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

Optimal. Leaf size=223 \[ \frac {(2 A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac {\log (x) (2 A b-a B)}{a^3}-\frac {-6 a A c-a b B+2 A b^2}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac {\left (a b B \left (b^2-6 a c\right )-2 A \left (6 a^2 c^2-6 a b^2 c+b^4\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {-A \left (b^2-2 a c\right )-\left (c x^2 (A b-2 a B)\right )+a b B}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

[Out]

1/2*(6*A*a*c-2*A*b^2+B*a*b)/a^2/(-4*a*c+b^2)/x^2+1/2*(-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)+1/2*(a*b*B*(-6*a*c+b^2)-2*A*(6*a^2*c^2-6*a*b^2*c+b^4))*arctanh((2*c*x^2+b)/(-4*a*c+b^2)^(1
/2))/a^3/(-4*a*c+b^2)^(3/2)-(2*A*b-B*a)*ln(x)/a^3+1/4*(2*A*b-B*a)*ln(c*x^4+b*x^2+a)/a^3

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Rubi [A]  time = 0.42, antiderivative size = 223, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ \frac {\left (a b B \left (b^2-6 a c\right )-2 A \left (6 a^2 c^2-6 a b^2 c+b^4\right )\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 a^3 \left (b^2-4 a c\right )^{3/2}}-\frac {-6 a A c-a b B+2 A b^2}{2 a^2 x^2 \left (b^2-4 a c\right )}+\frac {(2 A b-a B) \log \left (a+b x^2+c x^4\right )}{4 a^3}-\frac {\log (x) (2 A b-a B)}{a^3}-\frac {-A \left (b^2-2 a c\right )+c x^2 (-(A b-2 a B))+a b B}{2 a x^2 \left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(2*A*b^2 - a*b*B - 6*a*A*c)/(2*a^2*(b^2 - 4*a*c)*x^2) - (a*b*B - A*(b^2 - 2*a*c) - (A*b - 2*a*B)*c*x^2)/(2*a*
(b^2 - 4*a*c)*x^2*(a + b*x^2 + c*x^4)) + ((a*b*B*(b^2 - 6*a*c) - 2*A*(b^4 - 6*a*b^2*c + 6*a^2*c^2))*ArcTanh[(b
+ 2*c*x^2)/Sqrt[b^2 - 4*a*c]])/(2*a^3*(b^2 - 4*a*c)^(3/2)) - ((2*A*b - a*B)*Log[x])/a^3 + ((2*A*b - a*B)*Log[
a + b*x^2 + c*x^4])/(4*a^3)

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

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Mathematica [A]  time = 0.56, size = 379, normalized size = 1.70 \[ \frac {\frac {\left (2 A \left (6 a^2 c^2-6 a b^2 c-4 a b c \sqrt {b^2-4 a c}+b^3 \sqrt {b^2-4 a c}+b^4\right )+a B \left (-b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}+6 a b c-b^3\right )\right ) \log \left (-\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}+\frac {\left (2 A \left (-6 a^2 c^2+6 a b^2 c-4 a b c \sqrt {b^2-4 a c}+b^3 \sqrt {b^2-4 a c}-b^4\right )+a B \left (-b^2 \sqrt {b^2-4 a c}+4 a c \sqrt {b^2-4 a c}-6 a b c+b^3\right )\right ) \log \left (\sqrt {b^2-4 a c}+b+2 c x^2\right )}{\left (b^2-4 a c\right )^{3/2}}-\frac {2 a \left (A \left (-3 a b c-2 a c^2 x^2+b^3+b^2 c x^2\right )+a B \left (2 a c-b^2-b c x^2\right )\right )}{\left (b^2-4 a c\right ) \left (a+b x^2+c x^4\right )}+4 \log (x) (a B-2 A b)-\frac {2 a A}{x^2}}{4 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a*A)/x^2 - (2*a*(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)) + 4*(-2*A*b + a*B)*Log[x] + ((a*B*(-b^3 + 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sqr
t[b^2 - 4*a*c]) + 2*A*(b^4 - 6*a*b^2*c + 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c]))*Log[b
- Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2) + ((a*B*(b^3 - 6*a*b*c - b^2*Sqrt[b^2 - 4*a*c] + 4*a*c*Sq
rt[b^2 - 4*a*c]) + 2*A*(-b^4 + 6*a*b^2*c - 6*a^2*c^2 + b^3*Sqrt[b^2 - 4*a*c] - 4*a*b*c*Sqrt[b^2 - 4*a*c]))*Log
[b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/(b^2 - 4*a*c)^(3/2))/(4*a^3)

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fricas [B]  time = 3.38, size = 1635, normalized size = 7.33 \[ \text {result too large to display} \]

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)^2,x, algorithm="fricas")

[Out]

[-1/4*(2*A*a^2*b^4 - 16*A*a^3*b^2*c + 32*A*a^4*c^2 + 2*(24*A*a^3*c^3 + 2*(2*B*a^3*b - 7*A*a^2*b^2)*c^2 - (B*a^
2*b^3 - 2*A*a*b^4)*c)*x^4 - 2*(B*a^2*b^4 - 2*A*a*b^5 + 4*(2*B*a^4 - 7*A*a^3*b)*c^2 - 3*(2*B*a^3*b^2 - 5*A*a^2*
b^3)*c)*x^2 + ((12*A*a^2*c^3 + 6*(B*a^2*b - 2*A*a*b^2)*c^2 - (B*a*b^3 - 2*A*b^4)*c)*x^6 - (B*a*b^4 - 2*A*b^5 -
12*A*a^2*b*c^2 - 6*(B*a^2*b^2 - 2*A*a*b^3)*c)*x^4 - (B*a^2*b^3 - 2*A*a*b^4 - 12*A*a^3*c^2 - 6*(B*a^3*b - 2*A*
a^2*b^2)*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)) + ((16*(B*a^3 - 2*A*a^2*b)*c^3 - 8*(B*a^2*b^2 - 2*A*a*b^3)*c^2 + (B*a*b^4 - 2*A*b^5)*c)*
x^6 + (B*a*b^5 - 2*A*b^6 + 16*(B*a^3*b - 2*A*a^2*b^2)*c^2 - 8*(B*a^2*b^3 - 2*A*a*b^4)*c)*x^4 + (B*a^2*b^4 - 2*
A*a*b^5 + 16*(B*a^4 - 2*A*a^3*b)*c^2 - 8*(B*a^3*b^2 - 2*A*a^2*b^3)*c)*x^2)*log(c*x^4 + b*x^2 + a) - 4*((16*(B*
a^3 - 2*A*a^2*b)*c^3 - 8*(B*a^2*b^2 - 2*A*a*b^3)*c^2 + (B*a*b^4 - 2*A*b^5)*c)*x^6 + (B*a*b^5 - 2*A*b^6 + 16*(B
*a^3*b - 2*A*a^2*b^2)*c^2 - 8*(B*a^2*b^3 - 2*A*a*b^4)*c)*x^4 + (B*a^2*b^4 - 2*A*a*b^5 + 16*(B*a^4 - 2*A*a^3*b)
*c^2 - 8*(B*a^3*b^2 - 2*A*a^2*b^3)*c)*x^2)*log(x))/((a^3*b^4*c - 8*a^4*b^2*c^2 + 16*a^5*c^3)*x^6 + (a^3*b^5 -
8*a^4*b^3*c + 16*a^5*b*c^2)*x^4 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*x^2), -1/4*(2*A*a^2*b^4 - 16*A*a^3*b^2*
c + 32*A*a^4*c^2 + 2*(24*A*a^3*c^3 + 2*(2*B*a^3*b - 7*A*a^2*b^2)*c^2 - (B*a^2*b^3 - 2*A*a*b^4)*c)*x^4 - 2*(B*a
^2*b^4 - 2*A*a*b^5 + 4*(2*B*a^4 - 7*A*a^3*b)*c^2 - 3*(2*B*a^3*b^2 - 5*A*a^2*b^3)*c)*x^2 + 2*((12*A*a^2*c^3 + 6
*(B*a^2*b - 2*A*a*b^2)*c^2 - (B*a*b^3 - 2*A*b^4)*c)*x^6 - (B*a*b^4 - 2*A*b^5 - 12*A*a^2*b*c^2 - 6*(B*a^2*b^2 -
2*A*a*b^3)*c)*x^4 - (B*a^2*b^3 - 2*A*a*b^4 - 12*A*a^3*c^2 - 6*(B*a^3*b - 2*A*a^2*b^2)*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)) + ((16*(B*a^3 - 2*A*a^2*b)*c^3 - 8*(B*a^2*b^2 - 2*
A*a*b^3)*c^2 + (B*a*b^4 - 2*A*b^5)*c)*x^6 + (B*a*b^5 - 2*A*b^6 + 16*(B*a^3*b - 2*A*a^2*b^2)*c^2 - 8*(B*a^2*b^3
- 2*A*a*b^4)*c)*x^4 + (B*a^2*b^4 - 2*A*a*b^5 + 16*(B*a^4 - 2*A*a^3*b)*c^2 - 8*(B*a^3*b^2 - 2*A*a^2*b^3)*c)*x^
2)*log(c*x^4 + b*x^2 + a) - 4*((16*(B*a^3 - 2*A*a^2*b)*c^3 - 8*(B*a^2*b^2 - 2*A*a*b^3)*c^2 + (B*a*b^4 - 2*A*b^
5)*c)*x^6 + (B*a*b^5 - 2*A*b^6 + 16*(B*a^3*b - 2*A*a^2*b^2)*c^2 - 8*(B*a^2*b^3 - 2*A*a*b^4)*c)*x^4 + (B*a^2*b^
4 - 2*A*a*b^5 + 16*(B*a^4 - 2*A*a^3*b)*c^2 - 8*(B*a^3*b^2 - 2*A*a^2*b^3)*c)*x^2)*log(x))/((a^3*b^4*c - 8*a^4*b
^2*c^2 + 16*a^5*c^3)*x^6 + (a^3*b^5 - 8*a^4*b^3*c + 16*a^5*b*c^2)*x^4 + (a^4*b^4 - 8*a^5*b^2*c + 16*a^6*c^2)*x
^2)]

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giac [A]  time = 1.65, size = 250, normalized size = 1.12 \[ -\frac {{\left (B a b^{3} - 2 \, A b^{4} - 6 \, B a^{2} b c + 12 \, A a b^{2} c - 12 \, A a^{2} c^{2}\right )} \arctan \left (\frac {2 \, c x^{2} + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {B a b c x^{4} - 2 \, A b^{2} c x^{4} + 6 \, A a c^{2} x^{4} + B a b^{2} x^{2} - 2 \, A b^{3} x^{2} - 2 \, B a^{2} c x^{2} + 7 \, A a b c x^{2} - A a b^{2} + 4 \, A a^{2} c}{2 \, {\left (c x^{6} + b x^{4} + a x^{2}\right )} {\left (a^{2} b^{2} - 4 \, a^{3} c\right )}} - \frac {{\left (B a - 2 \, A b\right )} \log \left (c x^{4} + b x^{2} + a\right )}{4 \, a^{3}} + \frac {{\left (B a - 2 \, A b\right )} \log \left (x^{2}\right )}{2 \, a^{3}} \]

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)^2,x, algorithm="giac")

[Out]

-1/2*(B*a*b^3 - 2*A*b^4 - 6*B*a^2*b*c + 12*A*a*b^2*c - 12*A*a^2*c^2)*arctan((2*c*x^2 + b)/sqrt(-b^2 + 4*a*c))/
((a^3*b^2 - 4*a^4*c)*sqrt(-b^2 + 4*a*c)) + 1/2*(B*a*b*c*x^4 - 2*A*b^2*c*x^4 + 6*A*a*c^2*x^4 + B*a*b^2*x^2 - 2*
A*b^3*x^2 - 2*B*a^2*c*x^2 + 7*A*a*b*c*x^2 - A*a*b^2 + 4*A*a^2*c)/((c*x^6 + b*x^4 + a*x^2)*(a^2*b^2 - 4*a^3*c))
- 1/4*(B*a - 2*A*b)*log(c*x^4 + b*x^2 + a)/a^3 + 1/2*(B*a - 2*A*b)*log(x^2)/a^3

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maple [B]  time = 0.02, size = 622, normalized size = 2.79 \[ -\frac {A \,c^{2} x^{2}}{\left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a}+\frac {A \,b^{2} c \,x^{2}}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a^{2}}-\frac {B b c \,x^{2}}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a}-\frac {6 A \,c^{2} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a}+\frac {6 A \,b^{2} c \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{2}}-\frac {A \,b^{4} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{3}}-\frac {3 B b c \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} a}+\frac {B \,b^{3} \arctan \left (\frac {2 c \,x^{2}+b}{\sqrt {4 a c -b^{2}}}\right )}{2 \left (4 a c -b^{2}\right )^{\frac {3}{2}} a^{2}}-\frac {3 A b c}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a}+\frac {A \,b^{3}}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a^{2}}+\frac {2 A b c \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{\left (4 a c -b^{2}\right ) a^{2}}-\frac {A \,b^{3} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{2 \left (4 a c -b^{2}\right ) a^{3}}-\frac {B \,b^{2}}{2 \left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right ) a}-\frac {B c \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{\left (4 a c -b^{2}\right ) a}+\frac {B \,b^{2} \ln \left (c \,x^{4}+b \,x^{2}+a \right )}{4 \left (4 a c -b^{2}\right ) a^{2}}+\frac {B c}{\left (c \,x^{4}+b \,x^{2}+a \right ) \left (4 a c -b^{2}\right )}-\frac {2 A b \ln \relax (x )}{a^{3}}+\frac {B \ln \relax (x )}{a^{2}}-\frac {A}{2 a^{2} x^{2}} \]

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)^2,x)

[Out]

-1/2*A/a^2/x^2-2/a^3*ln(x)*A*b+1/a^2*ln(x)*B-1/a/(c*x^4+b*x^2+a)*c^2/(4*a*c-b^2)*x^2*A+1/2/a^2/(c*x^4+b*x^2+a)
*c/(4*a*c-b^2)*x^2*A*b^2-1/2/a/(c*x^4+b*x^2+a)*c/(4*a*c-b^2)*x^2*b*B-3/2/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*A*b*c+1
/2/a^2/(c*x^4+b*x^2+a)/(4*a*c-b^2)*A*b^3+1/(c*x^4+b*x^2+a)/(4*a*c-b^2)*B*c-1/2/a/(c*x^4+b*x^2+a)/(4*a*c-b^2)*B
*b^2+2/a^2/(4*a*c-b^2)*c*ln(c*x^4+b*x^2+a)*A*b-1/2/a^3/(4*a*c-b^2)*ln(c*x^4+b*x^2+a)*A*b^3-1/a/(4*a*c-b^2)*c*l
n(c*x^4+b*x^2+a)*B+1/4/a^2/(4*a*c-b^2)*ln(c*x^4+b*x^2+a)*b^2*B-6/a/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c
-b^2)^(1/2))*A*c^2+6/a^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A*b^2*c-1/a^3/(4*a*c-b^2)^(3/
2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*A*b^4-3/a/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*b*B
*c+1/2/a^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x^2+b)/(4*a*c-b^2)^(1/2))*B*b^3

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?

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mupad [B]  time = 9.09, size = 10034, normalized size = 45.00 \[ \text {result too large to display} \]

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)^2),x)

[Out]

(log(((c^4*(2*A*b - B*a)*(6*A*a*c - 2*A*b^2 + B*a*b)^2)/(a^6*(4*a*c - b^2)^2) - ((((B*a - 2*A*b + a^3*(-(2*A*b
^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)^2/(a^6*(4*a*c - b^2)^3))^(1/2))*((b*c^2*(B*a - 2*A*b
+ a^3*(-(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)^2/(a^6*(4*a*c - b^2)^3))^(1/2))*(a*b
+ 3*b^2*x^2 - 10*a*c*x^2))/a^3 + (4*b*c^2*(2*A*b^4 + 6*A*a^2*c^2 - B*a*b^3 - 10*A*a*b^2*c + 5*B*a^2*b*c))/(a^2
*(4*a*c - b^2)) + (2*c^3*x^2*(2*A*b^4 - 60*A*a^2*c^2 - B*a*b^3 + 4*A*a*b^2*c + 10*B*a^2*b*c))/(a^2*(4*a*c - b^
2))))/(4*a^3) + (c^3*(36*A^2*a^3*c^3 - 16*A^2*b^6 - 4*B^2*a^2*b^4 + 16*A*B*a*b^5 - 216*A^2*a^2*b^2*c^2 + 116*A
^2*a*b^4*c + 17*B^2*a^3*b^2*c - 92*A*B*a^2*b^3*c + 108*A*B*a^3*b*c^2))/(a^4*(4*a*c - b^2)^2) - (2*c^4*x^2*(12*
A^2*b^5 + 3*B^2*a^2*b^3 - 12*A*B*a*b^4 - 60*A*B*a^3*c^2 - 82*A^2*a*b^3*c - 10*B^2*a^3*b*c + 138*A^2*a^2*b*c^2
+ 61*A*B*a^2*b^2*c))/(a^4*(4*a*c - b^2)^2))*(B*a - 2*A*b + a^3*(-(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^
2*c + 6*B*a^2*b*c)^2/(a^6*(4*a*c - b^2)^3))^(1/2)))/(4*a^3) + (c^5*x^2*(6*A*a*c - 2*A*b^2 + B*a*b)^3)/(a^6*(4*
a*c - b^2)^3))*((c^4*(2*A*b - B*a)*(6*A*a*c - 2*A*b^2 + B*a*b)^2)/(a^6*(4*a*c - b^2)^2) - ((((2*A*b - B*a + a^
3*(-(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)^2/(a^6*(4*a*c - b^2)^3))^(1/2))*((4*b*c^2*
(2*A*b^4 + 6*A*a^2*c^2 - B*a*b^3 - 10*A*a*b^2*c + 5*B*a^2*b*c))/(a^2*(4*a*c - b^2)) - (b*c^2*(2*A*b - B*a + a^
3*(-(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)^2/(a^6*(4*a*c - b^2)^3))^(1/2))*(a*b + 3*b
^2*x^2 - 10*a*c*x^2))/a^3 + (2*c^3*x^2*(2*A*b^4 - 60*A*a^2*c^2 - B*a*b^3 + 4*A*a*b^2*c + 10*B*a^2*b*c))/(a^2*(
4*a*c - b^2))))/(4*a^3) - (c^3*(36*A^2*a^3*c^3 - 16*A^2*b^6 - 4*B^2*a^2*b^4 + 16*A*B*a*b^5 - 216*A^2*a^2*b^2*c
^2 + 116*A^2*a*b^4*c + 17*B^2*a^3*b^2*c - 92*A*B*a^2*b^3*c + 108*A*B*a^3*b*c^2))/(a^4*(4*a*c - b^2)^2) + (2*c^
4*x^2*(12*A^2*b^5 + 3*B^2*a^2*b^3 - 12*A*B*a*b^4 - 60*A*B*a^3*c^2 - 82*A^2*a*b^3*c - 10*B^2*a^3*b*c + 138*A^2*
a^2*b*c^2 + 61*A*B*a^2*b^2*c))/(a^4*(4*a*c - b^2)^2))*(2*A*b - B*a + a^3*(-(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 -
12*A*a*b^2*c + 6*B*a^2*b*c)^2/(a^6*(4*a*c - b^2)^3))^(1/2)))/(4*a^3) + (c^5*x^2*(6*A*a*c - 2*A*b^2 + B*a*b)^3
)/(a^6*(4*a*c - b^2)^3)))*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4
*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)) - (
log(x)*(2*A*b - B*a))/a^3 - (A/(2*a) - (x^2*(2*A*b^3 - B*a*b^2 + 2*B*a^2*c - 7*A*a*b*c))/(2*a^2*(4*a*c - b^2))
+ (c*x^4*(6*A*a*c - 2*A*b^2 + B*a*b))/(2*a^2*(4*a*c - b^2)))/(a*x^2 + b*x^4 + c*x^6) + (atan((x^2*((((216*A^3
*a^3*c^8 - 8*A^3*b^6*c^5 - 216*A^3*a^2*b^2*c^7 + B^3*a^3*b^3*c^5 + 72*A^3*a*b^4*c^6 + 12*A^2*B*a*b^5*c^5 + 108
*A^2*B*a^3*b*c^7 - 6*A*B^2*a^2*b^4*c^5 + 18*A*B^2*a^3*b^2*c^6 - 72*A^2*B*a^2*b^3*c^6)/(a^6*b^6 - 64*a^9*c^3 -
12*a^7*b^4*c + 48*a^8*b^2*c^2) + (((24*A^2*a^2*b^7*c^4 - 260*A^2*a^3*b^5*c^5 + 932*A^2*a^4*b^3*c^6 + 6*B^2*a^4
*b^5*c^4 - 44*B^2*a^5*b^3*c^5 + 480*A*B*a^6*c^7 - 1104*A^2*a^5*b*c^7 + 80*B^2*a^6*b*c^6 - 24*A*B*a^3*b^6*c^4 +
218*A*B*a^4*b^4*c^5 - 608*A*B*a^5*b^2*c^6)/(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2) + (((1920*A
*a^8*c^7 - 320*B*a^8*b*c^6 - 4*A*a^4*b^8*c^3 + 24*A*a^5*b^6*c^4 + 120*A*a^6*b^4*c^5 - 1088*A*a^7*b^2*c^6 + 2*B
*a^5*b^7*c^3 - 36*B*a^6*b^5*c^4 + 192*B*a^7*b^3*c^5)/(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2) -
((2560*a^10*b*c^6 + 12*a^6*b^9*c^2 - 184*a^7*b^7*c^3 + 1056*a^8*b^5*c^4 - 2688*a^9*b^3*c^5)*(4*A*b^7 + 128*B*a
^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/
(2*(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2)*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^
2*c^2)))*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^
3*c^2 - 96*B*a^3*b^2*c^2))/(2*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(4*A*b^7 + 128*B*a^
4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(
2*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)) - (((((1920*A*a^8*c^7 - 320*B*a^8*b*c^6 - 4*A*a^
4*b^8*c^3 + 24*A*a^5*b^6*c^4 + 120*A*a^6*b^4*c^5 - 1088*A*a^7*b^2*c^6 + 2*B*a^5*b^7*c^3 - 36*B*a^6*b^5*c^4 + 1
92*B*a^7*b^3*c^5)/(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2) - ((2560*a^10*b*c^6 + 12*a^6*b^9*c^2
- 184*a^7*b^7*c^3 + 1056*a^8*b^5*c^4 - 2688*a^9*b^3*c^5)*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c -
256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b
^4*c + 48*a^8*b^2*c^2)*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(2*A*b^4 + 12*A*a^2*c^2 -
B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c))/(4*a^3*(4*a*c - b^2)^(3/2)) - ((2560*a^10*b*c^6 + 12*a^6*b^9*c^2 - 184*
a^7*b^7*c^3 + 1056*a^8*b^5*c^4 - 2688*a^9*b^3*c^5)*(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*
b*c)*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^
2 - 96*B*a^3*b^2*c^2))/(8*a^3*(4*a*c - b^2)^(3/2)*(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2)*(4*a^
3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B
*a^2*b*c))/(4*a^3*(4*a*c - b^2)^(3/2)) + ((2560*a^10*b*c^6 + 12*a^6*b^9*c^2 - 184*a^7*b^7*c^3 + 1056*a^8*b^5*c
^4 - 2688*a^9*b^3*c^5)*(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)^2*(4*A*b^7 + 128*B*a^4*
c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(32
*a^6*(4*a*c - b^2)^3*(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2)*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*
b^4*c + 192*a^5*b^2*c^2)))*(6*A*a^3*c^3 - 6*A*b^6 + 3*B*a*b^5 + 42*A*a*b^4*c - 21*B*a^2*b^3*c + 33*B*a^3*b*c^2
- 72*A*a^2*b^2*c^2))/(8*a^3*c^2*(4*a*c - b^2)^3*(36*A^2*a^4*c^4 - 24*A^2*b^8 - 6*B^2*a^2*b^6 + 400*B^2*a^5*c^
3 + 24*A*B*a*b^7 - 1152*A^2*a^2*b^4*c^2 + 1528*A^2*a^3*b^2*c^3 - 291*B^2*a^4*b^2*c^2 + 288*A^2*a*b^6*c + 72*B^
2*a^3*b^4*c + 1158*A*B*a^3*b^3*c^2 - 288*A*B*a^2*b^5*c - 1564*A*B*a^4*b*c^3)) + (((((((1920*A*a^8*c^7 - 320*B*
a^8*b*c^6 - 4*A*a^4*b^8*c^3 + 24*A*a^5*b^6*c^4 + 120*A*a^6*b^4*c^5 - 1088*A*a^7*b^2*c^6 + 2*B*a^5*b^7*c^3 - 36
*B*a^6*b^5*c^4 + 192*B*a^7*b^3*c^5)/(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2) - ((2560*a^10*b*c^6
+ 12*a^6*b^9*c^2 - 184*a^7*b^7*c^3 + 1056*a^8*b^5*c^4 - 2688*a^9*b^3*c^5)*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^
6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*(a^6*b^6 - 64*
a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2)*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(2*A*b^4
+ 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c))/(4*a^3*(4*a*c - b^2)^(3/2)) - ((2560*a^10*b*c^6 + 12*
a^6*b^9*c^2 - 184*a^7*b^7*c^3 + 1056*a^8*b^5*c^4 - 2688*a^9*b^3*c^5)*(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*
a*b^2*c + 6*B*a^2*b*c)*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c
+ 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(8*a^3*(4*a*c - b^2)^(3/2)*(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*
a^8*b^2*c^2)*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6
- 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*(4*a^3*b^6 - 25
6*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)) + (((24*A^2*a^2*b^7*c^4 - 260*A^2*a^3*b^5*c^5 + 932*A^2*a^4*b^3*c
^6 + 6*B^2*a^4*b^5*c^4 - 44*B^2*a^5*b^3*c^5 + 480*A*B*a^6*c^7 - 1104*A^2*a^5*b*c^7 + 80*B^2*a^6*b*c^6 - 24*A*B
*a^3*b^6*c^4 + 218*A*B*a^4*b^4*c^5 - 608*A*B*a^5*b^2*c^6)/(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^
2) + (((1920*A*a^8*c^7 - 320*B*a^8*b*c^6 - 4*A*a^4*b^8*c^3 + 24*A*a^5*b^6*c^4 + 120*A*a^6*b^4*c^5 - 1088*A*a^7
*b^2*c^6 + 2*B*a^5*b^7*c^3 - 36*B*a^6*b^5*c^4 + 192*B*a^7*b^3*c^5)/(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a
^8*b^2*c^2) - ((2560*a^10*b*c^6 + 12*a^6*b^9*c^2 - 184*a^7*b^7*c^3 + 1056*a^8*b^5*c^4 - 2688*a^9*b^3*c^5)*(4*A
*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*
a^3*b^2*c^2))/(2*(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2)*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*
c + 192*a^5*b^2*c^2)))*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c
+ 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(2*A*
b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c))/(4*a^3*(4*a*c - b^2)^(3/2)) + ((2560*a^10*b*c^6 +
12*a^6*b^9*c^2 - 184*a^7*b^7*c^3 + 1056*a^8*b^5*c^4 - 2688*a^9*b^3*c^5)*(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12
*A*a*b^2*c + 6*B*a^2*b*c)^3)/(64*a^9*(4*a*c - b^2)^(9/2)*(a^6*b^6 - 64*a^9*c^3 - 12*a^7*b^4*c + 48*a^8*b^2*c^2
)))*(768*A*b^7 + 5120*B*a^4*c^3 - 384*B*a*b^6 - 6912*A*a*b^5*c - 12544*A*a^3*b*c^3 + 3456*B*a^2*b^4*c + 18432*
A*a^2*b^3*c^2 - 8832*B*a^3*b^2*c^2))/(1024*a^3*c^2*(4*a*c - b^2)^(7/2)*(36*A^2*a^4*c^4 - 24*A^2*b^8 - 6*B^2*a^
2*b^6 + 400*B^2*a^5*c^3 + 24*A*B*a*b^7 - 1152*A^2*a^2*b^4*c^2 + 1528*A^2*a^3*b^2*c^3 - 291*B^2*a^4*b^2*c^2 + 2
88*A^2*a*b^6*c + 72*B^2*a^3*b^4*c + 1158*A*B*a^3*b^3*c^2 - 288*A*B*a^2*b^5*c - 1564*A*B*a^4*b*c^3)))*(16*a^9*b
^6*(4*a*c - b^2)^(9/2) - 1024*a^12*c^3*(4*a*c - b^2)^(9/2) - 192*a^10*b^4*c*(4*a*c - b^2)^(9/2) + 768*a^11*b^2
*c^2*(4*a*c - b^2)^(9/2)))/(144*A^2*a^4*c^6 + 4*A^2*b^8*c^2 + 192*A^2*a^2*b^4*c^4 - 288*A^2*a^3*b^2*c^5 + B^2*
a^2*b^6*c^2 - 12*B^2*a^3*b^4*c^3 + 36*B^2*a^4*b^2*c^4 - 48*A^2*a*b^6*c^3 + 48*A*B*a^2*b^5*c^3 - 168*A*B*a^3*b^
3*c^4 - 4*A*B*a*b^7*c^2 + 144*A*B*a^4*b*c^5) + (((((((96*A*a^7*b*c^5 - 8*A*a^4*b^7*c^2 + 72*A*a^5*b^5*c^3 - 18
4*A*a^6*b^3*c^4 + 4*B*a^5*b^6*c^2 - 36*B*a^6*b^4*c^3 + 80*B*a^7*b^2*c^4)/(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)
- ((4*a^7*b^6*c^2 - 32*a^8*b^4*c^3 + 64*a^9*b^2*c^4)*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256
*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)
*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c
+ 6*B*a^2*b*c))/(4*a^3*(4*a*c - b^2)^(3/2)) - ((4*a^7*b^6*c^2 - 32*a^8*b^4*c^3 + 64*a^9*b^2*c^4)*(2*A*b^4 + 1
2*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*
A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(8*a^3*(4*a*c - b^2)^(3/2)*(a^6*b^4 + 16
*a^8*c^2 - 8*a^7*b^2*c)*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(4*A*b^7 + 128*B*a^4*c^3
- 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*(4*a
^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)) - (((36*A^2*a^5*c^6 - 16*A^2*a^2*b^6*c^3 + 116*A^2*a^3
*b^4*c^4 - 216*A^2*a^4*b^2*c^5 - 4*B^2*a^4*b^4*c^3 + 17*B^2*a^5*b^2*c^4 + 16*A*B*a^3*b^5*c^3 - 92*A*B*a^4*b^3*
c^4 + 108*A*B*a^5*b*c^5)/(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c) - (((96*A*a^7*b*c^5 - 8*A*a^4*b^7*c^2 + 72*A*a^5
*b^5*c^3 - 184*A*a^6*b^3*c^4 + 4*B*a^5*b^6*c^2 - 36*B*a^6*b^4*c^3 + 80*B*a^7*b^2*c^4)/(a^6*b^4 + 16*a^8*c^2 -
8*a^7*b^2*c) - ((4*a^7*b^6*c^2 - 32*a^8*b^4*c^3 + 64*a^9*b^2*c^4)*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*
a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*(a^6*b^4 + 16*a^8*c^2 -
8*a^7*b^2*c)*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^
6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*(4*a^3*b^6 - 2
56*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)
)/(4*a^3*(4*a*c - b^2)^(3/2)) + ((4*a^7*b^6*c^2 - 32*a^8*b^4*c^3 + 64*a^9*b^2*c^4)*(2*A*b^4 + 12*A*a^2*c^2 - B
*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)^3)/(64*a^9*(4*a*c - b^2)^(9/2)*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)))*(16
*a^9*b^6*(4*a*c - b^2)^(9/2) - 1024*a^12*c^3*(4*a*c - b^2)^(9/2) - 192*a^10*b^4*c*(4*a*c - b^2)^(9/2) + 768*a^
11*b^2*c^2*(4*a*c - b^2)^(9/2))*(768*A*b^7 + 5120*B*a^4*c^3 - 384*B*a*b^6 - 6912*A*a*b^5*c - 12544*A*a^3*b*c^3
+ 3456*B*a^2*b^4*c + 18432*A*a^2*b^3*c^2 - 8832*B*a^3*b^2*c^2))/(1024*a^3*c^2*(4*a*c - b^2)^(7/2)*(144*A^2*a^
4*c^6 + 4*A^2*b^8*c^2 + 192*A^2*a^2*b^4*c^4 - 288*A^2*a^3*b^2*c^5 + B^2*a^2*b^6*c^2 - 12*B^2*a^3*b^4*c^3 + 36*
B^2*a^4*b^2*c^4 - 48*A^2*a*b^6*c^3 + 48*A*B*a^2*b^5*c^3 - 168*A*B*a^3*b^3*c^4 - 4*A*B*a*b^7*c^2 + 144*A*B*a^4*
b*c^5)*(36*A^2*a^4*c^4 - 24*A^2*b^8 - 6*B^2*a^2*b^6 + 400*B^2*a^5*c^3 + 24*A*B*a*b^7 - 1152*A^2*a^2*b^4*c^2 +
1528*A^2*a^3*b^2*c^3 - 291*B^2*a^4*b^2*c^2 + 288*A^2*a*b^6*c + 72*B^2*a^3*b^4*c + 1158*A*B*a^3*b^3*c^2 - 288*A
*B*a^2*b^5*c - 1564*A*B*a^4*b*c^3)) + ((16*a^9*b^6*(4*a*c - b^2)^(9/2) - 1024*a^12*c^3*(4*a*c - b^2)^(9/2) - 1
92*a^10*b^4*c*(4*a*c - b^2)^(9/2) + 768*a^11*b^2*c^2*(4*a*c - b^2)^(9/2))*((B^3*a^3*b^2*c^4 - 8*A^3*b^5*c^4 +
36*A^2*B*a^3*c^6 + 48*A^3*a*b^3*c^5 - 72*A^3*a^2*b*c^6 + 12*A*B^2*a^3*b*c^5 + 12*A^2*B*a*b^4*c^4 - 6*A*B^2*a^2
*b^3*c^4 - 48*A^2*B*a^2*b^2*c^5)/(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c) - (((36*A^2*a^5*c^6 - 16*A^2*a^2*b^6*c^3
+ 116*A^2*a^3*b^4*c^4 - 216*A^2*a^4*b^2*c^5 - 4*B^2*a^4*b^4*c^3 + 17*B^2*a^5*b^2*c^4 + 16*A*B*a^3*b^5*c^3 - 9
2*A*B*a^4*b^3*c^4 + 108*A*B*a^5*b*c^5)/(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c) - (((96*A*a^7*b*c^5 - 8*A*a^4*b^7*
c^2 + 72*A*a^5*b^5*c^3 - 184*A*a^6*b^3*c^4 + 4*B*a^5*b^6*c^2 - 36*B*a^6*b^4*c^3 + 80*B*a^7*b^2*c^4)/(a^6*b^4 +
16*a^8*c^2 - 8*a^7*b^2*c) - ((4*a^7*b^6*c^2 - 32*a^8*b^4*c^3 + 64*a^9*b^2*c^4)*(4*A*b^7 + 128*B*a^4*c^3 - 2*B
*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*(a^6*b^4
+ 16*a^8*c^2 - 8*a^7*b^2*c)*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(4*A*b^7 + 128*B*a^4*
c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*
(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5
*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*(4*a^3*b^6 - 256*a^6*c^3 - 4
8*a^4*b^4*c + 192*a^5*b^2*c^2)) - (((((96*A*a^7*b*c^5 - 8*A*a^4*b^7*c^2 + 72*A*a^5*b^5*c^3 - 184*A*a^6*b^3*c^4
+ 4*B*a^5*b^6*c^2 - 36*B*a^6*b^4*c^3 + 80*B*a^7*b^2*c^4)/(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c) - ((4*a^7*b^6*c
^2 - 32*a^8*b^4*c^3 + 64*a^9*b^2*c^4)*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 +
24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(2*(a^6*b^4 + 16*a^8*c^2 - 8*a^7*b^2*c)*(4*a^3*b^6 - 2
56*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)
)/(4*a^3*(4*a*c - b^2)^(3/2)) - ((4*a^7*b^6*c^2 - 32*a^8*b^4*c^3 + 64*a^9*b^2*c^4)*(2*A*b^4 + 12*A*a^2*c^2 - B
*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^5*c - 256*A*a^3*b*c^3 + 2
4*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(8*a^3*(4*a*c - b^2)^(3/2)*(a^6*b^4 + 16*a^8*c^2 - 8*a^
7*b^2*c)*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A
*a*b^2*c + 6*B*a^2*b*c))/(4*a^3*(4*a*c - b^2)^(3/2)) + ((4*a^7*b^6*c^2 - 32*a^8*b^4*c^3 + 64*a^9*b^2*c^4)*(2*A
*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2*b*c)^2*(4*A*b^7 + 128*B*a^4*c^3 - 2*B*a*b^6 - 48*A*a*b^
5*c - 256*A*a^3*b*c^3 + 24*B*a^2*b^4*c + 192*A*a^2*b^3*c^2 - 96*B*a^3*b^2*c^2))/(32*a^6*(4*a*c - b^2)^3*(a^6*b
^4 + 16*a^8*c^2 - 8*a^7*b^2*c)*(4*a^3*b^6 - 256*a^6*c^3 - 48*a^4*b^4*c + 192*a^5*b^2*c^2)))*(6*A*a^3*c^3 - 6*A
*b^6 + 3*B*a*b^5 + 42*A*a*b^4*c - 21*B*a^2*b^3*c + 33*B*a^3*b*c^2 - 72*A*a^2*b^2*c^2))/(8*a^3*c^2*(4*a*c - b^2
)^3*(144*A^2*a^4*c^6 + 4*A^2*b^8*c^2 + 192*A^2*a^2*b^4*c^4 - 288*A^2*a^3*b^2*c^5 + B^2*a^2*b^6*c^2 - 12*B^2*a^
3*b^4*c^3 + 36*B^2*a^4*b^2*c^4 - 48*A^2*a*b^6*c^3 + 48*A*B*a^2*b^5*c^3 - 168*A*B*a^3*b^3*c^4 - 4*A*B*a*b^7*c^2
+ 144*A*B*a^4*b*c^5)*(36*A^2*a^4*c^4 - 24*A^2*b^8 - 6*B^2*a^2*b^6 + 400*B^2*a^5*c^3 + 24*A*B*a*b^7 - 1152*A^2
*a^2*b^4*c^2 + 1528*A^2*a^3*b^2*c^3 - 291*B^2*a^4*b^2*c^2 + 288*A^2*a*b^6*c + 72*B^2*a^3*b^4*c + 1158*A*B*a^3*
b^3*c^2 - 288*A*B*a^2*b^5*c - 1564*A*B*a^4*b*c^3)))*(2*A*b^4 + 12*A*a^2*c^2 - B*a*b^3 - 12*A*a*b^2*c + 6*B*a^2
*b*c))/(2*a^3*(4*a*c - b^2)^(3/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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