∫ A+Bx2 ________________________

3.131 \(\int \frac {A+B x^2}{x^3 (a+b x^2+c x^4)^3} \, dx\)

Optimal. Leaf size=363 \[ \frac {(3 A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac {\log (x) (3 A b-a B)}{a^4}-\frac {-A \left (20 a^2 c^2-20 a b^2 c+3 b^4\right )+c x^2 \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right )+a b B \left (b^2-10 a c\right )}{4 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {a b B \left (b^2-7 a c\right )-3 A \left (10 a^2 c^2-7 a b^2 c+b^4\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}+\frac {\left (a b B \left (30 a^2 c^2-10 a b^2 c+b^4\right )-3 A \left (-20 a^3 c^3+30 a^2 b^2 c^2-10 a b^4 c+b^6\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}-\frac {-A \left (b^2-2 a c\right )-\left (c x^2 (A b-2 a B)\right )+a b B}{4 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

[Out]

1/2*(a*b*B*(-7*a*c+b^2)-3*A*(10*a^2*c^2-7*a*b^2*c+b^4))/a^3/(-4*a*c+b^2)^2/x^2+1/4*(-a*b*B+A*(-2*a*c+b^2)+(A*b

-2*B*a)*c*x^2)/a/(-4*a*c+b^2)/x^2/(c*x^4+b*x^2+a)^2+1/4*(-a*b*B*(-10*a*c+b^2)+A*(20*a^2*c^2-20*a*b^2*c+3*b^4)-

c*(a*B*(-16*a*c+b^2)-3*A*(-6*a*b*c+b^3))*x^2)/a^2/(-4*a*c+b^2)^2/x^2/(c*x^4+b*x^2+a)+1/2*(a*b*B*(30*a^2*c^2-10

*a*b^2*c+b^4)-3*A*(-20*a^3*c^3+30*a^2*b^2*c^2-10*a*b^4*c+b^6))*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1/2))/a^4/(-4

*a*c+b^2)^(5/2)-(3*A*b-B*a)*ln(x)/a^4+1/4*(3*A*b-B*a)*ln(c*x^4+b*x^2+a)/a^4

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Rubi [A] time = 0.77, antiderivative size = 363, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {1251, 822, 800, 634, 618, 206, 628} \[ \frac {a b B \left (b^2-7 a c\right )-3 A \left (10 a^2 c^2-7 a b^2 c+b^4\right )}{2 a^3 x^2 \left (b^2-4 a c\right )^2}-\frac {-A \left (20 a^2 c^2-20 a b^2 c+3 b^4\right )+c x^2 \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right )+a b B \left (b^2-10 a c\right )}{4 a^2 x^2 \left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (a b B \left (30 a^2 c^2-10 a b^2 c+b^4\right )-3 A \left (30 a^2 b^2 c^2-20 a^3 c^3-10 a b^4 c+b^6\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}+\frac {(3 A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^4}-\frac {\log (x) (3 A b-a B)}{a^4}-\frac {-A \left (b^2-2 a c\right )+c x^2 (-(A b-2 a B))+a b B}{4 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2} \]

Antiderivative was successfully veriﬁed.

[In]

Int[(A + B*x^2)/(x^3*(a + b*x^2 + c*x^4)^3),x]

[Out]

(a*b*B*(b^2 - 7*a*c) - 3*A*(b^4 - 7*a*b^2*c + 10*a^2*c^2))/(2*a^3*(b^2 - 4*a*c)^2*x^2) - (a*b*B - A*(b^2 - 2*a

*c) - (A*b - 2*a*B)*c*x^2)/(4*a*(b^2 - 4*a*c)*x^2*(a + b*x^2 + c*x^4)^2) - (a*b*B*(b^2 - 10*a*c) - A*(3*b^4 -

20*a*b^2*c + 20*a^2*c^2) + c*(a*B*(b^2 - 16*a*c) - 3*A*(b^3 - 6*a*b*c))*x^2)/(4*a^2*(b^2 - 4*a*c)^2*x^2*(a + b

*x^2 + c*x^4)) + ((a*b*B*(b^4 - 10*a*b^2*c + 30*a^2*c^2) - 3*A*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^3

))*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^4*(b^2 - 4*a*c)^(5/2)) - ((3*A*b - a*B)*Log[x])/a^4 + ((3*A*

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

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 800

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 822

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp

[((d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x

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

c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2

*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d -

b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b,

c, d, e, f, g, m}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] ||

IntegerQ[p] || IntegersQ[2*m, 2*p])

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 {A+B x^2}{x^3 \left (a+b x^2+c x^4\right )^3} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {A+B x}{x^2 \left (a+b x+c x^2\right )^3} \, dx,x,x^2\right )\\ &=-\frac {a b B-A \left (b^2-2 a c\right )-(A b-2 a B) c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}-\frac {\operatorname {Subst}\left (\int \frac {-3 A b^2+a b B+10 a A c-4 (A b-2 a B) c x}{x^2 \left (a+b x+c x^2\right )^2} \, dx,x,x^2\right )}{4 a \left (b^2-4 a c\right )}\\ &=-\frac {a b B-A \left (b^2-2 a c\right )-(A b-2 a B) c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}-\frac {a b B \left (b^2-10 a c\right )-A \left (3 b^4-20 a b^2 c+20 a^2 c^2\right )+c \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}+\frac {\operatorname {Subst}\left (\int \frac {-2 \left (a b B \left (b^2-7 a c\right )-3 A \left (b^4-7 a b^2 c+10 a^2 c^2\right )\right )-2 c \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right ) x}{x^2 \left (a+b x+c x^2\right )} \, dx,x,x^2\right )}{4 a^2 \left (b^2-4 a c\right )^2}\\ &=-\frac {a b B-A \left (b^2-2 a c\right )-(A b-2 a B) c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}-\frac {a b B \left (b^2-10 a c\right )-A \left (3 b^4-20 a b^2 c+20 a^2 c^2\right )+c \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}+\frac {\operatorname {Subst}\left (\int \left (\frac {2 \left (-a b B \left (b^2-7 a c\right )+3 A \left (b^4-7 a b^2 c+10 a^2 c^2\right )\right )}{a x^2}+\frac {2 (-3 A b+a B) \left (-b^2+4 a c\right )^2}{a^2 x}+\frac {2 \left (-a b B \left (b^4-9 a b^2 c+23 a^2 c^2\right )+3 A \left (b^6-9 a b^4 c+23 a^2 b^2 c^2-10 a^3 c^3\right )+(3 A b-a B) c \left (b^2-4 a c\right )^2 x\right )}{a^2 \left (a+b x+c x^2\right )}\right ) \, dx,x,x^2\right )}{4 a^2 \left (b^2-4 a c\right )^2}\\ &=\frac {a b B \left (b^2-7 a c\right )-3 A \left (b^4-7 a b^2 c+10 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac {a b B-A \left (b^2-2 a c\right )-(A b-2 a B) c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}-\frac {a b B \left (b^2-10 a c\right )-A \left (3 b^4-20 a b^2 c+20 a^2 c^2\right )+c \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}-\frac {(3 A b-a B) \log (x)}{a^4}+\frac {\operatorname {Subst}\left (\int \frac {-a b B \left (b^4-9 a b^2 c+23 a^2 c^2\right )+3 A \left (b^6-9 a b^4 c+23 a^2 b^2 c^2-10 a^3 c^3\right )+(3 A b-a B) c \left (b^2-4 a c\right )^2 x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 a^4 \left (b^2-4 a c\right )^2}\\ &=\frac {a b B \left (b^2-7 a c\right )-3 A \left (b^4-7 a b^2 c+10 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac {a b B-A \left (b^2-2 a c\right )-(A b-2 a B) c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}-\frac {a b B \left (b^2-10 a c\right )-A \left (3 b^4-20 a b^2 c+20 a^2 c^2\right )+c \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}-\frac {(3 A b-a B) \log (x)}{a^4}+\frac {(3 A b-a B) \operatorname {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4}-\frac {\left (a b B \left (b^4-10 a b^2 c+30 a^2 c^2\right )-3 A \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 a^4 \left (b^2-4 a c\right )^2}\\ &=\frac {a b B \left (b^2-7 a c\right )-3 A \left (b^4-7 a b^2 c+10 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac {a b B-A \left (b^2-2 a c\right )-(A b-2 a B) c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}-\frac {a b B \left (b^2-10 a c\right )-A \left (3 b^4-20 a b^2 c+20 a^2 c^2\right )+c \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}-\frac {(3 A b-a B) \log (x)}{a^4}+\frac {(3 A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^4}+\frac {\left (a b B \left (b^4-10 a b^2 c+30 a^2 c^2\right )-3 A \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 a^4 \left (b^2-4 a c\right )^2}\\ &=\frac {a b B \left (b^2-7 a c\right )-3 A \left (b^4-7 a b^2 c+10 a^2 c^2\right )}{2 a^3 \left (b^2-4 a c\right )^2 x^2}-\frac {a b B-A \left (b^2-2 a c\right )-(A b-2 a B) c x^2}{4 a \left (b^2-4 a c\right ) x^2 \left (a+b x^2+c x^4\right )^2}-\frac {a b B \left (b^2-10 a c\right )-A \left (3 b^4-20 a b^2 c+20 a^2 c^2\right )+c \left (a B \left (b^2-16 a c\right )-3 A \left (b^3-6 a b c\right )\right ) x^2}{4 a^2 \left (b^2-4 a c\right )^2 x^2 \left (a+b x^2+c x^4\right )}+\frac {\left (a b B \left (b^4-10 a b^2 c+30 a^2 c^2\right )-3 A \left (b^6-10 a b^4 c+30 a^2 b^2 c^2-20 a^3 c^3\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^4 \left (b^2-4 a c\right )^{5/2}}-\frac {(3 A b-a B) \log (x)}{a^4}+\frac {(3 A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^4}\\ \end {align*}

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Mathematica [A] time = 1.50, size = 642, normalized size = 1.77 \[ \frac {-\frac {a^2 \left (A \left (-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right )+a B \left (2 a c-b^2-b c x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )^2}+\frac {a \left (a B \left (16 a^2 c^2-15 a b^2 c-14 a b c^2 x^2+2 b^4+2 b^3 c x^2\right )-A \left (46 a^2 b c^2+28 a^2 c^3 x^2-29 a b^3 c-26 a b^2 c^2 x^2+4 b^5+4 b^4 c x^2\right )\right )}{\left (b^2-4 a c\right )^2 \left (a+b x^2+c x^4\right )}+\frac {\left (3 A \left (-20 a^3 c^3+30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt {b^2-4 a c}-10 a b^4 c+b^5 \sqrt {b^2-4 a c}-8 a b^3 c \sqrt {b^2-4 a c}+b^6\right )-a B \left (16 a^2 c^2 \sqrt {b^2-4 a c}+30 a^2 b c^2-10 a b^3 c-8 a b^2 c \sqrt {b^2-4 a c}+b^4 \sqrt {b^2-4 a c}+b^5\right )\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+\frac {\left (a B \left (-16 a^2 c^2 \sqrt {b^2-4 a c}+30 a^2 b c^2-10 a b^3 c+8 a b^2 c \sqrt {b^2-4 a c}-b^4 \sqrt {b^2-4 a c}+b^5\right )+3 A \left (20 a^3 c^3-30 a^2 b^2 c^2+16 a^2 b c^2 \sqrt {b^2-4 a c}+10 a b^4 c+b^5 \sqrt {b^2-4 a c}-8 a b^3 c \sqrt {b^2-4 a c}-b^6\right )\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{5/2}}+4 \log (x) (a B-3 A b)-\frac {2 a A}{x^2}}{4 a^4} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[(A + B*x^2)/(x^3*(a + b*x^2 + c*x^4)^3),x]

[Out]

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

*c)*(a + b*x^2 + c*x^4)^2) + (a*(a*B*(2*b^4 - 15*a*b^2*c + 16*a^2*c^2 + 2*b^3*c*x^2 - 14*a*b*c^2*x^2) - A*(4*b

^5 - 29*a*b^3*c + 46*a^2*b*c^2 + 4*b^4*c*x^2 - 26*a*b^2*c^2*x^2 + 28*a^2*c^3*x^2)))/((b^2 - 4*a*c)^2*(a + b*x^

2 + c*x^4)) + 4*(-3*A*b + a*B)*Log[x] + ((-(a*B*(b^5 - 10*a*b^3*c + 30*a^2*b*c^2 + b^4*Sqrt[b^2 - 4*a*c] - 8*a

*b^2*c*Sqrt[b^2 - 4*a*c] + 16*a^2*c^2*Sqrt[b^2 - 4*a*c])) + 3*A*(b^6 - 10*a*b^4*c + 30*a^2*b^2*c^2 - 20*a^3*c^

3 + b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c]))*Log[b - Sqrt[b^2 -

4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2) + ((a*B*(b^5 - 10*a*b^3*c + 30*a^2*b*c^2 - b^4*Sqrt[b^2 - 4*a*c] + 8*a*

b^2*c*Sqrt[b^2 - 4*a*c] - 16*a^2*c^2*Sqrt[b^2 - 4*a*c]) + 3*A*(-b^6 + 10*a*b^4*c - 30*a^2*b^2*c^2 + 20*a^3*c^3

+ b^5*Sqrt[b^2 - 4*a*c] - 8*a*b^3*c*Sqrt[b^2 - 4*a*c] + 16*a^2*b*c^2*Sqrt[b^2 - 4*a*c]))*Log[b + Sqrt[b^2 - 4

*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(5/2))/(4*a^4)

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fricas [B] time = 10.17, size = 3956, normalized size = 10.90 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^3/(c*x^4+b*x^2+a)^3,x, algorithm="fricas")

[Out]

[-1/4*(2*A*a^3*b^6 - 24*A*a^4*b^4*c + 96*A*a^5*b^2*c^2 - 128*A*a^6*c^3 - 2*(120*A*a^4*c^5 + 2*(14*B*a^4*b - 57

*A*a^3*b^2)*c^4 - 11*(B*a^3*b^3 - 3*A*a^2*b^4)*c^3 + (B*a^2*b^5 - 3*A*a*b^6)*c^2)*x^8 + (8*(8*B*a^5 - 69*A*a^4

*b)*c^4 - 6*(22*B*a^4*b^2 - 81*A*a^3*b^3)*c^3 + 45*(B*a^3*b^4 - 3*A*a^2*b^5)*c^2 - 4*(B*a^2*b^6 - 3*A*a*b^7)*c

)*x^6 - 2*(B*a^2*b^7 - 3*A*a*b^8 + 200*A*a^5*c^4 + 2*(2*B*a^5*b - 11*A*a^4*b^2)*c^3 + (23*B*a^4*b^3 - 79*A*a^3

*b^4)*c^2 - 10*(B*a^3*b^5 - 3*A*a^2*b^6)*c)*x^4 - (3*B*a^3*b^6 - 9*A*a^2*b^7 - 8*(12*B*a^6 - 61*A*a^5*b)*c^3 +

2*(54*B*a^5*b^2 - 197*A*a^4*b^3)*c^2 - (33*B*a^4*b^4 - 104*A*a^3*b^5)*c)*x^2 - ((60*A*a^3*c^5 + 30*(B*a^3*b -

3*A*a^2*b^2)*c^4 - 10*(B*a^2*b^3 - 3*A*a*b^4)*c^3 + (B*a*b^5 - 3*A*b^6)*c^2)*x^10 + 2*(60*A*a^3*b*c^4 + 30*(B

*a^3*b^2 - 3*A*a^2*b^3)*c^3 - 10*(B*a^2*b^4 - 3*A*a*b^5)*c^2 + (B*a*b^6 - 3*A*b^7)*c)*x^8 + (B*a*b^7 - 3*A*b^8

+ 120*A*a^4*c^4 + 60*(B*a^4*b - 2*A*a^3*b^2)*c^3 + 10*(B*a^3*b^3 - 3*A*a^2*b^4)*c^2 - 8*(B*a^2*b^5 - 3*A*a*b^

6)*c)*x^6 + 2*(B*a^2*b^6 - 3*A*a*b^7 + 60*A*a^4*b*c^3 + 30*(B*a^4*b^2 - 3*A*a^3*b^3)*c^2 - 10*(B*a^3*b^4 - 3*A

*a^2*b^5)*c)*x^4 + (B*a^3*b^5 - 3*A*a^2*b^6 + 60*A*a^5*c^3 + 30*(B*a^5*b - 3*A*a^4*b^2)*c^2 - 10*(B*a^4*b^3 -

3*A*a^3*b^4)*c)*x^2)*sqrt(b^2 - 4*a*c)*log((2*c^2*x^4 + 2*b*c*x^2 + b^2 - 2*a*c + (2*c*x^2 + b)*sqrt(b^2 - 4*a

*c))/(c*x^4 + b*x^2 + a)) - ((64*(B*a^4 - 3*A*a^3*b)*c^5 - 48*(B*a^3*b^2 - 3*A*a^2*b^3)*c^4 + 12*(B*a^2*b^4 -

3*A*a*b^5)*c^3 - (B*a*b^6 - 3*A*b^7)*c^2)*x^10 + 2*(64*(B*a^4*b - 3*A*a^3*b^2)*c^4 - 48*(B*a^3*b^3 - 3*A*a^2*b

^4)*c^3 + 12*(B*a^2*b^5 - 3*A*a*b^6)*c^2 - (B*a*b^7 - 3*A*b^8)*c)*x^8 - (B*a*b^8 - 3*A*b^9 - 128*(B*a^5 - 3*A*

a^4*b)*c^4 + 32*(B*a^4*b^2 - 3*A*a^3*b^3)*c^3 + 24*(B*a^3*b^4 - 3*A*a^2*b^5)*c^2 - 10*(B*a^2*b^6 - 3*A*a*b^7)*

c)*x^6 - 2*(B*a^2*b^7 - 3*A*a*b^8 - 64*(B*a^5*b - 3*A*a^4*b^2)*c^3 + 48*(B*a^4*b^3 - 3*A*a^3*b^4)*c^2 - 12*(B*

a^3*b^5 - 3*A*a^2*b^6)*c)*x^4 - (B*a^3*b^6 - 3*A*a^2*b^7 - 64*(B*a^6 - 3*A*a^5*b)*c^3 + 48*(B*a^5*b^2 - 3*A*a^

4*b^3)*c^2 - 12*(B*a^4*b^4 - 3*A*a^3*b^5)*c)*x^2)*log(c*x^4 + b*x^2 + a) + 4*((64*(B*a^4 - 3*A*a^3*b)*c^5 - 48

*(B*a^3*b^2 - 3*A*a^2*b^3)*c^4 + 12*(B*a^2*b^4 - 3*A*a*b^5)*c^3 - (B*a*b^6 - 3*A*b^7)*c^2)*x^10 + 2*(64*(B*a^4

*b - 3*A*a^3*b^2)*c^4 - 48*(B*a^3*b^3 - 3*A*a^2*b^4)*c^3 + 12*(B*a^2*b^5 - 3*A*a*b^6)*c^2 - (B*a*b^7 - 3*A*b^8

)*c)*x^8 - (B*a*b^8 - 3*A*b^9 - 128*(B*a^5 - 3*A*a^4*b)*c^4 + 32*(B*a^4*b^2 - 3*A*a^3*b^3)*c^3 + 24*(B*a^3*b^4

- 3*A*a^2*b^5)*c^2 - 10*(B*a^2*b^6 - 3*A*a*b^7)*c)*x^6 - 2*(B*a^2*b^7 - 3*A*a*b^8 - 64*(B*a^5*b - 3*A*a^4*b^2

)*c^3 + 48*(B*a^4*b^3 - 3*A*a^3*b^4)*c^2 - 12*(B*a^3*b^5 - 3*A*a^2*b^6)*c)*x^4 - (B*a^3*b^6 - 3*A*a^2*b^7 - 64

*(B*a^6 - 3*A*a^5*b)*c^3 + 48*(B*a^5*b^2 - 3*A*a^4*b^3)*c^2 - 12*(B*a^4*b^4 - 3*A*a^3*b^5)*c)*x^2)*log(x))/((a

^4*b^6*c^2 - 12*a^5*b^4*c^3 + 48*a^6*b^2*c^4 - 64*a^7*c^5)*x^10 + 2*(a^4*b^7*c - 12*a^5*b^5*c^2 + 48*a^6*b^3*c

^3 - 64*a^7*b*c^4)*x^8 + (a^4*b^8 - 10*a^5*b^6*c + 24*a^6*b^4*c^2 + 32*a^7*b^2*c^3 - 128*a^8*c^4)*x^6 + 2*(a^5

*b^7 - 12*a^6*b^5*c + 48*a^7*b^3*c^2 - 64*a^8*b*c^3)*x^4 + (a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c

^3)*x^2), -1/4*(2*A*a^3*b^6 - 24*A*a^4*b^4*c + 96*A*a^5*b^2*c^2 - 128*A*a^6*c^3 - 2*(120*A*a^4*c^5 + 2*(14*B*a

^4*b - 57*A*a^3*b^2)*c^4 - 11*(B*a^3*b^3 - 3*A*a^2*b^4)*c^3 + (B*a^2*b^5 - 3*A*a*b^6)*c^2)*x^8 + (8*(8*B*a^5 -

69*A*a^4*b)*c^4 - 6*(22*B*a^4*b^2 - 81*A*a^3*b^3)*c^3 + 45*(B*a^3*b^4 - 3*A*a^2*b^5)*c^2 - 4*(B*a^2*b^6 - 3*A

*a*b^7)*c)*x^6 - 2*(B*a^2*b^7 - 3*A*a*b^8 + 200*A*a^5*c^4 + 2*(2*B*a^5*b - 11*A*a^4*b^2)*c^3 + (23*B*a^4*b^3 -

79*A*a^3*b^4)*c^2 - 10*(B*a^3*b^5 - 3*A*a^2*b^6)*c)*x^4 - (3*B*a^3*b^6 - 9*A*a^2*b^7 - 8*(12*B*a^6 - 61*A*a^5

*b)*c^3 + 2*(54*B*a^5*b^2 - 197*A*a^4*b^3)*c^2 - (33*B*a^4*b^4 - 104*A*a^3*b^5)*c)*x^2 - 2*((60*A*a^3*c^5 + 30

*(B*a^3*b - 3*A*a^2*b^2)*c^4 - 10*(B*a^2*b^3 - 3*A*a*b^4)*c^3 + (B*a*b^5 - 3*A*b^6)*c^2)*x^10 + 2*(60*A*a^3*b*

c^4 + 30*(B*a^3*b^2 - 3*A*a^2*b^3)*c^3 - 10*(B*a^2*b^4 - 3*A*a*b^5)*c^2 + (B*a*b^6 - 3*A*b^7)*c)*x^8 + (B*a*b^

7 - 3*A*b^8 + 120*A*a^4*c^4 + 60*(B*a^4*b - 2*A*a^3*b^2)*c^3 + 10*(B*a^3*b^3 - 3*A*a^2*b^4)*c^2 - 8*(B*a^2*b^5

- 3*A*a*b^6)*c)*x^6 + 2*(B*a^2*b^6 - 3*A*a*b^7 + 60*A*a^4*b*c^3 + 30*(B*a^4*b^2 - 3*A*a^3*b^3)*c^2 - 10*(B*a^

3*b^4 - 3*A*a^2*b^5)*c)*x^4 + (B*a^3*b^5 - 3*A*a^2*b^6 + 60*A*a^5*c^3 + 30*(B*a^5*b - 3*A*a^4*b^2)*c^2 - 10*(B

*a^4*b^3 - 3*A*a^3*b^4)*c)*x^2)*sqrt(-b^2 + 4*a*c)*arctan(-(2*c*x^2 + b)*sqrt(-b^2 + 4*a*c)/(b^2 - 4*a*c)) - (

(64*(B*a^4 - 3*A*a^3*b)*c^5 - 48*(B*a^3*b^2 - 3*A*a^2*b^3)*c^4 + 12*(B*a^2*b^4 - 3*A*a*b^5)*c^3 - (B*a*b^6 - 3

*A*b^7)*c^2)*x^10 + 2*(64*(B*a^4*b - 3*A*a^3*b^2)*c^4 - 48*(B*a^3*b^3 - 3*A*a^2*b^4)*c^3 + 12*(B*a^2*b^5 - 3*A

*a*b^6)*c^2 - (B*a*b^7 - 3*A*b^8)*c)*x^8 - (B*a*b^8 - 3*A*b^9 - 128*(B*a^5 - 3*A*a^4*b)*c^4 + 32*(B*a^4*b^2 -

3*A*a^3*b^3)*c^3 + 24*(B*a^3*b^4 - 3*A*a^2*b^5)*c^2 - 10*(B*a^2*b^6 - 3*A*a*b^7)*c)*x^6 - 2*(B*a^2*b^7 - 3*A*a

*b^8 - 64*(B*a^5*b - 3*A*a^4*b^2)*c^3 + 48*(B*a^4*b^3 - 3*A*a^3*b^4)*c^2 - 12*(B*a^3*b^5 - 3*A*a^2*b^6)*c)*x^4

- (B*a^3*b^6 - 3*A*a^2*b^7 - 64*(B*a^6 - 3*A*a^5*b)*c^3 + 48*(B*a^5*b^2 - 3*A*a^4*b^3)*c^2 - 12*(B*a^4*b^4 -

3*A*a^3*b^5)*c)*x^2)*log(c*x^4 + b*x^2 + a) + 4*((64*(B*a^4 - 3*A*a^3*b)*c^5 - 48*(B*a^3*b^2 - 3*A*a^2*b^3)*c^

4 + 12*(B*a^2*b^4 - 3*A*a*b^5)*c^3 - (B*a*b^6 - 3*A*b^7)*c^2)*x^10 + 2*(64*(B*a^4*b - 3*A*a^3*b^2)*c^4 - 48*(B

*a^3*b^3 - 3*A*a^2*b^4)*c^3 + 12*(B*a^2*b^5 - 3*A*a*b^6)*c^2 - (B*a*b^7 - 3*A*b^8)*c)*x^8 - (B*a*b^8 - 3*A*b^9

- 128*(B*a^5 - 3*A*a^4*b)*c^4 + 32*(B*a^4*b^2 - 3*A*a^3*b^3)*c^3 + 24*(B*a^3*b^4 - 3*A*a^2*b^5)*c^2 - 10*(B*a

^2*b^6 - 3*A*a*b^7)*c)*x^6 - 2*(B*a^2*b^7 - 3*A*a*b^8 - 64*(B*a^5*b - 3*A*a^4*b^2)*c^3 + 48*(B*a^4*b^3 - 3*A*a

^3*b^4)*c^2 - 12*(B*a^3*b^5 - 3*A*a^2*b^6)*c)*x^4 - (B*a^3*b^6 - 3*A*a^2*b^7 - 64*(B*a^6 - 3*A*a^5*b)*c^3 + 48

*(B*a^5*b^2 - 3*A*a^4*b^3)*c^2 - 12*(B*a^4*b^4 - 3*A*a^3*b^5)*c)*x^2)*log(x))/((a^4*b^6*c^2 - 12*a^5*b^4*c^3 +

48*a^6*b^2*c^4 - 64*a^7*c^5)*x^10 + 2*(a^4*b^7*c - 12*a^5*b^5*c^2 + 48*a^6*b^3*c^3 - 64*a^7*b*c^4)*x^8 + (a^4

*b^8 - 10*a^5*b^6*c + 24*a^6*b^4*c^2 + 32*a^7*b^2*c^3 - 128*a^8*c^4)*x^6 + 2*(a^5*b^7 - 12*a^6*b^5*c + 48*a^7*

b^3*c^2 - 64*a^8*b*c^3)*x^4 + (a^6*b^6 - 12*a^7*b^4*c + 48*a^8*b^2*c^2 - 64*a^9*c^3)*x^2)]

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giac [A] time = 6.37, size = 648, normalized size = 1.79 \[ -\frac {{\left (B a b^{5} - 3 \, A b^{6} - 10 \, B a^{2} b^{3} c + 30 \, A a b^{4} c + 30 \, B a^{3} b c^{2} - 90 \, A a^{2} b^{2} c^{2} + 60 \, A a^{3} c^{3}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {3 \, B a b^{4} c^{2} x^{8} - 9 \, A b^{5} c^{2} x^{8} - 24 \, B a^{2} b^{2} c^{3} x^{8} + 72 \, A a b^{3} c^{3} x^{8} + 48 \, B a^{3} c^{4} x^{8} - 144 \, A a^{2} b c^{4} x^{8} + 6 \, B a b^{5} c x^{6} - 18 \, A b^{6} c x^{6} - 44 \, B a^{2} b^{3} c^{2} x^{6} + 136 \, A a b^{4} c^{2} x^{6} + 68 \, B a^{3} b c^{3} x^{6} - 236 \, A a^{2} b^{2} c^{3} x^{6} - 56 \, A a^{3} c^{4} x^{6} + 3 \, B a b^{6} x^{4} - 9 \, A b^{7} x^{4} - 10 \, B a^{2} b^{4} c x^{4} + 38 \, A a b^{5} c x^{4} - 58 \, B a^{3} b^{2} c^{2} x^{4} + 110 \, A a^{2} b^{3} c^{2} x^{4} + 128 \, B a^{4} c^{3} x^{4} - 436 \, A a^{3} b c^{3} x^{4} + 10 \, B a^{2} b^{5} x^{2} - 26 \, A a b^{6} x^{2} - 72 \, B a^{3} b^{3} c x^{2} + 192 \, A a^{2} b^{4} c x^{2} + 92 \, B a^{4} b c^{2} x^{2} - 316 \, A a^{3} b^{2} c^{2} x^{2} - 72 \, A a^{4} c^{3} x^{2} + 9 \, B a^{3} b^{4} - 19 \, A a^{2} b^{5} - 66 \, B a^{4} b^{2} c + 144 \, A a^{3} b^{3} c + 96 \, B a^{5} c^{2} - 260 \, A a^{4} b c^{2}}{8 \, {\left (a^{4} b^{4} - 8 \, a^{5} b^{2} c + 16 \, a^{6} c^{2}\right )} {\left (c x^{4} + b x^{2} + a\right )}^{2}} - \frac {{\left (B a - 3 \, A b\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{4}} + \frac {{\left (B a - 3 \, A b\right )} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac {B a x^{2} - 3 \, A b x^{2} + A a}{2 \, a^{4} x^{2}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x^2+A)/x^3/(c*x^4+b*x^2+a)^3,x, algorithm="giac")

[Out]

-1/2*(B*a*b^5 - 3*A*b^6 - 10*B*a^2*b^3*c + 30*A*a*b^4*c + 30*B*a^3*b*c^2 - 90*A*a^2*b^2*c^2 + 60*A*a^3*c^3)*ar

ctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/((a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*sqrt(-b^2 + 4*a*c)) + 1/8*(3*B*a*

b^4*c^2*x^8 - 9*A*b^5*c^2*x^8 - 24*B*a^2*b^2*c^3*x^8 + 72*A*a*b^3*c^3*x^8 + 48*B*a^3*c^4*x^8 - 144*A*a^2*b*c^4

*x^8 + 6*B*a*b^5*c*x^6 - 18*A*b^6*c*x^6 - 44*B*a^2*b^3*c^2*x^6 + 136*A*a*b^4*c^2*x^6 + 68*B*a^3*b*c^3*x^6 - 23

6*A*a^2*b^2*c^3*x^6 - 56*A*a^3*c^4*x^6 + 3*B*a*b^6*x^4 - 9*A*b^7*x^4 - 10*B*a^2*b^4*c*x^4 + 38*A*a*b^5*c*x^4 -

58*B*a^3*b^2*c^2*x^4 + 110*A*a^2*b^3*c^2*x^4 + 128*B*a^4*c^3*x^4 - 436*A*a^3*b*c^3*x^4 + 10*B*a^2*b^5*x^2 - 2

6*A*a*b^6*x^2 - 72*B*a^3*b^3*c*x^2 + 192*A*a^2*b^4*c*x^2 + 92*B*a^4*b*c^2*x^2 - 316*A*a^3*b^2*c^2*x^2 - 72*A*a

^4*c^3*x^2 + 9*B*a^3*b^4 - 19*A*a^2*b^5 - 66*B*a^4*b^2*c + 144*A*a^3*b^3*c + 96*B*a^5*c^2 - 260*A*a^4*b*c^2)/(

(a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*(c*x^4 + b*x^2 + a)^2) - 1/4*(B*a - 3*A*b)*log(c*x^4 + b*x^2 + a)/a^4 + 1

/2*(B*a - 3*A*b)*log(x^2)/a^4 - 1/2*(B*a*x^2 - 3*A*b*x^2 + A*a)/(a^4*x^2)

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maple [B] time = 0.04, size = 1862, normalized size = 5.13 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((B*x^2+A)/x^3/(c*x^4+b*x^2+a)^3,x)

[Out]

6/a^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b^4*c-3/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4

)*x^2*B*b^3*c+13/2/a^2/(c*x^4+b*x^2+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*A*b^2-1/a^3/(c*x^4+b*x^2+a)^2*c^2/

(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*A*b^4-7/2/a/(c*x^4+b*x^2+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*b*B+1/2/a^2/(c

*x^4+b*x^2+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6*B*b^3-37/2/a/(c*x^4+b*x^2+a)^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^

4)*x^4*A*b+55/4/a^2/(c*x^4+b*x^2+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*A*b^3+45/a^2/(16*a^2*c^2-8*a*b^2*c+b^

4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A*b^2*c^2-15/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-

b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A*b^4*c-15/a/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arc

tan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*B*b*c^2+5/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+

b)/(4*a*c-b^2)^(1/2))*B*b^3*c-2/a^3/(c*x^4+b*x^2+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*A*b^5-29/4/a/(c*x^4+b*x

^2+a)^2*c^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*B*b^2+1/a^2/(c*x^4+b*x^2+a)^2*c/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*B*b^

4-7/2/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b^2*c^2-3/a^4*ln(x)*A*b+3/4/a/(c*x^4+b*x^2+a)^2/(16

*a^2*c^2-8*a*b^2*c+b^4)*B*b^4+3/4/a^4/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*A*b^5-4/a/(16*a^2*c^2-8*a*b

^2*c+b^4)*c^2*ln(c*x^4+b*x^2+a)*B-1/4/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*ln(c*x^4+b*x^2+a)*B*b^4+4/(c*x^4+b*x^2+a)

^2*c^3/(16*a^2*c^2-8*a*b^2*c+b^4)*x^4*B-9/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*c^3-29/2/(c*x^4+b

*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*A*b*c^2-21/4/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*B*b^2*c-1/2/(c*

x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*B*b*c^2-7/a/(c*x^4+b*x^2+a)^2*c^4/(16*a^2*c^2-8*a*b^2*c+b^4)*x^6

*A-1/a^3/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*x^2*A*b^6+1/2/a^2/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*

c+b^4)*x^2*B*b^5+9/a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*A*b^3*c+12/a^2/(16*a^2*c^2-8*a*b^2*c+b^4)*c^

2*ln(c*x^4+b*x^2+a)*A*b-6/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)*c*ln(c*x^4+b*x^2+a)*A*b^3+2/a^2/(16*a^2*c^2-8*a*b^2*c

+b^4)*c*ln(c*x^4+b*x^2+a)*B*b^2-30/a/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^

2)^(1/2))*A*c^3-1/2/a^3/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*B*b

^5+3/2/a^4/(16*a^2*c^2-8*a*b^2*c+b^4)/(4*a*c-b^2)^(1/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A*b^6-5/4/a^2/(c

*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*A*b^5+6*a/(c*x^4+b*x^2+a)^2/(16*a^2*c^2-8*a*b^2*c+b^4)*B*c^2-1/2*A/

a^3/x^2+1/a^3*ln(x)*B

________________________________________________________________________________________

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((B*x^2+A)/x^3/(c*x^4+b*x^2+a)^3,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 = 15.91, size = 16265, normalized size = 44.81 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x^2)/(x^3*(a + b*x^2 + c*x^4)^3),x)

[Out]

(log(((c^5*x^2*(3*A*b^4 + 30*A*a^2*c^2 - B*a*b^3 - 21*A*a*b^2*c + 7*B*a^2*b*c)^3)/(a^9*(4*a*c - b^2)^6) - ((B*

a - 3*A*b + a^4*(-(60*A*a^3*c^3 - 3*A*b^6 + B*a*b^5 + 30*A*a*b^4*c - 10*B*a^2*b^3*c + 30*B*a^3*b*c^2 - 90*A*a^

2*b^2*c^2)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*(((B*a - 3*A*b + a^4*(-(60*A*a^3*c^3 - 3*A*b^6 + B*a*b^5 + 30*A*a*b

^4*c - 10*B*a^2*b^3*c + 30*B*a^3*b*c^2 - 90*A*a^2*b^2*c^2)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*((4*b*c^2*(30*A*a^3

*c^3 - 3*A*b^6 + B*a*b^5 + 27*A*a*b^4*c - 9*B*a^2*b^3*c + 23*B*a^3*b*c^2 - 69*A*a^2*b^2*c^2))/(a^3*(4*a*c - b^

2)^2) + (2*c^3*x^2*(B*a*b^5 - 300*A*a^3*c^3 - 3*A*b^6 + 6*A*a*b^4*c - 2*B*a^2*b^3*c + 10*B*a^3*b*c^2 + 90*A*a^

2*b^2*c^2))/(a^3*(4*a*c - b^2)^2) + (b*c^2*(B*a - 3*A*b + a^4*(-(60*A*a^3*c^3 - 3*A*b^6 + B*a*b^5 + 30*A*a*b^4

*c - 10*B*a^2*b^3*c + 30*B*a^3*b*c^2 - 90*A*a^2*b^2*c^2)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*(a*b + 3*b^2*x^2 - 10

*a*c*x^2))/a^4))/(4*a^4) + (c^3*(900*A^2*a^5*c^5 - 36*A^2*b^10 - 4*B^2*a^2*b^8 + 24*A*B*a*b^9 - 3078*A^2*a^2*b

^6*c^2 + 7533*A^2*a^3*b^4*c^3 - 7020*A^2*a^4*b^2*c^4 - 302*B^2*a^4*b^4*c^2 + 497*B^2*a^5*b^2*c^3 + 549*A^2*a*b

^8*c + 61*B^2*a^3*b^6*c + 1932*A*B*a^3*b^5*c^2 - 4002*A*B*a^4*b^3*c^3 - 366*A*B*a^2*b^7*c + 2340*A*B*a^5*b*c^4

))/(a^6*(4*a*c - b^2)^4) - (c^4*x^2*(54*A^2*b^9 + 6*B^2*a^2*b^7 - 36*A*B*a*b^8 + 4311*A^2*a^2*b^5*c^2 - 9900*A

^2*a^3*b^3*c^3 + 409*B^2*a^4*b^3*c^2 - 2400*A*B*a^5*c^4 - 801*A^2*a*b^7*c + 8100*A^2*a^4*b*c^4 - 89*B^2*a^3*b^

5*c - 560*B^2*a^5*b*c^3 - 2664*A*B*a^3*b^4*c^2 + 4980*A*B*a^4*b^2*c^3 + 534*A*B*a^2*b^6*c))/(a^6*(4*a*c - b^2)

^4)))/(4*a^4) + (c^4*(3*A*b - B*a)*(3*A*b^4 + 30*A*a^2*c^2 - B*a*b^3 - 21*A*a*b^2*c + 7*B*a^2*b*c)^2)/(a^9*(4*

a*c - b^2)^4))*((c^5*x^2*(3*A*b^4 + 30*A*a^2*c^2 - B*a*b^3 - 21*A*a*b^2*c + 7*B*a^2*b*c)^3)/(a^9*(4*a*c - b^2)

^6) - ((3*A*b - B*a + a^4*(-(60*A*a^3*c^3 - 3*A*b^6 + B*a*b^5 + 30*A*a*b^4*c - 10*B*a^2*b^3*c + 30*B*a^3*b*c^2

- 90*A*a^2*b^2*c^2)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*(((3*A*b - B*a + a^4*(-(60*A*a^3*c^3 - 3*A*b^6 + B*a*b^5

+ 30*A*a*b^4*c - 10*B*a^2*b^3*c + 30*B*a^3*b*c^2 - 90*A*a^2*b^2*c^2)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*((4*b*c^2

*(30*A*a^3*c^3 - 3*A*b^6 + B*a*b^5 + 27*A*a*b^4*c - 9*B*a^2*b^3*c + 23*B*a^3*b*c^2 - 69*A*a^2*b^2*c^2))/(a^3*(

4*a*c - b^2)^2) + (2*c^3*x^2*(B*a*b^5 - 300*A*a^3*c^3 - 3*A*b^6 + 6*A*a*b^4*c - 2*B*a^2*b^3*c + 10*B*a^3*b*c^2

+ 90*A*a^2*b^2*c^2))/(a^3*(4*a*c - b^2)^2) - (b*c^2*(3*A*b - B*a + a^4*(-(60*A*a^3*c^3 - 3*A*b^6 + B*a*b^5 +

30*A*a*b^4*c - 10*B*a^2*b^3*c + 30*B*a^3*b*c^2 - 90*A*a^2*b^2*c^2)^2/(a^8*(4*a*c - b^2)^5))^(1/2))*(a*b + 3*b^

2*x^2 - 10*a*c*x^2))/a^4))/(4*a^4) - (c^3*(900*A^2*a^5*c^5 - 36*A^2*b^10 - 4*B^2*a^2*b^8 + 24*A*B*a*b^9 - 3078

*A^2*a^2*b^6*c^2 + 7533*A^2*a^3*b^4*c^3 - 7020*A^2*a^4*b^2*c^4 - 302*B^2*a^4*b^4*c^2 + 497*B^2*a^5*b^2*c^3 + 5

49*A^2*a*b^8*c + 61*B^2*a^3*b^6*c + 1932*A*B*a^3*b^5*c^2 - 4002*A*B*a^4*b^3*c^3 - 366*A*B*a^2*b^7*c + 2340*A*B

*a^5*b*c^4))/(a^6*(4*a*c - b^2)^4) + (c^4*x^2*(54*A^2*b^9 + 6*B^2*a^2*b^7 - 36*A*B*a*b^8 + 4311*A^2*a^2*b^5*c^

2 - 9900*A^2*a^3*b^3*c^3 + 409*B^2*a^4*b^3*c^2 - 2400*A*B*a^5*c^4 - 801*A^2*a*b^7*c + 8100*A^2*a^4*b*c^4 - 89*

B^2*a^3*b^5*c - 560*B^2*a^5*b*c^3 - 2664*A*B*a^3*b^4*c^2 + 4980*A*B*a^4*b^2*c^3 + 534*A*B*a^2*b^6*c))/(a^6*(4*

a*c - b^2)^4)))/(4*a^4) + (c^4*(3*A*b - B*a)*(3*A*b^4 + 30*A*a^2*c^2 - B*a*b^3 - 21*A*a*b^2*c + 7*B*a^2*b*c)^2

)/(a^9*(4*a*c - b^2)^4)))*(6*A*b^11 + 2048*B*a^6*c^5 - 2*B*a*b^10 - 120*A*a*b^9*c - 6144*A*a^5*b*c^5 + 40*B*a^

2*b^8*c + 960*A*a^2*b^7*c^2 - 3840*A*a^3*b^5*c^3 + 7680*A*a^4*b^3*c^4 - 320*B*a^3*b^6*c^2 + 1280*B*a^4*b^4*c^3

- 2560*B*a^5*b^2*c^4))/(2*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 51

20*a^8*b^2*c^4)) - (log(x)*(3*A*b - B*a))/a^4 - (A/(2*a) + (x^2*(9*A*b^5 - 24*B*a^3*c^2 - 3*B*a*b^4 - 68*A*a*b

^3*c + 122*A*a^2*b*c^2 + 21*B*a^2*b^2*c))/(4*a^2*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) - (x^6*(16*B*a^3*c^3 - 12*A*b

^5*c + 4*B*a*b^4*c + 87*A*a*b^3*c^2 - 138*A*a^2*b*c^3 - 29*B*a^2*b^2*c^2))/(4*a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*

c)) + (x^4*(3*A*b^6 + 50*A*a^3*c^3 - B*a*b^5 - 18*A*a*b^4*c + 6*B*a^2*b^3*c + B*a^3*b*c^2 + 7*A*a^2*b^2*c^2))/

(2*a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)) + (c^2*x^8*(3*A*b^4 + 30*A*a^2*c^2 - B*a*b^3 - 21*A*a*b^2*c + 7*B*a^2*b

*c))/(2*a^3*(b^4 + 16*a^2*c^2 - 8*a*b^2*c)))/(x^6*(2*a*c + b^2) + a^2*x^2 + c^2*x^10 + 2*a*b*x^4 + 2*b*c*x^8)

- (atan((x^2*((((((((153600*A*a^13*c^10 - 5120*B*a^13*b*c^9 + 6*A*a^6*b^14*c^3 - 108*A*a^7*b^12*c^4 + 588*A*a^

8*b^10*c^5 + 792*A*a^9*b^8*c^6 - 22272*A*a^10*b^6*c^7 + 100608*A*a^11*b^4*c^8 - 199680*A*a^12*b^2*c^9 - 2*B*a^

7*b^13*c^3 + 36*B*a^8*b^11*c^4 - 276*B*a^9*b^9*c^5 + 1216*B*a^10*b^7*c^6 - 3456*B*a^11*b^5*c^7 + 6144*B*a^12*b

^3*c^8)/(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4

- 6144*a^14*b^2*c^5) - ((163840*a^16*b*c^9 - 12*a^9*b^15*c^2 + 328*a^10*b^13*c^3 - 3840*a^11*b^11*c^4 + 24960*

a^12*b^9*c^5 - 97280*a^13*b^7*c^6 + 227328*a^14*b^5*c^7 - 294912*a^15*b^3*c^8)*(6*A*b^11 + 2048*B*a^6*c^5 - 2*

B*a*b^10 - 120*A*a*b^9*c - 6144*A*a^5*b*c^5 + 40*B*a^2*b^8*c + 960*A*a^2*b^7*c^2 - 3840*A*a^3*b^5*c^3 + 7680*A

*a^4*b^3*c^4 - 320*B*a^3*b^6*c^2 + 1280*B*a^4*b^4*c^3 - 2560*B*a^5*b^2*c^4))/(2*(4*a^4*b^10 - 4096*a^9*c^5 - 8

0*a^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)*(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*

c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5)))*(60*A*a^3*c^3 - 3*A*b^6 +

B*a*b^5 + 30*A*a*b^4*c - 10*B*a^2*b^3*c + 30*B*a^3*b*c^2 - 90*A*a^2*b^2*c^2))/(4*a^4*(4*a*c - b^2)^(5/2)) - ((

60*A*a^3*c^3 - 3*A*b^6 + B*a*b^5 + 30*A*a*b^4*c - 10*B*a^2*b^3*c + 30*B*a^3*b*c^2 - 90*A*a^2*b^2*c^2)*(163840*

a^16*b*c^9 - 12*a^9*b^15*c^2 + 328*a^10*b^13*c^3 - 3840*a^11*b^11*c^4 + 24960*a^12*b^9*c^5 - 97280*a^13*b^7*c^

6 + 227328*a^14*b^5*c^7 - 294912*a^15*b^3*c^8)*(6*A*b^11 + 2048*B*a^6*c^5 - 2*B*a*b^10 - 120*A*a*b^9*c - 6144*

A*a^5*b*c^5 + 40*B*a^2*b^8*c + 960*A*a^2*b^7*c^2 - 3840*A*a^3*b^5*c^3 + 7680*A*a^4*b^3*c^4 - 320*B*a^3*b^6*c^2

+ 1280*B*a^4*b^4*c^3 - 2560*B*a^5*b^2*c^4))/(8*a^4*(4*a*c - b^2)^(5/2)*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^

8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)*(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*

a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5)))*(6*A*b^11 + 2048*B*a^6*c^5 - 2*B*a

*b^10 - 120*A*a*b^9*c - 6144*A*a^5*b*c^5 + 40*B*a^2*b^8*c + 960*A*a^2*b^7*c^2 - 3840*A*a^3*b^5*c^3 + 7680*A*a^

4*b^3*c^4 - 320*B*a^3*b^6*c^2 + 1280*B*a^4*b^4*c^3 - 2560*B*a^5*b^2*c^4))/(2*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a

^5*b^8*c + 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)) - (((54*A^2*a^3*b^13*c^4 - 1233*A^2*a^4*b^1

1*c^5 + 11583*A^2*a^5*b^9*c^6 - 57204*A^2*a^6*b^7*c^7 + 156276*A^2*a^7*b^5*c^8 - 223200*A^2*a^8*b^3*c^9 + 6*B^

2*a^5*b^11*c^4 - 137*B^2*a^6*b^9*c^5 + 1217*B^2*a^7*b^7*c^6 - 5256*B^2*a^8*b^5*c^7 + 11024*B^2*a^9*b^3*c^8 - 3

8400*A*B*a^10*c^10 + 129600*A^2*a^9*b*c^10 - 8960*B^2*a^10*b*c^9 - 36*A*B*a^4*b^12*c^4 + 822*A*B*a^5*b^10*c^5

- 7512*A*B*a^6*b^8*c^6 + 34836*A*B*a^7*b^6*c^7 - 84864*A*B*a^8*b^4*c^8 + 98880*A*B*a^9*b^2*c^9)/(a^9*b^12 + 40

96*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5) -

(((153600*A*a^13*c^10 - 5120*B*a^13*b*c^9 + 6*A*a^6*b^14*c^3 - 108*A*a^7*b^12*c^4 + 588*A*a^8*b^10*c^5 + 792*

A*a^9*b^8*c^6 - 22272*A*a^10*b^6*c^7 + 100608*A*a^11*b^4*c^8 - 199680*A*a^12*b^2*c^9 - 2*B*a^7*b^13*c^3 + 36*B

*a^8*b^11*c^4 - 276*B*a^9*b^9*c^5 + 1216*B*a^10*b^7*c^6 - 3456*B*a^11*b^5*c^7 + 6144*B*a^12*b^3*c^8)/(a^9*b^12

+ 4096*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c

^5) - ((163840*a^16*b*c^9 - 12*a^9*b^15*c^2 + 328*a^10*b^13*c^3 - 3840*a^11*b^11*c^4 + 24960*a^12*b^9*c^5 - 97

280*a^13*b^7*c^6 + 227328*a^14*b^5*c^7 - 294912*a^15*b^3*c^8)*(6*A*b^11 + 2048*B*a^6*c^5 - 2*B*a*b^10 - 120*A*

a*b^9*c - 6144*A*a^5*b*c^5 + 40*B*a^2*b^8*c + 960*A*a^2*b^7*c^2 - 3840*A*a^3*b^5*c^3 + 7680*A*a^4*b^3*c^4 - 32

0*B*a^3*b^6*c^2 + 1280*B*a^4*b^4*c^3 - 2560*B*a^5*b^2*c^4))/(2*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c + 640

*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)*(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b^8*

c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5)))*(6*A*b^11 + 2048*B*a^6*c^5 - 2*B*a*b^10 - 1

20*A*a*b^9*c - 6144*A*a^5*b*c^5 + 40*B*a^2*b^8*c + 960*A*a^2*b^7*c^2 - 3840*A*a^3*b^5*c^3 + 7680*A*a^4*b^3*c^4

- 320*B*a^3*b^6*c^2 + 1280*B*a^4*b^4*c^3 - 2560*B*a^5*b^2*c^4))/(2*(4*a^4*b^10 - 4096*a^9*c^5 - 80*a^5*b^8*c

+ 640*a^6*b^6*c^2 - 2560*a^7*b^4*c^3 + 5120*a^8*b^2*c^4)))*(60*A*a^3*c^3 - 3*A*b^6 + B*a*b^5 + 30*A*a*b^4*c -

10*B*a^2*b^3*c + 30*B*a^3*b*c^2 - 90*A*a^2*b^2*c^2))/(4*a^4*(4*a*c - b^2)^(5/2)) + ((60*A*a^3*c^3 - 3*A*b^6 +

B*a*b^5 + 30*A*a*b^4*c - 10*B*a^2*b^3*c + 30*B*a^3*b*c^2 - 90*A*a^2*b^2*c^2)^3*(163840*a^16*b*c^9 - 12*a^9*b^1

5*c^2 + 328*a^10*b^13*c^3 - 3840*a^11*b^11*c^4 + 24960*a^12*b^9*c^5 - 97280*a^13*b^7*c^6 + 227328*a^14*b^5*c^7

- 294912*a^15*b^3*c^8))/(64*a^12*(4*a*c - b^2)^(15/2)*(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b

^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5)))*(9*A*b^9 - 160*B*a^5*c^4 - 3*B*a*b^8 - 1

17*A*a*b^7*c + 570*A*a^4*b*c^4 + 39*B*a^2*b^6*c + 540*A*a^2*b^5*c^2 - 1005*A*a^3*b^3*c^3 - 180*B*a^3*b^4*c^2 +

325*B*a^4*b^2*c^3))/(8*(4*a*c - b^2)^(13/2)*(900*A^2*a^9*c^8 + 6400*B^2*a^10*c^7 - 54*A^2*a^3*b^12*c^2 - 960*

a^6*b^6*c^4*(B^2*a - 36*A^2*c) + 120*a^5*b^8*c^3*(B^2*a - 72*A^2*c) - 6*a^4*b^10*c^2*(B^2*a - 180*A^2*c) - 25*

a^8*b^2*c^6*(311*B^2*a - 2196*A^2*c) + 25*a^7*b^4*c^5*(154*B^2*a - 2763*A^2*c) + 36*A*B*a^4*b^11*c^2 - 720*A*B

*a^5*b^9*c^3 + 5760*A*B*a^6*b^7*c^4 - 23070*A*B*a^7*b^5*c^5 + 46350*A*B*a^8*b^3*c^6 - 37500*A*B*a^9*b*c^7)) -

(((27000*A^3*a^6*c^11 + 27*A^3*b^12*c^5 + 4779*A^3*a^2*b^8*c^7 - 20601*A^3*a^3*b^6*c^8 + 47790*A^3*a^4*b^4*c^9

- 56700*A^3*a^5*b^2*c^10 - B^3*a^3*b^9*c^5 + 21*B^3*a^4*b^7*c^6 - 147*B^3*a^5*b^5*c^7 + 343*B^3*a^6*b^3*c^8 -

567*A^3*a*b^10*c^6 - 27*A^2*B*a*b^11*c^5 + 18900*A^2*B*a^6*b*c^10 + 9*A*B^2*a^2*b^10*c^5 - 189*A*B^2*a^3*b^8*

c^6 + 1413*A*B^2*a^4*b^6*c^7 - 4347*A*B^2*a^5*b^4*c^8 + 4410*A*B^2*a^6*b^2*c^9 + 567*A^2*B*a^2*b^9*c^6 - 4509*

A^2*B*a^3*b^7*c^7 + 16821*A^2*B*a^4*b^5*c^8 - 29160*A^2*B*a^5*b^3*c^9)/(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^1

0*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 6144*a^14*b^2*c^5) - (((54*A^2*a^3*b^13*c^4 -

1233*A^2*a^4*b^11*c^5 + 11583*A^2*a^5*b^9*c^6 - 57204*A^2*a^6*b^7*c^7 + 156276*A^2*a^7*b^5*c^8 - 223200*A^2*a

^8*b^3*c^9 + 6*B^2*a^5*b^11*c^4 - 137*B^2*a^6*b^9*c^5 + 1217*B^2*a^7*b^7*c^6 - 5256*B^2*a^8*b^5*c^7 + 11024*B^

2*a^9*b^3*c^8 - 38400*A*B*a^10*c^10 + 129600*A^2*a^9*b*c^10 - 8960*B^2*a^10*b*c^9 - 36*A*B*a^4*b^12*c^4 + 822*

A*B*a^5*b^10*c^5 - 7512*A*B*a^6*b^8*c^6 + 34836*A*B*a^7*b^6*c^7 - 84864*A*B*a^8*b^4*c^8 + 98880*A*B*a^9*b^2*c^

9)/(a^9*b^12 + 4096*a^15*c^6 - 24*a^10*b^10*c + 240*a^11*b^8*c^2 - 1280*a^12*b^6*c^3 + 3840*a^13*b^4*c^4 - 614

4*a^14*b^2*c^5) - (((153600*A*a^13*c^10 - 5120*B*a^13*b*c^9 + 6*A*a^6*b^14*c^3 - 108*A*a^7*b^12*c^4 + 588*A*a^