3.1.26 \(\int \frac {A+B x+C x^2}{x (a+b x^2+c x^4)} \, dx\) [26]

Optimal. Leaf size=229 \[ \frac {\sqrt {2} B \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} B \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {(A b-2 a C) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c}}+\frac {A \log (x)}{a}-\frac {A \log \left (a+b x^2+c x^4\right )}{4 a} \]

[Out]

A*ln(x)/a-1/4*A*ln(c*x^4+b*x^2+a)/a+1/2*(A*b-2*C*a)*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/a/(-4*a*c+b^2)^(1/
2)+B*arctan(x*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)*c^(1/2)/(-4*a*c+b^2)^(1/2)/(b-(-4*a*c+b^2)
^(1/2))^(1/2)-B*arctan(x*2^(1/2)*c^(1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2))*2^(1/2)*c^(1/2)/(-4*a*c+b^2)^(1/2)/(b+(
-4*a*c+b^2)^(1/2))^(1/2)

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Rubi [A]
time = 0.19, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 10, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {1676, 1265, 814, 648, 632, 212, 642, 12, 1107, 211} \begin {gather*} \frac {(A b-2 a C) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c}}-\frac {A \log \left (a+b x^2+c x^4\right )}{4 a}+\frac {A \log (x)}{a}+\frac {\sqrt {2} B \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} B \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {b^2-4 a c} \sqrt {\sqrt {b^2-4 a c}+b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x + C*x^2)/(x*(a + b*x^2 + c*x^4)),x]

[Out]

(Sqrt[2]*B*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b
^2 - 4*a*c]]) - (Sqrt[2]*B*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]
*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((A*b - 2*a*C)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a*Sqrt[b^2 - 4*a*c
]) + (A*Log[x])/a - (A*Log[a + b*x^2 + c*x^4])/(4*a)

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 212

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] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

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 642

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 648

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 814

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[(d + e*x)^m*((f + g*x)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 -
4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

Rule 1107

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

Rule 1265

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]

Rule 1676

Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{q = Expon[Pq, x],
k}, Int[(d*x)^m*Sum[Coeff[Pq, x, 2*k]*x^(2*k), {k, 0, q/2 + 1}]*(a + b*x^2 + c*x^4)^p, x] + Dist[1/d, Int[(d*
x)^(m + 1)*Sum[Coeff[Pq, x, 2*k + 1]*x^(2*k), {k, 0, (q - 1)/2 + 1}]*(a + b*x^2 + c*x^4)^p, x], x]] /; FreeQ[{
a, b, c, d, m, p}, x] && PolyQ[Pq, x] &&  !PolyQ[Pq, x^2]

Rubi steps

\begin {align*} \int \frac {A+B x+C x^2}{x \left (a+b x^2+c x^4\right )} \, dx &=\int \frac {B}{a+b x^2+c x^4} \, dx+\int \frac {A+C x^2}{x \left (a+b x^2+c x^4\right )} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \frac {A+C x}{x \left (a+b x+c x^2\right )} \, dx,x,x^2\right )+B \int \frac {1}{a+b x^2+c x^4} \, dx\\ &=\frac {1}{2} \text {Subst}\left (\int \left (\frac {A}{a x}+\frac {-A b+a C-A c x}{a \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )+\frac {(B c) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{\sqrt {b^2-4 a c}}-\frac {(B c) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{\sqrt {b^2-4 a c}}\\ &=\frac {\sqrt {2} B \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} B \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {A \log (x)}{a}+\frac {\text {Subst}\left (\int \frac {-A b+a C-A c x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a}\\ &=\frac {\sqrt {2} B \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} B \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {A \log (x)}{a}-\frac {A \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a}+\frac {(-A b+2 a C) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a}\\ &=\frac {\sqrt {2} B \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} B \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {A \log (x)}{a}-\frac {A \log \left (a+b x^2+c x^4\right )}{4 a}-\frac {(-A b+2 a C) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a}\\ &=\frac {\sqrt {2} B \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} B \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {(A b-2 a C) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c}}+\frac {A \log (x)}{a}-\frac {A \log \left (a+b x^2+c x^4\right )}{4 a}\\ \end {align*}

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Mathematica [A]
time = 0.29, size = 285, normalized size = 1.24 \begin {gather*} \frac {\sqrt {2} B \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {2} B \sqrt {c} \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {A \log (x)}{a}-\frac {\left (A \left (b+\sqrt {b^2-4 a c}\right )-2 a C\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{4 a \sqrt {b^2-4 a c}}-\frac {\left (A \left (-b+\sqrt {b^2-4 a c}\right )+2 a C\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{4 a \sqrt {b^2-4 a c}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x + C*x^2)/(x*(a + b*x^2 + c*x^4)),x]

[Out]

(Sqrt[2]*B*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b
^2 - 4*a*c]]) - (Sqrt[2]*B*Sqrt[c]*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]
*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (A*Log[x])/a - ((A*(b + Sqrt[b^2 - 4*a*c]) - 2*a*C)*Log[-b + Sqrt[b^2 - 4*a*c]
- 2*c*x^2])/(4*a*Sqrt[b^2 - 4*a*c]) - ((A*(-b + Sqrt[b^2 - 4*a*c]) + 2*a*C)*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x
^2])/(4*a*Sqrt[b^2 - 4*a*c])

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Maple [A]
time = 0.06, size = 251, normalized size = 1.10

method	result	size
default	\(\frac {4 c \left (\frac {\sqrt {-4 a c +b^{2}}\, \left (-\frac {\left (-A \sqrt {-4 a c +b^{2}}-A b +2 a C \right ) \ln \left (-b -2 c \,x^{2}+\sqrt {-4 a c +b^{2}}\right )}{4 c}+\frac {a B \sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a c -4 b^{2}}+\frac {\sqrt {-4 a c +b^{2}}\, \left (\frac {\left (A \sqrt {-4 a c +b^{2}}-A b +2 a C \right ) \ln \left (b +2 c \,x^{2}+\sqrt {-4 a c +b^{2}}\right )}{4 c}+\frac {a B \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{16 a c -4 b^{2}}\right )}{a}+\frac {A \ln \left (x \right )}{a}\)	\(251\)
risch	\(\frac {A \ln \left (x \right )}{a}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (16 a^{4} c^{2}-8 a^{3} b^{2} c +a^{2} b^{4}\right ) \textit {\_Z}^{4}+\left (32 A \,a^{3} c^{2}-16 A \,a^{2} b^{2} c +2 A a \,b^{4}\right ) \textit {\_Z}^{3}+\left (24 a^{2} c^{2} A^{2}-10 a \,b^{2} c \,A^{2}+b^{4} A^{2}-8 A C \,a^{2} b c +2 A C a \,b^{3}-4 B^{2} a^{2} b c +B^{2} a \,b^{3}+8 C^{2} a^{3} c -2 C^{2} a^{2} b^{2}\right ) \textit {\_Z}^{2}+\left (8 A^{3} a \,c^{2}-2 A^{3} b^{2} c -8 A^{2} C a b c +2 A^{2} C \,b^{3}+8 A \,C^{2} a^{2} c -2 A \,C^{2} a \,b^{2}-8 B^{2} C \,a^{2} c +2 B^{2} C a \,b^{2}\right ) \textit {\_Z} +A^{4} c^{2}-2 A^{3} C b c +A^{2} B^{2} b c +2 A^{2} C^{2} a c +A^{2} C^{2} b^{2}-4 A \,B^{2} C a c -2 A \,C^{3} a b +B^{4} a c +B^{2} C^{2} a b +C^{4} a^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (40 a^{3} c^{2}-22 a^{2} b^{2} c +3 b^{4} a \right ) \textit {\_R}^{4}+\left (60 A \,a^{2} c^{2}-27 A a \,b^{2} c +3 A \,b^{4}-4 C \,a^{2} b c +C a \,b^{3}\right ) \textit {\_R}^{3}+\left (30 A^{2} a \,c^{2}-8 A^{2} b^{2} c -14 A C a b c +4 A C \,b^{3}-7 B^{2} a b c +2 B^{2} b^{3}+18 C^{2} a^{2} c -5 C^{2} a \,b^{2}\right ) \textit {\_R}^{2}+\left (5 A^{3} c^{2}-6 A^{2} C b c -A \,B^{2} b c +13 A \,C^{2} a c -A \,C^{2} b^{2}-17 B^{2} C a c +4 B^{2} C \,b^{2}-C^{3} a b \right ) \textit {\_R} +2 A^{2} C^{2} c -6 A \,B^{2} C c -2 A \,C^{3} b +2 B^{4} c +2 B^{2} C^{2} b +2 C^{4} a \right ) x +\left (4 a^{2} b B c -B a \,b^{3}\right ) \textit {\_R}^{3}+\left (-6 A B a b c +2 A B \,b^{3}+4 B C \,a^{2} c -2 B C a \,b^{2}\right ) \textit {\_R}^{2}+\left (-4 A^{2} B b c -6 A B C a c +4 A B C \,b^{2}-a c \,B^{3}-B \,C^{2} a b \right ) \textit {\_R} -4 A^{2} B C c +2 A \,B^{3} c +2 A B \,C^{2} b \right )\right )}{2}\)	\(706\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((C*x^2+B*x+A)/x/(c*x^4+b*x^2+a),x,method=_RETURNVERBOSE)

[Out]

4/a*c*((-4*a*c+b^2)^(1/2)/(16*a*c-4*b^2)*(-1/4*(-A*(-4*a*c+b^2)^(1/2)-A*b+2*a*C)/c*ln(-b-2*c*x^2+(-4*a*c+b^2)^
(1/2))+a*B*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)))+(
-4*a*c+b^2)^(1/2)/(16*a*c-4*b^2)*(1/4*(A*(-4*a*c+b^2)^(1/2)-A*b+2*a*C)/c*ln(b+2*c*x^2+(-4*a*c+b^2)^(1/2))+a*B*
2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))))+A*ln(x)/a

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/x/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

A*log(x)/a - integrate((A*c*x^3 - B*a - (C*a - A*b)*x)/(c*x^4 + b*x^2 + a), x)/a

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/x/(c*x^4+b*x^2+a),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x**2+B*x+A)/x/(c*x**4+b*x**2+a),x)

[Out]

Timed out

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2337 vs. \(2 (186) = 372\).
time = 6.43, size = 2337, normalized size = 10.21 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((C*x^2+B*x+A)/x/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

-1/4*A*log(abs(c*x^4 + b*x^2 + a))/a + A*log(abs(x))/a + 1/4*((sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4 - 8
*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c + 2*b^4*c +
16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt
(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^2 - 16*a*b^2*c^2 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^3 +
32*a^2*c^3 - 2*(b^2 - 4*a*c)*b^2*c + 8*(b^2 - 4*a*c)*a*c^2)*B*abs(c) + (2*b^3*c^3 - 8*a*b*c^4 - sqrt(2)*sqrt(
b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*
c)*a*b*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c - sqrt(b^2 - 4*a*c)*c)*b*c^3 - 2*(b^2 - 4*a*c)*b*c^3)*B)*arctan(2*sqrt(1/2)*x/sqrt((a^2*b*c + sqrt(a^4
*b^2*c^2 - 4*a^5*c^3))/(a^2*c^2)))/((a*b^4 - 8*a^2*b^2*c - 2*a*b^3*c + 16*a^3*c^2 + 8*a^2*b*c^2 + a*b^2*c^2 -
4*a^2*c^3)*c^2) + 1/4*((sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c
)*a*b^2*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c - 2*b^4*c + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c
)*c)*a^2*c^2 + 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2
*c^2 + 16*a*b^2*c^2 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^3 - 32*a^2*c^3 + 2*(b^2 - 4*a*c)*b^2*c - 8
*(b^2 - 4*a*c)*a*c^2)*B*abs(c) - (2*b^3*c^3 - 8*a*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*
c)*c)*b^3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c
)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c^3 -
2*(b^2 - 4*a*c)*b*c^3)*B)*arctan(2*sqrt(1/2)*x/sqrt((a^2*b*c - sqrt(a^4*b^2*c^2 - 4*a^5*c^3))/(a^2*c^2)))/((a*
b^4 - 8*a^2*b^2*c - 2*a*b^3*c + 16*a^3*c^2 + 8*a^2*b*c^2 + a*b^2*c^2 - 4*a^2*c^3)*c^2) - 1/16*((b^6*c - 8*a*b^
4*c^2 - 2*b^5*c^2 + 16*a^2*b^2*c^3 + 8*a*b^3*c^3 + b^4*c^3 - 4*a*b^2*c^4 - (b^5*c - 8*a*b^3*c^2 - 2*b^4*c^2 +
16*a^2*b*c^3 + 8*a*b^2*c^3 + b^3*c^3 - 4*a*b*c^4)*sqrt(b^2 - 4*a*c))*A*abs(c) - 2*(a*b^5*c - 8*a^2*b^3*c^2 - 2
*a*b^4*c^2 + 16*a^3*b*c^3 + 8*a^2*b^2*c^3 + a*b^3*c^3 - 4*a^2*b*c^4 + (a*b^4*c - 8*a^2*b^2*c^2 - 2*a*b^3*c^2 +
16*a^3*c^3 + 8*a^2*b*c^3 + a*b^2*c^3 - 4*a^2*c^4)*sqrt(b^2 - 4*a*c))*C*abs(c) + (b^6*c^2 - 8*a*b^4*c^3 - 2*b^
5*c^3 + 16*a^2*b^2*c^4 + 8*a*b^3*c^4 + b^4*c^4 - 4*a*b^2*c^5 + (b^5*c^2 - 4*a*b^3*c^3 - 2*b^4*c^3 + b^3*c^4)*s
qrt(b^2 - 4*a*c))*A - 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 - 2*a*b^4*c^3 + 16*a^3*b*c^4 + 8*a^2*b^2*c^4 + a*b^3*c^4 -
4*a^2*b*c^5 - (a*b^4*c^2 - 4*a^2*b^2*c^3 - 2*a*b^3*c^3 + a*b^2*c^4)*sqrt(b^2 - 4*a*c))*C)*log(x^2 + 1/2*(a^2*b
*c + sqrt(a^4*b^2*c^2 - 4*a^5*c^3))/(a^2*c^2))/((a^2*b^4 - 8*a^3*b^2*c - 2*a^2*b^3*c + 16*a^4*c^2 + 8*a^3*b*c^
2 + a^2*b^2*c^2 - 4*a^3*c^3)*c^2*abs(c)) - 1/16*((b^6*c - 8*a*b^4*c^2 - 2*b^5*c^2 + 16*a^2*b^2*c^3 + 8*a*b^3*c
^3 + b^4*c^3 - 4*a*b^2*c^4 + (b^5*c - 8*a*b^3*c^2 - 2*b^4*c^2 + 16*a^2*b*c^3 + 8*a*b^2*c^3 + b^3*c^3 - 4*a*b*c
^4)*sqrt(b^2 - 4*a*c))*A*abs(c) - 2*(a*b^5*c - 8*a^2*b^3*c^2 - 2*a*b^4*c^2 + 16*a^3*b*c^3 + 8*a^2*b^2*c^3 + a*
b^3*c^3 - 4*a^2*b*c^4 + (a*b^4*c - 8*a^2*b^2*c^2 - 2*a*b^3*c^2 + 16*a^3*c^3 + 8*a^2*b*c^3 + a*b^2*c^3 - 4*a^2*
c^4)*sqrt(b^2 - 4*a*c))*C*abs(c) + (b^6*c^2 - 8*a*b^4*c^3 - 2*b^5*c^3 + 16*a^2*b^2*c^4 + 8*a*b^3*c^4 + b^4*c^4
- 4*a*b^2*c^5 - (b^5*c^2 - 4*a*b^3*c^3 - 2*b^4*c^3 + b^3*c^4)*sqrt(b^2 - 4*a*c))*A - 2*(a*b^5*c^2 - 8*a^2*b^3
*c^3 - 2*a*b^4*c^3 + 16*a^3*b*c^4 + 8*a^2*b^2*c^4 + a*b^3*c^4 - 4*a^2*b*c^5 - (a*b^4*c^2 - 4*a^2*b^2*c^3 - 2*a
*b^3*c^3 + a*b^2*c^4)*sqrt(b^2 - 4*a*c))*C)*log(x^2 + 1/2*(a^2*b*c - sqrt(a^4*b^2*c^2 - 4*a^5*c^3))/(a^2*c^2))
/((a^2*b^4 - 8*a^3*b^2*c - 2*a^2*b^3*c + 16*a^4*c^2 + 8*a^3*b*c^2 + a^2*b^2*c^2 - 4*a^3*c^3)*c^2*abs(c))

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Mupad [B]
time = 1.49, size = 2258, normalized size = 9.86 \begin {gather*} \left (\sum _{k=1}^4\ln \left (x\,\left (A^2\,C^2\,c^3-3\,A\,B^2\,C\,c^3-b\,A\,C^3\,c^2+B^4\,c^3+b\,B^2\,C^2\,c^2+a\,C^4\,c^2\right )-\mathrm {root}\left (128\,a^3\,b^2\,c\,z^4-256\,a^4\,c^2\,z^4-16\,a^2\,b^4\,z^4+128\,A\,a^2\,b^2\,c\,z^3-256\,A\,a^3\,c^2\,z^3-16\,A\,a\,b^4\,z^3+32\,A\,C\,a^2\,b\,c\,z^2-8\,A\,C\,a\,b^3\,z^2+16\,B^2\,a^2\,b\,c\,z^2+40\,A^2\,a\,b^2\,c\,z^2-32\,C^2\,a^3\,c\,z^2-4\,B^2\,a\,b^3\,z^2+8\,C^2\,a^2\,b^2\,z^2-96\,A^2\,a^2\,c^2\,z^2-4\,A^2\,b^4\,z^2+16\,A^2\,C\,a\,b\,c\,z+16\,B^2\,C\,a^2\,c\,z-16\,A\,C^2\,a^2\,c\,z-4\,B^2\,C\,a\,b^2\,z+4\,A\,C^2\,a\,b^2\,z+4\,A^3\,b^2\,c\,z-16\,A^3\,a\,c^2\,z-4\,A^2\,C\,b^3\,z+4\,A\,B^2\,C\,a\,c-2\,A^2\,C^2\,a\,c+2\,A^3\,C\,b\,c+2\,A\,C^3\,a\,b-B^2\,C^2\,a\,b-A^2\,B^2\,b\,c-B^4\,a\,c-A^2\,C^2\,b^2-C^4\,a^2-A^4\,c^2,z,k\right )\,\left (x\,\left (-5\,A^3\,c^4+6\,A^2\,C\,b\,c^3+A\,B^2\,b\,c^3+A\,C^2\,b^2\,c^2-13\,a\,A\,C^2\,c^3-4\,B^2\,C\,b^2\,c^2+17\,a\,B^2\,C\,c^3+a\,C^3\,b\,c^2\right )-\mathrm {root}\left (128\,a^3\,b^2\,c\,z^4-256\,a^4\,c^2\,z^4-16\,a^2\,b^4\,z^4+128\,A\,a^2\,b^2\,c\,z^3-256\,A\,a^3\,c^2\,z^3-16\,A\,a\,b^4\,z^3+32\,A\,C\,a^2\,b\,c\,z^2-8\,A\,C\,a\,b^3\,z^2+16\,B^2\,a^2\,b\,c\,z^2+40\,A^2\,a\,b^2\,c\,z^2-32\,C^2\,a^3\,c\,z^2-4\,B^2\,a\,b^3\,z^2+8\,C^2\,a^2\,b^2\,z^2-96\,A^2\,a^2\,c^2\,z^2-4\,A^2\,b^4\,z^2+16\,A^2\,C\,a\,b\,c\,z+16\,B^2\,C\,a^2\,c\,z-16\,A\,C^2\,a^2\,c\,z-4\,B^2\,C\,a\,b^2\,z+4\,A\,C^2\,a\,b^2\,z+4\,A^3\,b^2\,c\,z-16\,A^3\,a\,c^2\,z-4\,A^2\,C\,b^3\,z+4\,A\,B^2\,C\,a\,c-2\,A^2\,C^2\,a\,c+2\,A^3\,C\,b\,c+2\,A\,C^3\,a\,b-B^2\,C^2\,a\,b-A^2\,B^2\,b\,c-B^4\,a\,c-A^2\,C^2\,b^2-C^4\,a^2-A^4\,c^2,z,k\right )\,\left (x\,\left (60\,A^2\,a\,c^4-16\,A^2\,b^2\,c^3-28\,A\,C\,a\,b\,c^3+8\,A\,C\,b^3\,c^2-14\,B^2\,a\,b\,c^3+4\,B^2\,b^3\,c^2+36\,C^2\,a^2\,c^3-10\,C^2\,a\,b^2\,c^2\right )+\mathrm {root}\left (128\,a^3\,b^2\,c\,z^4-256\,a^4\,c^2\,z^4-16\,a^2\,b^4\,z^4+128\,A\,a^2\,b^2\,c\,z^3-256\,A\,a^3\,c^2\,z^3-16\,A\,a\,b^4\,z^3+32\,A\,C\,a^2\,b\,c\,z^2-8\,A\,C\,a\,b^3\,z^2+16\,B^2\,a^2\,b\,c\,z^2+40\,A^2\,a\,b^2\,c\,z^2-32\,C^2\,a^3\,c\,z^2-4\,B^2\,a\,b^3\,z^2+8\,C^2\,a^2\,b^2\,z^2-96\,A^2\,a^2\,c^2\,z^2-4\,A^2\,b^4\,z^2+16\,A^2\,C\,a\,b\,c\,z+16\,B^2\,C\,a^2\,c\,z-16\,A\,C^2\,a^2\,c\,z-4\,B^2\,C\,a\,b^2\,z+4\,A\,C^2\,a\,b^2\,z+4\,A^3\,b^2\,c\,z-16\,A^3\,a\,c^2\,z-4\,A^2\,C\,b^3\,z+4\,A\,B^2\,C\,a\,c-2\,A^2\,C^2\,a\,c+2\,A^3\,C\,b\,c+2\,A\,C^3\,a\,b-B^2\,C^2\,a\,b-A^2\,B^2\,b\,c-B^4\,a\,c-A^2\,C^2\,b^2-C^4\,a^2-A^4\,c^2,z,k\right )\,\left (x\,\left (-16\,C\,a^2\,b\,c^3+240\,A\,a^2\,c^4+4\,C\,a\,b^3\,c^2-108\,A\,a\,b^2\,c^3+12\,A\,b^4\,c^2\right )+\mathrm {root}\left (128\,a^3\,b^2\,c\,z^4-256\,a^4\,c^2\,z^4-16\,a^2\,b^4\,z^4+128\,A\,a^2\,b^2\,c\,z^3-256\,A\,a^3\,c^2\,z^3-16\,A\,a\,b^4\,z^3+32\,A\,C\,a^2\,b\,c\,z^2-8\,A\,C\,a\,b^3\,z^2+16\,B^2\,a^2\,b\,c\,z^2+40\,A^2\,a\,b^2\,c\,z^2-32\,C^2\,a^3\,c\,z^2-4\,B^2\,a\,b^3\,z^2+8\,C^2\,a^2\,b^2\,z^2-96\,A^2\,a^2\,c^2\,z^2-4\,A^2\,b^4\,z^2+16\,A^2\,C\,a\,b\,c\,z+16\,B^2\,C\,a^2\,c\,z-16\,A\,C^2\,a^2\,c\,z-4\,B^2\,C\,a\,b^2\,z+4\,A\,C^2\,a\,b^2\,z+4\,A^3\,b^2\,c\,z-16\,A^3\,a\,c^2\,z-4\,A^2\,C\,b^3\,z+4\,A\,B^2\,C\,a\,c-2\,A^2\,C^2\,a\,c+2\,A^3\,C\,b\,c+2\,A\,C^3\,a\,b-B^2\,C^2\,a\,b-A^2\,B^2\,b\,c-B^4\,a\,c-A^2\,C^2\,b^2-C^4\,a^2-A^4\,c^2,z,k\right )\,x\,\left (320\,a^3\,c^4-176\,a^2\,b^2\,c^3+24\,a\,b^4\,c^2\right )-4\,B\,a\,b^3\,c^2+16\,B\,a^2\,b\,c^3\right )+4\,A\,B\,b^3\,c^2+8\,B\,C\,a^2\,c^3-12\,A\,B\,a\,b\,c^3-4\,B\,C\,a\,b^2\,c^2\right )+B^3\,a\,c^3+4\,A^2\,B\,b\,c^3+6\,A\,B\,C\,a\,c^3-4\,A\,B\,C\,b^2\,c^2+B\,C^2\,a\,b\,c^2\right )+A\,B^3\,c^3-2\,A^2\,B\,C\,c^3+A\,B\,C^2\,b\,c^2\right )\,\mathrm {root}\left (128\,a^3\,b^2\,c\,z^4-256\,a^4\,c^2\,z^4-16\,a^2\,b^4\,z^4+128\,A\,a^2\,b^2\,c\,z^3-256\,A\,a^3\,c^2\,z^3-16\,A\,a\,b^4\,z^3+32\,A\,C\,a^2\,b\,c\,z^2-8\,A\,C\,a\,b^3\,z^2+16\,B^2\,a^2\,b\,c\,z^2+40\,A^2\,a\,b^2\,c\,z^2-32\,C^2\,a^3\,c\,z^2-4\,B^2\,a\,b^3\,z^2+8\,C^2\,a^2\,b^2\,z^2-96\,A^2\,a^2\,c^2\,z^2-4\,A^2\,b^4\,z^2+16\,A^2\,C\,a\,b\,c\,z+16\,B^2\,C\,a^2\,c\,z-16\,A\,C^2\,a^2\,c\,z-4\,B^2\,C\,a\,b^2\,z+4\,A\,C^2\,a\,b^2\,z+4\,A^3\,b^2\,c\,z-16\,A^3\,a\,c^2\,z-4\,A^2\,C\,b^3\,z+4\,A\,B^2\,C\,a\,c-2\,A^2\,C^2\,a\,c+2\,A^3\,C\,b\,c+2\,A\,C^3\,a\,b-B^2\,C^2\,a\,b-A^2\,B^2\,b\,c-B^4\,a\,c-A^2\,C^2\,b^2-C^4\,a^2-A^4\,c^2,z,k\right )\right )+\frac {A\,\ln \left (x\right )}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x + C*x^2)/(x*(a + b*x^2 + c*x^4)),x)

[Out]

symsum(log(x*(B^4*c^3 + C^4*a*c^2 + A^2*C^2*c^3 - 3*A*B^2*C*c^3 - A*C^3*b*c^2 + B^2*C^2*b*c^2) - root(128*a^3*
b^2*c*z^4 - 256*a^4*c^2*z^4 - 16*a^2*b^4*z^4 + 128*A*a^2*b^2*c*z^3 - 256*A*a^3*c^2*z^3 - 16*A*a*b^4*z^3 + 32*A
*C*a^2*b*c*z^2 - 8*A*C*a*b^3*z^2 + 16*B^2*a^2*b*c*z^2 + 40*A^2*a*b^2*c*z^2 - 32*C^2*a^3*c*z^2 - 4*B^2*a*b^3*z^
2 + 8*C^2*a^2*b^2*z^2 - 96*A^2*a^2*c^2*z^2 - 4*A^2*b^4*z^2 + 16*A^2*C*a*b*c*z + 16*B^2*C*a^2*c*z - 16*A*C^2*a^
2*c*z - 4*B^2*C*a*b^2*z + 4*A*C^2*a*b^2*z + 4*A^3*b^2*c*z - 16*A^3*a*c^2*z - 4*A^2*C*b^3*z + 4*A*B^2*C*a*c - 2
*A^2*C^2*a*c + 2*A^3*C*b*c + 2*A*C^3*a*b - B^2*C^2*a*b - A^2*B^2*b*c - B^4*a*c - A^2*C^2*b^2 - C^4*a^2 - A^4*c
^2, z, k)*(x*(A*B^2*b*c^3 - 5*A^3*c^4 - 13*A*C^2*a*c^3 + 6*A^2*C*b*c^3 + 17*B^2*C*a*c^3 + C^3*a*b*c^2 + A*C^2*
b^2*c^2 - 4*B^2*C*b^2*c^2) - root(128*a^3*b^2*c*z^4 - 256*a^4*c^2*z^4 - 16*a^2*b^4*z^4 + 128*A*a^2*b^2*c*z^3 -
256*A*a^3*c^2*z^3 - 16*A*a*b^4*z^3 + 32*A*C*a^2*b*c*z^2 - 8*A*C*a*b^3*z^2 + 16*B^2*a^2*b*c*z^2 + 40*A^2*a*b^2
*c*z^2 - 32*C^2*a^3*c*z^2 - 4*B^2*a*b^3*z^2 + 8*C^2*a^2*b^2*z^2 - 96*A^2*a^2*c^2*z^2 - 4*A^2*b^4*z^2 + 16*A^2*
C*a*b*c*z + 16*B^2*C*a^2*c*z - 16*A*C^2*a^2*c*z - 4*B^2*C*a*b^2*z + 4*A*C^2*a*b^2*z + 4*A^3*b^2*c*z - 16*A^3*a
*c^2*z - 4*A^2*C*b^3*z + 4*A*B^2*C*a*c - 2*A^2*C^2*a*c + 2*A^3*C*b*c + 2*A*C^3*a*b - B^2*C^2*a*b - A^2*B^2*b*c
- B^4*a*c - A^2*C^2*b^2 - C^4*a^2 - A^4*c^2, z, k)*(x*(60*A^2*a*c^4 - 16*A^2*b^2*c^3 + 4*B^2*b^3*c^2 + 36*C^2
*a^2*c^3 + 8*A*C*b^3*c^2 - 14*B^2*a*b*c^3 - 10*C^2*a*b^2*c^2 - 28*A*C*a*b*c^3) + root(128*a^3*b^2*c*z^4 - 256*
a^4*c^2*z^4 - 16*a^2*b^4*z^4 + 128*A*a^2*b^2*c*z^3 - 256*A*a^3*c^2*z^3 - 16*A*a*b^4*z^3 + 32*A*C*a^2*b*c*z^2 -
8*A*C*a*b^3*z^2 + 16*B^2*a^2*b*c*z^2 + 40*A^2*a*b^2*c*z^2 - 32*C^2*a^3*c*z^2 - 4*B^2*a*b^3*z^2 + 8*C^2*a^2*b^
2*z^2 - 96*A^2*a^2*c^2*z^2 - 4*A^2*b^4*z^2 + 16*A^2*C*a*b*c*z + 16*B^2*C*a^2*c*z - 16*A*C^2*a^2*c*z - 4*B^2*C*
a*b^2*z + 4*A*C^2*a*b^2*z + 4*A^3*b^2*c*z - 16*A^3*a*c^2*z - 4*A^2*C*b^3*z + 4*A*B^2*C*a*c - 2*A^2*C^2*a*c + 2
*A^3*C*b*c + 2*A*C^3*a*b - B^2*C^2*a*b - A^2*B^2*b*c - B^4*a*c - A^2*C^2*b^2 - C^4*a^2 - A^4*c^2, z, k)*(x*(24
0*A*a^2*c^4 + 12*A*b^4*c^2 - 108*A*a*b^2*c^3 + 4*C*a*b^3*c^2 - 16*C*a^2*b*c^3) + root(128*a^3*b^2*c*z^4 - 256*
a^4*c^2*z^4 - 16*a^2*b^4*z^4 + 128*A*a^2*b^2*c*z^3 - 256*A*a^3*c^2*z^3 - 16*A*a*b^4*z^3 + 32*A*C*a^2*b*c*z^2 -
8*A*C*a*b^3*z^2 + 16*B^2*a^2*b*c*z^2 + 40*A^2*a*b^2*c*z^2 - 32*C^2*a^3*c*z^2 - 4*B^2*a*b^3*z^2 + 8*C^2*a^2*b^
2*z^2 - 96*A^2*a^2*c^2*z^2 - 4*A^2*b^4*z^2 + 16*A^2*C*a*b*c*z + 16*B^2*C*a^2*c*z - 16*A*C^2*a^2*c*z - 4*B^2*C*
a*b^2*z + 4*A*C^2*a*b^2*z + 4*A^3*b^2*c*z - 16*A^3*a*c^2*z - 4*A^2*C*b^3*z + 4*A*B^2*C*a*c - 2*A^2*C^2*a*c + 2
*A^3*C*b*c + 2*A*C^3*a*b - B^2*C^2*a*b - A^2*B^2*b*c - B^4*a*c - A^2*C^2*b^2 - C^4*a^2 - A^4*c^2, z, k)*x*(320
*a^3*c^4 + 24*a*b^4*c^2 - 176*a^2*b^2*c^3) - 4*B*a*b^3*c^2 + 16*B*a^2*b*c^3) + 4*A*B*b^3*c^2 + 8*B*C*a^2*c^3 -
12*A*B*a*b*c^3 - 4*B*C*a*b^2*c^2) + B^3*a*c^3 + 4*A^2*B*b*c^3 + 6*A*B*C*a*c^3 - 4*A*B*C*b^2*c^2 + B*C^2*a*b*c
^2) + A*B^3*c^3 - 2*A^2*B*C*c^3 + A*B*C^2*b*c^2)*root(128*a^3*b^2*c*z^4 - 256*a^4*c^2*z^4 - 16*a^2*b^4*z^4 + 1
28*A*a^2*b^2*c*z^3 - 256*A*a^3*c^2*z^3 - 16*A*a*b^4*z^3 + 32*A*C*a^2*b*c*z^2 - 8*A*C*a*b^3*z^2 + 16*B^2*a^2*b*
c*z^2 + 40*A^2*a*b^2*c*z^2 - 32*C^2*a^3*c*z^2 - 4*B^2*a*b^3*z^2 + 8*C^2*a^2*b^2*z^2 - 96*A^2*a^2*c^2*z^2 - 4*A
^2*b^4*z^2 + 16*A^2*C*a*b*c*z + 16*B^2*C*a^2*c*z - 16*A*C^2*a^2*c*z - 4*B^2*C*a*b^2*z + 4*A*C^2*a*b^2*z + 4*A^
3*b^2*c*z - 16*A^3*a*c^2*z - 4*A^2*C*b^3*z + 4*A*B^2*C*a*c - 2*A^2*C^2*a*c + 2*A^3*C*b*c + 2*A*C^3*a*b - B^2*C
^2*a*b - A^2*B^2*b*c - B^4*a*c - A^2*C^2*b^2 - C^4*a^2 - A^4*c^2, z, k), k, 1, 4) + (A*log(x))/a

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