3.301 \(\int \frac {1}{x^3 (d+e x^2) (a+b x^2+c x^4)} \, dx\)

Optimal. Leaf size=205 \[ \frac {\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x^2+c x^4\right )}{4 a^2 \left (a e^2-b d e+c d^2\right )}-\frac {\left (3 a b c e-2 a c^2 d+b^3 (-e)+b^2 c d\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac {\log (x) (a e+b d)}{a^2 d^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2-b d e+c d^2\right )}-\frac {1}{2 a d x^2} \]

[Out]

-1/2/a/d/x^2-(a*e+b*d)*ln(x)/a^2/d^2+1/2*e^3*ln(e*x^2+d)/d^2/(a*e^2-b*d*e+c*d^2)+1/4*(a*c*e-b^2*e+b*c*d)*ln(c*
x^4+b*x^2+a)/a^2/(a*e^2-b*d*e+c*d^2)-1/2*(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^
(1/2))/a^2/(a*e^2-b*d*e+c*d^2)/(-4*a*c+b^2)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.47, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {1251, 893, 634, 618, 206, 628} \[ -\frac {\left (3 a b c e-2 a c^2 d+b^2 c d+b^3 (-e)\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^2 \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac {\left (a c e+b^2 (-e)+b c d\right ) \log \left (a+b x^2+c x^4\right )}{4 a^2 \left (a e^2-b d e+c d^2\right )}-\frac {\log (x) (a e+b d)}{a^2 d^2}+\frac {e^3 \log \left (d+e x^2\right )}{2 d^2 \left (a e^2-b d e+c d^2\right )}-\frac {1}{2 a d x^2} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^3*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

-1/(2*a*d*x^2) - ((b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^2*S
qrt[b^2 - 4*a*c]*(c*d^2 - b*d*e + a*e^2)) - ((b*d + a*e)*Log[x])/(a^2*d^2) + (e^3*Log[d + e*x^2])/(2*d^2*(c*d^
2 - b*d*e + a*e^2)) + ((b*c*d - b^2*e + a*c*e)*Log[a + b*x^2 + c*x^4])/(4*a^2*(c*d^2 - b*d*e + a*e^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 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rule 1251

Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Dist[1/2,
Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] &&
IntegerQ[(m - 1)/2]

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 0.34, size = 331, normalized size = 1.61 \[ \frac {1}{4} \left (\frac {\left (b^2 \left (e \sqrt {b^2-4 a c}-c d\right )-b c \left (d \sqrt {b^2-4 a c}+3 a e\right )+a c \left (2 c d-e \sqrt {b^2-4 a c}\right )+b^3 e\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right )}{a^2 \sqrt {b^2-4 a c} \left (e (b d-a e)-c d^2\right )}+\frac {\left (b^2 \left (e \sqrt {b^2-4 a c}+c d\right )+b c \left (3 a e-d \sqrt {b^2-4 a c}\right )-a c \left (e \sqrt {b^2-4 a c}+2 c d\right )+b^3 (-e)\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right )}{a^2 \sqrt {b^2-4 a c} \left (e (b d-a e)-c d^2\right )}-\frac {4 \log (x) (a e+b d)}{a^2 d^2}+\frac {2 e^3 \log \left (d+e x^2\right )}{d^2 e (a e-b d)+c d^4}-\frac {2}{a d x^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^3*(d + e*x^2)*(a + b*x^2 + c*x^4)),x]

[Out]

(-2/(a*d*x^2) - (4*(b*d + a*e)*Log[x])/(a^2*d^2) + ((b^3*e - b*c*(Sqrt[b^2 - 4*a*c]*d + 3*a*e) + a*c*(2*c*d -
Sqrt[b^2 - 4*a*c]*e) + b^2*(-(c*d) + Sqrt[b^2 - 4*a*c]*e))*Log[b - Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(a^2*Sqrt[b^2
- 4*a*c]*(-(c*d^2) + e*(b*d - a*e))) + ((-(b^3*e) + b*c*(-(Sqrt[b^2 - 4*a*c]*d) + 3*a*e) + b^2*(c*d + Sqrt[b^
2 - 4*a*c]*e) - a*c*(2*c*d + Sqrt[b^2 - 4*a*c]*e))*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(a^2*Sqrt[b^2 - 4*a*c
]*(-(c*d^2) + e*(b*d - a*e))) + (2*e^3*Log[d + e*x^2])/(c*d^4 + d^2*e*(-(b*d) + a*e)))/4

________________________________________________________________________________________

fricas [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(1/x^3/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [A]  time = 1.96, size = 237, normalized size = 1.16 \[ \frac {{\left (b c d - b^{2} e + a c e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )}} + \frac {e^{4} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c d^{4} e - b d^{3} e^{2} + a d^{2} e^{3}\right )}} + \frac {{\left (b^{2} c d - 2 \, a c^{2} d - b^{3} e + 3 \, a b c e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{2} c d^{2} - a^{2} b d e + a^{3} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} - \frac {{\left (b d + a e\right )} \log \left (x^{2}\right )}{2 \, a^{2} d^{2}} + \frac {b d x^{2} + a x^{2} e - a d}{2 \, a^{2} d^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^3/(e*x^2+d)/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

1/4*(b*c*d - b^2*e + a*c*e)*log(c*x^4 + b*x^2 + a)/(a^2*c*d^2 - a^2*b*d*e + a^3*e^2) + 1/2*e^4*log(abs(x^2*e +
d))/(c*d^4*e - b*d^3*e^2 + a*d^2*e^3) + 1/2*(b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)*arctan((2*c*x^2 + b)/sq
rt(-b^2 + 4*a*c))/((a^2*c*d^2 - a^2*b*d*e + a^3*e^2)*sqrt(-b^2 + 4*a*c)) - 1/2*(b*d + a*e)*log(x^2)/(a^2*d^2)
+ 1/2*(b*d*x^2 + a*x^2*e - a*d)/(a^2*d^2*x^2)

________________________________________________________________________________________

maple [B]  time = 0.02, size = 430, normalized size = 2.10 \[ \frac {3 b c e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right ) \sqrt {4 a c -b^{2}}\, a}-\frac {c^{2} d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-d e b +c \,d^{2}\right ) \sqrt {4 a c -b^{2}}\, a}-\frac {b^{3} e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right ) \sqrt {4 a c -b^{2}}\, a^{2}}+\frac {b^{2} c d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right ) \sqrt {4 a c -b^{2}}\, a^{2}}+\frac {c e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 \left (a \,e^{2}-d e b +c \,d^{2}\right ) a}-\frac {b^{2} e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 \left (a \,e^{2}-d e b +c \,d^{2}\right ) a^{2}}+\frac {b c d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 \left (a \,e^{2}-d e b +c \,d^{2}\right ) a^{2}}+\frac {e^{3} \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right ) d^{2}}-\frac {e \ln \relax (x )}{a \,d^{2}}-\frac {b \ln \relax (x )}{a^{2} d}-\frac {1}{2 a d \,x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^3/(e*x^2+d)/(c*x^4+b*x^2+a),x)

[Out]

-1/2/a/d/x^2-1/a/d^2*e*ln(x)-1/d/a^2*ln(x)*b+1/4/(a*e^2-b*d*e+c*d^2)/a*c*ln(c*x^4+b*x^2+a)*e-1/4/(a*e^2-b*d*e+
c*d^2)/a^2*ln(c*x^4+b*x^2+a)*b^2*e+1/4/(a*e^2-b*d*e+c*d^2)/a^2*c*ln(c*x^4+b*x^2+a)*b*d+3/2/(a*e^2-b*d*e+c*d^2)
/a/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*c*e-1/(a*e^2-b*d*e+c*d^2)/a/(4*a*c-b^2)^(1/2)*arc
tan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*c^2*d-1/2/(a*e^2-b*d*e+c*d^2)/a^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a
*c-b^2)^(1/2))*b^3*e+1/2/(a*e^2-b*d*e+c*d^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*c
*d+1/2*e^3*ln(e*x^2+d)/d^2/(a*e^2-b*d*e+c*d^2)

________________________________________________________________________________________

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(1/x^3/(e*x^2+d)/(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 = 62.95, size = 5368, normalized size = 26.19 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^3*(d + e*x^2)*(a + b*x^2 + c*x^4)),x)

[Out]

(log((((((4*c^2*e^2*(a*c^6*d^7 - 4*a^2*b^5*e^7 - 4*b^2*c^5*d^7 - 4*b^7*d^2*e^5 + 28*a^3*b^3*c*e^7 - 48*a^4*b*c
^2*e^7 + 8*b^3*c^4*d^6*e + 8*b^6*c*d^3*e^4 - 16*a^2*c^5*d^5*e^2 + 16*a^3*c^4*d^3*e^4 - 4*b^4*c^3*d^5*e^2 - 4*b
^5*c^2*d^4*e^3 - 7*a*b^6*d*e^6 - 20*a*b*c^5*d^6*e + 56*a^2*b^2*c^3*d^3*e^4 - 76*a^2*b^3*c^2*d^2*e^5 + 32*a*b^5
*c*d^2*e^5 + 46*a^2*b^4*c*d*e^6 + 20*a*b^2*c^4*d^5*e^2 + 6*a*b^3*c^3*d^4*e^3 - 44*a*b^4*c^2*d^3*e^4 + 22*a^2*b
*c^4*d^4*e^3 + 48*a^3*b*c^3*d^2*e^5 - 75*a^3*b^2*c^2*d*e^6))/(a^2*d^2) + (((16*c^2*e^2*(a^3*b^4*e^7 + 16*a^5*c
^2*e^7 + b^3*c^4*d^7 + b^7*d^3*e^4 - 8*a^4*b^2*c*e^7 + 2*a*b^6*d^2*e^5 + 2*a^2*b^5*d*e^6 - 4*a^2*c^5*d^6*e - 4
*b^4*c^3*d^6*e - 4*b^6*c*d^4*e^3 + 20*a^3*c^4*d^4*e^3 - 32*a^4*c^3*d^2*e^5 + 6*b^5*c^2*d^5*e^2 - a*b*c^5*d^7 -
52*a^2*b^2*c^3*d^4*e^3 + 45*a^2*b^3*c^2*d^3*e^4 + 48*a^3*b^2*c^2*d^2*e^5 + 11*a*b^2*c^4*d^6*e - 12*a*b^5*c*d^
3*e^4 - 15*a^3*b^3*c*d*e^6 + 28*a^4*b*c^2*d*e^6 - 27*a*b^3*c^3*d^5*e^2 + 27*a*b^4*c^2*d^4*e^3 + 27*a^2*b*c^4*d
^5*e^2 - 18*a^2*b^4*c*d^2*e^5 - 52*a^3*b*c^3*d^3*e^4))/(a*d) + (8*c^2*e^2*x^2*(10*a*c^6*d^7 + a^2*b^5*e^7 + b^
2*c^5*d^7 + b^7*d^2*e^5 - 11*a^3*b^3*c*e^7 + 28*a^4*b*c^2*e^7 - 88*a^4*c^3*d*e^6 - 6*b^3*c^4*d^6*e - 6*b^6*c*d
^3*e^4 + 26*a^2*c^5*d^5*e^2 + 88*a^3*c^4*d^3*e^4 + 5*b^4*c^3*d^5*e^2 + 5*b^5*c^2*d^4*e^3 + 12*a*b^6*d*e^6 - 3*
a*b*c^5*d^6*e - 110*a^2*b^2*c^3*d^3*e^4 + 155*a^2*b^3*c^2*d^2*e^5 - 28*a*b^5*c*d^2*e^5 - 93*a^2*b^4*c*d*e^6 -
10*a*b^2*c^4*d^5*e^2 - 27*a*b^3*c^3*d^4*e^3 + 46*a*b^4*c^2*d^3*e^4 + 15*a^2*b*c^4*d^4*e^3 - 236*a^3*b*c^3*d^2*
e^5 + 202*a^3*b^2*c^2*d*e^6))/(a*d) + (4*c^2*e^2*(a*b^2*e^3 + b*c^2*d^3 - 4*a^2*c*e^3 + b^3*d*e^2 + 4*a*c^2*d^
2*e - 2*b^2*c*d^2*e - 3*a*b*c*d*e^2)*(b^4*e + b^3*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2*e - b^3*c*d + 4*a*b*c^2*d
- 5*a*b^2*c*e + 2*a*c^2*d*(b^2 - 4*a*c)^(1/2) - b^2*c*d*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e*(b^2 - 4*a*c)^(1/2))*(
a*b^3*d^2*e^2 + a^2*b^2*d*e^3 + 4*a^2*c^2*d^3*e - 10*a*c^3*d^4*x^2 - 12*a^3*c*e^4*x^2 + 3*a^2*b^2*e^4*x^2 + 3*
b^2*c^2*d^4*x^2 + 3*b^4*d^2*e^2*x^2 + a*b*c^2*d^4 - 4*a^3*c*d*e^3 - 2*a*b^2*c*d^3*e - 14*a^2*c^2*d^2*e^2*x^2 -
3*a^2*b*c*d^2*e^2 - 4*a*b^3*d*e^3*x^2 - 6*b^3*c*d^3*e*x^2 - 8*a*b^2*c*d^2*e^2*x^2 + 22*a*b*c^2*d^3*e*x^2 + 16
*a^2*b*c*d*e^3*x^2))/(a^2*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)))*(b^4*e + b^3*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c
^2*e - b^3*c*d + 4*a*b*c^2*d - 5*a*b^2*c*e + 2*a*c^2*d*(b^2 - 4*a*c)^(1/2) - b^2*c*d*(b^2 - 4*a*c)^(1/2) - 3*a
*b*c*e*(b^2 - 4*a*c)^(1/2)))/(4*a^2*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)) - (4*c^2*e^2*x^2*(6*a*b^6*e^7 + 6*b
*c^6*d^7 + 6*b^7*d*e^6 - 16*a^4*c^3*e^7 - 44*a^2*b^4*c*e^7 - 8*b^2*c^5*d^6*e - 8*b^6*c*d^2*e^5 + 84*a^3*b^2*c^
2*e^7 + 30*a^2*c^5*d^4*e^3 - 2*b^3*c^4*d^5*e^2 + 8*b^4*c^3*d^4*e^3 - 2*b^5*c^2*d^3*e^4 + 11*a*c^6*d^6*e - 47*a
*b^5*c*d*e^6 - 96*a^2*b^2*c^3*d^2*e^5 + 14*a*b*c^5*d^5*e^2 - 94*a^3*b*c^3*d*e^6 - 35*a*b^2*c^4*d^4*e^3 + 7*a*b
^3*c^3*d^3*e^4 + 56*a*b^4*c^2*d^2*e^5 - 17*a^2*b*c^4*d^3*e^4 + 117*a^2*b^3*c^2*d*e^6))/(a^2*d^2))*(b^4*e + b^3
*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2*e - b^3*c*d + 4*a*b*c^2*d - 5*a*b^2*c*e + 2*a*c^2*d*(b^2 - 4*a*c)^(1/2) - b
^2*c*d*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e*(b^2 - 4*a*c)^(1/2)))/(4*a^2*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)) + (
4*c^2*e^2*x^2*(b^7*e^7 + c^7*d^7 - 6*a^3*b*c^3*e^7 + 2*a*c^6*d^5*e^2 - 4*a^3*c^4*d*e^6 + 14*a^2*b^3*c^2*e^7 +
6*a^2*c^5*d^3*e^4 + b^3*c^4*d^4*e^3 + b^4*c^3*d^3*e^4 - 7*a*b^5*c*e^7 + 2*a*b^4*c^2*d*e^6 - 6*a*b^2*c^4*d^3*e^
4 + 3*a*b^3*c^3*d^2*e^5 - 9*a^2*b*c^4*d^2*e^5 - 5*a^2*b^2*c^3*d*e^6))/(a^3*d^3) + (4*c^2*e^2*(a*e + b*d)*(b^3*
e^3 + c^3*d^3 - 3*a*b*c*e^3)^2)/(a^3*d^3))*(b^4*e + b^3*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2*e - b^3*c*d + 4*a*b*
c^2*d - 5*a*b^2*c*e + 2*a*c^2*d*(b^2 - 4*a*c)^(1/2) - b^2*c*d*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e*(b^2 - 4*a*c)^(1
/2)))/(4*a^2*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)) - (2*c^5*e^5*x^2*(b^3*e^3 + c^3*d^3 - 3*a*b*c*e^3))/(a^3*d
^3))*(b^4*e + b^3*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2*e - b^3*c*d + 4*a*b*c^2*d - 5*a*b^2*c*e + 2*a*c^2*d*(b^2 -
4*a*c)^(1/2) - b^2*c*d*(b^2 - 4*a*c)^(1/2) - 3*a*b*c*e*(b^2 - 4*a*c)^(1/2)))/(4*(4*a^4*c*e^2 - a^3*b^2*e^2 +
4*a^3*c^2*d^2 - a^2*b^2*c*d^2 + a^2*b^3*d*e - 4*a^3*b*c*d*e)) + (log((((((4*c^2*e^2*(a*c^6*d^7 - 4*a^2*b^5*e^7
- 4*b^2*c^5*d^7 - 4*b^7*d^2*e^5 + 28*a^3*b^3*c*e^7 - 48*a^4*b*c^2*e^7 + 8*b^3*c^4*d^6*e + 8*b^6*c*d^3*e^4 - 1
6*a^2*c^5*d^5*e^2 + 16*a^3*c^4*d^3*e^4 - 4*b^4*c^3*d^5*e^2 - 4*b^5*c^2*d^4*e^3 - 7*a*b^6*d*e^6 - 20*a*b*c^5*d^
6*e + 56*a^2*b^2*c^3*d^3*e^4 - 76*a^2*b^3*c^2*d^2*e^5 + 32*a*b^5*c*d^2*e^5 + 46*a^2*b^4*c*d*e^6 + 20*a*b^2*c^4
*d^5*e^2 + 6*a*b^3*c^3*d^4*e^3 - 44*a*b^4*c^2*d^3*e^4 + 22*a^2*b*c^4*d^4*e^3 + 48*a^3*b*c^3*d^2*e^5 - 75*a^3*b
^2*c^2*d*e^6))/(a^2*d^2) + (((16*c^2*e^2*(a^3*b^4*e^7 + 16*a^5*c^2*e^7 + b^3*c^4*d^7 + b^7*d^3*e^4 - 8*a^4*b^2
*c*e^7 + 2*a*b^6*d^2*e^5 + 2*a^2*b^5*d*e^6 - 4*a^2*c^5*d^6*e - 4*b^4*c^3*d^6*e - 4*b^6*c*d^4*e^3 + 20*a^3*c^4*
d^4*e^3 - 32*a^4*c^3*d^2*e^5 + 6*b^5*c^2*d^5*e^2 - a*b*c^5*d^7 - 52*a^2*b^2*c^3*d^4*e^3 + 45*a^2*b^3*c^2*d^3*e
^4 + 48*a^3*b^2*c^2*d^2*e^5 + 11*a*b^2*c^4*d^6*e - 12*a*b^5*c*d^3*e^4 - 15*a^3*b^3*c*d*e^6 + 28*a^4*b*c^2*d*e^
6 - 27*a*b^3*c^3*d^5*e^2 + 27*a*b^4*c^2*d^4*e^3 + 27*a^2*b*c^4*d^5*e^2 - 18*a^2*b^4*c*d^2*e^5 - 52*a^3*b*c^3*d
^3*e^4))/(a*d) + (8*c^2*e^2*x^2*(10*a*c^6*d^7 + a^2*b^5*e^7 + b^2*c^5*d^7 + b^7*d^2*e^5 - 11*a^3*b^3*c*e^7 + 2
8*a^4*b*c^2*e^7 - 88*a^4*c^3*d*e^6 - 6*b^3*c^4*d^6*e - 6*b^6*c*d^3*e^4 + 26*a^2*c^5*d^5*e^2 + 88*a^3*c^4*d^3*e
^4 + 5*b^4*c^3*d^5*e^2 + 5*b^5*c^2*d^4*e^3 + 12*a*b^6*d*e^6 - 3*a*b*c^5*d^6*e - 110*a^2*b^2*c^3*d^3*e^4 + 155*
a^2*b^3*c^2*d^2*e^5 - 28*a*b^5*c*d^2*e^5 - 93*a^2*b^4*c*d*e^6 - 10*a*b^2*c^4*d^5*e^2 - 27*a*b^3*c^3*d^4*e^3 +
46*a*b^4*c^2*d^3*e^4 + 15*a^2*b*c^4*d^4*e^3 - 236*a^3*b*c^3*d^2*e^5 + 202*a^3*b^2*c^2*d*e^6))/(a*d) + (4*c^2*e
^2*(a*b^2*e^3 + b*c^2*d^3 - 4*a^2*c*e^3 + b^3*d*e^2 + 4*a*c^2*d^2*e - 2*b^2*c*d^2*e - 3*a*b*c*d*e^2)*(b^4*e -
b^3*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2*e - b^3*c*d + 4*a*b*c^2*d - 5*a*b^2*c*e - 2*a*c^2*d*(b^2 - 4*a*c)^(1/2)
+ b^2*c*d*(b^2 - 4*a*c)^(1/2) + 3*a*b*c*e*(b^2 - 4*a*c)^(1/2))*(a*b^3*d^2*e^2 + a^2*b^2*d*e^3 + 4*a^2*c^2*d^3*
e - 10*a*c^3*d^4*x^2 - 12*a^3*c*e^4*x^2 + 3*a^2*b^2*e^4*x^2 + 3*b^2*c^2*d^4*x^2 + 3*b^4*d^2*e^2*x^2 + a*b*c^2*
d^4 - 4*a^3*c*d*e^3 - 2*a*b^2*c*d^3*e - 14*a^2*c^2*d^2*e^2*x^2 - 3*a^2*b*c*d^2*e^2 - 4*a*b^3*d*e^3*x^2 - 6*b^3
*c*d^3*e*x^2 - 8*a*b^2*c*d^2*e^2*x^2 + 22*a*b*c^2*d^3*e*x^2 + 16*a^2*b*c*d*e^3*x^2))/(a^2*(4*a*c - b^2)*(a*e^2
+ c*d^2 - b*d*e)))*(b^4*e - b^3*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2*e - b^3*c*d + 4*a*b*c^2*d - 5*a*b^2*c*e - 2
*a*c^2*d*(b^2 - 4*a*c)^(1/2) + b^2*c*d*(b^2 - 4*a*c)^(1/2) + 3*a*b*c*e*(b^2 - 4*a*c)^(1/2)))/(4*a^2*(4*a*c - b
^2)*(a*e^2 + c*d^2 - b*d*e)) - (4*c^2*e^2*x^2*(6*a*b^6*e^7 + 6*b*c^6*d^7 + 6*b^7*d*e^6 - 16*a^4*c^3*e^7 - 44*a
^2*b^4*c*e^7 - 8*b^2*c^5*d^6*e - 8*b^6*c*d^2*e^5 + 84*a^3*b^2*c^2*e^7 + 30*a^2*c^5*d^4*e^3 - 2*b^3*c^4*d^5*e^2
+ 8*b^4*c^3*d^4*e^3 - 2*b^5*c^2*d^3*e^4 + 11*a*c^6*d^6*e - 47*a*b^5*c*d*e^6 - 96*a^2*b^2*c^3*d^2*e^5 + 14*a*b
*c^5*d^5*e^2 - 94*a^3*b*c^3*d*e^6 - 35*a*b^2*c^4*d^4*e^3 + 7*a*b^3*c^3*d^3*e^4 + 56*a*b^4*c^2*d^2*e^5 - 17*a^2
*b*c^4*d^3*e^4 + 117*a^2*b^3*c^2*d*e^6))/(a^2*d^2))*(b^4*e - b^3*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2*e - b^3*c*d
+ 4*a*b*c^2*d - 5*a*b^2*c*e - 2*a*c^2*d*(b^2 - 4*a*c)^(1/2) + b^2*c*d*(b^2 - 4*a*c)^(1/2) + 3*a*b*c*e*(b^2 -
4*a*c)^(1/2)))/(4*a^2*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*d*e)) + (4*c^2*e^2*x^2*(b^7*e^7 + c^7*d^7 - 6*a^3*b*c^3
*e^7 + 2*a*c^6*d^5*e^2 - 4*a^3*c^4*d*e^6 + 14*a^2*b^3*c^2*e^7 + 6*a^2*c^5*d^3*e^4 + b^3*c^4*d^4*e^3 + b^4*c^3*
d^3*e^4 - 7*a*b^5*c*e^7 + 2*a*b^4*c^2*d*e^6 - 6*a*b^2*c^4*d^3*e^4 + 3*a*b^3*c^3*d^2*e^5 - 9*a^2*b*c^4*d^2*e^5
- 5*a^2*b^2*c^3*d*e^6))/(a^3*d^3) + (4*c^2*e^2*(a*e + b*d)*(b^3*e^3 + c^3*d^3 - 3*a*b*c*e^3)^2)/(a^3*d^3))*(b^
4*e - b^3*e*(b^2 - 4*a*c)^(1/2) + 4*a^2*c^2*e - b^3*c*d + 4*a*b*c^2*d - 5*a*b^2*c*e - 2*a*c^2*d*(b^2 - 4*a*c)^
(1/2) + b^2*c*d*(b^2 - 4*a*c)^(1/2) + 3*a*b*c*e*(b^2 - 4*a*c)^(1/2)))/(4*a^2*(4*a*c - b^2)*(a*e^2 + c*d^2 - b*
d*e)) - (2*c^5*e^5*x^2*(b^3*e^3 + c^3*d^3 - 3*a*b*c*e^3))/(a^3*d^3))*(b^4*e - b^3*e*(b^2 - 4*a*c)^(1/2) + 4*a^
2*c^2*e - b^3*c*d + 4*a*b*c^2*d - 5*a*b^2*c*e - 2*a*c^2*d*(b^2 - 4*a*c)^(1/2) + b^2*c*d*(b^2 - 4*a*c)^(1/2) +
3*a*b*c*e*(b^2 - 4*a*c)^(1/2)))/(4*(4*a^4*c*e^2 - a^3*b^2*e^2 + 4*a^3*c^2*d^2 - a^2*b^2*c*d^2 + a^2*b^3*d*e -
4*a^3*b*c*d*e)) + (e^3*log(d + e*x^2))/(2*c*d^4 + 2*a*d^2*e^2 - 2*b*d^3*e) - 1/(2*a*d*x^2) - (log(x)*(a*e + b*
d))/(a^2*d^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(1/x**3/(e*x**2+d)/(c*x**4+b*x**2+a),x)

[Out]

Timed out

________________________________________________________________________________________