Integrand size = 24, antiderivative size = 563 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {x \left (c \left (b^2 d^3-2 a d \left (c d^2-3 a e^2\right )-\frac {a b e \left (3 c d^2+a e^2\right )}{c}\right )-\left (a b^2 e^3+2 a c e \left (3 c d^2-a e^2\right )-b c d \left (c d^2+3 a e^2\right )\right ) x^2\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\left (a b^3 e^3+6 a c \left (2 c d+\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2+a \sqrt {b^2-4 a c} e^3\right )-b c \left (a e^2 \left (3 \sqrt {b^2-4 a c} d+8 a e\right )+c d^2 \left (\sqrt {b^2-4 a c} d+12 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (a b^3 e^3+6 a c \left (2 c d-\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )-b^2 \left (c^2 d^3-3 a c d e^2-a \sqrt {b^2-4 a c} e^3\right )+b c \left (c d^2 \left (\sqrt {b^2-4 a c} d-12 a e\right )+a e^2 \left (3 \sqrt {b^2-4 a c} d-8 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} a c^{3/2} \left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}} \]
1/2*x*(c*(b^2*d^3-2*a*d*(-3*a*e^2+c*d^2)-a*b*e*(a*e^2+3*c*d^2)/c)-(a*b^2*e ^3+2*a*c*e*(-a*e^2+3*c*d^2)-b*c*d*(3*a*e^2+c*d^2))*x^2)/a/c/(-4*a*c+b^2)/( c*x^4+b*x^2+a)-1/4*arctan(x*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))* (a*b^3*e^3+6*a*c*(a*e^2+c*d^2)*(2*c*d+e*(-4*a*c+b^2)^(1/2))-b^2*(c^2*d^3-3 *a*c*d*e^2+a*e^3*(-4*a*c+b^2)^(1/2))-b*c*(c*d^2*(12*a*e+d*(-4*a*c+b^2)^(1/ 2))+a*e^2*(8*a*e+3*d*(-4*a*c+b^2)^(1/2))))/a/c^(3/2)/(-4*a*c+b^2)^(3/2)*2^ (1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2)+1/4*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c +b^2)^(1/2))^(1/2))*(a*b^3*e^3+6*a*c*(a*e^2+c*d^2)*(2*c*d-e*(-4*a*c+b^2)^( 1/2))-b^2*(c^2*d^3-3*a*c*d*e^2-a*e^3*(-4*a*c+b^2)^(1/2))+b*c*(c*d^2*(-12*a *e+d*(-4*a*c+b^2)^(1/2))+a*e^2*(-8*a*e+3*d*(-4*a*c+b^2)^(1/2))))/a/c^(3/2) /(-4*a*c+b^2)^(3/2)*2^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.97 (sec) , antiderivative size = 540, normalized size of antiderivative = 0.96 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {\frac {2 \sqrt {c} x \left (b^2 \left (c d^3-a e^3 x^2\right )+b \left (-a^2 e^3+c^2 d^3 x^2-3 a c d e \left (d-e x^2\right )\right )+2 a c \left (a e^2 \left (3 d+e x^2\right )-c d^2 \left (d+3 e x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\sqrt {2} \left (-a b^3 e^3-6 a c \left (2 c d+\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )+b^2 \left (c^2 d^3-3 a c d e^2+a \sqrt {b^2-4 a c} e^3\right )+b c \left (a e^2 \left (3 \sqrt {b^2-4 a c} d+8 a e\right )+c d^2 \left (\sqrt {b^2-4 a c} d+12 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\sqrt {2} \left (a b^3 e^3+6 a c \left (2 c d-\sqrt {b^2-4 a c} e\right ) \left (c d^2+a e^2\right )+b^2 \left (-c^2 d^3+3 a c d e^2+a \sqrt {b^2-4 a c} e^3\right )+b c \left (c d^2 \left (\sqrt {b^2-4 a c} d-12 a e\right )+a e^2 \left (3 \sqrt {b^2-4 a c} d-8 a e\right )\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\left (b^2-4 a c\right )^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}}{4 a c^{3/2}} \]
((2*Sqrt[c]*x*(b^2*(c*d^3 - a*e^3*x^2) + b*(-(a^2*e^3) + c^2*d^3*x^2 - 3*a *c*d*e*(d - e*x^2)) + 2*a*c*(a*e^2*(3*d + e*x^2) - c*d^2*(d + 3*e*x^2))))/ ((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (Sqrt[2]*(-(a*b^3*e^3) - 6*a*c*(2*c* d + Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) + b^2*(c^2*d^3 - 3*a*c*d*e^2 + a* Sqrt[b^2 - 4*a*c]*e^3) + b*c*(a*e^2*(3*Sqrt[b^2 - 4*a*c]*d + 8*a*e) + c*d^ 2*(Sqrt[b^2 - 4*a*c]*d + 12*a*e)))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqr t[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqr t[2]*(a*b^3*e^3 + 6*a*c*(2*c*d - Sqrt[b^2 - 4*a*c]*e)*(c*d^2 + a*e^2) + b^ 2*(-(c^2*d^3) + 3*a*c*d*e^2 + a*Sqrt[b^2 - 4*a*c]*e^3) + b*c*(c*d^2*(Sqrt[ b^2 - 4*a*c]*d - 12*a*e) + a*e^2*(3*Sqrt[b^2 - 4*a*c]*d - 8*a*e)))*ArcTan[ (Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/((b^2 - 4*a*c)^(3/2)*Sqr t[b + Sqrt[b^2 - 4*a*c]]))/(4*a*c^(3/2))
Time = 1.06 (sec) , antiderivative size = 521, normalized size of antiderivative = 0.93, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1517, 25, 1480, 218}
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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 1517 |
| \(\displaystyle \frac {x \left (c \left (-\frac {a b e \left (a e^2+3 c d^2\right )}{c}-2 a d \left (c d^2-3 a e^2\right )+b^2 d^3\right )-x^2 \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int -\frac {b^2 d^3-6 a \left (c d^2+a e^2\right ) d+\left (\frac {a b^2 e^3}{c}-6 a \left (c d^2+a e^2\right ) e+b \left (c d^3+3 a e^2 d\right )\right ) x^2+\frac {a b e \left (3 c d^2+a e^2\right )}{c}}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b^2 d^3-6 a \left (c d^2+a e^2\right ) d+\left (\frac {a b^2 e^3}{c}-6 a \left (c d^2+a e^2\right ) e+b \left (c d^3+3 a e^2 d\right )\right ) x^2+\frac {a b e \left (3 c d^2+a e^2\right )}{c}}{c x^4+b x^2+a}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c \left (-\frac {a b e \left (a e^2+3 c d^2\right )}{c}-2 a d \left (c d^2-3 a e^2\right )+b^2 d^3\right )-x^2 \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 1480 |
| \(\displaystyle \frac {\frac {1}{2} \left (\frac {a b^2 e^3}{c}-\frac {a b^3 e^3-b^2 c d \left (c d^2-3 a e^2\right )-4 a b c e \left (2 a e^2+3 c d^2\right )+12 a c^2 d \left (a e^2+c d^2\right )}{c \sqrt {b^2-4 a c}}+b \left (3 a d e^2+c d^3\right )-6 a e \left (a e^2+c d^2\right )\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b-\sqrt {b^2-4 a c}\right )}dx+\frac {1}{2} \left (\frac {a b^2 e^3}{c}+\frac {a b^3 e^3-b^2 c d \left (c d^2-3 a e^2\right )-4 a b c e \left (2 a e^2+3 c d^2\right )+12 a c^2 d \left (a e^2+c d^2\right )}{c \sqrt {b^2-4 a c}}+b \left (3 a d e^2+c d^3\right )-6 a e \left (a e^2+c d^2\right )\right ) \int \frac {1}{c x^2+\frac {1}{2} \left (b+\sqrt {b^2-4 a c}\right )}dx}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c \left (-\frac {a b e \left (a e^2+3 c d^2\right )}{c}-2 a d \left (c d^2-3 a e^2\right )+b^2 d^3\right )-x^2 \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {a b^2 e^3}{c}-\frac {a b^3 e^3-b^2 c d \left (c d^2-3 a e^2\right )-4 a b c e \left (2 a e^2+3 c d^2\right )+12 a c^2 d \left (a e^2+c d^2\right )}{c \sqrt {b^2-4 a c}}+b \left (3 a d e^2+c d^3\right )-6 a e \left (a e^2+c d^2\right )\right )}{\sqrt {2} \sqrt {c} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {a b^2 e^3}{c}+\frac {a b^3 e^3-b^2 c d \left (c d^2-3 a e^2\right )-4 a b c e \left (2 a e^2+3 c d^2\right )+12 a c^2 d \left (a e^2+c d^2\right )}{c \sqrt {b^2-4 a c}}+b \left (3 a d e^2+c d^3\right )-6 a e \left (a e^2+c d^2\right )\right )}{\sqrt {2} \sqrt {c} \sqrt {\sqrt {b^2-4 a c}+b}}}{2 a \left (b^2-4 a c\right )}+\frac {x \left (c \left (-\frac {a b e \left (a e^2+3 c d^2\right )}{c}-2 a d \left (c d^2-3 a e^2\right )+b^2 d^3\right )-x^2 \left (a b^2 e^3-b c d \left (3 a e^2+c d^2\right )+2 a c e \left (3 c d^2-a e^2\right )\right )\right )}{2 a c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}\) |
(x*(c*(b^2*d^3 - 2*a*d*(c*d^2 - 3*a*e^2) - (a*b*e*(3*c*d^2 + a*e^2))/c) - (a*b^2*e^3 + 2*a*c*e*(3*c*d^2 - a*e^2) - b*c*d*(c*d^2 + 3*a*e^2))*x^2))/(2 *a*c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + ((((a*b^2*e^3)/c - 6*a*e*(c*d^2 + a*e^2) + b*(c*d^3 + 3*a*d*e^2) - (a*b^3*e^3 - b^2*c*d*(c*d^2 - 3*a*e^2) + 12*a*c^2*d*(c*d^2 + a*e^2) - 4*a*b*c*e*(3*c*d^2 + 2*a*e^2))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[ 2]*Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (((a*b^2*e^3)/c - 6*a*e*(c*d^2 + a*e^2) + b*(c*d^3 + 3*a*d*e^2) + (a*b^3*e^3 - b^2*c*d*(c*d^2 - 3*a*e^2) + 12*a*c^2*d*(c*d^2 + a*e^2) - 4*a*b*c*e*(3*c*d^2 + 2*a*e^2))/(c*Sqrt[b^2 - 4*a*c]))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2 ]*Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]))/(2*a*(b^2 - 4*a*c))
3.3.70.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : > With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q)) Int[1/( b/2 - q/2 + c*x^2), x], x] + Simp[(e/2 - (2*c*d - b*e)/(2*q)) Int[1/(b/2 + q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^2 - 4*a*c]
Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x _Symbol] :> With[{f = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 0], g = Coeff[PolynomialRemainder[(d + e*x^2)^q, a + b*x^2 + c*x^4, x], x, 2]}, Simp[x*(a + b*x^2 + c*x^4)^(p + 1)*((a*b*g - f*(b^2 - 2* a*c) - c*(b*f - 2*a*g)*x^2)/(2*a*(p + 1)*(b^2 - 4*a*c))), x] + Simp[1/(2*a* (p + 1)*(b^2 - 4*a*c)) Int[(a + b*x^2 + c*x^4)^(p + 1)*ExpandToSum[2*a*(p + 1)*(b^2 - 4*a*c)*PolynomialQuotient[(d + e*x^2)^q, a + b*x^2 + c*x^4, x] + b^2*f*(2*p + 3) - 2*a*c*f*(4*p + 5) - a*b*g + c*(4*p + 7)*(b*f - 2*a*g)* x^2, x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ [c*d^2 - b*d*e + a*e^2, 0] && IGtQ[q, 1] && LtQ[p, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.26 (sec) , antiderivative size = 320, normalized size of antiderivative = 0.57
| method | result | size |
| risch | \(\frac {-\frac {\left (2 a^{2} c \,e^{3}-a \,b^{2} e^{3}+3 a b c d \,e^{2}-6 a \,c^{2} d^{2} e +b \,c^{2} d^{3}\right ) x^{3}}{2 a c \left (4 a c -b^{2}\right )}+\frac {\left (a^{2} b \,e^{3}-6 a^{2} c d \,e^{2}+3 a b c \,d^{2} e +2 a \,c^{2} d^{3}-b^{2} c \,d^{3}\right ) x}{2 c \left (4 a c -b^{2}\right ) a}}{c \,x^{4}+b \,x^{2}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\frac {\left (6 a^{2} c \,e^{3}-a \,b^{2} e^{3}-3 a b c d \,e^{2}+6 a \,c^{2} d^{2} e -b \,c^{2} d^{3}\right ) \textit {\_R}^{2}}{4 a c -b^{2}}-\frac {a^{2} b \,e^{3}-6 a^{2} c d \,e^{2}+3 a b c \,d^{2} e -6 a \,c^{2} d^{3}+b^{2} c \,d^{3}}{4 a c -b^{2}}\right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{4 a c}\) | \(320\) |
| default | \(\frac {-\frac {\left (2 a^{2} c \,e^{3}-a \,b^{2} e^{3}+3 a b c d \,e^{2}-6 a \,c^{2} d^{2} e +b \,c^{2} d^{3}\right ) x^{3}}{2 a c \left (4 a c -b^{2}\right )}+\frac {\left (a^{2} b \,e^{3}-6 a^{2} c d \,e^{2}+3 a b c \,d^{2} e +2 a \,c^{2} d^{3}-b^{2} c \,d^{3}\right ) x}{2 c \left (4 a c -b^{2}\right ) a}}{c \,x^{4}+b \,x^{2}+a}+\frac {\frac {\left (6 a^{2} c \,e^{3} \sqrt {-4 a c +b^{2}}-a \,b^{2} e^{3} \sqrt {-4 a c +b^{2}}-3 a b c d \,e^{2} \sqrt {-4 a c +b^{2}}+6 a \,c^{2} d^{2} e \sqrt {-4 a c +b^{2}}-b \,c^{2} d^{3} \sqrt {-4 a c +b^{2}}+8 a^{2} b \,e^{3} c -12 a^{2} c^{2} d \,e^{2}-a \,b^{3} e^{3}-3 a \,b^{2} c d \,e^{2}+12 a b \,c^{2} d^{2} e -12 a \,c^{3} d^{3}+b^{2} c^{2} d^{3}\right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (6 a^{2} c \,e^{3} \sqrt {-4 a c +b^{2}}-a \,b^{2} e^{3} \sqrt {-4 a c +b^{2}}-3 a b c d \,e^{2} \sqrt {-4 a c +b^{2}}+6 a \,c^{2} d^{2} e \sqrt {-4 a c +b^{2}}-b \,c^{2} d^{3} \sqrt {-4 a c +b^{2}}-8 a^{2} b \,e^{3} c +12 a^{2} c^{2} d \,e^{2}+a \,b^{3} e^{3}+3 a \,b^{2} c d \,e^{2}-12 a b \,c^{2} d^{2} e +12 a \,c^{3} d^{3}-b^{2} c^{2} d^{3}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{4 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{a \left (4 a c -b^{2}\right )}\) | \(628\) |
(-1/2*(2*a^2*c*e^3-a*b^2*e^3+3*a*b*c*d*e^2-6*a*c^2*d^2*e+b*c^2*d^3)/a/c/(4 *a*c-b^2)*x^3+1/2/c*(a^2*b*e^3-6*a^2*c*d*e^2+3*a*b*c*d^2*e+2*a*c^2*d^3-b^2 *c*d^3)/(4*a*c-b^2)/a*x)/(c*x^4+b*x^2+a)+1/4/a/c*sum(((6*a^2*c*e^3-a*b^2*e ^3-3*a*b*c*d*e^2+6*a*c^2*d^2*e-b*c^2*d^3)/(4*a*c-b^2)*_R^2-(a^2*b*e^3-6*a^ 2*c*d*e^2+3*a*b*c*d^2*e-6*a*c^2*d^3+b^2*c*d^3)/(4*a*c-b^2))/(2*_R^3*c+_R*b )*ln(x-_R),_R=RootOf(_Z^4*c+_Z^2*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 12117 vs. \(2 (507) = 1014\).
Time = 107.28 (sec) , antiderivative size = 12117, normalized size of antiderivative = 21.52 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
\[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\int { \frac {{\left (e x^{2} + d\right )}^{3}}{{\left (c x^{4} + b x^{2} + a\right )}^{2}} \,d x } \]
1/2*((b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 - (a*b^2 - 2*a^2*c)*e^3)*x ^3 - (3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 - (b^2*c - 2*a*c^2)*d^3)*x )/(a^2*b^2*c - 4*a^3*c^2 + (a*b^2*c^2 - 4*a^2*c^3)*x^4 + (a*b^3*c - 4*a^2* b*c^2)*x^2) - 1/2*integrate(-(3*a*b*c*d^2*e - 6*a^2*c*d*e^2 + a^2*b*e^3 + (b^2*c - 6*a*c^2)*d^3 + (b*c^2*d^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e^2 + (a*b^ 2 - 6*a^2*c)*e^3)*x^2)/(c*x^4 + b*x^2 + a), x)/(a*b^2*c - 4*a^2*c^2)
Leaf count of result is larger than twice the leaf count of optimal. 8992 vs. \(2 (507) = 1014\).
Time = 1.77 (sec) , antiderivative size = 8992, normalized size of antiderivative = 15.97 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
1/2*(b*c^2*d^3*x^3 - 6*a*c^2*d^2*e*x^3 + 3*a*b*c*d*e^2*x^3 - a*b^2*e^3*x^3 + 2*a^2*c*e^3*x^3 + b^2*c*d^3*x - 2*a*c^2*d^3*x - 3*a*b*c*d^2*e*x + 6*a^2 *c*d*e^2*x - a^2*b*e^3*x)/((c*x^4 + b*x^2 + a)*(a*b^2*c - 4*a^2*c^2)) + 1/ 16*((2*b^3*c^4 - 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 + sqrt(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*c^4)*(a*b^2*c - 4*a^2*c^2)^2*d^3 - 6*(2*a*b^2*c^4 - 8*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a *b^2*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2 *c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^4 - 2*(b^ 2 - 4*a*c)*a*c^4)*(a*b^2*c - 4*a^2*c^2)^2*d^2*e + 3*(2*a*b^3*c^3 - 8*a^2*b *c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 + 2 *sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - sqr t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*a*b*c^3)*(a*b^2*c - 4*a^2*c^2)^2*d*e^2 + (2*a*b^4*c^2 - 20*a^2*b^2* c^3 + 48*a^3*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)* c)*a*b^4 + 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)...
Time = 10.27 (sec) , antiderivative size = 29030, normalized size of antiderivative = 51.56 \[ \int \frac {\left (d+e x^2\right )^3}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
- ((x^3*(b*c^2*d^3 - a*b^2*e^3 + 2*a^2*c*e^3 - 6*a*c^2*d^2*e + 3*a*b*c*d*e ^2))/(2*a*c*(4*a*c - b^2)) - (x*(2*a*c^2*d^3 + a^2*b*e^3 - b^2*c*d^3 - 6*a ^2*c*d*e^2 + 3*a*b*c*d^2*e))/(2*a*c*(4*a*c - b^2)))/(a + b*x^2 + c*x^4) - atan(((((6144*a^5*c^7*d^3 + 16*a*b^8*c^3*d^3 - 1024*a^6*b*c^5*e^3 + 6144*a ^6*c^6*d*e^2 - 288*a^2*b^6*c^4*d^3 + 1920*a^3*b^4*c^5*d^3 - 5632*a^4*b^2*c ^6*d^3 + 16*a^3*b^7*c^2*e^3 - 192*a^4*b^5*c^3*e^3 + 768*a^5*b^3*c^4*e^3 - 3072*a^5*b*c^6*d^2*e + 48*a^2*b^7*c^3*d^2*e - 576*a^3*b^5*c^4*d^2*e - 96*a ^3*b^6*c^3*d*e^2 + 2304*a^4*b^3*c^5*d^2*e + 1152*a^4*b^4*c^4*d*e^2 - 4608* a^5*b^2*c^5*d*e^2)/(8*(64*a^5*c^4 - a^2*b^6*c + 12*a^3*b^4*c^2 - 48*a^4*b^ 2*c^3)) - (x*((27*a*b^9*c^4*d^6 - b^11*c^3*d^6 - a^3*b^11*e^6 + 3840*a^5*b *c^8*d^6 - 9*a*c^4*d^6*(-(4*a*c - b^2)^9)^(1/2) + 27*a^4*b^9*c*e^6 + 3840* a^8*b*c^5*e^6 + 9*a^4*c*e^6*(-(4*a*c - b^2)^9)^(1/2) - 9216*a^6*c^8*d^5*e - 9216*a^8*c^6*d*e^5 - 288*a^2*b^7*c^5*d^6 + 1504*a^3*b^5*c^6*d^6 - 3840*a ^4*b^3*c^7*d^6 - a^3*b^2*e^6*(-(4*a*c - b^2)^9)^(1/2) - 288*a^5*b^7*c^2*e^ 6 + 1504*a^6*b^5*c^3*e^6 - 3840*a^7*b^3*c^4*e^6 + b^2*c^3*d^6*(-(4*a*c - b ^2)^9)^(1/2) - 18432*a^7*c^7*d^3*e^3 + 9*a^2*b^9*c^3*d^4*e^2 - 384*a^3*b^7 *c^4*d^4*e^2 + 88*a^3*b^8*c^3*d^3*e^3 + 9*a^3*b^9*c^2*d^2*e^4 + 3744*a^4*b ^5*c^5*d^4*e^2 - 768*a^4*b^6*c^4*d^3*e^3 - 384*a^4*b^7*c^3*d^2*e^4 - 13824 *a^5*b^3*c^6*d^4*e^2 + 768*a^5*b^4*c^5*d^3*e^3 + 3744*a^5*b^5*c^4*d^2*e^4 + 8192*a^6*b^2*c^6*d^3*e^3 - 13824*a^6*b^3*c^5*d^2*e^4 - 9*a^2*c^3*d^4*...