3.51 \(\int \frac {d+e x^2+f x^4}{x (a+b x^2+c x^4)} \, dx\)

Optimal. Leaf size=97 \[ \frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) (a b f-2 a c e+b c d)}{2 a c \sqrt {b^2-4 a c}}-\frac {(c d-a f) \log \left (a+b x^2+c x^4\right )}{4 a c}+\frac {d \log (x)}{a} \]

[Out]

d*ln(x)/a-1/4*(-a*f+c*d)*ln(c*x^4+b*x^2+a)/a/c+1/2*(a*b*f-2*a*c*e+b*c*d)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2
))/a/c/(-4*a*c+b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.20, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {1663, 1628, 634, 618, 206, 628} \[ \frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) (a b f-2 a c e+b c d)}{2 a c \sqrt {b^2-4 a c}}-\frac {(c d-a f) \log \left (a+b x^2+c x^4\right )}{4 a c}+\frac {d \log (x)}{a} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2 + f*x^4)/(x*(a + b*x^2 + c*x^4)),x]

[Out]

((b*c*d - 2*a*c*e + a*b*f)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a*c*Sqrt[b^2 - 4*a*c]) + (d*Log[x])/a
- ((c*d - a*f)*Log[a + b*x^2 + c*x^4])/(4*a*c)

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(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]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1663

Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Dist[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] && Inte
gerQ[(m - 1)/2]

Rubi steps

\begin {align*} \int \frac {d+e x^2+f x^4}{x \left (a+b x^2+c x^4\right )} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {d+e x+f x^2}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {d}{a x}+\frac {-b d+a e-(c d-a f) x}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=\frac {d \log (x)}{a}+\frac {\operatorname {Subst}\left (\int \frac {-b d+a e-(c d-a f) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a}\\ &=\frac {d \log (x)}{a}-\frac {(c d-a f) \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a c}-\frac {(b c d-2 a c e+a b f) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a c}\\ &=\frac {d \log (x)}{a}-\frac {(c d-a f) \log \left (a+b x^2+c x^4\right )}{4 a c}+\frac {(b c d-2 a c e+a b f) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a c}\\ &=\frac {(b c d-2 a c e+a b f) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a c \sqrt {b^2-4 a c}}+\frac {d \log (x)}{a}-\frac {(c d-a f) \log \left (a+b x^2+c x^4\right )}{4 a c}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.14, size = 178, normalized size = 1.84 \[ \frac {-\log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right ) \left (c d \sqrt {b^2-4 a c}-a f \sqrt {b^2-4 a c}+a b f-2 a c e+b c d\right )+\log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right ) \left (-c d \sqrt {b^2-4 a c}+a f \sqrt {b^2-4 a c}+a b f-2 a c e+b c d\right )+4 c d \log (x) \sqrt {b^2-4 a c}}{4 a c \sqrt {b^2-4 a c}} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2 + f*x^4)/(x*(a + b*x^2 + c*x^4)),x]

[Out]

(4*c*Sqrt[b^2 - 4*a*c]*d*Log[x] - (b*c*d + c*Sqrt[b^2 - 4*a*c]*d - 2*a*c*e + a*b*f - a*Sqrt[b^2 - 4*a*c]*f)*Lo
g[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2] + (b*c*d - c*Sqrt[b^2 - 4*a*c]*d - 2*a*c*e + a*b*f + a*Sqrt[b^2 - 4*a*c]*f)
*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(4*a*c*Sqrt[b^2 - 4*a*c])

________________________________________________________________________________________

fricas [A]  time = 1.41, size = 309, normalized size = 3.19 \[ \left [\frac {4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d \log \relax (x) + {\left (b c d - 2 \, a c e + a b f\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c + {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )}}, \frac {4 \, {\left (b^{2} c - 4 \, a c^{2}\right )} d \log \relax (x) + 2 \, {\left (b c d - 2 \, a c e + a b f\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) - {\left ({\left (b^{2} c - 4 \, a c^{2}\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a b^{2} c - 4 \, a^{2} c^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

[1/4*(4*(b^2*c - 4*a*c^2)*d*log(x) + (b*c*d - 2*a*c*e + a*b*f)*sqrt(b^2 - 4*a*c)*log((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)) - ((b^2*c - 4*a*c^2)*d - (a*b^2 - 4*a^2*c)
*f)*log(c*x^4 + b*x^2 + a))/(a*b^2*c - 4*a^2*c^2), 1/4*(4*(b^2*c - 4*a*c^2)*d*log(x) + 2*(b*c*d - 2*a*c*e + a*
b*f)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - ((b^2*c - 4*a*c^2)*d - (a*b^
2 - 4*a^2*c)*f)*log(c*x^4 + b*x^2 + a))/(a*b^2*c - 4*a^2*c^2)]

________________________________________________________________________________________

giac [A]  time = 1.90, size = 97, normalized size = 1.00 \[ \frac {d \log \left (x^{2}\right )}{2 \, a} - \frac {{\left (c d - a f\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a c} - \frac {{\left (b c d + a b f - 2 \, a c e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

1/2*d*log(x^2)/a - 1/4*(c*d - a*f)*log(c*x^4 + b*x^2 + a)/(a*c) - 1/2*(b*c*d + a*b*f - 2*a*c*e)*arctan((2*c*x^
2 + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*a*c)

________________________________________________________________________________________

maple [A]  time = 0.01, size = 165, normalized size = 1.70 \[ -\frac {b d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a}-\frac {b f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, c}+\frac {e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+\frac {d \ln \relax (x )}{a}-\frac {d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a}+\frac {f \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^4+e*x^2+d)/x/(c*x^4+b*x^2+a),x)

[Out]

d*ln(x)/a+1/4/c*ln(c*x^4+b*x^2+a)*f-1/4/a*ln(c*x^4+b*x^2+a)*d+1/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^
2)^(1/2))*e-1/2/a/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*d-1/2/(4*a*c-b^2)^(1/2)*arctan((2*
c*x^2+b)/(4*a*c-b^2)^(1/2))*b/c*f

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 8.88, size = 3927, normalized size = 40.48 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2 + f*x^4)/(x*(a + b*x^2 + c*x^4)),x)

[Out]

(d*log(x))/a - (log((b^2*d*f^2 + c^2*d*e^2 - x^2*(b*f - c*e)*(a*f^2 + c*e^2 - b*e*f - c*d*f) + ((c*d - a*f + a
*c*(-(a*b*f - 2*a*c*e + b*c*d)^2/(a^2*c^2*(4*a*c - b^2)))^(1/2))*(a*b^2*f^2 - x^2*(b*c^2*e^2 - 3*b^3*f^2 + 5*c
^3*d*e + 11*a*b*c*f^2 - 9*a*c^2*e*f - 7*b*c^2*d*f + 2*b^2*c*e*f) + a*c^2*e^2 - 4*b*c^2*d*e + 4*b^2*c*d*f + ((c
*d - a*f + a*c*(-(a*b*f - 2*a*c*e + b*c*d)^2/(a^2*c^2*(4*a*c - b^2)))^(1/2))*(2*c*x^2*(6*b^3*f + 10*a*c^2*e +
5*b*c^2*d - 4*b^2*c*e - 19*a*b*c*f) + 4*b^2*c^2*d - 4*a*b*c^2*e + 4*a*b^2*c*f + (b*c*(c*d - a*f + a*c*(-(a*b*f
- 2*a*c*e + b*c*d)^2/(a^2*c^2*(4*a*c - b^2)))^(1/2))*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a))/(4*a*c) - 2*a*b*c*e*
f))/(4*a*c) - 2*b*c*d*e*f)*(b^2*d*f^2 + c^2*d*e^2 - x^2*(b*f - c*e)*(a*f^2 + c*e^2 - b*e*f - c*d*f) + ((a*f -
c*d + a*c*(-(a*b*f - 2*a*c*e + b*c*d)^2/(a^2*c^2*(4*a*c - b^2)))^(1/2))*(x^2*(b*c^2*e^2 - 3*b^3*f^2 + 5*c^3*d*
e + 11*a*b*c*f^2 - 9*a*c^2*e*f - 7*b*c^2*d*f + 2*b^2*c*e*f) - a*b^2*f^2 - a*c^2*e^2 + 4*b*c^2*d*e - 4*b^2*c*d*
f + ((a*f - c*d + a*c*(-(a*b*f - 2*a*c*e + b*c*d)^2/(a^2*c^2*(4*a*c - b^2)))^(1/2))*(2*c*x^2*(6*b^3*f + 10*a*c
^2*e + 5*b*c^2*d - 4*b^2*c*e - 19*a*b*c*f) + 4*b^2*c^2*d - 4*a*b*c^2*e + 4*a*b^2*c*f - (b*c*(a*f - c*d + a*c*(
-(a*b*f - 2*a*c*e + b*c*d)^2/(a^2*c^2*(4*a*c - b^2)))^(1/2))*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a))/(4*a*c) + 2*a
*b*c*e*f))/(4*a*c) - 2*b*c*d*e*f))*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(2*(16*a^2*c^2 - 4*a*b^2*c
)) + (atan(((4*a*c - b^2)*(((((a*b*f - 2*a*c*e + b*c*d)*(4*b^2*c^2*d - 4*a*b*c^2*e + 4*a*b^2*c*f + (2*a*b^2*c^
2*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(16*a^2*c^2 - 4*a*b^2*c)))/(4*a*c*(4*a*c - b^2)^(1/2)) + (b
^2*c*(a*b*f - 2*a*c*e + b*c*d)*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(2*(16*a^2*c^2 - 4*a*b^2*c)*(4
*a*c - b^2)^(1/2)))*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(2*(16*a^2*c^2 - 4*a*b^2*c)) + ((a*b*f -
2*a*c*e + b*c*d)*(a*b^2*f^2 + a*c^2*e^2 + ((4*b^2*c^2*d - 4*a*b*c^2*e + 4*a*b^2*c*f + (2*a*b^2*c^2*(8*a*c^2*d
+ 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(16*a^2*c^2 - 4*a*b^2*c))*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f
))/(2*(16*a^2*c^2 - 4*a*b^2*c)) - 4*b*c^2*d*e + 4*b^2*c*d*f - 2*a*b*c*e*f))/(4*a*c*(4*a*c - b^2)^(1/2)) - (b^2
*(a*b*f - 2*a*c*e + b*c*d)^3)/(16*a^2*c*(4*a*c - b^2)^(3/2)))*(6*b^4*d + 20*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*
e - 4*a^3*c*f - 28*a*b^2*c*d + 6*a^2*b*c*e))/(c*(a^2*b^2*f^2 + 4*a^2*c^2*e^2 + b^2*c^2*d^2 - 4*a*b*c^2*d*e + 2
*a*b^2*c*d*f - 4*a^2*b*c*e*f)*(a^3*f^2 + 25*a*c^2*d^2 + a^2*c*e^2 - 6*b^2*c*d^2 + 3*a*b^2*d*f - a^2*b*e*f - 10
*a^2*c*d*f - a*b*c*d*e)) + (16*a^3*c*x^2*(((3*b^3*d - a*b^2*e + a^2*b*f + a^2*c*e - 8*a*b*c*d)*(c^2*e^3 + ((8*
a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f)*(3*b^3*f^2 - b*c^2*e^2 + ((8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*
a^2*c*f)*(((12*b^3*c^2 - 40*a*b*c^3)*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(2*(16*a^2*c^2 - 4*a*b^2
*c)) - 8*b^2*c^2*e + 20*a*c^3*e + 10*b*c^3*d + 12*b^3*c*f - 38*a*b*c^2*f))/(2*(16*a^2*c^2 - 4*a*b^2*c)) - 5*c^
3*d*e - 11*a*b*c*f^2 + 9*a*c^2*e*f + 7*b*c^2*d*f - 2*b^2*c*e*f))/(2*(16*a^2*c^2 - 4*a*b^2*c)) + b^2*e*f^2 - a*
b*f^3 + a*c*e*f^2 + b*c*d*f^2 - 2*b*c*e^2*f - c^2*d*e*f - ((((a*b*f - 2*a*c*e + b*c*d)*(((12*b^3*c^2 - 40*a*b*
c^3)*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(2*(16*a^2*c^2 - 4*a*b^2*c)) - 8*b^2*c^2*e + 20*a*c^3*e
+ 10*b*c^3*d + 12*b^3*c*f - 38*a*b*c^2*f))/(4*a*c*(4*a*c - b^2)^(1/2)) + ((12*b^3*c^2 - 40*a*b*c^3)*(a*b*f - 2
*a*c*e + b*c*d)*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(8*a*c*(16*a^2*c^2 - 4*a*b^2*c)*(4*a*c - b^2)
^(1/2)))*(a*b*f - 2*a*c*e + b*c*d))/(4*a*c*(4*a*c - b^2)^(1/2)) - ((12*b^3*c^2 - 40*a*b*c^3)*(a*b*f - 2*a*c*e
+ b*c*d)^2*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(32*a^2*c^2*(16*a^2*c^2 - 4*a*b^2*c)*(4*a*c - b^2)
)))/(8*a^3*c^2*(a^3*f^2 + 25*a*c^2*d^2 + a^2*c*e^2 - 6*b^2*c*d^2 + 3*a*b^2*d*f - a^2*b*e*f - 10*a^2*c*d*f - a*
b*c*d*e)) + ((((((a*b*f - 2*a*c*e + b*c*d)*(((12*b^3*c^2 - 40*a*b*c^3)*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*
a^2*c*f))/(2*(16*a^2*c^2 - 4*a*b^2*c)) - 8*b^2*c^2*e + 20*a*c^3*e + 10*b*c^3*d + 12*b^3*c*f - 38*a*b*c^2*f))/(
4*a*c*(4*a*c - b^2)^(1/2)) + ((12*b^3*c^2 - 40*a*b*c^3)*(a*b*f - 2*a*c*e + b*c*d)*(8*a*c^2*d + 2*a*b^2*f - 2*b
^2*c*d - 8*a^2*c*f))/(8*a*c*(16*a^2*c^2 - 4*a*b^2*c)*(4*a*c - b^2)^(1/2)))*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d
- 8*a^2*c*f))/(2*(16*a^2*c^2 - 4*a*b^2*c)) + ((a*b*f - 2*a*c*e + b*c*d)*(3*b^3*f^2 - b*c^2*e^2 + ((8*a*c^2*d +
2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f)*(((12*b^3*c^2 - 40*a*b*c^3)*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f
))/(2*(16*a^2*c^2 - 4*a*b^2*c)) - 8*b^2*c^2*e + 20*a*c^3*e + 10*b*c^3*d + 12*b^3*c*f - 38*a*b*c^2*f))/(2*(16*a
^2*c^2 - 4*a*b^2*c)) - 5*c^3*d*e - 11*a*b*c*f^2 + 9*a*c^2*e*f + 7*b*c^2*d*f - 2*b^2*c*e*f))/(4*a*c*(4*a*c - b^
2)^(1/2)) - ((12*b^3*c^2 - 40*a*b*c^3)*(a*b*f - 2*a*c*e + b*c*d)^3)/(64*a^3*c^3*(4*a*c - b^2)^(3/2)))*(6*b^4*d
+ 20*a^2*c^2*d + 2*a^2*b^2*f - 2*a*b^3*e - 4*a^3*c*f - 28*a*b^2*c*d + 6*a^2*b*c*e))/(16*a^3*c^2*(4*a*c - b^2)
^(1/2)*(a^3*f^2 + 25*a*c^2*d^2 + a^2*c*e^2 - 6*b^2*c*d^2 + 3*a*b^2*d*f - a^2*b*e*f - 10*a^2*c*d*f - a*b*c*d*e)
))*(4*a*c - b^2)^(3/2))/(a^2*b^2*f^2 + 4*a^2*c^2*e^2 + b^2*c^2*d^2 - 4*a*b*c^2*d*e + 2*a*b^2*c*d*f - 4*a^2*b*c
*e*f) + (2*(4*a*c - b^2)^(3/2)*(3*b^3*d - a*b^2*e + a^2*b*f + a^2*c*e - 8*a*b*c*d)*(b^2*d*f^2 + c^2*d*e^2 + ((
8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f)*(a*b^2*f^2 + a*c^2*e^2 + ((4*b^2*c^2*d - 4*a*b*c^2*e + 4*a*b^2*
c*f + (2*a*b^2*c^2*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(16*a^2*c^2 - 4*a*b^2*c))*(8*a*c^2*d + 2*a
*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(2*(16*a^2*c^2 - 4*a*b^2*c)) - 4*b*c^2*d*e + 4*b^2*c*d*f - 2*a*b*c*e*f))/(2*(
16*a^2*c^2 - 4*a*b^2*c)) - ((((a*b*f - 2*a*c*e + b*c*d)*(4*b^2*c^2*d - 4*a*b*c^2*e + 4*a*b^2*c*f + (2*a*b^2*c^
2*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(16*a^2*c^2 - 4*a*b^2*c)))/(4*a*c*(4*a*c - b^2)^(1/2)) + (b
^2*c*(a*b*f - 2*a*c*e + b*c*d)*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(2*(16*a^2*c^2 - 4*a*b^2*c)*(4
*a*c - b^2)^(1/2)))*(a*b*f - 2*a*c*e + b*c*d))/(4*a*c*(4*a*c - b^2)^(1/2)) - 2*b*c*d*e*f - (b^2*(a*b*f - 2*a*c
*e + b*c*d)^2*(8*a*c^2*d + 2*a*b^2*f - 2*b^2*c*d - 8*a^2*c*f))/(8*a*(16*a^2*c^2 - 4*a*b^2*c)*(4*a*c - b^2))))/
(c*(a^2*b^2*f^2 + 4*a^2*c^2*e^2 + b^2*c^2*d^2 - 4*a*b*c^2*d*e + 2*a*b^2*c*d*f - 4*a^2*b*c*e*f)*(a^3*f^2 + 25*a
*c^2*d^2 + a^2*c*e^2 - 6*b^2*c*d^2 + 3*a*b^2*d*f - a^2*b*e*f - 10*a^2*c*d*f - a*b*c*d*e)))*(a*b*f - 2*a*c*e +
b*c*d))/(2*a*c*(4*a*c - b^2)^(1/2))

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x**4+e*x**2+d)/x/(c*x**4+b*x**2+a),x)

[Out]

Timed out

________________________________________________________________________________________