3.1.21 \(\int \frac {x^4 (A+B x+C x^2)}{a+b x^2+c x^4} \, dx\) [21]

Optimal. Leaf size=339 \[ \frac {(A c-b C) x}{c^2}+\frac {B x^2}{2 c}+\frac {C x^3}{3 c}-\frac {\left (A b c-b^2 C+a c C-\frac {A c \left (b^2-2 a c\right )-b \left (b^2-3 a c\right ) C}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (A b c-b^2 C+a c C+\frac {A c \left (b^2-2 a c\right )-b \left (b^2-3 a c\right ) C}{\sqrt {b^2-4 a c}}\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {B \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}-\frac {b B \log \left (a+b x^2+c x^4\right )}{4 c^2} \]

[Out]

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

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Rubi [A]
time = 1.21, antiderivative size = 339, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {1676, 1293, 1180, 211, 12, 1128, 717, 648, 632, 212, 642} \begin {gather*} -\frac {\left (-\frac {A c \left (b^2-2 a c\right )-b C \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}+a c C+A b c+b^2 (-C)\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (\frac {A c \left (b^2-2 a c\right )-b C \left (b^2-3 a c\right )}{\sqrt {b^2-4 a c}}+a c C+A b c+b^2 (-C)\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right )}{\sqrt {2} c^{5/2} \sqrt {\sqrt {b^2-4 a c}+b}}-\frac {B \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}-\frac {b B \log \left (a+b x^2+c x^4\right )}{4 c^2}+\frac {x (A c-b C)}{c^2}+\frac {B x^2}{2 c}+\frac {C x^3}{3 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 12

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

Rule 211

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

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

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

Rule 642

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

Rule 648

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

Rule 717

Int[((d_.) + (e_.)*(x_))^(m_)/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[e*((d + e*x)^(m - 1)/(c*(
m - 1))), x] + Dist[1/c, Int[(d + e*x)^(m - 2)*(Simp[c*d^2 - a*e^2 + e*(2*c*d - b*e)*x, 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] && GtQ[m, 1]

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 1293

Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Simp[e*f*
(f*x)^(m - 1)*((a + b*x^2 + c*x^4)^(p + 1)/(c*(m + 4*p + 3))), x] - Dist[f^2/(c*(m + 4*p + 3)), Int[(f*x)^(m -
2)*(a + b*x^2 + c*x^4)^p*Simp[a*e*(m - 1) + (b*e*(m + 2*p + 1) - c*d*(m + 4*p + 3))*x^2, x], x], x] /; FreeQ[
{a, b, c, d, e, f, p}, x] && NeQ[b^2 - 4*a*c, 0] && GtQ[m, 1] && NeQ[m + 4*p + 3, 0] && IntegerQ[2*p] && (Inte
gerQ[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 {x^4 \left (A+B x+C x^2\right )}{a+b x^2+c x^4} \, dx &=\int \frac {B x^5}{a+b x^2+c x^4} \, dx+\int \frac {x^4 \left (A+C x^2\right )}{a+b x^2+c x^4} \, dx\\ &=\frac {C x^3}{3 c}+B \int \frac {x^5}{a+b x^2+c x^4} \, dx-\frac {\int \frac {x^2 \left (3 a C-3 (A c-b C) x^2\right )}{a+b x^2+c x^4} \, dx}{3 c}\\ &=\frac {(A c-b C) x}{c^2}+\frac {C x^3}{3 c}+\frac {1}{2} B \text {Subst}\left (\int \frac {x^2}{a+b x+c x^2} \, dx,x,x^2\right )+\frac {\int \frac {-3 a (A c-b C)-3 \left (A b c-b^2 C+a c C\right ) x^2}{a+b x^2+c x^4} \, dx}{3 c^2}\\ &=\frac {(A c-b C) x}{c^2}+\frac {B x^2}{2 c}+\frac {C x^3}{3 c}+\frac {B \text {Subst}\left (\int \frac {-a-b x}{a+b x+c x^2} \, dx,x,x^2\right )}{2 c}-\frac {\left (A b c-b^2 C+a c C-\frac {A c \left (b^2-2 a c\right )-b \left (b^2-3 a c\right ) C}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {b}{2}-\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{2 c^2}-\frac {\left (A b c-b^2 C+a c C+\frac {A c \left (b^2-2 a c\right )-b \left (b^2-3 a c\right ) C}{\sqrt {b^2-4 a c}}\right ) \int \frac {1}{\frac {b}{2}+\frac {1}{2} \sqrt {b^2-4 a c}+c x^2} \, dx}{2 c^2}\\ &=\frac {(A c-b C) x}{c^2}+\frac {B x^2}{2 c}+\frac {C x^3}{3 c}-\frac {\left (A b c-b^2 C+a c C-\frac {A c \left (b^2-2 a c\right )-b \left (b^2-3 a c\right ) 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} c^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (A b c-b^2 C+a c C+\frac {A c \left (b^2-2 a c\right )-b \left (b^2-3 a c\right ) 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} c^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {(b B) \text {Subst}\left (\int \frac {b+2 c x}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2}+\frac {\left (B \left (b^2-2 a c\right )\right ) \text {Subst}\left (\int \frac {1}{a+b x+c x^2} \, dx,x,x^2\right )}{4 c^2}\\ &=\frac {(A c-b C) x}{c^2}+\frac {B x^2}{2 c}+\frac {C x^3}{3 c}-\frac {\left (A b c-b^2 C+a c C-\frac {A c \left (b^2-2 a c\right )-b \left (b^2-3 a c\right ) 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} c^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (A b c-b^2 C+a c C+\frac {A c \left (b^2-2 a c\right )-b \left (b^2-3 a c\right ) 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} c^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {b B \log \left (a+b x^2+c x^4\right )}{4 c^2}-\frac {\left (B \left (b^2-2 a c\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x^2\right )}{2 c^2}\\ &=\frac {(A c-b C) x}{c^2}+\frac {B x^2}{2 c}+\frac {C x^3}{3 c}-\frac {\left (A b c-b^2 C+a c C-\frac {A c \left (b^2-2 a c\right )-b \left (b^2-3 a c\right ) 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} c^{5/2} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (A b c-b^2 C+a c C+\frac {A c \left (b^2-2 a c\right )-b \left (b^2-3 a c\right ) 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} c^{5/2} \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {B \left (b^2-2 a c\right ) \tanh ^{-1}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}-\frac {b B \log \left (a+b x^2+c x^4\right )}{4 c^2}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 460, normalized size = 1.36 \begin {gather*} \frac {12 \sqrt {c} (A c-b C) x+6 B c^{3/2} x^2+4 c^{3/2} C x^3+\frac {6 \sqrt {2} \left (A c \left (b^2-2 a c-b \sqrt {b^2-4 a c}\right )+\left (-b^3+3 a b c+b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}\right ) 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 {6 \sqrt {2} \left (-A c \left (b^2-2 a c+b \sqrt {b^2-4 a c}\right )+\left (b^3-3 a b c+b^2 \sqrt {b^2-4 a c}-a c \sqrt {b^2-4 a c}\right ) 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 {3 B \sqrt {c} \left (-b^2+2 a c+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 {3 B \sqrt {c} \left (b^2-2 a c+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}}}{12 c^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(12*Sqrt[c]*(A*c - b*C)*x + 6*B*c^(3/2)*x^2 + 4*c^(3/2)*C*x^3 + (6*Sqrt[2]*(A*c*(b^2 - 2*a*c - b*Sqrt[b^2 - 4*
a*c]) + (-b^3 + 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*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]]) + (6*Sqrt[2]*(-(A*c*(b^2 - 2*a*c + b*Sq
rt[b^2 - 4*a*c])) + (b^3 - 3*a*b*c + b^2*Sqrt[b^2 - 4*a*c] - a*c*Sqrt[b^2 - 4*a*c])*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]]) - (3*B*Sqrt[c]*(-b^2 + 2*a*c
+ 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] - (3*B*Sqrt[c]*(b^2 - 2*a*c +
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])/(12*c^(5/2))

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Maple [A]
time = 0.07, size = 331, normalized size = 0.98

method	result	size
risch	\(\frac {C \,x^{3}}{3 c}+\frac {B \,x^{2}}{2 c}+\frac {A x}{c}-\frac {b C x}{c^{2}}+\frac {\munderset {\textit {\_R} =\RootOf \left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (-b B c \,\textit {\_R}^{3}+\left (-b c A -a c C +C \,b^{2}\right ) \textit {\_R}^{2}-a c B \textit {\_R} -a c A +a b C \right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{2 c^{2}}\)	\(118\)
default	\(\frac {\frac {1}{3} c C \,x^{3}+\frac {1}{2} B c \,x^{2}+A c x -b C x}{c^{2}}+\frac {\frac {\left (2 a c \sqrt {-4 a c +b^{2}}-b^{2} \sqrt {-4 a c +b^{2}}-4 a b c +b^{3}\right ) \left (\frac {B \ln \left (-b -2 c \,x^{2}+\sqrt {-4 a c +b^{2}}\right )}{2}+\frac {\left (-2 A c -C \sqrt {-4 a c +b^{2}}+b C \right ) \sqrt {2}\, \arctanh \left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 c \left (4 a c -b^{2}\right )}-\frac {\left (2 a c \sqrt {-4 a c +b^{2}}-b^{2} \sqrt {-4 a c +b^{2}}+4 a b c -b^{3}\right ) \left (\frac {B \ln \left (b +2 c \,x^{2}+\sqrt {-4 a c +b^{2}}\right )}{2}+\frac {\left (2 A c -C \sqrt {-4 a c +b^{2}}-b C \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 c \left (4 a c -b^{2}\right )}}{c}\)	\(331\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

1/6*(2*C*c*x^3 + 3*B*c*x^2 - 6*(C*b - A*c)*x)/c^2 - integrate((B*b*c*x^3 + B*a*c*x - C*a*b + A*a*c - (C*b^2 -
(C*a + A*b)*c)*x^2)/(c*x^4 + b*x^2 + a), x)/c^2

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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(x^4*(C*x^2+B*x+A)/(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(x**4*(C*x**2+B*x+A)/(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. 5305 vs. \(2 (294) = 588\).
time = 6.58, size = 5305, normalized size = 15.65 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*B*b*log(abs(c*x^4 + b*x^2 + a))/c^2 - 1/8*((2*b^5*c^3 - 16*a*b^3*c^4 + 32*a^2*b*c^5 - sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b
^3*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sq
rt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 - 8*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)*b^3*c^3 + 4*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)*b^3*c^3 + 8*(b^2 - 4*a*c)*a*b*c^4)*A*c^2 - (2*b^6*c^2 - 18*a*b
^4*c^3 + 48*a^2*b^2*c^4 - 32*a^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^6 + 9*sqrt(
2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2
- 4*a*c)*c)*b^5*c - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^2 - 10*sqrt(2)*sqr
t(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a
*c)*c)*b^4*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*c^3 + 8*sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 + 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*
a*b^2*c^3 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*c^4 - 2*(b^2 - 4*a*c)*b^4*c^2 + 10
*(b^2 - 4*a*c)*a*b^2*c^3 - 8*(b^2 - 4*a*c)*a^2*c^4)*C*c^2 + 2*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c
^3 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 - 2*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c
^4 + 2*a*b^4*c^4 + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*c^5 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)
*c)*a^2*b*c^5 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^5 - 16*a^2*b^2*c^5 - 4*sqrt(2)*sqrt(b*c - sqrt
(b^2 - 4*a*c)*c)*a^2*c^6 + 32*a^3*c^6 - 2*(b^2 - 4*a*c)*a*b^2*c^4 + 8*(b^2 - 4*a*c)*a^2*c^5)*A*abs(c) - 2*(sqr
t(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^5*c^2 - 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^3*c^3 - 2*sqr
t(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c^3 + 2*a*b^5*c^3 + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^3*
b*c^4 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 + sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*
c^4 - 16*a^2*b^3*c^4 - 4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^5 + 32*a^3*b*c^5 - 2*(b^2 - 4*a*c)*a*
b^3*c^3 + 8*(b^2 - 4*a*c)*a^2*b*c^4)*C*abs(c) - (2*b^5*c^5 - 12*a*b^3*c^6 + 16*a^2*b*c^7 - sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^5*c^3 + 6*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a
*b^3*c^4 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4*c^4 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*s
qrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b*c^5 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c
^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3*c^5 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c
- sqrt(b^2 - 4*a*c)*c)*a*b*c^6 - 2*(b^2 - 4*a*c)*b^3*c^5 + 4*(b^2 - 4*a*c)*a*b*c^6)*A + (2*b^6*c^4 - 14*a*b^4*
c^5 + 24*a^2*b^2*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^6*c^2 + 7*sqrt(2)*sqrt(b^2
- 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^4*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*
c)*b^5*c^3 - 12*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^4 - 6*sqrt(2)*sqrt(b^2 - 4
*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^3*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^
4*c^4 + 3*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c^5 - 2*(b^2 - 4*a*c)*b^4*c^4 + 6*(b
^2 - 4*a*c)*a*b^2*c^5)*C)*arctan(2*sqrt(1/2)*x/sqrt((b*c^7 + sqrt(b^2*c^14 - 4*a*c^15))/c^8))/((a*b^4*c^4 - 8*
a^2*b^2*c^5 - 2*a*b^3*c^5 + 16*a^3*c^6 + 8*a^2*b*c^6 + a*b^2*c^6 - 4*a^2*c^7)*c^2) - 1/8*((2*b^5*c^3 - 16*a*b^
3*c^4 + 32*a^2*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c + 8*sqrt(2)*sqrt(b^2 -
4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)
*b^4*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^3 - 8*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)*b^3*c^
3 + 4*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)*b^3*c^3 + 8*(b^2 - 4
*a*c)*a*b*c^4)*A*c^2 - (2*b^6*c^2 - 18*a*b^4*c^3 + 48*a^2*b^2*c^4 - 32*a^3*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqr
t(b*c + sqrt(b^2 - 4*a*c)*c)*b^6 + 9*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c + 2*sqr
t(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^
2 - 4*a*c)*c)*a^2*b^2*c^2 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^2 - sqrt(2)*s
qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^2 + 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 -
4*a*c)*c)*a^3*c^3 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)...

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Mupad [B]
time = 0.96, size = 2588, normalized size = 7.63 \begin {gather*} x\,\left (\frac {A}{c}-\frac {C\,b}{c^2}\right )+\left (\sum _{k=1}^4\ln \left (-\mathrm {root}\left (128\,a\,b^2\,c^6\,z^4-16\,b^4\,c^5\,z^4-256\,a^2\,c^7\,z^4-256\,B\,a^2\,b\,c^5\,z^3+128\,B\,a\,b^3\,c^4\,z^3-16\,B\,b^5\,c^3\,z^3-64\,A\,C\,a\,b^4\,c^2\,z^2+144\,A\,C\,a^2\,b^2\,c^3\,z^2+8\,A\,C\,b^6\,c\,z^2+80\,C^2\,a^3\,b\,c^3\,z^2+32\,B^2\,a\,b^4\,c^2\,z^2-48\,A^2\,a^2\,b\,c^4\,z^2+28\,A^2\,a\,b^3\,c^3\,z^2+36\,C^2\,a\,b^5\,c\,z^2-64\,A\,C\,a^3\,c^4\,z^2-100\,C^2\,a^2\,b^3\,c^2\,z^2-56\,B^2\,a^2\,b^2\,c^3\,z^2-4\,B^2\,b^6\,c\,z^2-32\,B^2\,a^3\,c^4\,z^2-4\,A^2\,b^5\,c^2\,z^2-4\,C^2\,b^7\,z^2+32\,A\,B\,C\,a^3\,b\,c^2\,z-8\,A\,B\,C\,a^2\,b^3\,c\,z-20\,B\,C^2\,a^3\,b^2\,c\,z+4\,A^2\,B\,a^2\,b^2\,c^2\,z-16\,B^3\,a^3\,b\,c^2\,z+4\,B^3\,a^2\,b^3\,c\,z+16\,B\,C^2\,a^4\,c^2\,z+4\,B\,C^2\,a^2\,b^4\,z-16\,A^2\,B\,a^3\,c^3\,z+2\,A^3\,C\,a^3\,b\,c+4\,A\,B^2\,C\,a^4\,c-2\,A^2\,C^2\,a^4\,c+2\,A\,C^3\,a^4\,b-A^2\,B^2\,a^3\,b\,c-B^2\,C^2\,a^4\,b-A^2\,C^2\,a^3\,b^2-A^4\,a^3\,c^2-B^4\,a^4\,c-C^4\,a^5,z,k\right )\,\left (\mathrm {root}\left (128\,a\,b^2\,c^6\,z^4-16\,b^4\,c^5\,z^4-256\,a^2\,c^7\,z^4-256\,B\,a^2\,b\,c^5\,z^3+128\,B\,a\,b^3\,c^4\,z^3-16\,B\,b^5\,c^3\,z^3-64\,A\,C\,a\,b^4\,c^2\,z^2+144\,A\,C\,a^2\,b^2\,c^3\,z^2+8\,A\,C\,b^6\,c\,z^2+80\,C^2\,a^3\,b\,c^3\,z^2+32\,B^2\,a\,b^4\,c^2\,z^2-48\,A^2\,a^2\,b\,c^4\,z^2+28\,A^2\,a\,b^3\,c^3\,z^2+36\,C^2\,a\,b^5\,c\,z^2-64\,A\,C\,a^3\,c^4\,z^2-100\,C^2\,a^2\,b^3\,c^2\,z^2-56\,B^2\,a^2\,b^2\,c^3\,z^2-4\,B^2\,b^6\,c\,z^2-32\,B^2\,a^3\,c^4\,z^2-4\,A^2\,b^5\,c^2\,z^2-4\,C^2\,b^7\,z^2+32\,A\,B\,C\,a^3\,b\,c^2\,z-8\,A\,B\,C\,a^2\,b^3\,c\,z-20\,B\,C^2\,a^3\,b^2\,c\,z+4\,A^2\,B\,a^2\,b^2\,c^2\,z-16\,B^3\,a^3\,b\,c^2\,z+4\,B^3\,a^2\,b^3\,c\,z+16\,B\,C^2\,a^4\,c^2\,z+4\,B\,C^2\,a^2\,b^4\,z-16\,A^2\,B\,a^3\,c^3\,z+2\,A^3\,C\,a^3\,b\,c+4\,A\,B^2\,C\,a^4\,c-2\,A^2\,C^2\,a^4\,c+2\,A\,C^3\,a^4\,b-A^2\,B^2\,a^3\,b\,c-B^2\,C^2\,a^4\,b-A^2\,C^2\,a^3\,b^2-A^4\,a^3\,c^2-B^4\,a^4\,c-C^4\,a^5,z,k\right )\,\left (-\frac {-16\,C\,a^2\,b\,c^4+16\,A\,a^2\,c^5+4\,C\,a\,b^3\,c^3-4\,A\,a\,b^2\,c^4}{c^3}+\frac {x\,\left (16\,B\,a^2\,c^5-36\,B\,a\,b^2\,c^4+8\,B\,b^4\,c^3\right )}{c^3}+\frac {\mathrm {root}\left (128\,a\,b^2\,c^6\,z^4-16\,b^4\,c^5\,z^4-256\,a^2\,c^7\,z^4-256\,B\,a^2\,b\,c^5\,z^3+128\,B\,a\,b^3\,c^4\,z^3-16\,B\,b^5\,c^3\,z^3-64\,A\,C\,a\,b^4\,c^2\,z^2+144\,A\,C\,a^2\,b^2\,c^3\,z^2+8\,A\,C\,b^6\,c\,z^2+80\,C^2\,a^3\,b\,c^3\,z^2+32\,B^2\,a\,b^4\,c^2\,z^2-48\,A^2\,a^2\,b\,c^4\,z^2+28\,A^2\,a\,b^3\,c^3\,z^2+36\,C^2\,a\,b^5\,c\,z^2-64\,A\,C\,a^3\,c^4\,z^2-100\,C^2\,a^2\,b^3\,c^2\,z^2-56\,B^2\,a^2\,b^2\,c^3\,z^2-4\,B^2\,b^6\,c\,z^2-32\,B^2\,a^3\,c^4\,z^2-4\,A^2\,b^5\,c^2\,z^2-4\,C^2\,b^7\,z^2+32\,A\,B\,C\,a^3\,b\,c^2\,z-8\,A\,B\,C\,a^2\,b^3\,c\,z-20\,B\,C^2\,a^3\,b^2\,c\,z+4\,A^2\,B\,a^2\,b^2\,c^2\,z-16\,B^3\,a^3\,b\,c^2\,z+4\,B^3\,a^2\,b^3\,c\,z+16\,B\,C^2\,a^4\,c^2\,z+4\,B\,C^2\,a^2\,b^4\,z-16\,A^2\,B\,a^3\,c^3\,z+2\,A^3\,C\,a^3\,b\,c+4\,A\,B^2\,C\,a^4\,c-2\,A^2\,C^2\,a^4\,c+2\,A\,C^3\,a^4\,b-A^2\,B^2\,a^3\,b\,c-B^2\,C^2\,a^4\,b-A^2\,C^2\,a^3\,b^2-A^4\,a^3\,c^2-B^4\,a^4\,c-C^4\,a^5,z,k\right )\,x\,\left (8\,b^3\,c^5-32\,a\,b\,c^6\right )}{c^3}\right )+\frac {8\,B\,C\,a^3\,c^3-4\,A\,B\,a^2\,b\,c^3}{c^3}+\frac {x\,\left (4\,A^2\,a^2\,c^4-8\,A^2\,a\,b^2\,c^3+2\,A^2\,b^4\,c^2-20\,A\,C\,a^2\,b\,c^3+20\,A\,C\,a\,b^3\,c^2-4\,A\,C\,b^5\,c+6\,B^2\,a^2\,b\,c^3-10\,B^2\,a\,b^3\,c^2+2\,B^2\,b^5\,c-4\,C^2\,a^3\,c^3+18\,C^2\,a^2\,b^2\,c^2-12\,C^2\,a\,b^4\,c+2\,C^2\,b^6\right )}{c^3}\right )+\frac {A^3\,a^2\,b\,c^2+A^2\,C\,a^3\,c^2-2\,A^2\,C\,a^2\,b^2\,c-A\,B^2\,a^3\,c^2+A\,B^2\,a^2\,b^2\,c+A\,C^2\,a^2\,b^3-B^2\,C\,a^3\,b\,c+C^3\,a^4\,c-C^3\,a^3\,b^2}{c^3}+\frac {x\,\left (A^2\,B\,a^2\,b\,c^2+2\,A\,B\,C\,a^3\,c^2-2\,A\,B\,C\,a^2\,b^2\,c-B^3\,a^3\,c^2+B^3\,a^2\,b^2\,c-2\,B\,C^2\,a^3\,b\,c+B\,C^2\,a^2\,b^3\right )}{c^3}\right )\,\mathrm {root}\left (128\,a\,b^2\,c^6\,z^4-16\,b^4\,c^5\,z^4-256\,a^2\,c^7\,z^4-256\,B\,a^2\,b\,c^5\,z^3+128\,B\,a\,b^3\,c^4\,z^3-16\,B\,b^5\,c^3\,z^3-64\,A\,C\,a\,b^4\,c^2\,z^2+144\,A\,C\,a^2\,b^2\,c^3\,z^2+8\,A\,C\,b^6\,c\,z^2+80\,C^2\,a^3\,b\,c^3\,z^2+32\,B^2\,a\,b^4\,c^2\,z^2-48\,A^2\,a^2\,b\,c^4\,z^2+28\,A^2\,a\,b^3\,c^3\,z^2+36\,C^2\,a\,b^5\,c\,z^2-64\,A\,C\,a^3\,c^4\,z^2-100\,C^2\,a^2\,b^3\,c^2\,z^2-56\,B^2\,a^2\,b^2\,c^3\,z^2-4\,B^2\,b^6\,c\,z^2-32\,B^2\,a^3\,c^4\,z^2-4\,A^2\,b^5\,c^2\,z^2-4\,C^2\,b^7\,z^2+32\,A\,B\,C\,a^3\,b\,c^2\,z-8\,A\,B\,C\,a^2\,b^3\,c\,z-20\,B\,C^2\,a^3\,b^2\,c\,z+4\,A^2\,B\,a^2\,b^2\,c^2\,z-16\,B^3\,a^3\,b\,c^2\,z+4\,B^3\,a^2\,b^3\,c\,z+16\,B\,C^2\,a^4\,c^2\,z+4\,B\,C^2\,a^2\,b^4\,z-16\,A^2\,B\,a^3\,c^3\,z+2\,A^3\,C\,a^3\,b\,c+4\,A\,B^2\,C\,a^4\,c-2\,A^2\,C^2\,a^4\,c+2\,A\,C^3\,a^4\,b-A^2\,B^2\,a^3\,b\,c-B^2\,C^2\,a^4\,b-A^2\,C^2\,a^3\,b^2-A^4\,a^3\,c^2-B^4\,a^4\,c-C^4\,a^5,z,k\right )\right )+\frac {B\,x^2}{2\,c}+\frac {C\,x^3}{3\,c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x*(A/c - (C*b)/c^2) + symsum(log((C^3*a^4*c - C^3*a^3*b^2 - A*B^2*a^3*c^2 + A*C^2*a^2*b^3 + A^2*C*a^3*c^2 + A^
3*a^2*b*c^2 + A*B^2*a^2*b^2*c - 2*A^2*C*a^2*b^2*c - B^2*C*a^3*b*c)/c^3 - root(128*a*b^2*c^6*z^4 - 16*b^4*c^5*z
^4 - 256*a^2*c^7*z^4 - 256*B*a^2*b*c^5*z^3 + 128*B*a*b^3*c^4*z^3 - 16*B*b^5*c^3*z^3 - 64*A*C*a*b^4*c^2*z^2 + 1
44*A*C*a^2*b^2*c^3*z^2 + 8*A*C*b^6*c*z^2 + 80*C^2*a^3*b*c^3*z^2 + 32*B^2*a*b^4*c^2*z^2 - 48*A^2*a^2*b*c^4*z^2
+ 28*A^2*a*b^3*c^3*z^2 + 36*C^2*a*b^5*c*z^2 - 64*A*C*a^3*c^4*z^2 - 100*C^2*a^2*b^3*c^2*z^2 - 56*B^2*a^2*b^2*c^
3*z^2 - 4*B^2*b^6*c*z^2 - 32*B^2*a^3*c^4*z^2 - 4*A^2*b^5*c^2*z^2 - 4*C^2*b^7*z^2 + 32*A*B*C*a^3*b*c^2*z - 8*A*
B*C*a^2*b^3*c*z - 20*B*C^2*a^3*b^2*c*z + 4*A^2*B*a^2*b^2*c^2*z - 16*B^3*a^3*b*c^2*z + 4*B^3*a^2*b^3*c*z + 16*B
*C^2*a^4*c^2*z + 4*B*C^2*a^2*b^4*z - 16*A^2*B*a^3*c^3*z + 2*A^3*C*a^3*b*c + 4*A*B^2*C*a^4*c - 2*A^2*C^2*a^4*c
+ 2*A*C^3*a^4*b - A^2*B^2*a^3*b*c - B^2*C^2*a^4*b - A^2*C^2*a^3*b^2 - A^4*a^3*c^2 - B^4*a^4*c - C^4*a^5, z, k)
*(root(128*a*b^2*c^6*z^4 - 16*b^4*c^5*z^4 - 256*a^2*c^7*z^4 - 256*B*a^2*b*c^5*z^3 + 128*B*a*b^3*c^4*z^3 - 16*B
*b^5*c^3*z^3 - 64*A*C*a*b^4*c^2*z^2 + 144*A*C*a^2*b^2*c^3*z^2 + 8*A*C*b^6*c*z^2 + 80*C^2*a^3*b*c^3*z^2 + 32*B^
2*a*b^4*c^2*z^2 - 48*A^2*a^2*b*c^4*z^2 + 28*A^2*a*b^3*c^3*z^2 + 36*C^2*a*b^5*c*z^2 - 64*A*C*a^3*c^4*z^2 - 100*
C^2*a^2*b^3*c^2*z^2 - 56*B^2*a^2*b^2*c^3*z^2 - 4*B^2*b^6*c*z^2 - 32*B^2*a^3*c^4*z^2 - 4*A^2*b^5*c^2*z^2 - 4*C^
2*b^7*z^2 + 32*A*B*C*a^3*b*c^2*z - 8*A*B*C*a^2*b^3*c*z - 20*B*C^2*a^3*b^2*c*z + 4*A^2*B*a^2*b^2*c^2*z - 16*B^3
*a^3*b*c^2*z + 4*B^3*a^2*b^3*c*z + 16*B*C^2*a^4*c^2*z + 4*B*C^2*a^2*b^4*z - 16*A^2*B*a^3*c^3*z + 2*A^3*C*a^3*b
*c + 4*A*B^2*C*a^4*c - 2*A^2*C^2*a^4*c + 2*A*C^3*a^4*b - A^2*B^2*a^3*b*c - B^2*C^2*a^4*b - A^2*C^2*a^3*b^2 - A
^4*a^3*c^2 - B^4*a^4*c - C^4*a^5, z, k)*((x*(16*B*a^2*c^5 + 8*B*b^4*c^3 - 36*B*a*b^2*c^4))/c^3 - (16*A*a^2*c^5
- 4*A*a*b^2*c^4 + 4*C*a*b^3*c^3 - 16*C*a^2*b*c^4)/c^3 + (root(128*a*b^2*c^6*z^4 - 16*b^4*c^5*z^4 - 256*a^2*c^
7*z^4 - 256*B*a^2*b*c^5*z^3 + 128*B*a*b^3*c^4*z^3 - 16*B*b^5*c^3*z^3 - 64*A*C*a*b^4*c^2*z^2 + 144*A*C*a^2*b^2*
c^3*z^2 + 8*A*C*b^6*c*z^2 + 80*C^2*a^3*b*c^3*z^2 + 32*B^2*a*b^4*c^2*z^2 - 48*A^2*a^2*b*c^4*z^2 + 28*A^2*a*b^3*
c^3*z^2 + 36*C^2*a*b^5*c*z^2 - 64*A*C*a^3*c^4*z^2 - 100*C^2*a^2*b^3*c^2*z^2 - 56*B^2*a^2*b^2*c^3*z^2 - 4*B^2*b
^6*c*z^2 - 32*B^2*a^3*c^4*z^2 - 4*A^2*b^5*c^2*z^2 - 4*C^2*b^7*z^2 + 32*A*B*C*a^3*b*c^2*z - 8*A*B*C*a^2*b^3*c*z
- 20*B*C^2*a^3*b^2*c*z + 4*A^2*B*a^2*b^2*c^2*z - 16*B^3*a^3*b*c^2*z + 4*B^3*a^2*b^3*c*z + 16*B*C^2*a^4*c^2*z
+ 4*B*C^2*a^2*b^4*z - 16*A^2*B*a^3*c^3*z + 2*A^3*C*a^3*b*c + 4*A*B^2*C*a^4*c - 2*A^2*C^2*a^4*c + 2*A*C^3*a^4*b
- A^2*B^2*a^3*b*c - B^2*C^2*a^4*b - A^2*C^2*a^3*b^2 - A^4*a^3*c^2 - B^4*a^4*c - C^4*a^5, z, k)*x*(8*b^3*c^5 -
32*a*b*c^6))/c^3) + (8*B*C*a^3*c^3 - 4*A*B*a^2*b*c^3)/c^3 + (x*(2*C^2*b^6 + 2*B^2*b^5*c + 4*A^2*a^2*c^4 + 2*A
^2*b^4*c^2 - 4*C^2*a^3*c^3 - 4*A*C*b^5*c + 18*C^2*a^2*b^2*c^2 - 12*C^2*a*b^4*c - 8*A^2*a*b^2*c^3 - 10*B^2*a*b^
3*c^2 + 6*B^2*a^2*b*c^3 + 20*A*C*a*b^3*c^2 - 20*A*C*a^2*b*c^3))/c^3) + (x*(B*C^2*a^2*b^3 - B^3*a^3*c^2 + B^3*a
^2*b^2*c + A^2*B*a^2*b*c^2 + 2*A*B*C*a^3*c^2 - 2*B*C^2*a^3*b*c - 2*A*B*C*a^2*b^2*c))/c^3)*root(128*a*b^2*c^6*z
^4 - 16*b^4*c^5*z^4 - 256*a^2*c^7*z^4 - 256*B*a^2*b*c^5*z^3 + 128*B*a*b^3*c^4*z^3 - 16*B*b^5*c^3*z^3 - 64*A*C*
a*b^4*c^2*z^2 + 144*A*C*a^2*b^2*c^3*z^2 + 8*A*C*b^6*c*z^2 + 80*C^2*a^3*b*c^3*z^2 + 32*B^2*a*b^4*c^2*z^2 - 48*A
^2*a^2*b*c^4*z^2 + 28*A^2*a*b^3*c^3*z^2 + 36*C^2*a*b^5*c*z^2 - 64*A*C*a^3*c^4*z^2 - 100*C^2*a^2*b^3*c^2*z^2 -
56*B^2*a^2*b^2*c^3*z^2 - 4*B^2*b^6*c*z^2 - 32*B^2*a^3*c^4*z^2 - 4*A^2*b^5*c^2*z^2 - 4*C^2*b^7*z^2 + 32*A*B*C*a
^3*b*c^2*z - 8*A*B*C*a^2*b^3*c*z - 20*B*C^2*a^3*b^2*c*z + 4*A^2*B*a^2*b^2*c^2*z - 16*B^3*a^3*b*c^2*z + 4*B^3*a
^2*b^3*c*z + 16*B*C^2*a^4*c^2*z + 4*B*C^2*a^2*b^4*z - 16*A^2*B*a^3*c^3*z + 2*A^3*C*a^3*b*c + 4*A*B^2*C*a^4*c -
2*A^2*C^2*a^4*c + 2*A*C^3*a^4*b - A^2*B^2*a^3*b*c - B^2*C^2*a^4*b - A^2*C^2*a^3*b^2 - A^4*a^3*c^2 - B^4*a^4*c
- C^4*a^5, z, k), k, 1, 4) + (B*x^2)/(2*c) + (C*x^3)/(3*c)

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