x^2+b) /(-4*a*c+b^2)^(1/2))/(-4*a*c+b^2)^(3/2)/(a*e^4+b*d^2*e^2+c*d^4)+e^7*ln(e*x +d)/(a*e^4+b*d^2*e^2+c*d^4)^2-1/4*e^7*ln(c*x^4+b*x^2+a)/(a*e^4+b*d^2*e^2+c *d^4)^2
Time = 5.35 (sec) , antiderivative size = 1031, normalized size of antiderivative = 1.02 \[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^2} \, dx=\frac {-\frac {2 \left (c d^4+b d^2 e^2+a e^4\right ) \left (-2 a^2 c e^3+b d \left (c d^2+b e^2\right ) x \left (b+c x^2\right )+a \left (b^2 e^3+b c e \left (d^2-3 d e x+e^2 x^2\right )-2 c^2 d x \left (d^2-d e x+e^2 x^2\right )\right )\right )}{a \left (-b^2+4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\sqrt {2} \sqrt {c} d \left (-b^4 d^2 e^4-b^3 \left (2 c d^4 e^2+\sqrt {b^2-4 a c} d^2 e^4+3 a e^6\right )+b c \left (-c \sqrt {b^2-4 a c} d^6+20 a c d^4 e^2+a \sqrt {b^2-4 a c} d^2 e^4+16 a^2 e^6\right )-b^2 \left (c^2 d^6+2 c \sqrt {b^2-4 a c} d^4 e^2-3 a c d^2 e^4+3 a \sqrt {b^2-4 a c} e^6\right )+2 a c \left (6 c^2 d^6+c \sqrt {b^2-4 a c} d^4 e^2+14 a c d^2 e^4+5 a \sqrt {b^2-4 a c} e^6\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} d \left (-b^4 d^2 e^4+b^3 \left (-2 c d^4 e^2+\sqrt {b^2-4 a c} d^2 e^4-3 a e^6\right )+b c \left (c \sqrt {b^2-4 a c} d^6+20 a c d^4 e^2-a \sqrt {b^2-4 a c} d^2 e^4+16 a^2 e^6\right )+2 a c \left (6 c^2 d^6-c \sqrt {b^2-4 a c} d^4 e^2+14 a c d^2 e^4-5 a \sqrt {b^2-4 a c} e^6\right )+b^2 \left (-c^2 d^6+2 c \sqrt {b^2-4 a c} d^4 e^2+3 a c d^2 e^4+3 a \sqrt {b^2-4 a c} e^6\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{a \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}+4 e^7 \log (d+e x)+\frac {\left (4 c^3 d^6 e-b^2 \left (b+\sqrt {b^2-4 a c}\right ) e^7+2 a c \left (3 b+2 \sqrt {b^2-4 a c}\right ) e^7+6 c^2 \left (b d^4 e^3+2 a d^2 e^5\right )\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {\left (4 c^3 d^6 e+2 a c \left (3 b-2 \sqrt {b^2-4 a c}\right ) e^7+b^2 \left (-b+\sqrt {b^2-4 a c}\right ) e^7+6 c^2 \left (b d^4 e^3+2 a d^2 e^5\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}}{4 \left (c d^4+b d^2 e^2+a e^4\right )^2} \] Input:
Integrate[1/((d + e*x)*(a + b*x^2 + c*x^4)^2),x]
Output:
((-2*(c*d^4 + b*d^2*e^2 + a*e^4)*(-2*a^2*c*e^3 + b*d*(c*d^2 + b*e^2)*x*(b + c*x^2) + a*(b^2*e^3 + b*c*e*(d^2 - 3*d*e*x + e^2*x^2) - 2*c^2*d*x*(d^2 - d*e*x + e^2*x^2))))/(a*(-b^2 + 4*a*c)*(a + b*x^2 + c*x^4)) - (Sqrt[2]*Sqr t[c]*d*(-(b^4*d^2*e^4) - b^3*(2*c*d^4*e^2 + Sqrt[b^2 - 4*a*c]*d^2*e^4 + 3* a*e^6) + b*c*(-(c*Sqrt[b^2 - 4*a*c]*d^6) + 20*a*c*d^4*e^2 + a*Sqrt[b^2 - 4 *a*c]*d^2*e^4 + 16*a^2*e^6) - b^2*(c^2*d^6 + 2*c*Sqrt[b^2 - 4*a*c]*d^4*e^2 - 3*a*c*d^2*e^4 + 3*a*Sqrt[b^2 - 4*a*c]*e^6) + 2*a*c*(6*c^2*d^6 + c*Sqrt[ b^2 - 4*a*c]*d^4*e^2 + 14*a*c*d^2*e^4 + 5*a*Sqrt[b^2 - 4*a*c]*e^6))*ArcTan [(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)* Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*d*(-(b^4*d^2*e^4) + b^3*(- 2*c*d^4*e^2 + Sqrt[b^2 - 4*a*c]*d^2*e^4 - 3*a*e^6) + b*c*(c*Sqrt[b^2 - 4*a *c]*d^6 + 20*a*c*d^4*e^2 - a*Sqrt[b^2 - 4*a*c]*d^2*e^4 + 16*a^2*e^6) + 2*a *c*(6*c^2*d^6 - c*Sqrt[b^2 - 4*a*c]*d^4*e^2 + 14*a*c*d^2*e^4 - 5*a*Sqrt[b^ 2 - 4*a*c]*e^6) + b^2*(-(c^2*d^6) + 2*c*Sqrt[b^2 - 4*a*c]*d^4*e^2 + 3*a*c* d^2*e^4 + 3*a*Sqrt[b^2 - 4*a*c]*e^6))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(a*(b^2 - 4*a*c)^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + 4*e^7*Log[d + e*x] + ((4*c^3*d^6*e - b^2*(b + Sqrt[b^2 - 4*a*c])*e^7 + 2* a*c*(3*b + 2*Sqrt[b^2 - 4*a*c])*e^7 + 6*c^2*(b*d^4*e^3 + 2*a*d^2*e^5))*Log [-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/(b^2 - 4*a*c)^(3/2) - ((4*c^3*d^6*e + 2*a*c*(3*b - 2*Sqrt[b^2 - 4*a*c])*e^7 + b^2*(-b + Sqrt[b^2 - 4*a*c])*e^...
Time = 5.06 (sec) , antiderivative size = 1006, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, 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.
| \(\displaystyle \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {e^4 (e x-d) \left (b e^2+c d^2+c e^2 x^2\right )}{\left (a+b x^2+c x^4\right ) \left (a e^4+b d^2 e^2+c d^4\right )^2}+\frac {(d-e x) \left (b e^2+c d^2+c e^2 x^2\right )}{\left (a+b x^2+c x^4\right )^2 \left (a e^4+b d^2 e^2+c d^4\right )}+\frac {e^8}{(d+e x) \left (a e^4+b d^2 e^2+c d^4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log (d+e x) e^7}{\left (c d^4+b e^2 d^2+a e^4\right )^2}-\frac {\log \left (c x^4+b x^2+a\right ) e^7}{4 \left (c d^4+b e^2 d^2+a e^4\right )^2}+\frac {\left (2 c d^2+b e^2\right ) \text {arctanh}\left (\frac {2 c x^2+b}{\sqrt {b^2-4 a c}}\right ) e^5}{2 \sqrt {b^2-4 a c} \left (c d^4+b e^2 d^2+a e^4\right )^2}+\frac {\sqrt {c} d \left (e^2+\frac {2 c d^2+b e^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) e^4}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^4+b e^2 d^2+a e^4\right )^2}+\frac {\sqrt {c} d \left (e^2-\frac {2 c d^2+b e^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right ) e^4}{\sqrt {2} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^4+b e^2 d^2+a e^4\right )^2}-\frac {c \left (2 c d^2+b e^2\right ) \text {arctanh}\left (\frac {2 c x^2+b}{\sqrt {b^2-4 a c}}\right ) e}{\left (b^2-4 a c\right )^{3/2} \left (c d^4+b e^2 d^2+a e^4\right )}+\frac {\left (b c d^2+b^2 e^2-2 a c e^2+c \left (2 c d^2+b e^2\right ) x^2\right ) e}{2 \left (b^2-4 a c\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \left (c x^4+b x^2+a\right )}+\frac {\sqrt {c} d \left (e^2 b^3+\left (c d^2+\sqrt {b^2-4 a c} e^2\right ) b^2+c \left (\sqrt {b^2-4 a c} d^2-8 a e^2\right ) b-2 a c \left (6 c d^2+\sqrt {b^2-4 a c} e^2\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^4+b e^2 d^2+a e^4\right )}+\frac {\sqrt {c} d \left (b c d^2+b^2 e^2-2 a c e^2-\frac {e^2 b^3+c d^2 b^2-8 a c e^2 b-12 a c^2 d^2}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a \left (b^2-4 a c\right ) \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^4+b e^2 d^2+a e^4\right )}-\frac {d x \left (a b c e^2-c \left (b c d^2+b^2 e^2-2 a c e^2\right ) x^2-\left (b^2-2 a c\right ) \left (c d^2+b e^2\right )\right )}{2 a \left (b^2-4 a c\right ) \left (c d^4+b e^2 d^2+a e^4\right ) \left (c x^4+b x^2+a\right )}\) |
Input:
Int[1/((d + e*x)*(a + b*x^2 + c*x^4)^2),x]
Output:
(e*(b*c*d^2 + b^2*e^2 - 2*a*c*e^2 + c*(2*c*d^2 + b*e^2)*x^2))/(2*(b^2 - 4* a*c)*(c*d^4 + b*d^2*e^2 + a*e^4)*(a + b*x^2 + c*x^4)) - (d*x*(a*b*c*e^2 - (b^2 - 2*a*c)*(c*d^2 + b*e^2) - c*(b*c*d^2 + b^2*e^2 - 2*a*c*e^2)*x^2))/(2 *a*(b^2 - 4*a*c)*(c*d^4 + b*d^2*e^2 + a*e^4)*(a + b*x^2 + c*x^4)) + (Sqrt[ c]*d*e^4*(e^2 + (2*c*d^2 + b*e^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[ c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*( c*d^4 + b*d^2*e^2 + a*e^4)^2) + (Sqrt[c]*d*(b^3*e^2 + b*c*(Sqrt[b^2 - 4*a* c]*d^2 - 8*a*e^2) + b^2*(c*d^2 + Sqrt[b^2 - 4*a*c]*e^2) - 2*a*c*(6*c*d^2 + Sqrt[b^2 - 4*a*c]*e^2))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4* a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^ 4 + b*d^2*e^2 + a*e^4)) + (Sqrt[c]*d*e^4*(e^2 - (2*c*d^2 + b*e^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[ 2]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^4 + b*d^2*e^2 + a*e^4)^2) + (Sqrt[c]*d *(b*c*d^2 + b^2*e^2 - 2*a*c*e^2 - (b^2*c*d^2 - 12*a*c^2*d^2 + b^3*e^2 - 8* a*b*c*e^2)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*a*(b^2 - 4*a*c)*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^4 + b*d^2*e^2 + a*e^4)) + (e^5*(2*c*d^2 + b*e^2)*ArcTanh[(b + 2*c*x^2)/Sqrt [b^2 - 4*a*c]])/(2*Sqrt[b^2 - 4*a*c]*(c*d^4 + b*d^2*e^2 + a*e^4)^2) - (c*e *(2*c*d^2 + b*e^2)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/((b^2 - 4*a*c )^(3/2)*(c*d^4 + b*d^2*e^2 + a*e^4)) + (e^7*Log[d + e*x])/(c*d^4 + b*d^...
Time = 0.42 (sec) , antiderivative size = 1568, normalized size of antiderivative = 1.56
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1568\) |
| risch | \(\text {Expression too large to display}\) | \(7157\) |
Input:
int(1/(e*x+d)/(c*x^4+b*x^2+a)^2,x,method=_RETURNVERBOSE)
Output:
1/(a*e^4+b*d^2*e^2+c*d^4)^2*((1/2*c*d*(2*a^2*c*e^6-a*b^2*e^6+a*b*c*d^2*e^4 +2*a*c^2*d^4*e^2-b^3*d^2*e^4-2*b^2*c*d^4*e^2-b*c^2*d^6)/a/(4*a*c-b^2)*x^3- 1/2*c*e*(a*b*e^6+2*a*c*d^2*e^4+b^2*d^2*e^4+3*b*c*d^4*e^2+2*c^2*d^6)/(4*a*c -b^2)*x^2+1/2*d*(3*a^2*b*c*e^6+2*a^2*c^2*d^2*e^4-a*b^3*e^6+2*a*b^2*c*d^2*e ^4+5*a*b*c^2*d^4*e^2+2*a*c^3*d^6-b^4*d^2*e^4-2*b^3*c*d^4*e^2-b^2*c^2*d^6)/ a/(4*a*c-b^2)*x+1/2*e*(2*a^2*c*e^6-a*b^2*e^6+a*b*c*d^2*e^4+2*a*c^2*d^4*e^2 -b^3*d^2*e^4-2*b^2*c*d^4*e^2-b*c^2*d^6)/(4*a*c-b^2))/(c*x^4+b*x^2+a)+2/a/( 4*a*c-b^2)*c*(1/(16*a*c-4*b^2)*(-1/4*(-12*(-4*a*c+b^2)^(1/2)*a^2*b*c*e^7-2 4*(-4*a*c+b^2)^(1/2)*a^2*c^2*d^2*e^5+2*(-4*a*c+b^2)^(1/2)*a*b^3*e^7-12*(-4 *a*c+b^2)^(1/2)*a*b*c^2*d^4*e^3-8*(-4*a*c+b^2)^(1/2)*a*c^3*d^6*e+32*a^3*e^ 7*c^2-16*a^2*e^7*c*b^2+2*a*e^7*b^4)/c*ln(-2*c*x^2+(-4*a*c+b^2)^(1/2)-b)+1/ 2*(16*(-4*a*c+b^2)^(1/2)*a^2*b*c*d*e^6+28*(-4*a*c+b^2)^(1/2)*a^2*c^2*d^3*e ^4-3*(-4*a*c+b^2)^(1/2)*a*b^3*d*e^6+3*(-4*a*c+b^2)^(1/2)*a*b^2*c*d^3*e^4+2 0*(-4*a*c+b^2)^(1/2)*a*b*c^2*d^5*e^2+12*(-4*a*c+b^2)^(1/2)*a*c^3*d^7-(-4*a *c+b^2)^(1/2)*b^4*d^3*e^4-2*(-4*a*c+b^2)^(1/2)*b^3*c*d^5*e^2-(-4*a*c+b^2)^ (1/2)*b^2*c^2*d^7-40*a^3*c^2*d*e^6+22*a^2*b^2*c*d*e^6-4*a^2*b*c^2*d^3*e^4- 8*a^2*c^3*d^5*e^2-3*a*b^4*d*e^6+5*a*b^3*c*d^3*e^4+10*a*b^2*c^2*d^5*e^2+4*a *b*c^3*d^7-b^5*d^3*e^4-2*b^4*c*d^5*e^2-b^3*c^2*d^7)*2^(1/2)/((-b+(-4*a*c+b ^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)) )+1/(16*a*c-4*b^2)*(1/4*(-12*(-4*a*c+b^2)^(1/2)*a^2*b*c*e^7-24*(-4*a*c+...
Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x+d)/(c*x**4+b*x**2+a)**2,x)
Output:
Timed out
\[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {1}{{\left (c x^{4} + b x^{2} + a\right )}^{2} {\left (e x + d\right )}} \,d x } \] Input:
integrate(1/(e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm="maxima")
Output:
e^7*log(e*x + d)/(c^2*d^8 + 2*b*c*d^6*e^2 + 2*a*b*d^2*e^6 + a^2*e^8 + (b^2 + 2*a*c)*d^4*e^4) + 1/2*(a*b*c*d^2*e + (a*b^2 - 2*a^2*c)*e^3 + (b*c^2*d^3 + (b^2*c - 2*a*c^2)*d*e^2)*x^3 + (2*a*c^2*d^2*e + a*b*c*e^3)*x^2 + ((b^2* c - 2*a*c^2)*d^3 + (b^3 - 3*a*b*c)*d*e^2)*x)/((a^2*b^2*c - 4*a^3*c^2)*d^4 + (a^2*b^3 - 4*a^3*b*c)*d^2*e^2 + (a^3*b^2 - 4*a^4*c)*e^4 + ((a*b^2*c^2 - 4*a^2*c^3)*d^4 + (a*b^3*c - 4*a^2*b*c^2)*d^2*e^2 + (a^2*b^2*c - 4*a^3*c^2) *e^4)*x^4 + ((a*b^3*c - 4*a^2*b*c^2)*d^4 + (a*b^4 - 4*a^2*b^2*c)*d^2*e^2 + (a^2*b^3 - 4*a^3*b*c)*e^4)*x^2) + 1/2*integrate(-(2*(a*b^2*c - 4*a^2*c^2) *e^7*x^3 - (b^2*c^2 - 6*a*c^3)*d^7 - (2*b^3*c - 11*a*b*c^2)*d^5*e^2 - (b^4 - 2*a*b^2*c - 14*a^2*c^2)*d^3*e^4 - (3*a*b^3 - 13*a^2*b*c)*d*e^6 - (b*c^3 *d^7 + 2*(b^2*c^2 - a*c^3)*d^5*e^2 + (b^3*c - a*b*c^2)*d^3*e^4 + (3*a*b^2* c - 10*a^2*c^2)*d*e^6)*x^2 - 2*(2*a*c^3*d^6*e + 3*a*b*c^2*d^4*e^3 + 6*a^2* c^2*d^2*e^5 - (a*b^3 - 5*a^2*b*c)*e^7)*x)/(c*x^4 + b*x^2 + a), x)/((a*b^2* c^2 - 4*a^2*c^3)*d^8 + 2*(a*b^3*c - 4*a^2*b*c^2)*d^6*e^2 + (a*b^4 - 2*a^2* b^2*c - 8*a^3*c^2)*d^4*e^4 + 2*(a^2*b^3 - 4*a^3*b*c)*d^2*e^6 + (a^3*b^2 - 4*a^4*c)*e^8)
Leaf count of result is larger than twice the leaf count of optimal. 130635 vs. \(2 (912) = 1824\).
Time = 15.97 (sec) , antiderivative size = 130635, normalized size of antiderivative = 129.86 \[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)/(c*x^4+b*x^2+a)^2,x, algorithm="giac")
Output:
e^8*log(abs(e*x + d))/(c^2*d^8*e + 2*b*c*d^6*e^3 + b^2*d^4*e^5 + 2*a*c*d^4 *e^5 + 2*a*b*d^2*e^7 + a^2*e^9) - 1/4*e^7*log(abs(c*x^4 + b*x^2 + a))/(c^2 *d^8 + 2*b*c*d^6*e^2 + b^2*d^4*e^4 + 2*a*c*d^4*e^4 + 2*a*b*d^2*e^6 + a^2*e ^8) + 1/16*((2*a^4*b^11*c^14 - 56*a^5*b^9*c^15 + 576*a^6*b^7*c^16 - 2816*a ^7*b^5*c^17 + 6656*a^8*b^3*c^18 - 6144*a^9*b*c^19 - sqrt(2)*sqrt(b^2 - 4*a *c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^11*c^12 + 28*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^9*c^13 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^10*c^13 - 288*sqrt(2)*sqrt(b ^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^7*c^14 - 48*sqrt(2)*sqrt (b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^8*c^14 - sqrt(2)*sqrt( b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^4*b^9*c^14 + 1408*sqrt(2)*s qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^5*c^15 + 384*sqrt(2 )*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^6*c^15 + 24*sqrt (2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^5*b^7*c^15 - 3328* sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8*b^3*c^16 - 1 280*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^7*b^4*c^16 - 192*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^6*b^5*c ^16 + 3072*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^9*b *c^17 + 1536*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^8 *b^2*c^17 + 640*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*...
Time = 28.71 (sec) , antiderivative size = 27673, normalized size of antiderivative = 27.51 \[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \] Input:
int(1/((d + e*x)*(a + b*x^2 + c*x^4)^2),x)
Output:
symsum(log(root(12582912*a^11*b*c^7*d^6*e^10*z^5 + 12582912*a^10*b*c^8*d^1 0*e^6*z^5 + 4194304*a^12*b*c^6*d^2*e^14*z^5 + 4194304*a^9*b*c^9*d^14*e^2*z ^5 - 35840*a^6*b^12*c*d^4*e^12*z^5 - 24576*a^7*b^11*c*d^2*e^14*z^5 - 21504 *a^5*b^13*c*d^6*e^10*z^5 - 3072*a^4*b^14*c*d^8*e^8*z^5 + 1024*a^3*b^15*c*d ^10*e^6*z^5 - 14680064*a^10*b^3*c^6*d^6*e^10*z^5 - 14680064*a^9*b^3*c^7*d^ 10*e^6*z^5 - 11927552*a^9*b^4*c^6*d^8*e^8*z^5 + 8257536*a^8*b^6*c^5*d^8*e^ 8*z^5 - 6291456*a^11*b^3*c^5*d^2*e^14*z^5 - 6291456*a^8*b^3*c^8*d^14*e^2*z ^5 - 5505024*a^10*b^4*c^5*d^4*e^12*z^5 + 5505024*a^9*b^5*c^5*d^6*e^10*z^5 + 5505024*a^8*b^5*c^6*d^10*e^6*z^5 - 5505024*a^8*b^4*c^7*d^12*e^4*z^5 + 45 87520*a^9*b^6*c^4*d^4*e^12*z^5 + 4587520*a^7*b^6*c^6*d^12*e^4*z^5 + 393216 0*a^10*b^5*c^4*d^2*e^14*z^5 + 3932160*a^7*b^5*c^7*d^14*e^2*z^5 + 3145728*a ^10*b^2*c^7*d^8*e^8*z^5 - 2580480*a^7*b^8*c^4*d^8*e^8*z^5 - 1720320*a^8*b^ 8*c^3*d^4*e^12*z^5 - 1720320*a^6*b^8*c^5*d^12*e^4*z^5 - 1310720*a^9*b^7*c^ 3*d^2*e^14*z^5 - 1310720*a^6*b^7*c^6*d^14*e^2*z^5 - 573440*a^7*b^9*c^3*d^6 *e^10*z^5 - 573440*a^6*b^9*c^4*d^10*e^6*z^5 + 372736*a^6*b^10*c^3*d^8*e^8* z^5 + 344064*a^7*b^10*c^2*d^4*e^12*z^5 + 344064*a^5*b^10*c^4*d^12*e^4*z^5 + 245760*a^8*b^9*c^2*d^2*e^14*z^5 + 245760*a^5*b^9*c^5*d^14*e^2*z^5 + 1720 32*a^6*b^11*c^2*d^6*e^10*z^5 + 172032*a^5*b^11*c^3*d^10*e^6*z^5 - 35840*a^ 4*b^12*c^3*d^12*e^4*z^5 - 24576*a^4*b^11*c^4*d^14*e^2*z^5 - 21504*a^4*b^13 *c^2*d^10*e^6*z^5 - 10752*a^5*b^12*c^2*d^8*e^8*z^5 + 1536*a^3*b^14*c^2*...
Time = 4.33 (sec) , antiderivative size = 17358, normalized size of antiderivative = 17.25 \[ \int \frac {1}{(d+e x) \left (a+b x^2+c x^4\right )^2} \, dx =\text {Too large to display} \] Input:
int(1/(e*x+d)/(c*x^4+b*x^2+a)^2,x)
Output:
(24*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*s qrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**4*b**2* c*e**7 + 48*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan(( sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a* *4*b*c**2*d**2*e**5 - 4*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt (a) + b))*a**3*b**4*e**7 + 24*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*s qrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c )*sqrt(a) + b))*a**3*b**3*c*e**7*x**2 + 24*sqrt(2*sqrt(c)*sqrt(a) + b)*sqr t(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/ sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b**2*c**2*d**4*e**3 + 48*sqrt(2*sqrt(c)* sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b**2*c**2*d**2*e**5*x**2 + 24*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2 *sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a) + b))*a**3*b** 2*c**2*e**7*x**4 + 16*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c)*sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt(c)*sqrt(a ) + b))*a**3*b*c**3*d**6*e + 48*sqrt(2*sqrt(c)*sqrt(a) + b)*sqrt(2*sqrt(c) *sqrt(a) - b)*atan((sqrt(2*sqrt(c)*sqrt(a) - b) - 2*sqrt(c)*x)/sqrt(2*sqrt (c)*sqrt(a) + b))*a**3*b*c**3*d**2*e**5*x**4 - 4*sqrt(2*sqrt(c)*sqrt(a)...