Integrand size = 24, antiderivative size = 429 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )} \, dx=\frac {e^2 x}{2 d \left (c d^2-b d e+a e^2\right ) \left (d+e x^2\right )}+\frac {\sqrt {c} \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}-\frac {\sqrt {c} \left (2 c^2 d^2+b \left (b-\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )^2}+\frac {e^{3/2} (2 c d-b e) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{\sqrt {d} \left (c d^2-b d e+a e^2\right )^2}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (c d^2-b d e+a e^2\right )} \]
1/2*e^2*x/d/(a*e^2-b*d*e+c*d^2)/(e*x^2+d)+1/2*e^(3/2)*arctan(x*e^(1/2)/d^( 1/2))/d^(3/2)/(a*e^2-b*d*e+c*d^2)+e^(3/2)*(-b*e+2*c*d)*arctan(x*e^(1/2)/d^ (1/2))/(a*e^2-b*d*e+c*d^2)^2/d^(1/2)+1/2*arctan(x*2^(1/2)*c^(1/2)/(b-(-4*a *c+b^2)^(1/2))^(1/2))*c^(1/2)*(2*c^2*d^2+b*e^2*(b+(-4*a*c+b^2)^(1/2))-2*c* e*(b*d+a*e+d*(-4*a*c+b^2)^(1/2)))/(a*e^2-b*d*e+c*d^2)^2*2^(1/2)/(-4*a*c+b^ 2)^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)-1/2*arctan(x*2^(1/2)*c^(1/2)/(b+(-4* a*c+b^2)^(1/2))^(1/2))*c^(1/2)*(2*c^2*d^2+b*e^2*(b-(-4*a*c+b^2)^(1/2))-2*c *e*(b*d+a*e-d*(-4*a*c+b^2)^(1/2)))/(a*e^2-b*d*e+c*d^2)^2*2^(1/2)/(-4*a*c+b ^2)^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.49 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.83 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )} \, dx=\frac {\frac {e^2 \left (c d^2+e (-b d+a e)\right ) x}{d \left (d+e x^2\right )}+\frac {\sqrt {2} \sqrt {c} \left (2 c^2 d^2+b \left (b+\sqrt {b^2-4 a c}\right ) e^2-2 c e \left (b d+\sqrt {b^2-4 a c} d+a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \sqrt {c} \left (-2 c^2 d^2+b \left (-b+\sqrt {b^2-4 a c}\right ) e^2+2 c e \left (b d-\sqrt {b^2-4 a c} d+a e\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {e^{3/2} \left (5 c d^2+e (-3 b d+a e)\right ) \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{d^{3/2}}}{2 \left (c d^2+e (-b d+a e)\right )^2} \]
((e^2*(c*d^2 + e*(-(b*d) + a*e))*x)/(d*(d + e*x^2)) + (Sqrt[2]*Sqrt[c]*(2* c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^ 2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*Sqrt[c]*(-2*c^2*d^2 + b *(-b + Sqrt[b^2 - 4*a*c])*e^2 + 2*c*e*(b*d - Sqrt[b^2 - 4*a*c]*d + a*e))*A rcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c] *Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (e^(3/2)*(5*c*d^2 + e*(-3*b*d + a*e))*ArcT an[(Sqrt[e]*x)/Sqrt[d]])/d^(3/2))/(2*(c*d^2 + e*(-(b*d) + a*e))^2)
Time = 1.19 (sec) , antiderivative size = 429, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {1484, 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.
\(\displaystyle \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )} \, dx\) |
\(\Big \downarrow \) 1484 |
\(\displaystyle \int \left (\frac {-c e (a e+2 b d)+b^2 e^2-c e x^2 (2 c d-b e)+c^2 d^2}{\left (a+b x^2+c x^4\right ) \left (a e^2-b d e+c d^2\right )^2}-\frac {e^2 (b e-2 c d)}{\left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )^2}+\frac {e^2}{\left (d+e x^2\right )^2 \left (a e^2-b d e+c d^2\right )}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (-2 c e \left (d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (\sqrt {b^2-4 a c}+b\right )+2 c^2 d^2\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} \left (a e^2-b d e+c d^2\right )^2}-\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-2 c e \left (-d \sqrt {b^2-4 a c}+a e+b d\right )+b e^2 \left (b-\sqrt {b^2-4 a c}\right )+2 c^2 d^2\right )}{\sqrt {2} \sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )^2}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right ) (2 c d-b e)}{\sqrt {d} \left (a e^2-b d e+c d^2\right )^2}+\frac {e^{3/2} \arctan \left (\frac {\sqrt {e} x}{\sqrt {d}}\right )}{2 d^{3/2} \left (a e^2-b d e+c d^2\right )}+\frac {e^2 x}{2 d \left (d+e x^2\right ) \left (a e^2-b d e+c d^2\right )}\) |
(e^2*x)/(2*d*(c*d^2 - b*d*e + a*e^2)*(d + e*x^2)) + (Sqrt[c]*(2*c^2*d^2 + b*(b + Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d + Sqrt[b^2 - 4*a*c]*d + a*e))*A rcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)^2) - (Sqrt[c] *(2*c^2*d^2 + b*(b - Sqrt[b^2 - 4*a*c])*e^2 - 2*c*e*(b*d - Sqrt[b^2 - 4*a* c]*d + a*e))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqr t[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2) ^2) + (e^(3/2)*(2*c*d - b*e)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(Sqrt[d]*(c*d^2 - b*d*e + a*e^2)^2) + (e^(3/2)*ArcTan[(Sqrt[e]*x)/Sqrt[d]])/(2*d^(3/2)*(c* d^2 - b*d*e + a*e^2))
3.3.69.3.1 Defintions of rubi rules used
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symb ol] :> Int[ExpandIntegrand[(d + e*x^2)^q/(a + b*x^2 + c*x^4), x], x] /; Fre eQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[q]
Time = 0.52 (sec) , antiderivative size = 345, normalized size of antiderivative = 0.80
method | result | size |
default | \(\frac {4 c \left (\frac {\left (b \,e^{2} \sqrt {-4 a c +b^{2}}-2 d c e \sqrt {-4 a c +b^{2}}+2 e^{2} a c -b^{2} e^{2}+2 b c d e -2 c^{2} d^{2}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (b \,e^{2} \sqrt {-4 a c +b^{2}}-2 d c e \sqrt {-4 a c +b^{2}}-2 e^{2} a c +b^{2} e^{2}-2 b c d e +2 c^{2} d^{2}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}+\frac {e^{2} \left (\frac {\left (a \,e^{2}-b d e +c \,d^{2}\right ) x}{2 d \left (e \,x^{2}+d \right )}+\frac {\left (a \,e^{2}-3 b d e +5 c \,d^{2}\right ) \arctan \left (\frac {e x}{\sqrt {e d}}\right )}{2 d \sqrt {e d}}\right )}{\left (a \,e^{2}-b d e +c \,d^{2}\right )^{2}}\) | \(345\) |
risch | \(\text {Expression too large to display}\) | \(22770\) |
4/(a*e^2-b*d*e+c*d^2)^2*c*(1/8*(b*e^2*(-4*a*c+b^2)^(1/2)-2*d*c*e*(-4*a*c+b ^2)^(1/2)+2*e^2*a*c-b^2*e^2+2*b*c*d*e-2*c^2*d^2)/(-4*a*c+b^2)^(1/2)*2^(1/2 )/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/ 2))*c)^(1/2))-1/8*(b*e^2*(-4*a*c+b^2)^(1/2)-2*d*c*e*(-4*a*c+b^2)^(1/2)-2*e ^2*a*c+b^2*e^2-2*b*c*d*e+2*c^2*d^2)/(-4*a*c+b^2)^(1/2)*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)))+e^2/(a*e^2-b*d*e+c*d^2)^2*(1/2*(a*e^2-b*d*e+c*d^2)/d*x/(e*x^2+d)+1/2* (a*e^2-3*b*d*e+5*c*d^2)/d/(e*d)^(1/2)*arctan(e*x/(e*d)^(1/2)))
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
Leaf count of result is larger than twice the leaf count of optimal. 2394 vs. \(2 (366) = 732\).
Time = 1.83 (sec) , antiderivative size = 2394, normalized size of antiderivative = 5.58 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
1/2*e^2*x/((c*d^3 - b*d^2*e + a*d*e^2)*(e*x^2 + d)) + 1/2*(sqrt(2)*sqrt(b* c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)* c)*a*b^2*c^3 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^3 - 2*b^4*c ^3 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^4 + 8*sqrt(2)*sqrt(b *c + sqrt(b^2 - 4*a*c)*c)*a*b*c^4 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c )*b^2*c^4 + 16*a*b^2*c^4 + 2*b^3*c^4 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a *c)*c)*a*c^5 - 32*a^2*c^5 - 8*a*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + s qrt(b^2 - 4*a*c)*c)*a*b*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt( b^2 - 4*a*c)*c)*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^4 + 2*(b^2 - 4*a*c)*b^2*c^3 - 8*(b^2 - 4*a*c)*a*c^4 - 2*(b^2 - 4*a*c)*b*c^4)*arctan(2*sqrt(1/2)*x/sqrt((b*c^2*d^4 - 2*b^2*c*d^3*e + b^ 3*d^2*e^2 + 2*a*b*c*d^2*e^2 - 2*a*b^2*d*e^3 + a^2*b*e^4 + sqrt((b*c^2*d^4 - 2*b^2*c*d^3*e + b^3*d^2*e^2 + 2*a*b*c*d^2*e^2 - 2*a*b^2*d*e^3 + a^2*b*e^ 4)^2 - 4*(a*c^2*d^4 - 2*a*b*c*d^3*e + a*b^2*d^2*e^2 + 2*a^2*c*d^2*e^2 - 2* a^2*b*d*e^3 + a^3*e^4)*(c^3*d^4 - 2*b*c^2*d^3*e + b^2*c*d^2*e^2 + 2*a*c^2* d^2*e^2 - 2*a*b*c*d*e^3 + a^2*c*e^4)))/(c^3*d^4 - 2*b*c^2*d^3*e + b^2*c*d^ 2*e^2 + 2*a*c^2*d^2*e^2 - 2*a*b*c*d*e^3 + a^2*c*e^4)))/(2*(a*b^4*c^2 - 8*a ^2*b^2*c^3 - 2*a*b^3*c^3 + 16*a^3*c^4 + 8*a^2*b*c^4 + a*b^2*c^4 - 4*a^2*c^ 5)*d^2*abs(c) - 2*(a*b^5*c - 8*a^2*b^3*c^2 - 2*a*b^4*c^2 + 16*a^3*b*c^3...
Time = 11.41 (sec) , antiderivative size = 91169, normalized size of antiderivative = 212.52 \[ \int \frac {1}{\left (d+e x^2\right )^2 \left (a+b x^2+c x^4\right )} \, dx=\text {Too large to display} \]
(atan(((((x*(54*c^9*d^6*e^5 - 2*a^3*c^6*e^11 - 22*a*c^8*d^4*e^7 - 118*b*c^ 8*d^5*e^6 + a^2*b^2*c^5*e^11 - 14*a^2*c^7*d^2*e^9 + 107*b^2*c^7*d^4*e^7 - 48*b^3*c^6*d^3*e^8 + 9*b^4*c^5*d^2*e^9 + 20*a*b*c^7*d^3*e^8 - 6*a*b^3*c^5* d*e^10 + 10*a^2*b*c^6*d*e^10 + 4*a*b^2*c^6*d^2*e^9))/(2*(c^4*d^10 + a^4*d^ 2*e^8 + b^4*d^6*e^4 - 4*a*b^3*d^5*e^5 - 4*a^3*b*d^3*e^7 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 - 4*b^3*c*d^7*e^3 + 6*a^2*b^2*d^4*e^6 + 6*a^2*c^2*d^6*e^ 4 + 6*b^2*c^2*d^8*e^2 - 4*b*c^3*d^9*e - 12*a*b*c^2*d^7*e^3 + 12*a*b^2*c*d^ 6*e^4 - 12*a^2*b*c*d^5*e^5)) - (((2*a^2*b^6*c^2*e^13 - 200*a*c^9*d^8*e^5 - 8*a^5*c^5*e^13 - 14*a^3*b^4*c^3*e^13 + 26*a^4*b^2*c^4*e^13 + 480*a^2*c^8* d^6*e^7 + 784*a^3*c^7*d^4*e^9 + 96*a^4*c^6*d^2*e^11 + 50*b^2*c^8*d^8*e^5 - 240*b^3*c^7*d^7*e^6 + 466*b^4*c^6*d^6*e^7 - 464*b^5*c^5*d^5*e^8 + 246*b^6 *c^4*d^4*e^9 - 64*b^7*c^3*d^3*e^10 + 6*b^8*c^2*d^2*e^11 + 4*a^2*b^2*c^6*d^ 4*e^9 + 672*a^2*b^3*c^5*d^3*e^10 - 354*a^2*b^4*c^4*d^2*e^11 + 464*a^3*b^2* c^5*d^2*e^11 + 960*a*b*c^8*d^7*e^6 - 8*a*b^7*c^2*d*e^12 - 96*a^4*b*c^5*d*e ^12 - 1984*a*b^2*c^7*d^6*e^7 + 2072*a*b^3*c^6*d^5*e^8 - 1034*a*b^4*c^5*d^4 *e^9 + 160*a*b^5*c^4*d^3*e^10 + 34*a*b^6*c^3*d^2*e^11 - 864*a^2*b*c^7*d^5* e^8 + 40*a^2*b^5*c^3*d*e^12 - 1152*a^3*b*c^6*d^3*e^10 - 8*a^3*b^3*c^4*d*e^ 12)/(2*(c^4*d^10 + a^4*d^2*e^8 + b^4*d^6*e^4 - 4*a*b^3*d^5*e^5 - 4*a^3*b*d ^3*e^7 + 4*a*c^3*d^8*e^2 + 4*a^3*c*d^4*e^6 - 4*b^3*c*d^7*e^3 + 6*a^2*b^2*d ^4*e^6 + 6*a^2*c^2*d^6*e^4 + 6*b^2*c^2*d^8*e^2 - 4*b*c^3*d^9*e - 12*a*b...