\(\int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx\) [227]

Optimal result

Integrand size = 38, antiderivative size = 605 \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\frac {\left (4 a^2 B+b \left (b-\sqrt {8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b-\sqrt {8 a^2+b^2-4 a c}\right )+b C-\sqrt {8 a^2+b^2-4 a c} C+2 c D\right )\right ) \arctan \left (\frac {b-\sqrt {8 a^2+b^2-4 a c}+4 a x}{\sqrt {2} \sqrt {4 a^2+2 a c-b \left (b-\sqrt {8 a^2+b^2-4 a c}\right )}}\right )}{\sqrt {2} a \sqrt {8 a^2+b^2-4 a c} \sqrt {4 a^2+2 a c-b \left (b-\sqrt {8 a^2+b^2-4 a c}\right )}}-\frac {\left (4 a^2 B+b \left (b+\sqrt {8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b+\sqrt {8 a^2+b^2-4 a c}\right )+b C+\sqrt {8 a^2+b^2-4 a c} C+2 c D\right )\right ) \arctan \left (\frac {b+\sqrt {8 a^2+b^2-4 a c}+4 a x}{\sqrt {2} \sqrt {4 a^2+2 a c-b \left (b+\sqrt {8 a^2+b^2-4 a c}\right )}}\right )}{\sqrt {2} a \sqrt {8 a^2+b^2-4 a c} \sqrt {4 a^2+2 a c-b \left (b+\sqrt {8 a^2+b^2-4 a c}\right )}}-\frac {\left (2 a (A-C)+\left (b-\sqrt {8 a^2+b^2-4 a c}\right ) D\right ) \log \left (2 a+\left (b-\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2\right )}{4 a \sqrt {8 a^2+b^2-4 a c}}+\frac {\left (2 a (A-C)+\left (b+\sqrt {8 a^2+b^2-4 a c}\right ) D\right ) \log \left (2 a+\left (b+\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2\right )}{4 a \sqrt {8 a^2+b^2-4 a c}} \]

[Out]

-1/4*ln(2*a+2*a*x^2+x*(b-(8*a^2-4*a*c+b^2)^(1/2)))*(2*a*(A-C)+D*(b-(8*a^2-4*a*c+b^2)^(1/2)))/a/(8*a^2-4*a*c+b^
2)^(1/2)+1/4*ln(2*a+2*a*x^2+x*(b+(8*a^2-4*a*c+b^2)^(1/2)))*(2*a*(A-C)+D*(b+(8*a^2-4*a*c+b^2)^(1/2)))/a/(8*a^2-
4*a*c+b^2)^(1/2)+1/2*arctan(1/2*(b+4*a*x-(8*a^2-4*a*c+b^2)^(1/2))*2^(1/2)/(4*a^2+2*a*c-b*(b-(8*a^2-4*a*c+b^2)^
(1/2)))^(1/2))*(4*a^2*B+b*D*(b-(8*a^2-4*a*c+b^2)^(1/2))-a*(b*C+2*c*D+A*(b-(8*a^2-4*a*c+b^2)^(1/2))-C*(8*a^2-4*
a*c+b^2)^(1/2)))/a*2^(1/2)/(8*a^2-4*a*c+b^2)^(1/2)/(4*a^2+2*a*c-b*(b-(8*a^2-4*a*c+b^2)^(1/2)))^(1/2)-1/2*arcta
n(1/2*(b+4*a*x+(8*a^2-4*a*c+b^2)^(1/2))*2^(1/2)/(4*a^2+2*a*c-b*(b+(8*a^2-4*a*c+b^2)^(1/2)))^(1/2))*(4*a^2*B+b*
D*(b+(8*a^2-4*a*c+b^2)^(1/2))-a*(b*C+2*c*D+C*(8*a^2-4*a*c+b^2)^(1/2)+A*(b+(8*a^2-4*a*c+b^2)^(1/2))))/a*2^(1/2)
/(8*a^2-4*a*c+b^2)^(1/2)/(4*a^2+2*a*c-b*(b+(8*a^2-4*a*c+b^2)^(1/2)))^(1/2)

Rubi [A] (verified)

Time = 3.07 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2111, 648, 632, 210, 642} \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\frac {\arctan \left (\frac {-\sqrt {8 a^2-4 a c+b^2}+4 a x+b}{\sqrt {2} \sqrt {-b \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+4 a^2+2 a c}}\right ) \left (-a \left (A \left (b-\sqrt {8 a^2-4 a c+b^2}\right )-C \sqrt {8 a^2-4 a c+b^2}+b C+2 c D\right )+b D \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+4 a^2 B\right )}{\sqrt {2} a \sqrt {8 a^2-4 a c+b^2} \sqrt {-b \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+4 a^2+2 a c}}-\frac {\arctan \left (\frac {\sqrt {8 a^2-4 a c+b^2}+4 a x+b}{\sqrt {2} \sqrt {-b \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+4 a^2+2 a c}}\right ) \left (-a \left (A \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+C \sqrt {8 a^2-4 a c+b^2}+b C+2 c D\right )+b D \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+4 a^2 B\right )}{\sqrt {2} a \sqrt {8 a^2-4 a c+b^2} \sqrt {-b \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+4 a^2+2 a c}}-\frac {\log \left (x \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+2 a x^2+2 a\right ) \left (D \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+2 a (A-C)\right )}{4 a \sqrt {8 a^2-4 a c+b^2}}+\frac {\log \left (x \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+2 a x^2+2 a\right ) \left (D \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+2 a (A-C)\right )}{4 a \sqrt {8 a^2-4 a c+b^2}} \]

[In]

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

[Out]

((4*a^2*B + b*(b - Sqrt[8*a^2 + b^2 - 4*a*c])*D - a*(A*(b - Sqrt[8*a^2 + b^2 - 4*a*c]) + b*C - Sqrt[8*a^2 + b^
2 - 4*a*c]*C + 2*c*D))*ArcTan[(b - Sqrt[8*a^2 + b^2 - 4*a*c] + 4*a*x)/(Sqrt[2]*Sqrt[4*a^2 + 2*a*c - b*(b - Sqr
t[8*a^2 + b^2 - 4*a*c])])])/(Sqrt[2]*a*Sqrt[8*a^2 + b^2 - 4*a*c]*Sqrt[4*a^2 + 2*a*c - b*(b - Sqrt[8*a^2 + b^2
- 4*a*c])]) - ((4*a^2*B + b*(b + Sqrt[8*a^2 + b^2 - 4*a*c])*D - a*(A*(b + Sqrt[8*a^2 + b^2 - 4*a*c]) + b*C + S
qrt[8*a^2 + b^2 - 4*a*c]*C + 2*c*D))*ArcTan[(b + Sqrt[8*a^2 + b^2 - 4*a*c] + 4*a*x)/(Sqrt[2]*Sqrt[4*a^2 + 2*a*
c - b*(b + Sqrt[8*a^2 + b^2 - 4*a*c])])])/(Sqrt[2]*a*Sqrt[8*a^2 + b^2 - 4*a*c]*Sqrt[4*a^2 + 2*a*c - b*(b + Sqr
t[8*a^2 + b^2 - 4*a*c])]) - ((2*a*(A - C) + (b - Sqrt[8*a^2 + b^2 - 4*a*c])*D)*Log[2*a + (b - Sqrt[8*a^2 + b^2
- 4*a*c])*x + 2*a*x^2])/(4*a*Sqrt[8*a^2 + b^2 - 4*a*c]) + ((2*a*(A - C) + (b + Sqrt[8*a^2 + b^2 - 4*a*c])*D)*
Log[2*a + (b + Sqrt[8*a^2 + b^2 - 4*a*c])*x + 2*a*x^2])/(4*a*Sqrt[8*a^2 + b^2 - 4*a*c])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[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 2111

Int[(P3_)/((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4), x_Symbol] :> With[{q = Sqrt[8*a^2
+ b^2 - 4*a*c], A = Coeff[P3, x, 0], B = Coeff[P3, x, 1], C = Coeff[P3, x, 2], D = Coeff[P3, x, 3]}, Dist[1/q,
Int[(b*A - 2*a*B + 2*a*D + A*q + (2*a*A - 2*a*C + b*D + D*q)*x)/(2*a + (b + q)*x + 2*a*x^2), x], x] - Dist[1/
q, Int[(b*A - 2*a*B + 2*a*D - A*q + (2*a*A - 2*a*C + b*D - D*q)*x)/(2*a + (b - q)*x + 2*a*x^2), x], x]] /; Fre
eQ[{a, b, c}, x] && PolyQ[P3, x, 3] && EqQ[a, e] && EqQ[b, d]

Rubi steps \begin{align*} \text {integral}& = -\frac {\int \frac {A b-2 a B-A \sqrt {8 a^2+b^2-4 a c}+2 a D+\left (2 a A-2 a C+b D-\sqrt {8 a^2+b^2-4 a c} D\right ) x}{2 a+\left (b-\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2} \, dx}{\sqrt {8 a^2+b^2-4 a c}}+\frac {\int \frac {A b-2 a B+A \sqrt {8 a^2+b^2-4 a c}+2 a D+\left (2 a A-2 a C+b D+\sqrt {8 a^2+b^2-4 a c} D\right ) x}{2 a+\left (b+\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2} \, dx}{\sqrt {8 a^2+b^2-4 a c}} \\ & = -\frac {\left (2 a (A-C)+\left (b-\sqrt {8 a^2+b^2-4 a c}\right ) D\right ) \int \frac {b-\sqrt {8 a^2+b^2-4 a c}+4 a x}{2 a+\left (b-\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2} \, dx}{4 a \sqrt {8 a^2+b^2-4 a c}}+\frac {\left (2 a (A-C)+\left (b+\sqrt {8 a^2+b^2-4 a c}\right ) D\right ) \int \frac {b+\sqrt {8 a^2+b^2-4 a c}+4 a x}{2 a+\left (b+\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2} \, dx}{4 a \sqrt {8 a^2+b^2-4 a c}}+\frac {\left (4 a^2 B+b \left (b-\sqrt {8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b-\sqrt {8 a^2+b^2-4 a c}\right )+b C-\sqrt {8 a^2+b^2-4 a c} C+2 c D\right )\right ) \int \frac {1}{2 a+\left (b-\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2} \, dx}{2 a \sqrt {8 a^2+b^2-4 a c}}-\frac {\left (4 a^2 B+b \left (b+\sqrt {8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b+\sqrt {8 a^2+b^2-4 a c}\right )+b C+\sqrt {8 a^2+b^2-4 a c} C+2 c D\right )\right ) \int \frac {1}{2 a+\left (b+\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2} \, dx}{2 a \sqrt {8 a^2+b^2-4 a c}} \\ & = -\frac {\left (2 a (A-C)+\left (b-\sqrt {8 a^2+b^2-4 a c}\right ) D\right ) \log \left (2 a+\left (b-\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2\right )}{4 a \sqrt {8 a^2+b^2-4 a c}}+\frac {\left (2 a (A-C)+\left (b+\sqrt {8 a^2+b^2-4 a c}\right ) D\right ) \log \left (2 a+\left (b+\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2\right )}{4 a \sqrt {8 a^2+b^2-4 a c}}-\frac {\left (4 a^2 B+b \left (b-\sqrt {8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b-\sqrt {8 a^2+b^2-4 a c}\right )+b C-\sqrt {8 a^2+b^2-4 a c} C+2 c D\right )\right ) \text {Subst}\left (\int \frac {1}{-16 a^2+\left (b-\sqrt {8 a^2+b^2-4 a c}\right )^2-x^2} \, dx,x,b-\sqrt {8 a^2+b^2-4 a c}+4 a x\right )}{a \sqrt {8 a^2+b^2-4 a c}}+\frac {\left (4 a^2 B+b \left (b+\sqrt {8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b+\sqrt {8 a^2+b^2-4 a c}\right )+b C+\sqrt {8 a^2+b^2-4 a c} C+2 c D\right )\right ) \text {Subst}\left (\int \frac {1}{-16 a^2+\left (b+\sqrt {8 a^2+b^2-4 a c}\right )^2-x^2} \, dx,x,b+\sqrt {8 a^2+b^2-4 a c}+4 a x\right )}{a \sqrt {8 a^2+b^2-4 a c}} \\ & = \frac {\left (4 a^2 B+b \left (b-\sqrt {8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b-\sqrt {8 a^2+b^2-4 a c}\right )+b C-\sqrt {8 a^2+b^2-4 a c} C+2 c D\right )\right ) \tan ^{-1}\left (\frac {b-\sqrt {8 a^2+b^2-4 a c}+4 a x}{\sqrt {2} \sqrt {4 a^2+2 a c-b \left (b-\sqrt {8 a^2+b^2-4 a c}\right )}}\right )}{\sqrt {2} a \sqrt {8 a^2+b^2-4 a c} \sqrt {4 a^2+2 a c-b \left (b-\sqrt {8 a^2+b^2-4 a c}\right )}}-\frac {\left (4 a^2 B+b \left (b+\sqrt {8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b+\sqrt {8 a^2+b^2-4 a c}\right )+b C+\sqrt {8 a^2+b^2-4 a c} C+2 c D\right )\right ) \tan ^{-1}\left (\frac {b+\sqrt {8 a^2+b^2-4 a c}+4 a x}{\sqrt {2} \sqrt {4 a^2+2 a c-b \left (b+\sqrt {8 a^2+b^2-4 a c}\right )}}\right )}{\sqrt {2} a \sqrt {8 a^2+b^2-4 a c} \sqrt {4 a^2+2 a c-b \left (b+\sqrt {8 a^2+b^2-4 a c}\right )}}-\frac {\left (2 a (A-C)+\left (b-\sqrt {8 a^2+b^2-4 a c}\right ) D\right ) \log \left (2 a+\left (b-\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2\right )}{4 a \sqrt {8 a^2+b^2-4 a c}}+\frac {\left (2 a (A-C)+\left (b+\sqrt {8 a^2+b^2-4 a c}\right ) D\right ) \log \left (2 a+\left (b+\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2\right )}{4 a \sqrt {8 a^2+b^2-4 a c}} \\ \end{align*}

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.16 \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\text {RootSum}\left [a+b \text {$\#$1}+c \text {$\#$1}^2+b \text {$\#$1}^3+a \text {$\#$1}^4\&,\frac {A \log (x-\text {$\#$1})+B \log (x-\text {$\#$1}) \text {$\#$1}+C \log (x-\text {$\#$1}) \text {$\#$1}^2+D \log (x-\text {$\#$1}) \text {$\#$1}^3}{b+2 c \text {$\#$1}+3 b \text {$\#$1}^2+4 a \text {$\#$1}^3}\&\right ] \]

[In]

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

[Out]

RootSum[a + b*#1 + c*#1^2 + b*#1^3 + a*#1^4 & , (A*Log[x - #1] + B*Log[x - #1]*#1 + C*Log[x - #1]*#1^2 + D*Log
[x - #1]*#1^3)/(b + 2*c*#1 + 3*b*#1^2 + 4*a*#1^3) & ]

Maple [A] (verified)

Time = 0.16 (sec) , antiderivative size = 535, normalized size of antiderivative = 0.88

method	result	size
default	\(4 a \left (\frac {-\frac {\left (2 A a -2 C a -\sqrt {8 a^{2}-4 a c +b^{2}}\, D+D b \right ) \ln \left (-2 a \,x^{2}+\sqrt {8 a^{2}-4 a c +b^{2}}\, x -b x -2 a \right )}{4 a}+\frac {2 \left (\frac {\left (2 A a -2 C a -\sqrt {8 a^{2}-4 a c +b^{2}}\, D+D b \right ) \left (\sqrt {8 a^{2}-4 a c +b^{2}}-b \right )}{4 a}-\sqrt {8 a^{2}-4 a c +b^{2}}\, A +A b -2 B a +2 D a \right ) \arctan \left (\frac {-4 a x +\sqrt {8 a^{2}-4 a c +b^{2}}-b}{\sqrt {8 a^{2}+4 a c -2 b^{2}+2 \sqrt {8 a^{2}-4 a c +b^{2}}\, b}}\right )}{\sqrt {8 a^{2}+4 a c -2 b^{2}+2 \sqrt {8 a^{2}-4 a c +b^{2}}\, b}}}{4 a \sqrt {8 a^{2}-4 a c +b^{2}}}+\frac {\frac {\left (2 A a -2 C a +\sqrt {8 a^{2}-4 a c +b^{2}}\, D+D b \right ) \ln \left (2 a \,x^{2}+\sqrt {8 a^{2}-4 a c +b^{2}}\, x +b x +2 a \right )}{4 a}+\frac {2 \left (-\frac {\left (2 A a -2 C a +\sqrt {8 a^{2}-4 a c +b^{2}}\, D+D b \right ) \left (b +\sqrt {8 a^{2}-4 a c +b^{2}}\right )}{4 a}+\sqrt {8 a^{2}-4 a c +b^{2}}\, A +A b -2 B a +2 D a \right ) \arctan \left (\frac {b +4 a x +\sqrt {8 a^{2}-4 a c +b^{2}}}{\sqrt {8 a^{2}+4 a c -2 b^{2}-2 \sqrt {8 a^{2}-4 a c +b^{2}}\, b}}\right )}{\sqrt {8 a^{2}+4 a c -2 b^{2}-2 \sqrt {8 a^{2}-4 a c +b^{2}}\, b}}}{4 a \sqrt {8 a^{2}-4 a c +b^{2}}}\right )\)	\(535\)

[In]

int((D*x^3+C*x^2+B*x+A)/(a*x^4+b*x^3+c*x^2+b*x+a),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{a x^{4} + b x^{3} + c x^{2} + b x + a} \,d x } \]

[In]

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

[Out]

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

Giac [F(-2)]

Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >> an error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:Not invertible Error: Bad Argument Value

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{a\,x^4+b\,x^3+c\,x^2+b\,x+a} \,d x \]

[In]

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

[Out]

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