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

Optimal. Leaf size=605 \[ \frac {\tan ^{-1}\left (\frac {-\sqrt {8 a^2-4 a c+b^2}+4 a x+b}{\sqrt {2} \sqrt {-b \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+4 a^2+2 a c}}\right ) \left (-a \left (A \left (b-\sqrt {8 a^2-4 a c+b^2}\right )-C \sqrt {8 a^2-4 a c+b^2}+b C+2 c D\right )+b D \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+4 a^2 B\right )}{\sqrt {2} a \sqrt {8 a^2-4 a c+b^2} \sqrt {-b \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+4 a^2+2 a c}}-\frac {\tan ^{-1}\left (\frac {\sqrt {8 a^2-4 a c+b^2}+4 a x+b}{\sqrt {2} \sqrt {-b \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+4 a^2+2 a c}}\right ) \left (-a \left (A \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+C \sqrt {8 a^2-4 a c+b^2}+b C+2 c D\right )+b D \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+4 a^2 B\right )}{\sqrt {2} a \sqrt {8 a^2-4 a c+b^2} \sqrt {-b \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+4 a^2+2 a c}}-\frac {\log \left (x \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+2 a x^2+2 a\right ) \left (D \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+2 a (A-C)\right )}{4 a \sqrt {8 a^2-4 a c+b^2}}+\frac {\log \left (x \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+2 a x^2+2 a\right ) \left (D \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+2 a (A-C)\right )}{4 a \sqrt {8 a^2-4 a c+b^2}} \]

[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]  time = 4.54, antiderivative size = 605, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 5, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.132, Rules used = {2086, 634, 618, 204, 628} \[ \frac {\tan ^{-1}\left (\frac {-\sqrt {8 a^2-4 a c+b^2}+4 a x+b}{\sqrt {2} \sqrt {-b \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+4 a^2+2 a c}}\right ) \left (-a \left (A \left (b-\sqrt {8 a^2-4 a c+b^2}\right )-C \sqrt {8 a^2-4 a c+b^2}+b C+2 c D\right )+b D \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+4 a^2 B\right )}{\sqrt {2} a \sqrt {8 a^2-4 a c+b^2} \sqrt {-b \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+4 a^2+2 a c}}-\frac {\tan ^{-1}\left (\frac {\sqrt {8 a^2-4 a c+b^2}+4 a x+b}{\sqrt {2} \sqrt {-b \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+4 a^2+2 a c}}\right ) \left (-a \left (A \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+C \sqrt {8 a^2-4 a c+b^2}+b C+2 c D\right )+b D \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+4 a^2 B\right )}{\sqrt {2} a \sqrt {8 a^2-4 a c+b^2} \sqrt {-b \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+4 a^2+2 a c}}-\frac {\log \left (x \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+2 a x^2+2 a\right ) \left (D \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+2 a (A-C)\right )}{4 a \sqrt {8 a^2-4 a c+b^2}}+\frac {\log \left (x \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+2 a x^2+2 a\right ) \left (D \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+2 a (A-C)\right )}{4 a \sqrt {8 a^2-4 a c+b^2}} \]

Antiderivative was successfully verified.

[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 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[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 2086

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*} \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 {\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 ) \operatorname {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 ) \operatorname {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]  time = 0.07, size = 98, normalized size = 0.16 \[ \text {RootSum}\left [\text {$\#$1}^4 a+\text {$\#$1}^3 b+\text {$\#$1}^2 c+\text {$\#$1} b+a\& ,\frac {\text {$\#$1}^3 D \log (x-\text {$\#$1})+\text {$\#$1}^2 C \log (x-\text {$\#$1})+A \log (x-\text {$\#$1})+\text {$\#$1} B \log (x-\text {$\#$1})}{4 \text {$\#$1}^3 a+3 \text {$\#$1}^2 b+2 \text {$\#$1} c+b}\& \right ] \]

Antiderivative was successfully verified.

[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) & ]

________________________________________________________________________________________

fricas [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((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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[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,x):;OUTPUT:Eval
uation time: 2.16Not invertible Error: Bad Argument Value

________________________________________________________________________________________

maple [B]  time = 0.09, size = 2105, normalized size = 3.48 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {D x^{3} + C x^{2} + B x + A}{a x^{4} + b x^{3} + c x^{2} + b x + a}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {A+B\,x+C\,x^2+x^3\,D}{a\,x^4+b\,x^3+c\,x^2+b\,x+a} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

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((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

________________________________________________________________________________________