3.1.27 \(\int \frac {A+B x+C x^2}{x^2 (a+b x^2+c x^4)} \, dx\) [27]

Optimal. Leaf size=260 \[ -\frac {A}{a x}-\frac {\sqrt {c} \left (A+\frac {A b-2 a C}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (A-\frac {A b-2 a C}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {b B \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a \sqrt {b^2-4 a c}}+\frac {B \log (x)}{a}-\frac {B \log \left (a+b x^2+c x^4\right )}{4 a} \]

[Out]

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

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Rubi [A]
time = 0.32, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 12, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {1676, 1295, 1180, 211, 12, 1128, 719, 29, 648, 632, 212, 642} \begin {gather*} -\frac {\sqrt {c} \left (\frac {A b-2 a C}{\sqrt {b^2-4 a c}}+A\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} a \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\sqrt {c} \left (A-\frac {A b-2 a C}{\sqrt {b^2-4 a c}}\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} a \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {A}{a x}+\frac {b B \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 {B \log \left (a+b x^2+c x^4\right )}{4 a}+\frac {B \log (x)}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(A/(a*x)) - (Sqrt[c]*(A + (A*b - 2*a*C)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a
*c]]])/(Sqrt[2]*a*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - (Sqrt[c]*(A - (A*b - 2*a*C)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2
]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*a*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + (b*B*ArcTanh[(b + 2*c*x^2
)/Sqrt[b^2 - 4*a*c]])/(2*a*Sqrt[b^2 - 4*a*c]) + (B*Log[x])/a - (B*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 29

Int[(x_)^(-1), x_Symbol] :> Simp[Log[x], 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 719

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

Rule 1128

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

Rule 1180

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

Rule 1295

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[d*(f
*x)^(m + 1)*((a + b*x^2 + c*x^4)^(p + 1)/(a*f*(m + 1))), x] + Dist[1/(a*f^2*(m + 1)), Int[(f*x)^(m + 2)*(a + b
*x^2 + c*x^4)^p*Simp[a*e*(m + 1) - b*d*(m + 2*p + 3) - c*d*(m + 4*p + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d,
e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && LtQ[m, -1] && IntegerQ[2*p] && (IntegerQ[p] || IntegerQ[m])

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

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Mathematica [A]
time = 0.69, size = 315, normalized size = 1.21 \begin {gather*} -\frac {\frac {4 A}{x}+\frac {2 \sqrt {2} \sqrt {c} \left (A \left (b+\sqrt {b^2-4 a c}\right )-2 a C\right ) \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 {2 \sqrt {2} \sqrt {c} \left (A \left (-b+\sqrt {b^2-4 a c}\right )+2 a C\right ) \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}}}-4 B \log (x)+\frac {B \left (b+\sqrt {b^2-4 a c}\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\sqrt {b^2-4 a c}}+\frac {B \left (-b+\sqrt {b^2-4 a c}\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\sqrt {b^2-4 a c}}}{4 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((4*A)/x + (2*Sqrt[2]*Sqrt[c]*(A*(b + Sqrt[b^2 - 4*a*c]) - 2*a*C)*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqr
t[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (2*Sqrt[2]*Sqrt[c]*(A*(-b + Sqrt[b^2 - 4*a
*c]) + 2*a*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]]) - 4*B*Log[x] + (B*(b + Sqrt[b^2 - 4*a*c])*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/Sqrt[b^2 - 4*a*c] +
(B*(-b + Sqrt[b^2 - 4*a*c])*Log[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/a

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Maple [A]
time = 0.07, size = 325, normalized size = 1.25

method	result	size
default	\(\frac {4 c \left (\frac {-\frac {\left (-B \sqrt {-4 a c +b^{2}}\, b +4 a c B -b^{2} B \right ) \ln \left (-b -2 c \,x^{2}+\sqrt {-4 a c +b^{2}}\right )}{4 c}+\frac {\left (-A b \sqrt {-4 a c +b^{2}}+4 a c A -A \,b^{2}+2 C \sqrt {-4 a c +b^{2}}\, a \right ) \sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{16 a c -4 b^{2}}+\frac {\frac {\left (-B \sqrt {-4 a c +b^{2}}\, b -4 a c B +b^{2} B \right ) \ln \left (b +2 c \,x^{2}+\sqrt {-4 a c +b^{2}}\right )}{4 c}+\frac {\left (-A b \sqrt {-4 a c +b^{2}}-4 a c A +A \,b^{2}+2 C \sqrt {-4 a c +b^{2}}\, a \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}}{16 a c -4 b^{2}}\right )}{a}-\frac {A}{a x}+\frac {B \ln \left (x \right )}{a}\)	\(325\)
risch	\(-\frac {A}{a x}+\frac {B \ln \left (x \right )}{a}+\frac {\left (\munderset {\textit {\_R} =\RootOf \left (\left (16 a^{5} c^{2}-8 a^{4} b^{2} c +a^{3} b^{4}\right ) \textit {\_Z}^{4}+\left (32 B \,a^{4} c^{2}-16 B \,a^{3} b^{2} c +2 B \,a^{2} b^{4}\right ) \textit {\_Z}^{3}+\left (12 A^{2} a^{2} b \,c^{2}-7 A^{2} a \,b^{3} c +A^{2} b^{5}-16 A C \,a^{3} c^{2}+12 A C \,a^{2} b^{2} c -2 A C a \,b^{4}+24 B^{2} a^{3} c^{2}-10 B^{2} a^{2} b^{2} c +B^{2} a \,b^{4}-4 C^{2} a^{3} b c +C^{2} a^{2} b^{3}\right ) \textit {\_Z}^{2}+\left (8 A^{2} B a b \,c^{2}-2 A^{2} B \,b^{3} c -16 A B C \,a^{2} c^{2}+4 A B C a \,b^{2} c +8 B^{3} a^{2} c^{2}-2 B^{3} a \,b^{2} c \right ) \textit {\_Z} +c^{3} A^{4}-2 A^{3} C b \,c^{2}+A^{2} B^{2} b \,c^{2}+2 A^{2} C^{2} a \,c^{2}+A^{2} C^{2} b^{2} c -4 A \,B^{2} C a \,c^{2}-2 A \,C^{3} a b c +B^{4} a \,c^{2}+B^{2} C^{2} a b c +C^{4} a^{2} c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (40 a^{5} c^{2}-22 a^{4} b^{2} c +3 a^{3} b^{4}\right ) \textit {\_R}^{4}+\left (60 B \,a^{4} c^{2}-27 B \,a^{3} b^{2} c +3 B \,a^{2} b^{4}\right ) \textit {\_R}^{3}+\left (25 A^{2} a^{2} b \,c^{2}-14 A^{2} a \,b^{3} c +2 A^{2} b^{5}-36 A C \,a^{3} c^{2}+24 A C \,a^{2} b^{2} c -4 A C a \,b^{4}+30 B^{2} a^{3} c^{2}-8 B^{2} a^{2} b^{2} c -7 C^{2} a^{3} b c +2 C^{2} a^{2} b^{3}\right ) \textit {\_R}^{2}+\left (14 A^{2} B a b \,c^{2}-4 A^{2} B \,b^{3} c -26 A B C \,a^{2} c^{2}+8 A B C a \,b^{2} c +5 B^{3} a^{2} c^{2}-B \,C^{2} a^{2} b c \right ) \textit {\_R} +2 c^{3} A^{4}-4 A^{3} C b \,c^{2}+2 A^{2} B^{2} b \,c^{2}+4 A^{2} C^{2} a \,c^{2}+2 A^{2} C^{2} b^{2} c -4 A \,B^{2} C a \,c^{2}-4 A \,C^{3} a b c +2 C^{4} a^{2} c \right ) x +\left (4 A \,a^{4} c^{2}-5 A \,a^{3} b^{2} c +A \,a^{2} b^{4}+4 C \,a^{4} b c -C \,a^{3} b^{3}\right ) \textit {\_R}^{3}+\left (-4 A B \,a^{3} c^{2}+8 A B \,a^{2} b^{2} c -2 A B a \,b^{4}-6 B C \,a^{3} b c +2 B C \,a^{2} b^{3}\right ) \textit {\_R}^{2}+\left (-A^{2} C \,a^{2} c^{2}-7 A \,B^{2} a^{2} c^{2}+4 A \,B^{2} a \,b^{2} c +A \,C^{2} a^{2} b c -4 B^{2} C \,a^{2} b c -C^{3} a^{3} c \right ) \textit {\_R} +2 A^{2} B C a \,c^{2}-2 A \,B^{3} a \,c^{2}-2 A B \,C^{2} a b c +2 B \,C^{3} a^{2} c \right )\right )}{2}\)	\(903\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4/a*c*(1/(16*a*c-4*b^2)*(-1/4*(-B*(-4*a*c+b^2)^(1/2)*b+4*a*c*B-b^2*B)/c*ln(-b-2*c*x^2+(-4*a*c+b^2)^(1/2))+1/2*
(-A*b*(-4*a*c+b^2)^(1/2)+4*a*c*A-A*b^2+2*C*(-4*a*c+b^2)^(1/2)*a)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arc
tanh(c*x*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)))+1/(16*a*c-4*b^2)*(1/4*(-B*(-4*a*c+b^2)^(1/2)*b-4*a*c*B+b^
2*B)/c*ln(b+2*c*x^2+(-4*a*c+b^2)^(1/2))+1/2*(-A*b*(-4*a*c+b^2)^(1/2)-4*a*c*A+A*b^2+2*C*(-4*a*c+b^2)^(1/2)*a)*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/a/x+B*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^2/(c*x^4+b*x^2+a),x, algorithm="maxima")

[Out]

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

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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^2/(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**2/(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. 3508 vs. \(2 (218) = 436\).
time = 7.46, size = 3508, normalized size = 13.49 \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^2/(c*x^4+b*x^2+a),x, algorithm="giac")

[Out]

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

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Mupad [B]
time = 1.02, size = 2588, normalized size = 9.95 \begin {gather*} \left (\sum _{k=1}^4\ln \left (\mathrm {root}\left (128\,a^4\,b^2\,c\,z^4-256\,a^5\,c^2\,z^4-16\,a^3\,b^4\,z^4+128\,B\,a^3\,b^2\,c\,z^3-256\,B\,a^4\,c^2\,z^3-16\,B\,a^2\,b^4\,z^3-48\,A\,C\,a^2\,b^2\,c\,z^2+8\,A\,C\,a\,b^4\,z^2+40\,B^2\,a^2\,b^2\,c\,z^2-48\,A^2\,a^2\,b\,c^2\,z^2+16\,C^2\,a^3\,b\,c\,z^2+28\,A^2\,a\,b^3\,c\,z^2+64\,A\,C\,a^3\,c^2\,z^2-4\,B^2\,a\,b^4\,z^2-96\,B^2\,a^3\,c^2\,z^2-4\,C^2\,a^2\,b^3\,z^2-4\,A^2\,b^5\,z^2-8\,A\,B\,C\,a\,b^2\,c\,z-16\,A^2\,B\,a\,b\,c^2\,z+32\,A\,B\,C\,a^2\,c^2\,z+4\,A^2\,B\,b^3\,c\,z+4\,B^3\,a\,b^2\,c\,z-16\,B^3\,a^2\,c^2\,z+4\,A\,B^2\,C\,a\,c^2+2\,A\,C^3\,a\,b\,c-B^2\,C^2\,a\,b\,c-2\,A^2\,C^2\,a\,c^2+2\,A^3\,C\,b\,c^2-A^2\,C^2\,b^2\,c-A^2\,B^2\,b\,c^2-C^4\,a^2\,c-B^4\,a\,c^2-A^4\,c^3,z,k\right )\,\left (-\frac {A^2\,C\,a\,c^4+7\,A\,B^2\,a\,c^4-4\,A\,B^2\,b^2\,c^3-A\,C^2\,a\,b\,c^3+4\,B^2\,C\,a\,b\,c^3+C^3\,a^2\,c^3}{a}+\mathrm {root}\left (128\,a^4\,b^2\,c\,z^4-256\,a^5\,c^2\,z^4-16\,a^3\,b^4\,z^4+128\,B\,a^3\,b^2\,c\,z^3-256\,B\,a^4\,c^2\,z^3-16\,B\,a^2\,b^4\,z^3-48\,A\,C\,a^2\,b^2\,c\,z^2+8\,A\,C\,a\,b^4\,z^2+40\,B^2\,a^2\,b^2\,c\,z^2-48\,A^2\,a^2\,b\,c^2\,z^2+16\,C^2\,a^3\,b\,c\,z^2+28\,A^2\,a\,b^3\,c\,z^2+64\,A\,C\,a^3\,c^2\,z^2-4\,B^2\,a\,b^4\,z^2-96\,B^2\,a^3\,c^2\,z^2-4\,C^2\,a^2\,b^3\,z^2-4\,A^2\,b^5\,z^2-8\,A\,B\,C\,a\,b^2\,c\,z-16\,A^2\,B\,a\,b\,c^2\,z+32\,A\,B\,C\,a^2\,c^2\,z+4\,A^2\,B\,b^3\,c\,z+4\,B^3\,a\,b^2\,c\,z-16\,B^3\,a^2\,c^2\,z+4\,A\,B^2\,C\,a\,c^2+2\,A\,C^3\,a\,b\,c-B^2\,C^2\,a\,b\,c-2\,A^2\,C^2\,a\,c^2+2\,A^3\,C\,b\,c^2-A^2\,C^2\,b^2\,c-A^2\,B^2\,b\,c^2-C^4\,a^2\,c-B^4\,a\,c^2-A^4\,c^3,z,k\right )\,\left (-\frac {12\,B\,C\,a^2\,b\,c^3+8\,A\,B\,a^2\,c^4-4\,B\,C\,a\,b^3\,c^2-16\,A\,B\,a\,b^2\,c^3+4\,A\,B\,b^4\,c^2}{a}+\mathrm {root}\left (128\,a^4\,b^2\,c\,z^4-256\,a^5\,c^2\,z^4-16\,a^3\,b^4\,z^4+128\,B\,a^3\,b^2\,c\,z^3-256\,B\,a^4\,c^2\,z^3-16\,B\,a^2\,b^4\,z^3-48\,A\,C\,a^2\,b^2\,c\,z^2+8\,A\,C\,a\,b^4\,z^2+40\,B^2\,a^2\,b^2\,c\,z^2-48\,A^2\,a^2\,b\,c^2\,z^2+16\,C^2\,a^3\,b\,c\,z^2+28\,A^2\,a\,b^3\,c\,z^2+64\,A\,C\,a^3\,c^2\,z^2-4\,B^2\,a\,b^4\,z^2-96\,B^2\,a^3\,c^2\,z^2-4\,C^2\,a^2\,b^3\,z^2-4\,A^2\,b^5\,z^2-8\,A\,B\,C\,a\,b^2\,c\,z-16\,A^2\,B\,a\,b\,c^2\,z+32\,A\,B\,C\,a^2\,c^2\,z+4\,A^2\,B\,b^3\,c\,z+4\,B^3\,a\,b^2\,c\,z-16\,B^3\,a^2\,c^2\,z+4\,A\,B^2\,C\,a\,c^2+2\,A\,C^3\,a\,b\,c-B^2\,C^2\,a\,b\,c-2\,A^2\,C^2\,a\,c^2+2\,A^3\,C\,b\,c^2-A^2\,C^2\,b^2\,c-A^2\,B^2\,b\,c^2-C^4\,a^2\,c-B^4\,a\,c^2-A^4\,c^3,z,k\right )\,\left (\frac {16\,C\,a^3\,b\,c^3+16\,A\,a^3\,c^4-4\,C\,a^2\,b^3\,c^2-20\,A\,a^2\,b^2\,c^3+4\,A\,a\,b^4\,c^2}{a}+\frac {x\,\left (240\,B\,a^4\,c^4-108\,B\,a^3\,b^2\,c^3+12\,B\,a^2\,b^4\,c^2\right )}{a^2}+\frac {\mathrm {root}\left (128\,a^4\,b^2\,c\,z^4-256\,a^5\,c^2\,z^4-16\,a^3\,b^4\,z^4+128\,B\,a^3\,b^2\,c\,z^3-256\,B\,a^4\,c^2\,z^3-16\,B\,a^2\,b^4\,z^3-48\,A\,C\,a^2\,b^2\,c\,z^2+8\,A\,C\,a\,b^4\,z^2+40\,B^2\,a^2\,b^2\,c\,z^2-48\,A^2\,a^2\,b\,c^2\,z^2+16\,C^2\,a^3\,b\,c\,z^2+28\,A^2\,a\,b^3\,c\,z^2+64\,A\,C\,a^3\,c^2\,z^2-4\,B^2\,a\,b^4\,z^2-96\,B^2\,a^3\,c^2\,z^2-4\,C^2\,a^2\,b^3\,z^2-4\,A^2\,b^5\,z^2-8\,A\,B\,C\,a\,b^2\,c\,z-16\,A^2\,B\,a\,b\,c^2\,z+32\,A\,B\,C\,a^2\,c^2\,z+4\,A^2\,B\,b^3\,c\,z+4\,B^3\,a\,b^2\,c\,z-16\,B^3\,a^2\,c^2\,z+4\,A\,B^2\,C\,a\,c^2+2\,A\,C^3\,a\,b\,c-B^2\,C^2\,a\,b\,c-2\,A^2\,C^2\,a\,c^2+2\,A^3\,C\,b\,c^2-A^2\,C^2\,b^2\,c-A^2\,B^2\,b\,c^2-C^4\,a^2\,c-B^4\,a\,c^2-A^4\,c^3,z,k\right )\,x\,\left (320\,a^5\,c^4-176\,a^4\,b^2\,c^3+24\,a^3\,b^4\,c^2\right )}{a^2}\right )+\frac {x\,\left (50\,A^2\,a^2\,b\,c^4-28\,A^2\,a\,b^3\,c^3+4\,A^2\,b^5\,c^2-72\,A\,C\,a^3\,c^4+48\,A\,C\,a^2\,b^2\,c^3-8\,A\,C\,a\,b^4\,c^2+60\,B^2\,a^3\,c^4-16\,B^2\,a^2\,b^2\,c^3-14\,C^2\,a^3\,b\,c^3+4\,C^2\,a^2\,b^3\,c^2\right )}{a^2}\right )+\frac {x\,\left (14\,A^2\,B\,a\,b\,c^4-4\,A^2\,B\,b^3\,c^3-26\,A\,B\,C\,a^2\,c^4+8\,A\,B\,C\,a\,b^2\,c^3+5\,B^3\,a^2\,c^4-B\,C^2\,a^2\,b\,c^3\right )}{a^2}\right )-\frac {-A^2\,B\,C\,c^4+A\,B^3\,c^4+b\,A\,B\,C^2\,c^3-a\,B\,C^3\,c^3}{a}+\frac {x\,\left (A^4\,c^5-2\,A^3\,C\,b\,c^4+A^2\,B^2\,b\,c^4+2\,A^2\,C^2\,a\,c^4+A^2\,C^2\,b^2\,c^3-2\,A\,B^2\,C\,a\,c^4-2\,A\,C^3\,a\,b\,c^3+C^4\,a^2\,c^3\right )}{a^2}\right )\,\mathrm {root}\left (128\,a^4\,b^2\,c\,z^4-256\,a^5\,c^2\,z^4-16\,a^3\,b^4\,z^4+128\,B\,a^3\,b^2\,c\,z^3-256\,B\,a^4\,c^2\,z^3-16\,B\,a^2\,b^4\,z^3-48\,A\,C\,a^2\,b^2\,c\,z^2+8\,A\,C\,a\,b^4\,z^2+40\,B^2\,a^2\,b^2\,c\,z^2-48\,A^2\,a^2\,b\,c^2\,z^2+16\,C^2\,a^3\,b\,c\,z^2+28\,A^2\,a\,b^3\,c\,z^2+64\,A\,C\,a^3\,c^2\,z^2-4\,B^2\,a\,b^4\,z^2-96\,B^2\,a^3\,c^2\,z^2-4\,C^2\,a^2\,b^3\,z^2-4\,A^2\,b^5\,z^2-8\,A\,B\,C\,a\,b^2\,c\,z-16\,A^2\,B\,a\,b\,c^2\,z+32\,A\,B\,C\,a^2\,c^2\,z+4\,A^2\,B\,b^3\,c\,z+4\,B^3\,a\,b^2\,c\,z-16\,B^3\,a^2\,c^2\,z+4\,A\,B^2\,C\,a\,c^2+2\,A\,C^3\,a\,b\,c-B^2\,C^2\,a\,b\,c-2\,A^2\,C^2\,a\,c^2+2\,A^3\,C\,b\,c^2-A^2\,C^2\,b^2\,c-A^2\,B^2\,b\,c^2-C^4\,a^2\,c-B^4\,a\,c^2-A^4\,c^3,z,k\right )\right )-\frac {A}{a\,x}+\frac {B\,\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^2*(a + b*x^2 + c*x^4)),x)

[Out]

symsum(log(root(128*a^4*b^2*c*z^4 - 256*a^5*c^2*z^4 - 16*a^3*b^4*z^4 + 128*B*a^3*b^2*c*z^3 - 256*B*a^4*c^2*z^3
- 16*B*a^2*b^4*z^3 - 48*A*C*a^2*b^2*c*z^2 + 8*A*C*a*b^4*z^2 + 40*B^2*a^2*b^2*c*z^2 - 48*A^2*a^2*b*c^2*z^2 + 1
6*C^2*a^3*b*c*z^2 + 28*A^2*a*b^3*c*z^2 + 64*A*C*a^3*c^2*z^2 - 4*B^2*a*b^4*z^2 - 96*B^2*a^3*c^2*z^2 - 4*C^2*a^2
*b^3*z^2 - 4*A^2*b^5*z^2 - 8*A*B*C*a*b^2*c*z - 16*A^2*B*a*b*c^2*z + 32*A*B*C*a^2*c^2*z + 4*A^2*B*b^3*c*z + 4*B
^3*a*b^2*c*z - 16*B^3*a^2*c^2*z + 4*A*B^2*C*a*c^2 + 2*A*C^3*a*b*c - B^2*C^2*a*b*c - 2*A^2*C^2*a*c^2 + 2*A^3*C*
b*c^2 - A^2*C^2*b^2*c - A^2*B^2*b*c^2 - C^4*a^2*c - B^4*a*c^2 - A^4*c^3, z, k)*(root(128*a^4*b^2*c*z^4 - 256*a
^5*c^2*z^4 - 16*a^3*b^4*z^4 + 128*B*a^3*b^2*c*z^3 - 256*B*a^4*c^2*z^3 - 16*B*a^2*b^4*z^3 - 48*A*C*a^2*b^2*c*z^
2 + 8*A*C*a*b^4*z^2 + 40*B^2*a^2*b^2*c*z^2 - 48*A^2*a^2*b*c^2*z^2 + 16*C^2*a^3*b*c*z^2 + 28*A^2*a*b^3*c*z^2 +
64*A*C*a^3*c^2*z^2 - 4*B^2*a*b^4*z^2 - 96*B^2*a^3*c^2*z^2 - 4*C^2*a^2*b^3*z^2 - 4*A^2*b^5*z^2 - 8*A*B*C*a*b^2*
c*z - 16*A^2*B*a*b*c^2*z + 32*A*B*C*a^2*c^2*z + 4*A^2*B*b^3*c*z + 4*B^3*a*b^2*c*z - 16*B^3*a^2*c^2*z + 4*A*B^2
*C*a*c^2 + 2*A*C^3*a*b*c - B^2*C^2*a*b*c - 2*A^2*C^2*a*c^2 + 2*A^3*C*b*c^2 - A^2*C^2*b^2*c - A^2*B^2*b*c^2 - C
^4*a^2*c - B^4*a*c^2 - A^4*c^3, z, k)*(root(128*a^4*b^2*c*z^4 - 256*a^5*c^2*z^4 - 16*a^3*b^4*z^4 + 128*B*a^3*b
^2*c*z^3 - 256*B*a^4*c^2*z^3 - 16*B*a^2*b^4*z^3 - 48*A*C*a^2*b^2*c*z^2 + 8*A*C*a*b^4*z^2 + 40*B^2*a^2*b^2*c*z^
2 - 48*A^2*a^2*b*c^2*z^2 + 16*C^2*a^3*b*c*z^2 + 28*A^2*a*b^3*c*z^2 + 64*A*C*a^3*c^2*z^2 - 4*B^2*a*b^4*z^2 - 96
*B^2*a^3*c^2*z^2 - 4*C^2*a^2*b^3*z^2 - 4*A^2*b^5*z^2 - 8*A*B*C*a*b^2*c*z - 16*A^2*B*a*b*c^2*z + 32*A*B*C*a^2*c
^2*z + 4*A^2*B*b^3*c*z + 4*B^3*a*b^2*c*z - 16*B^3*a^2*c^2*z + 4*A*B^2*C*a*c^2 + 2*A*C^3*a*b*c - B^2*C^2*a*b*c
- 2*A^2*C^2*a*c^2 + 2*A^3*C*b*c^2 - A^2*C^2*b^2*c - A^2*B^2*b*c^2 - C^4*a^2*c - B^4*a*c^2 - A^4*c^3, z, k)*((1
6*A*a^3*c^4 + 4*A*a*b^4*c^2 + 16*C*a^3*b*c^3 - 20*A*a^2*b^2*c^3 - 4*C*a^2*b^3*c^2)/a + (x*(240*B*a^4*c^4 + 12*
B*a^2*b^4*c^2 - 108*B*a^3*b^2*c^3))/a^2 + (root(128*a^4*b^2*c*z^4 - 256*a^5*c^2*z^4 - 16*a^3*b^4*z^4 + 128*B*a
^3*b^2*c*z^3 - 256*B*a^4*c^2*z^3 - 16*B*a^2*b^4*z^3 - 48*A*C*a^2*b^2*c*z^2 + 8*A*C*a*b^4*z^2 + 40*B^2*a^2*b^2*
c*z^2 - 48*A^2*a^2*b*c^2*z^2 + 16*C^2*a^3*b*c*z^2 + 28*A^2*a*b^3*c*z^2 + 64*A*C*a^3*c^2*z^2 - 4*B^2*a*b^4*z^2
- 96*B^2*a^3*c^2*z^2 - 4*C^2*a^2*b^3*z^2 - 4*A^2*b^5*z^2 - 8*A*B*C*a*b^2*c*z - 16*A^2*B*a*b*c^2*z + 32*A*B*C*a
^2*c^2*z + 4*A^2*B*b^3*c*z + 4*B^3*a*b^2*c*z - 16*B^3*a^2*c^2*z + 4*A*B^2*C*a*c^2 + 2*A*C^3*a*b*c - B^2*C^2*a*
b*c - 2*A^2*C^2*a*c^2 + 2*A^3*C*b*c^2 - A^2*C^2*b^2*c - A^2*B^2*b*c^2 - C^4*a^2*c - B^4*a*c^2 - A^4*c^3, z, k)
*x*(320*a^5*c^4 + 24*a^3*b^4*c^2 - 176*a^4*b^2*c^3))/a^2) - (8*A*B*a^2*c^4 + 4*A*B*b^4*c^2 - 16*A*B*a*b^2*c^3
- 4*B*C*a*b^3*c^2 + 12*B*C*a^2*b*c^3)/a + (x*(4*A^2*b^5*c^2 + 60*B^2*a^3*c^4 - 16*B^2*a^2*b^2*c^3 + 4*C^2*a^2*
b^3*c^2 - 72*A*C*a^3*c^4 - 28*A^2*a*b^3*c^3 + 50*A^2*a^2*b*c^4 - 14*C^2*a^3*b*c^3 + 48*A*C*a^2*b^2*c^3 - 8*A*C
*a*b^4*c^2))/a^2) - (C^3*a^2*c^3 + 7*A*B^2*a*c^4 + A^2*C*a*c^4 - 4*A*B^2*b^2*c^3 - A*C^2*a*b*c^3 + 4*B^2*C*a*b
*c^3)/a + (x*(5*B^3*a^2*c^4 - 4*A^2*B*b^3*c^3 - B*C^2*a^2*b*c^3 - 26*A*B*C*a^2*c^4 + 14*A^2*B*a*b*c^4 + 8*A*B*
C*a*b^2*c^3))/a^2) - (A*B^3*c^4 - A^2*B*C*c^4 - B*C^3*a*c^3 + A*B*C^2*b*c^3)/a + (x*(A^4*c^5 + C^4*a^2*c^3 + A
^2*C^2*b^2*c^3 - 2*A^3*C*b*c^4 + A^2*B^2*b*c^4 + 2*A^2*C^2*a*c^4 - 2*A*B^2*C*a*c^4 - 2*A*C^3*a*b*c^3))/a^2)*ro
ot(128*a^4*b^2*c*z^4 - 256*a^5*c^2*z^4 - 16*a^3*b^4*z^4 + 128*B*a^3*b^2*c*z^3 - 256*B*a^4*c^2*z^3 - 16*B*a^2*b
^4*z^3 - 48*A*C*a^2*b^2*c*z^2 + 8*A*C*a*b^4*z^2 + 40*B^2*a^2*b^2*c*z^2 - 48*A^2*a^2*b*c^2*z^2 + 16*C^2*a^3*b*c
*z^2 + 28*A^2*a*b^3*c*z^2 + 64*A*C*a^3*c^2*z^2 - 4*B^2*a*b^4*z^2 - 96*B^2*a^3*c^2*z^2 - 4*C^2*a^2*b^3*z^2 - 4*
A^2*b^5*z^2 - 8*A*B*C*a*b^2*c*z - 16*A^2*B*a*b*c^2*z + 32*A*B*C*a^2*c^2*z + 4*A^2*B*b^3*c*z + 4*B^3*a*b^2*c*z
- 16*B^3*a^2*c^2*z + 4*A*B^2*C*a*c^2 + 2*A*C^3*a*b*c - B^2*C^2*a*b*c - 2*A^2*C^2*a*c^2 + 2*A^3*C*b*c^2 - A^2*C
^2*b^2*c - A^2*B^2*b*c^2 - C^4*a^2*c - B^4*a*c^2 - A^4*c^3, z, k), k, 1, 4) - A/(a*x) + (B*log(x))/a

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