3.52 \(\int \frac {d+e x^2+f x^4}{x^3 (a+b x^2+c x^4)} \, dx\)

Optimal. Leaf size=118 \[ -\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-a b e-2 a (c d-a f)+b^2 d\right )}{2 a^2 \sqrt {b^2-4 a c}}+\frac {(b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac {\log (x) (b d-a e)}{a^2}-\frac {d}{2 a x^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.29, antiderivative size = 118, 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} \[ -\frac {\tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-a b e-2 a (c d-a f)+b^2 d\right )}{2 a^2 \sqrt {b^2-4 a c}}+\frac {(b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac {\log (x) (b d-a e)}{a^2}-\frac {d}{2 a x^2} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x^2 + f*x^4)/(x^3*(a + b*x^2 + c*x^4)),x]

[Out]

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

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^3 \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^2 \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^2}+\frac {-b d+a e}{a^2 x}+\frac {b^2 d-a b e-a (c d-a f)+c (b d-a e) x}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )\\ &=-\frac {d}{2 a x^2}-\frac {(b d-a e) \log (x)}{a^2}+\frac {\operatorname {Subst}\left (\int \frac {b^2 d-a b e-a (c d-a f)+c (b d-a e) x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^2}\\ &=-\frac {d}{2 a x^2}-\frac {(b d-a e) \log (x)}{a^2}+\frac {(b d-a e) \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2}+\frac {\left (b^2 d-a b e-2 a (c d-a f)\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^2}\\ &=-\frac {d}{2 a x^2}-\frac {(b d-a e) \log (x)}{a^2}+\frac {(b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^2}-\frac {\left (b^2 d-a b e-2 a (c d-a f)\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^2}\\ &=-\frac {d}{2 a x^2}-\frac {\left (b^2 d-a b e-2 a (c d-a f)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c}}-\frac {(b d-a e) \log (x)}{a^2}+\frac {(b d-a e) \log \left (a+b x^2+c x^4\right )}{4 a^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]  time = 0.15, size = 203, normalized size = 1.72 \[ \frac {\frac {\log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right ) \left (a \left (-e \sqrt {b^2-4 a c}+2 a f-2 c d\right )+b \left (d \sqrt {b^2-4 a c}-a e\right )+b^2 d\right )}{\sqrt {b^2-4 a c}}+\frac {\log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right ) \left (-a \left (e \sqrt {b^2-4 a c}+2 a f-2 c d\right )+b \left (d \sqrt {b^2-4 a c}+a e\right )+b^2 (-d)\right )}{\sqrt {b^2-4 a c}}+4 \log (x) (a e-b d)-\frac {2 a d}{x^2}}{4 a^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x^2 + f*x^4)/(x^3*(a + b*x^2 + c*x^4)),x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.64, size = 399, normalized size = 3.38 \[ \left [-\frac {{\left (a b e - 2 \, a^{2} f - {\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt {b^{2} - 4 \, a c} x^{2} \log \left (\frac {2 \, c^{2} x^{4} + 2 \, b c x^{2} + b^{2} - 2 \, a c - {\left (2 \, c x^{2} + b\right )} \sqrt {b^{2} - 4 \, a c}}{c x^{4} + b x^{2} + a}\right ) - {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{2} \log \left (c x^{4} + b x^{2} + a\right ) + 4 \, {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{2} \log \relax (x) + 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d}{4 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2}}, \frac {2 \, {\left (a b e - 2 \, a^{2} f - {\left (b^{2} - 2 \, a c\right )} d\right )} \sqrt {-b^{2} + 4 \, a c} x^{2} \arctan \left (-\frac {{\left (2 \, c x^{2} + b\right )} \sqrt {-b^{2} + 4 \, a c}}{b^{2} - 4 \, a c}\right ) + {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{2} \log \left (c x^{4} + b x^{2} + a\right ) - 4 \, {\left ({\left (b^{3} - 4 \, a b c\right )} d - {\left (a b^{2} - 4 \, a^{2} c\right )} e\right )} x^{2} \log \relax (x) - 2 \, {\left (a b^{2} - 4 \, a^{2} c\right )} d}{4 \, {\left (a^{2} b^{2} - 4 \, a^{3} c\right )} x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x^3/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

[-1/4*((a*b*e - 2*a^2*f - (b^2 - 2*a*c)*d)*sqrt(b^2 - 4*a*c)*x^2*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^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x^2*log(c*x^4
+ b*x^2 + a) + 4*((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x^2*log(x) + 2*(a*b^2 - 4*a^2*c)*d)/((a^2*b^2 - 4*a
^3*c)*x^2), 1/4*(2*(a*b*e - 2*a^2*f - (b^2 - 2*a*c)*d)*sqrt(-b^2 + 4*a*c)*x^2*arctan(-(2*c*x^2 + b)*sqrt(-b^2
+ 4*a*c)/(b^2 - 4*a*c)) + ((b^3 - 4*a*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x^2*log(c*x^4 + b*x^2 + a) - 4*((b^3 - 4*a
*b*c)*d - (a*b^2 - 4*a^2*c)*e)*x^2*log(x) - 2*(a*b^2 - 4*a^2*c)*d)/((a^2*b^2 - 4*a^3*c)*x^2)]

________________________________________________________________________________________

giac [A]  time = 1.78, size = 135, normalized size = 1.14 \[ \frac {{\left (b d - a e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{2}} - \frac {{\left (b d - a e\right )} \log \left (x^{2}\right )}{2 \, a^{2}} + \frac {{\left (b^{2} d - 2 \, a c d + 2 \, a^{2} f - a b e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt {-b^{2} + 4 \, a c} a^{2}} + \frac {b d x^{2} - a x^{2} e - a d}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x^4+e*x^2+d)/x^3/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [B]  time = 0.01, size = 227, normalized size = 1.92 \[ -\frac {b e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a}-\frac {c d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a}+\frac {b^{2} d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \sqrt {4 a c -b^{2}}\, a^{2}}+\frac {f \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}+\frac {e \ln \relax (x )}{a}-\frac {e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a}-\frac {b d \ln \relax (x )}{a^{2}}+\frac {b d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 a^{2}}-\frac {d}{2 a \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x^4+e*x^2+d)/x^3/(c*x^4+b*x^2+a),x)

[Out]

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

________________________________________________________________________________________

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^3/(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 = 7.86, size = 4437, normalized size = 37.60 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x^2 + f*x^4)/(x^3*(a + b*x^2 + c*x^4)),x)

[Out]

(log(x)*(a*e - b*d))/a^2 - d/(2*a*x^2) - (log(((c^2*(a*e - b*d)*(a*f - c*d)^2)/a^3 - ((b*d - a*e + a^2*(-(b^2*
d + 2*a^2*f - a*b*e - 2*a*c*d)^2/(a^4*(4*a*c - b^2)))^(1/2))*(((b*d - a*e + a^2*(-(b^2*d + 2*a^2*f - a*b*e - 2
*a*c*d)^2/(a^4*(4*a*c - b^2)))^(1/2))*((2*c^2*x^2*(10*a*c^2*d + 4*a*b^2*f + b^2*c*d - 10*a^2*c*f - 5*a*b*c*e))
/a + (4*b*c^2*(b^2*d + a^2*f - a*b*e - a*c*d))/a + (b*c^2*(b*d - a*e + a^2*(-(b^2*d + 2*a^2*f - a*b*e - 2*a*c*
d)^2/(a^4*(4*a*c - b^2)))^(1/2))*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a^2))/(4*a^2) + (c^2*(a*f - c*d)*(4*b^2*d + a
^2*f - 4*a*b*e - a*c*d))/a^2 - (c^2*x^2*(a*f - c*d)*(a*b*f + 5*a*c*e - 6*b*c*d))/a^2))/(4*a^2) + (c^2*x^2*(a*f
- c*d)^3)/a^3)*((c^2*(a*e - b*d)*(a*f - c*d)^2)/a^3 - ((a*e - b*d + a^2*(-(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d)
^2/(a^4*(4*a*c - b^2)))^(1/2))*(((a*e - b*d + a^2*(-(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d)^2/(a^4*(4*a*c - b^2)))
^(1/2))*((2*c^2*x^2*(10*a*c^2*d + 4*a*b^2*f + b^2*c*d - 10*a^2*c*f - 5*a*b*c*e))/a + (4*b*c^2*(b^2*d + a^2*f -
a*b*e - a*c*d))/a - (b*c^2*(a*e - b*d + a^2*(-(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d)^2/(a^4*(4*a*c - b^2)))^(1/2
))*(a*b + 3*b^2*x^2 - 10*a*c*x^2))/a^2))/(4*a^2) - (c^2*(a*f - c*d)*(4*b^2*d + a^2*f - 4*a*b*e - a*c*d))/a^2 +
(c^2*x^2*(a*f - c*d)*(a*b*f + 5*a*c*e - 6*b*c*d))/a^2))/(4*a^2) + (c^2*x^2*(a*f - c*d)^3)/a^3))*(2*b^3*d - 2*
a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(2*(16*a^3*c - 4*a^2*b^2)) - (atan((16*a^6*(4*a*c - b^2)^(3/2)*(x^2*((((c^5*
d^3 - a^3*c^2*f^3 + 3*a^2*c^3*d*f^2 - 3*a*c^4*d^2*f)/a^3 + (((a^3*b*c^2*f^2 + 6*a*b*c^4*d^2 - 5*a^2*c^4*d*e +
5*a^3*c^3*e*f - 7*a^2*b*c^3*d*f)/a^3 + (((20*a^3*c^4*d - 20*a^4*c^3*f + 2*a^2*b^2*c^3*d + 8*a^3*b^2*c^2*f - 10
*a^3*b*c^3*e)/a^3 + ((40*a^4*b*c^3 - 12*a^3*b^3*c^2)*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(2*a^3*(16
*a^3*c - 4*a^2*b^2)))*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(2*(16*a^3*c - 4*a^2*b^2)))*(2*b^3*d - 2*
a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(2*(16*a^3*c - 4*a^2*b^2)) - (((((20*a^3*c^4*d - 20*a^4*c^3*f + 2*a^2*b^2*c^
3*d + 8*a^3*b^2*c^2*f - 10*a^3*b*c^3*e)/a^3 + ((40*a^4*b*c^3 - 12*a^3*b^3*c^2)*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*
e - 8*a*b*c*d))/(2*a^3*(16*a^3*c - 4*a^2*b^2)))*(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d))/(4*a^2*(4*a*c - b^2)^(1/2
)) + ((40*a^4*b*c^3 - 12*a^3*b^3*c^2)*(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d)*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8
*a*b*c*d))/(8*a^5*(4*a*c - b^2)^(1/2)*(16*a^3*c - 4*a^2*b^2)))*(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d))/(4*a^2*(4*
a*c - b^2)^(1/2)) - ((40*a^4*b*c^3 - 12*a^3*b^3*c^2)*(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d)^2*(2*b^3*d - 2*a*b^2*
e + 8*a^2*c*e - 8*a*b*c*d))/(32*a^7*(4*a*c - b^2)*(16*a^3*c - 4*a^2*b^2)))*(3*b^4*d + a^2*c^2*d + a^2*b^2*f -
3*a*b^3*e - a^3*c*f - 9*a*b^2*c*d + 8*a^2*b*c*e))/(8*a^3*c^2*(a^4*f^2 - 6*b^4*d^2 + 25*a^3*c*e^2 - 6*a^2*b^2*e
^2 + a^2*c^2*d^2 + 12*a*b^3*d*e - a^3*b*e*f - 2*a^3*c*d*f + 24*a*b^2*c*d^2 + a^2*b^2*d*f - 49*a^2*b*c*d*e)) -
(((((((20*a^3*c^4*d - 20*a^4*c^3*f + 2*a^2*b^2*c^3*d + 8*a^3*b^2*c^2*f - 10*a^3*b*c^3*e)/a^3 + ((40*a^4*b*c^3
- 12*a^3*b^3*c^2)*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(2*a^3*(16*a^3*c - 4*a^2*b^2)))*(b^2*d + 2*a^
2*f - a*b*e - 2*a*c*d))/(4*a^2*(4*a*c - b^2)^(1/2)) + ((40*a^4*b*c^3 - 12*a^3*b^3*c^2)*(b^2*d + 2*a^2*f - a*b*
e - 2*a*c*d)*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(8*a^5*(4*a*c - b^2)^(1/2)*(16*a^3*c - 4*a^2*b^2))
)*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(2*(16*a^3*c - 4*a^2*b^2)) + (((a^3*b*c^2*f^2 + 6*a*b*c^4*d^2
- 5*a^2*c^4*d*e + 5*a^3*c^3*e*f - 7*a^2*b*c^3*d*f)/a^3 + (((20*a^3*c^4*d - 20*a^4*c^3*f + 2*a^2*b^2*c^3*d + 8
*a^3*b^2*c^2*f - 10*a^3*b*c^3*e)/a^3 + ((40*a^4*b*c^3 - 12*a^3*b^3*c^2)*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a
*b*c*d))/(2*a^3*(16*a^3*c - 4*a^2*b^2)))*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(2*(16*a^3*c - 4*a^2*b
^2)))*(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d))/(4*a^2*(4*a*c - b^2)^(1/2)) - ((40*a^4*b*c^3 - 12*a^3*b^3*c^2)*(b^2
*d + 2*a^2*f - a*b*e - 2*a*c*d)^3)/(64*a^9*(4*a*c - b^2)^(3/2)))*(6*b^5*d + 2*a^2*b^3*f - 20*a^3*c^2*e - 6*a*b
^4*e - 30*a*b^3*c*d - 6*a^3*b*c*f + 26*a^2*b*c^2*d + 28*a^2*b^2*c*e))/(16*a^3*c^2*(4*a*c - b^2)^(1/2)*(a^4*f^2
- 6*b^4*d^2 + 25*a^3*c*e^2 - 6*a^2*b^2*e^2 + a^2*c^2*d^2 + 12*a*b^3*d*e - a^3*b*e*f - 2*a^3*c*d*f + 24*a*b^2*
c*d^2 + a^2*b^2*d*f - 49*a^2*b*c*d*e))) + (((b*c^4*d^3 - a^3*c^2*e*f^2 - a*c^4*d^2*e - 2*a*b*c^3*d^2*f + 2*a^2
*c^3*d*e*f + a^2*b*c^2*d*f^2)/a^3 - (((a^2*c^4*d^2 + a^4*c^2*f^2 - 4*a*b^2*c^3*d^2 - 2*a^3*c^3*d*f + 4*a^2*b*c
^3*d*e - 4*a^3*b*c^2*e*f + 4*a^2*b^2*c^2*d*f)/a^3 - (((4*a^2*b^3*c^2*d - 4*a^3*b^2*c^2*e - 4*a^3*b*c^3*d + 4*a
^4*b*c^2*f)/a^3 - (2*a*b^2*c^2*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(16*a^3*c - 4*a^2*b^2))*(2*b^3*d
- 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(2*(16*a^3*c - 4*a^2*b^2)))*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*
d))/(2*(16*a^3*c - 4*a^2*b^2)) - (((((4*a^2*b^3*c^2*d - 4*a^3*b^2*c^2*e - 4*a^3*b*c^3*d + 4*a^4*b*c^2*f)/a^3 -
(2*a*b^2*c^2*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(16*a^3*c - 4*a^2*b^2))*(b^2*d + 2*a^2*f - a*b*e
- 2*a*c*d))/(4*a^2*(4*a*c - b^2)^(1/2)) - (b^2*c^2*(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d)*(2*b^3*d - 2*a*b^2*e +
8*a^2*c*e - 8*a*b*c*d))/(2*a*(4*a*c - b^2)^(1/2)*(16*a^3*c - 4*a^2*b^2)))*(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d))
/(4*a^2*(4*a*c - b^2)^(1/2)) + (b^2*c^2*(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d)^2*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e
- 8*a*b*c*d))/(8*a^3*(4*a*c - b^2)*(16*a^3*c - 4*a^2*b^2)))*(3*b^4*d + a^2*c^2*d + a^2*b^2*f - 3*a*b^3*e - a^
3*c*f - 9*a*b^2*c*d + 8*a^2*b*c*e))/(8*a^3*c^2*(a^4*f^2 - 6*b^4*d^2 + 25*a^3*c*e^2 - 6*a^2*b^2*e^2 + a^2*c^2*d
^2 + 12*a*b^3*d*e - a^3*b*e*f - 2*a^3*c*d*f + 24*a*b^2*c*d^2 + a^2*b^2*d*f - 49*a^2*b*c*d*e)) - (((((((4*a^2*b
^3*c^2*d - 4*a^3*b^2*c^2*e - 4*a^3*b*c^3*d + 4*a^4*b*c^2*f)/a^3 - (2*a*b^2*c^2*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*
e - 8*a*b*c*d))/(16*a^3*c - 4*a^2*b^2))*(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d))/(4*a^2*(4*a*c - b^2)^(1/2)) - (b^
2*c^2*(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d)*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(2*a*(4*a*c - b^2)^(1
/2)*(16*a^3*c - 4*a^2*b^2)))*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(2*(16*a^3*c - 4*a^2*b^2)) - (((a^
2*c^4*d^2 + a^4*c^2*f^2 - 4*a*b^2*c^3*d^2 - 2*a^3*c^3*d*f + 4*a^2*b*c^3*d*e - 4*a^3*b*c^2*e*f + 4*a^2*b^2*c^2*
d*f)/a^3 - (((4*a^2*b^3*c^2*d - 4*a^3*b^2*c^2*e - 4*a^3*b*c^3*d + 4*a^4*b*c^2*f)/a^3 - (2*a*b^2*c^2*(2*b^3*d -
2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(16*a^3*c - 4*a^2*b^2))*(2*b^3*d - 2*a*b^2*e + 8*a^2*c*e - 8*a*b*c*d))/(2
*(16*a^3*c - 4*a^2*b^2)))*(b^2*d + 2*a^2*f - a*b*e - 2*a*c*d))/(4*a^2*(4*a*c - b^2)^(1/2)) + (b^2*c^2*(b^2*d +
2*a^2*f - a*b*e - 2*a*c*d)^3)/(16*a^5*(4*a*c - b^2)^(3/2)))*(6*b^5*d + 2*a^2*b^3*f - 20*a^3*c^2*e - 6*a*b^4*e
- 30*a*b^3*c*d - 6*a^3*b*c*f + 26*a^2*b*c^2*d + 28*a^2*b^2*c*e))/(16*a^3*c^2*(4*a*c - b^2)^(1/2)*(a^4*f^2 - 6
*b^4*d^2 + 25*a^3*c*e^2 - 6*a^2*b^2*e^2 + a^2*c^2*d^2 + 12*a*b^3*d*e - a^3*b*e*f - 2*a^3*c*d*f + 24*a*b^2*c*d^
2 + a^2*b^2*d*f - 49*a^2*b*c*d*e))))/(4*a^2*c^4*d^2 + b^4*c^2*d^2 + 4*a^4*c^2*f^2 - 4*a*b^2*c^3*d^2 + a^2*b^2*
c^2*e^2 - 8*a^3*c^3*d*f - 2*a*b^3*c^2*d*e + 4*a^2*b*c^3*d*e - 4*a^3*b*c^2*e*f + 4*a^2*b^2*c^2*d*f))*(b^2*d + 2
*a^2*f - a*b*e - 2*a*c*d))/(2*a^2*(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**3/(c*x**4+b*x**2+a),x)

[Out]

Timed out

________________________________________________________________________________________