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

Optimal. Leaf size=167 \[ \frac {\left (2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac {e^2 \log \left (d+e x^2\right )}{2 d \left (a e^2-b d e+c d^2\right )}-\frac {(c d-b e) \log \left (a+b x^2+c x^4\right )}{4 a \left (a e^2-b d e+c d^2\right )}+\frac {\log (x)}{a d} \]

[Out]

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

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Rubi [A]  time = 0.31, antiderivative size = 167, 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} \[ \frac {\left (2 a c e+b^2 (-e)+b c d\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c} \left (a e^2-b d e+c d^2\right )}-\frac {e^2 \log \left (d+e x^2\right )}{2 d \left (a e^2-b d e+c d^2\right )}-\frac {(c d-b e) \log \left (a+b x^2+c x^4\right )}{4 a \left (a e^2-b d e+c d^2\right )}+\frac {\log (x)}{a d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Mathematica [A]  time = 0.32, size = 242, normalized size = 1.45 \[ \frac {4 \log (x) \sqrt {b^2-4 a c} \left (e (a e-b d)+c d^2\right )-2 a e^2 \sqrt {b^2-4 a c} \log \left (d+e x^2\right )-d \left (c d \sqrt {b^2-4 a c}-b e \sqrt {b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right )+d \left (-c d \sqrt {b^2-4 a c}+b e \sqrt {b^2-4 a c}+2 a c e+b^2 (-e)+b c d\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right )}{4 a d \sqrt {b^2-4 a c} \left (e (a e-b d)+c d^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

Timed out

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giac [A]  time = 1.92, size = 172, normalized size = 1.03 \[ -\frac {{\left (c d - b e\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, {\left (a c d^{2} - a b d e + a^{2} e^{2}\right )}} - \frac {e^{3} \log \left ({\left | x^{2} e + d \right |}\right )}{2 \, {\left (c d^{3} e - b d^{2} e^{2} + a d e^{3}\right )}} - \frac {{\left (b c d - b^{2} e + 2 \, a c e\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a c d^{2} - a b d e + a^{2} e^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {\log \left (x^{2}\right )}{2 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.01, size = 298, normalized size = 1.78 \[ \frac {b^{2} e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right ) \sqrt {4 a c -b^{2}}\, a}-\frac {b c d \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right ) \sqrt {4 a c -b^{2}}\, a}-\frac {c e \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (a \,e^{2}-d e b +c \,d^{2}\right ) \sqrt {4 a c -b^{2}}}+\frac {b e \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 \left (a \,e^{2}-d e b +c \,d^{2}\right ) a}-\frac {c d \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 \left (a \,e^{2}-d e b +c \,d^{2}\right ) a}-\frac {e^{2} \ln \left (e \,x^{2}+d \right )}{2 \left (a \,e^{2}-d e b +c \,d^{2}\right ) d}+\frac {\ln \relax (x )}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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/(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?

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mupad [B]  time = 17.20, size = 6285, normalized size = 37.63 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(log(256*a^4*e^8*(4*a*c - b^2)^4 - 80*c^4*d^8*(4*a*c - b^2)^4 - 61*d^4*e^4*(4*a*c - b^2)^6 + 160*b^3*c^4*d^8*(
b^2 - 4*a*c)^(5/2) + 16*b^5*c^4*d^8*(b^2 - 4*a*c)^(3/2) - 184*b^3*d^4*e^4*(b^2 - 4*a*c)^(9/2) + 370*b^5*d^4*e^
4*(b^2 - 4*a*c)^(7/2) + 128*b^7*d^4*e^4*(b^2 - 4*a*c)^(5/2) + 5*b^9*d^4*e^4*(b^2 - 4*a*c)^(3/2) + 128*a^3*e^8*
x^2*(b^2 - 4*a*c)^(9/2) + 160*c^5*d^8*x^2*(b^2 - 4*a*c)^(7/2) - 256*a^4*b^2*e^8*(4*a*c - b^2)^3 + 32*b^2*c^4*d
^8*(4*a*c - b^2)^3 + 112*b^4*c^4*d^8*(4*a*c - b^2)^2 - 144*a^2*d^2*e^6*(4*a*c - b^2)^5 + 544*b^2*d^4*e^4*(4*a*
c - b^2)^5 + 382*b^4*d^4*e^4*(4*a*c - b^2)^4 - 152*b^6*d^4*e^4*(4*a*c - b^2)^3 + 71*b^8*d^4*e^4*(4*a*c - b^2)^
2 + 200*c^2*d^6*e^2*(4*a*c - b^2)^5 - 13*d^3*e^5*x^2*(4*a*c - b^2)^6 + 512*a^4*b*e^8*(b^2 - 4*a*c)^(7/2) - 176
*b*c^4*d^8*(b^2 - 4*a*c)^(7/2) - 26*a*d^3*e^5*(b^2 - 4*a*c)^(11/2) + 352*a^3*d*e^7*(b^2 - 4*a*c)^(9/2) - 319*b
*d^4*e^4*(b^2 - 4*a*c)^(11/2) + 148*c*d^5*e^3*(b^2 - 4*a*c)^(11/2) + 168*c^3*d^7*e*(b^2 - 4*a*c)^(9/2) - 768*a
*b^3*d^3*e^5*(4*a*c - b^2)^4 - 368*a*b^5*d^3*e^5*(4*a*c - b^2)^3 + 128*a^3*b^3*d*e^7*(4*a*c - b^2)^3 - 32*a*b^
7*d^3*e^5*(4*a*c - b^2)^2 - 672*a^2*b^3*d^2*e^6*(b^2 - 4*a*c)^(7/2) - 272*a^2*b^5*d^2*e^6*(b^2 - 4*a*c)^(5/2)
+ 408*b^3*c*d^5*e^3*(4*a*c - b^2)^4 + 256*b^3*c^3*d^7*e*(4*a*c - b^2)^3 + 792*b^5*c*d^5*e^3*(4*a*c - b^2)^3 -
352*b^5*c^3*d^7*e*(4*a*c - b^2)^2 - 248*b^7*c*d^5*e^3*(4*a*c - b^2)^2 - 328*b^3*c^2*d^6*e^2*(b^2 - 4*a*c)^(7/2
) + 1064*b^5*c^2*d^6*e^2*(b^2 - 4*a*c)^(5/2) + 40*b^7*c^2*d^6*e^2*(b^2 - 4*a*c)^(3/2) + 384*a^3*b*e^8*x^2*(4*a
*c - b^2)^4 + 384*a^3*b^2*e^8*x^2*(b^2 - 4*a*c)^(7/2) - 512*b*c^5*d^8*x^2*(4*a*c - b^2)^3 + 576*b^2*c^5*d^8*x^
2*(b^2 - 4*a*c)^(5/2) + 32*b^4*c^5*d^8*x^2*(b^2 - 4*a*c)^(3/2) - 176*a^2*d*e^7*x^2*(4*a*c - b^2)^5 - 800*b^3*d
^3*e^5*x^2*(b^2 - 4*a*c)^(9/2) + 158*b^5*d^3*e^5*x^2*(b^2 - 4*a*c)^(7/2) + 56*b^7*d^3*e^5*x^2*(b^2 - 4*a*c)^(5
/2) - b^9*d^3*e^5*x^2*(b^2 - 4*a*c)^(3/2) - 336*c^4*d^7*e*x^2*(4*a*c - b^2)^4 + 400*c^3*d^6*e^2*x^2*(b^2 - 4*a
*c)^(9/2) - 608*a^2*b^2*d^2*e^6*(4*a*c - b^2)^4 + 560*a^2*b^4*d^2*e^6*(4*a*c - b^2)^3 - 1096*b^2*c^2*d^6*e^2*(
4*a*c - b^2)^4 - 872*b^4*c^2*d^6*e^2*(4*a*c - b^2)^3 + 424*b^6*c^2*d^6*e^2*(4*a*c - b^2)^2 - 128*a^3*b^3*e^8*x
^2*(4*a*c - b^2)^3 + 256*b^3*c^5*d^8*x^2*(4*a*c - b^2)^2 + 584*b^2*d^3*e^5*x^2*(4*a*c - b^2)^5 - 410*b^4*d^3*e
^5*x^2*(4*a*c - b^2)^4 - 256*b^6*d^3*e^5*x^2*(4*a*c - b^2)^3 - 17*b^8*d^3*e^5*x^2*(4*a*c - b^2)^2 + 296*c^2*d^
5*e^3*x^2*(4*a*c - b^2)^5 + 336*a*b*d^3*e^5*(4*a*c - b^2)^5 + 384*a^3*b*d*e^7*(4*a*c - b^2)^4 - 832*a*b^2*d^3*
e^5*(b^2 - 4*a*c)^(9/2) - 52*a*b^4*d^3*e^5*(b^2 - 4*a*c)^(7/2) + 144*a*b^6*d^3*e^5*(b^2 - 4*a*c)^(5/2) - 2*a*b
^8*d^3*e^5*(b^2 - 4*a*c)^(3/2) - 80*a^2*b*d^2*e^6*(b^2 - 4*a*c)^(9/2) - 192*a^3*b^2*d*e^7*(b^2 - 4*a*c)^(7/2)
+ 96*a^3*b^4*d*e^7*(b^2 - 4*a*c)^(5/2) - 632*b*c*d^5*e^3*(4*a*c - b^2)^5 + 608*b*c^3*d^7*e*(4*a*c - b^2)^4 - 7
76*b*c^2*d^6*e^2*(b^2 - 4*a*c)^(9/2) + 920*b^2*c*d^5*e^3*(b^2 - 4*a*c)^(9/2) + 584*b^2*c^3*d^7*e*(b^2 - 4*a*c)
^(7/2) - 384*b^4*c*d^5*e^3*(b^2 - 4*a*c)^(7/2) - 712*b^4*c^3*d^7*e*(b^2 - 4*a*c)^(5/2) - 664*b^6*c*d^5*e^3*(b^
2 - 4*a*c)^(5/2) - 40*b^6*c^3*d^7*e*(b^2 - 4*a*c)^(3/2) - 20*b^8*c*d^5*e^3*(b^2 - 4*a*c)^(3/2) + 72*a*d^2*e^6*
x^2*(b^2 - 4*a*c)^(11/2) - 181*b*d^3*e^5*x^2*(b^2 - 4*a*c)^(11/2) + 122*c*d^4*e^4*x^2*(b^2 - 4*a*c)^(11/2) + 3
68*a^2*b*d*e^7*x^2*(b^2 - 4*a*c)^(9/2) - 1552*b*c^4*d^7*e*x^2*(b^2 - 4*a*c)^(7/2) - 3400*b^2*c^2*d^5*e^3*x^2*(
4*a*c - b^2)^4 - 4800*b^3*c^3*d^6*e^2*x^2*(4*a*c - b^2)^3 + 3448*b^4*c^2*d^5*e^3*x^2*(4*a*c - b^2)^3 + 928*b^5
*c^3*d^6*e^2*x^2*(4*a*c - b^2)^2 - 536*b^6*c^2*d^5*e^3*x^2*(4*a*c - b^2)^2 - 32*a*b*d^2*e^6*x^2*(4*a*c - b^2)^
5 - 344*a*b^2*d^2*e^6*x^2*(b^2 - 4*a*c)^(9/2) - 616*a*b^4*d^2*e^6*x^2*(b^2 - 4*a*c)^(7/2) - 136*a*b^6*d^2*e^6*
x^2*(b^2 - 4*a*c)^(5/2) - 160*a^2*b^3*d*e^7*x^2*(b^2 - 4*a*c)^(7/2) + 48*a^2*b^5*d*e^7*x^2*(b^2 - 4*a*c)^(5/2)
- 760*b*c*d^4*e^4*x^2*(4*a*c - b^2)^5 - 1560*b*c^2*d^5*e^3*x^2*(b^2 - 4*a*c)^(9/2) + 1848*b^2*c*d^4*e^4*x^2*(
b^2 - 4*a*c)^(9/2) - 2208*b^3*c^4*d^7*e*x^2*(b^2 - 4*a*c)^(5/2) + 1452*b^4*c*d^4*e^4*x^2*(b^2 - 4*a*c)^(7/2) -
80*b^5*c^4*d^7*e*x^2*(b^2 - 4*a*c)^(3/2) + 408*b^6*c*d^4*e^4*x^2*(b^2 - 4*a*c)^(5/2) + 10*b^8*c*d^4*e^4*x^2*(
b^2 - 4*a*c)^(3/2) - 640*a*b^3*d^2*e^6*x^2*(4*a*c - b^2)^4 + 96*a^2*b^2*d*e^7*x^2*(4*a*c - b^2)^4 + 416*a*b^5*
d^2*e^6*x^2*(4*a*c - b^2)^3 + 16*a^2*b^4*d*e^7*x^2*(4*a*c - b^2)^3 + 1952*b*c^3*d^6*e^2*x^2*(4*a*c - b^2)^4 +
2216*b^3*c*d^4*e^4*x^2*(4*a*c - b^2)^4 + 2720*b^2*c^4*d^7*e*x^2*(4*a*c - b^2)^3 - 712*b^5*c*d^4*e^4*x^2*(4*a*c
- b^2)^3 - 784*b^4*c^4*d^7*e*x^2*(4*a*c - b^2)^2 + 152*b^7*c*d^4*e^4*x^2*(4*a*c - b^2)^2 + 4144*b^2*c^3*d^6*e
^2*x^2*(b^2 - 4*a*c)^(7/2) - 4216*b^3*c^2*d^5*e^3*x^2*(b^2 - 4*a*c)^(7/2) + 3056*b^4*c^3*d^6*e^2*x^2*(b^2 - 4*
a*c)^(5/2) - 1864*b^5*c^2*d^5*e^3*x^2*(b^2 - 4*a*c)^(5/2) + 80*b^6*c^3*d^6*e^2*x^2*(b^2 - 4*a*c)^(3/2) - 40*b^
7*c^2*d^5*e^3*x^2*(b^2 - 4*a*c)^(3/2))*(d*((b^2*c)/4 - a*c^2 + (b*c*(b^2 - 4*a*c)^(1/2))/4) - (b^3*e)/4 - (b^2
*e*(b^2 - 4*a*c)^(1/2))/4 + (a*c*e*(b^2 - 4*a*c)^(1/2))/2 + a*b*c*e))/(4*a^3*c*e^2 - a^2*b^2*e^2 + 4*a^2*c^2*d
^2 + a*b^3*d*e - a*b^2*c*d^2 - 4*a^2*b*c*d*e) - (log(80*c^4*d^8*(4*a*c - b^2)^4 - 256*a^4*e^8*(4*a*c - b^2)^4
+ 61*d^4*e^4*(4*a*c - b^2)^6 + 160*b^3*c^4*d^8*(b^2 - 4*a*c)^(5/2) + 16*b^5*c^4*d^8*(b^2 - 4*a*c)^(3/2) - 184*
b^3*d^4*e^4*(b^2 - 4*a*c)^(9/2) + 370*b^5*d^4*e^4*(b^2 - 4*a*c)^(7/2) + 128*b^7*d^4*e^4*(b^2 - 4*a*c)^(5/2) +
5*b^9*d^4*e^4*(b^2 - 4*a*c)^(3/2) + 128*a^3*e^8*x^2*(b^2 - 4*a*c)^(9/2) + 160*c^5*d^8*x^2*(b^2 - 4*a*c)^(7/2)
+ 256*a^4*b^2*e^8*(4*a*c - b^2)^3 - 32*b^2*c^4*d^8*(4*a*c - b^2)^3 - 112*b^4*c^4*d^8*(4*a*c - b^2)^2 + 144*a^2
*d^2*e^6*(4*a*c - b^2)^5 - 544*b^2*d^4*e^4*(4*a*c - b^2)^5 - 382*b^4*d^4*e^4*(4*a*c - b^2)^4 + 152*b^6*d^4*e^4
*(4*a*c - b^2)^3 - 71*b^8*d^4*e^4*(4*a*c - b^2)^2 - 200*c^2*d^6*e^2*(4*a*c - b^2)^5 + 13*d^3*e^5*x^2*(4*a*c -
b^2)^6 + 512*a^4*b*e^8*(b^2 - 4*a*c)^(7/2) - 176*b*c^4*d^8*(b^2 - 4*a*c)^(7/2) - 26*a*d^3*e^5*(b^2 - 4*a*c)^(1
1/2) + 352*a^3*d*e^7*(b^2 - 4*a*c)^(9/2) - 319*b*d^4*e^4*(b^2 - 4*a*c)^(11/2) + 148*c*d^5*e^3*(b^2 - 4*a*c)^(1
1/2) + 168*c^3*d^7*e*(b^2 - 4*a*c)^(9/2) + 768*a*b^3*d^3*e^5*(4*a*c - b^2)^4 + 368*a*b^5*d^3*e^5*(4*a*c - b^2)
^3 - 128*a^3*b^3*d*e^7*(4*a*c - b^2)^3 + 32*a*b^7*d^3*e^5*(4*a*c - b^2)^2 - 672*a^2*b^3*d^2*e^6*(b^2 - 4*a*c)^
(7/2) - 272*a^2*b^5*d^2*e^6*(b^2 - 4*a*c)^(5/2) - 408*b^3*c*d^5*e^3*(4*a*c - b^2)^4 - 256*b^3*c^3*d^7*e*(4*a*c
- b^2)^3 - 792*b^5*c*d^5*e^3*(4*a*c - b^2)^3 + 352*b^5*c^3*d^7*e*(4*a*c - b^2)^2 + 248*b^7*c*d^5*e^3*(4*a*c -
b^2)^2 - 328*b^3*c^2*d^6*e^2*(b^2 - 4*a*c)^(7/2) + 1064*b^5*c^2*d^6*e^2*(b^2 - 4*a*c)^(5/2) + 40*b^7*c^2*d^6*
e^2*(b^2 - 4*a*c)^(3/2) - 384*a^3*b*e^8*x^2*(4*a*c - b^2)^4 + 384*a^3*b^2*e^8*x^2*(b^2 - 4*a*c)^(7/2) + 512*b*
c^5*d^8*x^2*(4*a*c - b^2)^3 + 576*b^2*c^5*d^8*x^2*(b^2 - 4*a*c)^(5/2) + 32*b^4*c^5*d^8*x^2*(b^2 - 4*a*c)^(3/2)
+ 176*a^2*d*e^7*x^2*(4*a*c - b^2)^5 - 800*b^3*d^3*e^5*x^2*(b^2 - 4*a*c)^(9/2) + 158*b^5*d^3*e^5*x^2*(b^2 - 4*
a*c)^(7/2) + 56*b^7*d^3*e^5*x^2*(b^2 - 4*a*c)^(5/2) - b^9*d^3*e^5*x^2*(b^2 - 4*a*c)^(3/2) + 336*c^4*d^7*e*x^2*
(4*a*c - b^2)^4 + 400*c^3*d^6*e^2*x^2*(b^2 - 4*a*c)^(9/2) + 608*a^2*b^2*d^2*e^6*(4*a*c - b^2)^4 - 560*a^2*b^4*
d^2*e^6*(4*a*c - b^2)^3 + 1096*b^2*c^2*d^6*e^2*(4*a*c - b^2)^4 + 872*b^4*c^2*d^6*e^2*(4*a*c - b^2)^3 - 424*b^6
*c^2*d^6*e^2*(4*a*c - b^2)^2 + 128*a^3*b^3*e^8*x^2*(4*a*c - b^2)^3 - 256*b^3*c^5*d^8*x^2*(4*a*c - b^2)^2 - 584
*b^2*d^3*e^5*x^2*(4*a*c - b^2)^5 + 410*b^4*d^3*e^5*x^2*(4*a*c - b^2)^4 + 256*b^6*d^3*e^5*x^2*(4*a*c - b^2)^3 +
17*b^8*d^3*e^5*x^2*(4*a*c - b^2)^2 - 296*c^2*d^5*e^3*x^2*(4*a*c - b^2)^5 - 336*a*b*d^3*e^5*(4*a*c - b^2)^5 -
384*a^3*b*d*e^7*(4*a*c - b^2)^4 - 832*a*b^2*d^3*e^5*(b^2 - 4*a*c)^(9/2) - 52*a*b^4*d^3*e^5*(b^2 - 4*a*c)^(7/2)
+ 144*a*b^6*d^3*e^5*(b^2 - 4*a*c)^(5/2) - 2*a*b^8*d^3*e^5*(b^2 - 4*a*c)^(3/2) - 80*a^2*b*d^2*e^6*(b^2 - 4*a*c
)^(9/2) - 192*a^3*b^2*d*e^7*(b^2 - 4*a*c)^(7/2) + 96*a^3*b^4*d*e^7*(b^2 - 4*a*c)^(5/2) + 632*b*c*d^5*e^3*(4*a*
c - b^2)^5 - 608*b*c^3*d^7*e*(4*a*c - b^2)^4 - 776*b*c^2*d^6*e^2*(b^2 - 4*a*c)^(9/2) + 920*b^2*c*d^5*e^3*(b^2
- 4*a*c)^(9/2) + 584*b^2*c^3*d^7*e*(b^2 - 4*a*c)^(7/2) - 384*b^4*c*d^5*e^3*(b^2 - 4*a*c)^(7/2) - 712*b^4*c^3*d
^7*e*(b^2 - 4*a*c)^(5/2) - 664*b^6*c*d^5*e^3*(b^2 - 4*a*c)^(5/2) - 40*b^6*c^3*d^7*e*(b^2 - 4*a*c)^(3/2) - 20*b
^8*c*d^5*e^3*(b^2 - 4*a*c)^(3/2) + 72*a*d^2*e^6*x^2*(b^2 - 4*a*c)^(11/2) - 181*b*d^3*e^5*x^2*(b^2 - 4*a*c)^(11
/2) + 122*c*d^4*e^4*x^2*(b^2 - 4*a*c)^(11/2) + 368*a^2*b*d*e^7*x^2*(b^2 - 4*a*c)^(9/2) - 1552*b*c^4*d^7*e*x^2*
(b^2 - 4*a*c)^(7/2) + 3400*b^2*c^2*d^5*e^3*x^2*(4*a*c - b^2)^4 + 4800*b^3*c^3*d^6*e^2*x^2*(4*a*c - b^2)^3 - 34
48*b^4*c^2*d^5*e^3*x^2*(4*a*c - b^2)^3 - 928*b^5*c^3*d^6*e^2*x^2*(4*a*c - b^2)^2 + 536*b^6*c^2*d^5*e^3*x^2*(4*
a*c - b^2)^2 + 32*a*b*d^2*e^6*x^2*(4*a*c - b^2)^5 - 344*a*b^2*d^2*e^6*x^2*(b^2 - 4*a*c)^(9/2) - 616*a*b^4*d^2*
e^6*x^2*(b^2 - 4*a*c)^(7/2) - 136*a*b^6*d^2*e^6*x^2*(b^2 - 4*a*c)^(5/2) - 160*a^2*b^3*d*e^7*x^2*(b^2 - 4*a*c)^
(7/2) + 48*a^2*b^5*d*e^7*x^2*(b^2 - 4*a*c)^(5/2) + 760*b*c*d^4*e^4*x^2*(4*a*c - b^2)^5 - 1560*b*c^2*d^5*e^3*x^
2*(b^2 - 4*a*c)^(9/2) + 1848*b^2*c*d^4*e^4*x^2*(b^2 - 4*a*c)^(9/2) - 2208*b^3*c^4*d^7*e*x^2*(b^2 - 4*a*c)^(5/2
) + 1452*b^4*c*d^4*e^4*x^2*(b^2 - 4*a*c)^(7/2) - 80*b^5*c^4*d^7*e*x^2*(b^2 - 4*a*c)^(3/2) + 408*b^6*c*d^4*e^4*
x^2*(b^2 - 4*a*c)^(5/2) + 10*b^8*c*d^4*e^4*x^2*(b^2 - 4*a*c)^(3/2) + 640*a*b^3*d^2*e^6*x^2*(4*a*c - b^2)^4 - 9
6*a^2*b^2*d*e^7*x^2*(4*a*c - b^2)^4 - 416*a*b^5*d^2*e^6*x^2*(4*a*c - b^2)^3 - 16*a^2*b^4*d*e^7*x^2*(4*a*c - b^
2)^3 - 1952*b*c^3*d^6*e^2*x^2*(4*a*c - b^2)^4 - 2216*b^3*c*d^4*e^4*x^2*(4*a*c - b^2)^4 - 2720*b^2*c^4*d^7*e*x^
2*(4*a*c - b^2)^3 + 712*b^5*c*d^4*e^4*x^2*(4*a*c - b^2)^3 + 784*b^4*c^4*d^7*e*x^2*(4*a*c - b^2)^2 - 152*b^7*c*
d^4*e^4*x^2*(4*a*c - b^2)^2 + 4144*b^2*c^3*d^6*e^2*x^2*(b^2 - 4*a*c)^(7/2) - 4216*b^3*c^2*d^5*e^3*x^2*(b^2 - 4
*a*c)^(7/2) + 3056*b^4*c^3*d^6*e^2*x^2*(b^2 - 4*a*c)^(5/2) - 1864*b^5*c^2*d^5*e^3*x^2*(b^2 - 4*a*c)^(5/2) + 80
*b^6*c^3*d^6*e^2*x^2*(b^2 - 4*a*c)^(3/2) - 40*b^7*c^2*d^5*e^3*x^2*(b^2 - 4*a*c)^(3/2))*((b^3*e)/4 + d*(a*c^2 -
(b^2*c)/4 + (b*c*(b^2 - 4*a*c)^(1/2))/4) - (b^2*e*(b^2 - 4*a*c)^(1/2))/4 + (a*c*e*(b^2 - 4*a*c)^(1/2))/2 - a*
b*c*e))/(4*a^3*c*e^2 - a^2*b^2*e^2 + 4*a^2*c^2*d^2 + a*b^3*d*e - a*b^2*c*d^2 - 4*a^2*b*c*d*e) - (e^2*log(d + e
*x^2))/(2*c*d^3 + 2*a*d*e^2 - 2*b*d^2*e) + log(x)/(a*d)

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

[Out]

Timed out

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