Integrand size = 30, antiderivative size = 165 \[ \int \frac {x^3 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {x^2 \left (2 a c e-b (c d+a f)-\left (2 c^2 d-b c e+b^2 f-2 a c f\right ) x^2\right )}{2 c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {\left (4 a c^2 e+b^3 f-2 b c (c d+3 a f)\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {f \log \left (a+b x^2+c x^4\right )}{4 c^2} \]
1/2*x^2*(2*a*c*e-b*(a*f+c*d)-(-2*a*c*f+b^2*f-b*c*e+2*c^2*d)*x^2)/c/(-4*a*c +b^2)/(c*x^4+b*x^2+a)+1/2*(4*a*c^2*e+b^3*f-2*b*c*(3*a*f+c*d))*arctanh((2*c *x^2+b)/(-4*a*c+b^2)^(1/2))/c^2/(-4*a*c+b^2)^(3/2)+1/4*f*ln(c*x^4+b*x^2+a) /c^2
Time = 0.16 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.06 \[ \int \frac {x^3 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {\frac {2 \left (-2 a^2 c f+b \left (c^2 d-b c e+b^2 f\right ) x^2+a \left (b^2 f+2 c^2 \left (d+e x^2\right )-b c \left (e+3 f x^2\right )\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+\frac {2 \left (4 a c^2 e+b^3 f-2 b c (c d+3 a f)\right ) \arctan \left (\frac {b+2 c x^2}{\sqrt {-b^2+4 a c}}\right )}{\left (-b^2+4 a c\right )^{3/2}}+f \log \left (a+b x^2+c x^4\right )}{4 c^2} \]
((2*(-2*a^2*c*f + b*(c^2*d - b*c*e + b^2*f)*x^2 + a*(b^2*f + 2*c^2*(d + e* x^2) - b*c*(e + 3*f*x^2))))/((b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) + (2*(4*a* c^2*e + b^3*f - 2*b*c*(c*d + 3*a*f))*ArcTan[(b + 2*c*x^2)/Sqrt[-b^2 + 4*a* c]])/(-b^2 + 4*a*c)^(3/2) + f*Log[a + b*x^2 + c*x^4])/(4*c^2)
Time = 0.49 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.16, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.233, Rules used = {2194, 2175, 27, 1142, 1083, 219, 1103}
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.
| \(\displaystyle \int \frac {x^3 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle \frac {1}{2} \int \frac {x^2 \left (f x^4+e x^2+d\right )}{\left (c x^4+b x^2+a\right )^2}dx^2\) |
| \(\Big \downarrow \) 2175 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x^2 \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {c \left (2 a e-\frac {b (c d+a f)}{c}\right )-\left (b^2-4 a c\right ) f x^2}{c \left (c x^4+b x^2+a\right )}dx^2}{b^2-4 a c}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x^2 \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\int \frac {-\left (\left (b^2-4 a c\right ) f x^2\right )+2 a c e-b (c d+a f)}{c x^4+b x^2+a}dx^2}{c \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x^2 \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {\frac {\left (-2 b c (3 a f+c d)+4 a c^2 e+b^3 f\right ) \int \frac {1}{c x^4+b x^2+a}dx^2}{2 c}-\frac {f \left (b^2-4 a c\right ) \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c}}{c \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x^2 \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-\frac {f \left (b^2-4 a c\right ) \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c}-\frac {\left (-2 b c (3 a f+c d)+4 a c^2 e+b^3 f\right ) \int \frac {1}{-x^4+b^2-4 a c}d\left (2 c x^2+b\right )}{c}}{c \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x^2 \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-\frac {f \left (b^2-4 a c\right ) \int \frac {2 c x^2+b}{c x^4+b x^2+a}dx^2}{2 c}-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-2 b c (3 a f+c d)+4 a c^2 e+b^3 f\right )}{c \sqrt {b^2-4 a c}}}{c \left (b^2-4 a c\right )}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {1}{2} \left (\frac {x^2 \left (c \left (2 a e-b \left (\frac {a f}{c}+d\right )\right )-x^2 \left (-2 a c f+b^2 f-b c e+2 c^2 d\right )\right )}{c \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}-\frac {-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-2 b c (3 a f+c d)+4 a c^2 e+b^3 f\right )}{c \sqrt {b^2-4 a c}}-\frac {f \left (b^2-4 a c\right ) \log \left (a+b x^2+c x^4\right )}{2 c}}{c \left (b^2-4 a c\right )}\right )\) |
((x^2*(c*(2*a*e - b*(d + (a*f)/c)) - (2*c^2*d - b*c*e + b^2*f - 2*a*c*f)*x ^2))/(c*(b^2 - 4*a*c)*(a + b*x^2 + c*x^4)) - (-(((4*a*c^2*e + b^3*f - 2*b* c*(c*d + 3*a*f))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(c*Sqrt[b^2 - 4 *a*c])) - ((b^2 - 4*a*c)*f*Log[a + b*x^2 + c*x^4])/(2*c))/(c*(b^2 - 4*a*c) ))/2
3.1.63.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[(2*c*d - b*e)/(2*c) Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, a + b*x + c*x^2, x], R = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], S = Coeff[Polyno mialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*((R*b - 2*a*S + (2*c*R - b*S)*x)/((p + 1)*(b^2 - 4*a*c))), x ] + Simp[1/((p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2 )^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(d + e*x)*Qx + S*(2*a*e*m + b*d *(2*p + 3)) - R*(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*R - b*S)*(m + 2*p + 3)*x , x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a *c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (Inte gerQ[p] || !IntegerQ[m] || !RationalQ[a, b, c, d, e]) && !(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : > Simp[1/2 Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2) ^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ [(m - 1)/2]
Time = 0.21 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.38
| method | result | size |
| default | \(\frac {\frac {\left (3 a b c f -2 a \,c^{2} e -b^{3} f +b^{2} c e -b \,c^{2} d \right ) x^{2}}{\left (4 a c -b^{2}\right ) c^{2}}+\frac {a \left (2 a c f -b^{2} f +e b c -2 c^{2} d \right )}{\left (4 a c -b^{2}\right ) c^{2}}}{2 c \,x^{4}+2 b \,x^{2}+2 a}+\frac {\frac {\left (4 a c f -b^{2} f \right ) \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 c}+\frac {2 \left (-a b f +2 a c e -b c d -\frac {\left (4 a c f -b^{2} f \right ) b}{2 c}\right ) \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{2 c \left (4 a c -b^{2}\right )}\) | \(228\) |
| risch | \(\text {Expression too large to display}\) | \(1621\) |
1/2*((3*a*b*c*f-2*a*c^2*e-b^3*f+b^2*c*e-b*c^2*d)/(4*a*c-b^2)/c^2*x^2+a*(2* a*c*f-b^2*f+b*c*e-2*c^2*d)/(4*a*c-b^2)/c^2)/(c*x^4+b*x^2+a)+1/2/c/(4*a*c-b ^2)*(1/2*(4*a*c*f-b^2*f)/c*ln(c*x^4+b*x^2+a)+2*(-a*b*f+2*a*c*e-b*c*d-1/2*( 4*a*c*f-b^2*f)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2) ))
Leaf count of result is larger than twice the leaf count of optimal. 473 vs. \(2 (155) = 310\).
Time = 0.33 (sec) , antiderivative size = 970, normalized size of antiderivative = 5.88 \[ \int \frac {x^3 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\left [\frac {2 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 6 \, a b^{2} c^{2} + 8 \, a^{2} c^{3}\right )} e + {\left (b^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} f\right )} x^{2} - {\left (2 \, a b c^{2} d - 4 \, a^{2} c^{2} e + {\left (2 \, b c^{3} d - 4 \, a c^{3} e - {\left (b^{3} c - 6 \, a b c^{2}\right )} f\right )} x^{4} + {\left (2 \, b^{2} c^{2} d - 4 \, a b c^{2} e - {\left (b^{4} - 6 \, a b^{2} c\right )} f\right )} x^{2} - {\left (a b^{3} - 6 \, a^{2} b c\right )} f\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) + 4 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d - 2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} e + 2 \, {\left (a b^{4} - 6 \, a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} f + {\left ({\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} f x^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} f x^{2} + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} + {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{2}\right )}}, \frac {2 \, {\left ({\left (b^{3} c^{2} - 4 \, a b c^{3}\right )} d - {\left (b^{4} c - 6 \, a b^{2} c^{2} + 8 \, a^{2} c^{3}\right )} e + {\left (b^{5} - 7 \, a b^{3} c + 12 \, a^{2} b c^{2}\right )} f\right )} x^{2} - 2 \, {\left (2 \, a b c^{2} d - 4 \, a^{2} c^{2} e + {\left (2 \, b c^{3} d - 4 \, a c^{3} e - {\left (b^{3} c - 6 \, a b c^{2}\right )} f\right )} x^{4} + {\left (2 \, b^{2} c^{2} d - 4 \, a b c^{2} e - {\left (b^{4} - 6 \, a b^{2} c\right )} f\right )} x^{2} - {\left (a b^{3} - 6 \, a^{2} b c\right )} f\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + 4 \, {\left (a b^{2} c^{2} - 4 \, a^{2} c^{3}\right )} d - 2 \, {\left (a b^{3} c - 4 \, a^{2} b c^{2}\right )} e + 2 \, {\left (a b^{4} - 6 \, a^{2} b^{2} c + 8 \, a^{3} c^{2}\right )} f + {\left ({\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} f x^{4} + {\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} f x^{2} + {\left (a b^{4} - 8 \, a^{2} b^{2} c + 16 \, a^{3} c^{2}\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a b^{4} c^{2} - 8 \, a^{2} b^{2} c^{3} + 16 \, a^{3} c^{4} + {\left (b^{4} c^{3} - 8 \, a b^{2} c^{4} + 16 \, a^{2} c^{5}\right )} x^{4} + {\left (b^{5} c^{2} - 8 \, a b^{3} c^{3} + 16 \, a^{2} b c^{4}\right )} x^{2}\right )}}\right ] \]
[1/4*(2*((b^3*c^2 - 4*a*b*c^3)*d - (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*e + ( b^5 - 7*a*b^3*c + 12*a^2*b*c^2)*f)*x^2 - (2*a*b*c^2*d - 4*a^2*c^2*e + (2*b *c^3*d - 4*a*c^3*e - (b^3*c - 6*a*b*c^2)*f)*x^4 + (2*b^2*c^2*d - 4*a*b*c^2 *e - (b^4 - 6*a*b^2*c)*f)*x^2 - (a*b^3 - 6*a^2*b*c)*f)*sqrt(b^2 - 4*a*c)*l og((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a*c)) /(c*x^4 + b*x^2 + a)) + 4*(a*b^2*c^2 - 4*a^2*c^3)*d - 2*(a*b^3*c - 4*a^2*b *c^2)*e + 2*(a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*f + ((b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*f*x^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*f*x^2 + (a*b^4 - 8*a^ 2*b^2*c + 16*a^3*c^2)*f)*log(c*x^4 + b*x^2 + a))/(a*b^4*c^2 - 8*a^2*b^2*c^ 3 + 16*a^3*c^4 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + (b^5*c^2 - 8*a *b^3*c^3 + 16*a^2*b*c^4)*x^2), 1/4*(2*((b^3*c^2 - 4*a*b*c^3)*d - (b^4*c - 6*a*b^2*c^2 + 8*a^2*c^3)*e + (b^5 - 7*a*b^3*c + 12*a^2*b*c^2)*f)*x^2 - 2*( 2*a*b*c^2*d - 4*a^2*c^2*e + (2*b*c^3*d - 4*a*c^3*e - (b^3*c - 6*a*b*c^2)*f )*x^4 + (2*b^2*c^2*d - 4*a*b*c^2*e - (b^4 - 6*a*b^2*c)*f)*x^2 - (a*b^3 - 6 *a^2*b*c)*f)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/( b^2 - 4*a*c)) + 4*(a*b^2*c^2 - 4*a^2*c^3)*d - 2*(a*b^3*c - 4*a^2*b*c^2)*e + 2*(a*b^4 - 6*a^2*b^2*c + 8*a^3*c^2)*f + ((b^4*c - 8*a*b^2*c^2 + 16*a^2*c ^3)*f*x^4 + (b^5 - 8*a*b^3*c + 16*a^2*b*c^2)*f*x^2 + (a*b^4 - 8*a^2*b^2*c + 16*a^3*c^2)*f)*log(c*x^4 + b*x^2 + a))/(a*b^4*c^2 - 8*a^2*b^2*c^3 + 16*a ^3*c^4 + (b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*x^4 + (b^5*c^2 - 8*a*b^3*...
Timed out. \[ \int \frac {x^3 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {x^3 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` for more deta
Time = 0.62 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.16 \[ \int \frac {x^3 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\frac {{\left (2 \, b c^{2} d - 4 \, a c^{2} e - b^{3} f + 6 \, a b c f\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {f \log \left (c x^{4} + b x^{2} + a\right )}{4 \, c^{2}} + \frac {2 \, a c^{2} d - a b c e + a b^{2} f - 2 \, a^{2} c f + {\left (b c^{2} d - b^{2} c e + 2 \, a c^{2} e + b^{3} f - 3 \, a b c f\right )} x^{2}}{2 \, {\left (c x^{4} + b x^{2} + a\right )} {\left (b^{2} - 4 \, a c\right )} c^{2}} \]
1/2*(2*b*c^2*d - 4*a*c^2*e - b^3*f + 6*a*b*c*f)*arctan((2*c*x^2 + b)/sqrt( -b^2 + 4*a*c))/((b^2*c^2 - 4*a*c^3)*sqrt(-b^2 + 4*a*c)) + 1/4*f*log(c*x^4 + b*x^2 + a)/c^2 + 1/2*(2*a*c^2*d - a*b*c*e + a*b^2*f - 2*a^2*c*f + (b*c^2 *d - b^2*c*e + 2*a*c^2*e + b^3*f - 3*a*b*c*f)*x^2)/((c*x^4 + b*x^2 + a)*(b ^2 - 4*a*c)*c^2)
Time = 9.66 (sec) , antiderivative size = 1651, normalized size of antiderivative = 10.01 \[ \int \frac {x^3 \left (d+e x^2+f x^4\right )}{\left (a+b x^2+c x^4\right )^2} \, dx=\text {Too large to display} \]
- ((a*(2*c^2*d + b^2*f - 2*a*c*f - b*c*e))/(2*c^2*(4*a*c - b^2)) + (x^2*(b ^3*f + 2*a*c^2*e + b*c^2*d - b^2*c*e - 3*a*b*c*f))/(2*c^2*(4*a*c - b^2)))/ (a + b*x^2 + c*x^4) - (log(a + b*x^2 + c*x^4)*(2*b^6*f - 128*a^3*c^3*f + 9 6*a^2*b^2*c^2*f - 24*a*b^4*c*f))/(2*(256*a^3*c^5 - 4*b^6*c^2 + 48*a*b^4*c^ 3 - 192*a^2*b^2*c^4)) - (atan(((8*a*c^3*(4*a*c - b^2)^3 - 2*b^2*c^2*(4*a*c - b^2)^3)*((((8*a*f + (8*a*c^2*(2*b^6*f - 128*a^3*c^3*f + 96*a^2*b^2*c^2* f - 24*a*b^4*c*f))/(256*a^3*c^5 - 4*b^6*c^2 + 48*a*b^4*c^3 - 192*a^2*b^2*c ^4))*(b^3*f + 4*a*c^2*e - 2*b*c^2*d - 6*a*b*c*f))/(8*c^2*(4*a*c - b^2)^(3/ 2)) + (a*(2*b^6*f - 128*a^3*c^3*f + 96*a^2*b^2*c^2*f - 24*a*b^4*c*f)*(b^3* f + 4*a*c^2*e - 2*b*c^2*d - 6*a*b*c*f))/((4*a*c - b^2)^(3/2)*(256*a^3*c^5 - 4*b^6*c^2 + 48*a*b^4*c^3 - 192*a^2*b^2*c^4)))/(a*(4*a*c - b^2)) - x^2*(( (((6*b^3*c^2*f + 8*a*c^4*e - 4*b*c^4*d - 28*a*b*c^3*f)/(4*a*c^3 - b^2*c^2) + ((8*b^3*c^4 - 32*a*b*c^5)*(2*b^6*f - 128*a^3*c^3*f + 96*a^2*b^2*c^2*f - 24*a*b^4*c*f))/(2*(4*a*c^3 - b^2*c^2)*(256*a^3*c^5 - 4*b^6*c^2 + 48*a*b^4 *c^3 - 192*a^2*b^2*c^4)))*(b^3*f + 4*a*c^2*e - 2*b*c^2*d - 6*a*b*c*f))/(8* c^2*(4*a*c - b^2)^(3/2)) + ((8*b^3*c^4 - 32*a*b*c^5)*(2*b^6*f - 128*a^3*c^ 3*f + 96*a^2*b^2*c^2*f - 24*a*b^4*c*f)*(b^3*f + 4*a*c^2*e - 2*b*c^2*d - 6* a*b*c*f))/(16*c^2*(4*a*c - b^2)^(3/2)*(4*a*c^3 - b^2*c^2)*(256*a^3*c^5 - 4 *b^6*c^2 + 48*a*b^4*c^3 - 192*a^2*b^2*c^4)))/(a*(4*a*c - b^2)) + (b*((b^3* f^2 - 5*a*b*c*f^2 + 2*a*c^2*e*f - b*c^2*d*f)/(4*a*c^3 - b^2*c^2) + (((6...