Integrand size = 38, antiderivative size = 289 \[ \int \frac {A+B x+C x^2+D x^3}{88-402 x+855 x^2-837 x^3+324 x^4} \, dx=-\frac {\left (54 \left (1+\sqrt {177}\right ) A-36 \left (15-\sqrt {177}\right ) B-732 C+36 \sqrt {177} C-689 D+39 \sqrt {177} D\right ) \arctan \left (\frac {31-\sqrt {177}-48 x}{\sqrt {2 \left (167-\sqrt {177}\right )}}\right )}{324 \sqrt {354 \left (167-\sqrt {177}\right )}}+\frac {\left (54 \left (1-\sqrt {177}\right ) A-36 \left (15+\sqrt {177}\right ) B-732 C-36 \sqrt {177} C-689 D-39 \sqrt {177} D\right ) \arctan \left (\frac {31+\sqrt {177}-48 x}{\sqrt {2 \left (167+\sqrt {177}\right )}}\right )}{324 \sqrt {354 \left (167+\sqrt {177}\right )}}-\frac {\left (\sqrt {177} (36 A+24 B+8 C-5 D)-59 D\right ) \log \left (16-\left (1-\sqrt {177}\right ) (2-3 x)+8 (2-3 x)^2\right )}{76464}+\frac {\left (\sqrt {177} (36 A+24 B+8 C-5 D)+59 D\right ) \log \left (16-\left (1+\sqrt {177}\right ) (2-3 x)+8 (2-3 x)^2\right )}{76464} \] Output:
-1/324*(54*(1+177^(1/2))*A-36*(15-177^(1/2))*B-732*C+36*177^(1/2)*C-689*D+ 39*177^(1/2)*D)*arctan((31-177^(1/2)-48*x)/(334-2*177^(1/2))^(1/2))/(59118 -354*177^(1/2))^(1/2)+1/324*(54*(1-177^(1/2))*A-36*(15+177^(1/2))*B-732*C- 36*177^(1/2)*C-689*D-39*177^(1/2)*D)*arctan((31+177^(1/2)-48*x)/(334+2*177 ^(1/2))^(1/2))/(59118+354*177^(1/2))^(1/2)-1/76464*(177^(1/2)*(36*A+24*B+8 *C-5*D)-59*D)*ln(16-(1-177^(1/2))*(2-3*x)+8*(2-3*x)^2)+1/76464*(177^(1/2)* (36*A+24*B+8*C-5*D)+59*D)*ln(16-(1+177^(1/2))*(2-3*x)+8*(2-3*x)^2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.13 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.34 \[ \int \frac {A+B x+C x^2+D x^3}{88-402 x+855 x^2-837 x^3+324 x^4} \, dx=\frac {1}{3} \text {RootSum}\left [88-402 \text {$\#$1}+855 \text {$\#$1}^2-837 \text {$\#$1}^3+324 \text {$\#$1}^4\&,\frac {A \log (x-\text {$\#$1})+B \log (x-\text {$\#$1}) \text {$\#$1}+C \log (x-\text {$\#$1}) \text {$\#$1}^2+D \log (x-\text {$\#$1}) \text {$\#$1}^3}{-134+570 \text {$\#$1}-837 \text {$\#$1}^2+432 \text {$\#$1}^3}\&\right ] \] Input:
Integrate[(A + B*x + C*x^2 + D*x^3)/(88 - 402*x + 855*x^2 - 837*x^3 + 324* x^4),x]
Output:
RootSum[88 - 402*#1 + 855*#1^2 - 837*#1^3 + 324*#1^4 & , (A*Log[x - #1] + B*Log[x - #1]*#1 + C*Log[x - #1]*#1^2 + D*Log[x - #1]*#1^3)/(-134 + 570*#1 - 837*#1^2 + 432*#1^3) & ]/3
Time = 1.90 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.17, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2494, 2492, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{324 x^4-837 x^3+855 x^2-402 x+88} \, dx\) |
\(\Big \downarrow \) 2494 |
\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{324 \left (x-\frac {2}{3}\right )^4+27 \left (x-\frac {2}{3}\right )^3+45 \left (x-\frac {2}{3}\right )^2+6 \left (x-\frac {2}{3}\right )+16}d\left (x-\frac {2}{3}\right )\) |
\(\Big \downarrow \) 2492 |
\(\displaystyle \frac {1}{324} \int \left (\frac {2 \sqrt {\frac {3}{59}} \left (27 \left (1+\sqrt {177}\right ) A-18 \left (7-\sqrt {177}\right ) B-12 \left (15-\sqrt {177}\right ) C-8 \left (19-\sqrt {177}\right ) D+6 \left (108 A+72 B+24 C+\sqrt {177} D-15 D\right ) \left (x-\frac {2}{3}\right )\right )}{72 \left (x-\frac {2}{3}\right )^2+3 \left (1+\sqrt {177}\right ) \left (x-\frac {2}{3}\right )+16}-\frac {2 \sqrt {\frac {3}{59}} \left (27 \left (1-\sqrt {177}\right ) A-18 \left (7+\sqrt {177}\right ) B-12 \left (15+\sqrt {177}\right ) C-8 \left (19+\sqrt {177}\right ) D+6 \left (108 A+72 B+24 C-\sqrt {177} D-15 D\right ) \left (x-\frac {2}{3}\right )\right )}{72 \left (x-\frac {2}{3}\right )^2+3 \left (1-\sqrt {177}\right ) \left (x-\frac {2}{3}\right )+16}\right )d\left (x-\frac {2}{3}\right )\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{324} \left (-\frac {\arctan \left (\frac {48 \left (x-\frac {2}{3}\right )-\sqrt {177}+1}{\sqrt {2 \left (167+\sqrt {177}\right )}}\right ) \left (54 \left (1-\sqrt {177}\right ) A-36 \left (15+\sqrt {177}\right ) B-36 \sqrt {177} C-732 C-39 \sqrt {177} D-689 D\right )}{\sqrt {354 \left (167+\sqrt {177}\right )}}+\frac {\arctan \left (\frac {48 \left (x-\frac {2}{3}\right )+\sqrt {177}+1}{\sqrt {2 \left (167-\sqrt {177}\right )}}\right ) \left (54 \left (1+\sqrt {177}\right ) A-36 \left (15-\sqrt {177}\right ) B+36 \sqrt {177} C-732 C+39 \sqrt {177} D-689 D\right )}{\sqrt {354 \left (167-\sqrt {177}\right )}}+\frac {1}{236} \log \left (72 \left (x-\frac {2}{3}\right )^2+3 \left (1-\sqrt {177}\right ) \left (x-\frac {2}{3}\right )+16\right ) \left (59 D-\sqrt {59} \left (36 \sqrt {3} A+24 \sqrt {3} B+8 \sqrt {3} C-5 \sqrt {3} D\right )\right )+\frac {1}{236} \log \left (72 \left (x-\frac {2}{3}\right )^2+3 \left (1+\sqrt {177}\right ) \left (x-\frac {2}{3}\right )+16\right ) \left (\sqrt {59} \left (36 \sqrt {3} A+24 \sqrt {3} B+8 \sqrt {3} C-5 \sqrt {3} D\right )+59 D\right )\right )\) |
Input:
Int[(A + B*x + C*x^2 + D*x^3)/(88 - 402*x + 855*x^2 - 837*x^3 + 324*x^4),x ]
Output:
(-(((54*(1 - Sqrt[177])*A - 36*(15 + Sqrt[177])*B - 732*C - 36*Sqrt[177]*C - 689*D - 39*Sqrt[177]*D)*ArcTan[(1 - Sqrt[177] + 48*(-2/3 + x))/Sqrt[2*( 167 + Sqrt[177])]])/Sqrt[354*(167 + Sqrt[177])]) + ((54*(1 + Sqrt[177])*A - 36*(15 - Sqrt[177])*B - 732*C + 36*Sqrt[177]*C - 689*D + 39*Sqrt[177]*D) *ArcTan[(1 + Sqrt[177] + 48*(-2/3 + x))/Sqrt[2*(167 - Sqrt[177])]])/Sqrt[3 54*(167 - Sqrt[177])] + ((59*D - Sqrt[59]*(36*Sqrt[3]*A + 24*Sqrt[3]*B + 8 *Sqrt[3]*C - 5*Sqrt[3]*D))*Log[16 + 3*(1 - Sqrt[177])*(-2/3 + x) + 72*(-2/ 3 + x)^2])/236 + ((59*D + Sqrt[59]*(36*Sqrt[3]*A + 24*Sqrt[3]*B + 8*Sqrt[3 ]*C - 5*Sqrt[3]*D))*Log[16 + 3*(1 + Sqrt[177])*(-2/3 + x) + 72*(-2/3 + x)^ 2])/236)/324
Int[(Px_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4) ^(p_), x_Symbol] :> Simp[e^p Int[ExpandIntegrand[Px*(b/d + ((d + Sqrt[e*( (b^2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p*(b/d + ((d - Sqrt[e*((b^ 2 - 4*a*c)/a) + 8*a*d*(e/b)])/(2*e))*x + x^2)^p, x], x], x] /; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && ILtQ[p, 0] && EqQ[a*d^2 - b^2*e, 0]
Int[(Px_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2 + (d_.)*(x_)^3 + (e_.)*(x_)^4) ^(p_), x_Symbol] :> With[{S = Root[a*d^2 - b^2*e + (b*d^2 - 4*b*c*e + 8*a*d *e)*x + (c*d^2 - 4*c^2*e + 2*b*d*e + 16*a*e^2)*x^2 + (d^3 - 4*c*d*e + 8*b*e ^2)*x^3, 3]}, Subst[Int[(Px /. x -> x + S)*ExpandToSum[a + b*(x + S) + c*(x + S)^2 + d*(x + S)^3 + e*(x + S)^4, x]^p, x], x, x - S] /; RationalQ[S]] / ; FreeQ[{a, b, c, d, e}, x] && PolyQ[Px, x] && ILtQ[p, 0] && RationalQ[a, b , c, d, e] && NeQ[a*d^2 - b^2*e, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.14 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.23
method | result | size |
default | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (324 \textit {\_Z}^{4}-837 \textit {\_Z}^{3}+855 \textit {\_Z}^{2}-402 \textit {\_Z} +88\right )}{\sum }\frac {\left (D \textit {\_R}^{3}+C \,\textit {\_R}^{2}+B \textit {\_R} +A \right ) \ln \left (x -\textit {\_R} \right )}{432 \textit {\_R}^{3}-837 \textit {\_R}^{2}+570 \textit {\_R} -134}\right )}{3}\) | \(66\) |
Input:
int((D*x^3+C*x^2+B*x+A)/(324*x^4-837*x^3+855*x^2-402*x+88),x,method=_RETUR NVERBOSE)
Output:
1/3*sum((D*_R^3+C*_R^2+B*_R+A)/(432*_R^3-837*_R^2+570*_R-134)*ln(x-_R),_R= RootOf(324*_Z^4-837*_Z^3+855*_Z^2-402*_Z+88))
Result contains complex when optimal does not.
Time = 10.78 (sec) , antiderivative size = 778293, normalized size of antiderivative = 2693.06 \[ \int \frac {A+B x+C x^2+D x^3}{88-402 x+855 x^2-837 x^3+324 x^4} \, dx=\text {Too large to display} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(324*x^4-837*x^3+855*x^2-402*x+88),x, algori thm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{88-402 x+855 x^2-837 x^3+324 x^4} \, dx=\text {Timed out} \] Input:
integrate((D*x**3+C*x**2+B*x+A)/(324*x**4-837*x**3+855*x**2-402*x+88),x)
Output:
Timed out
\[ \int \frac {A+B x+C x^2+D x^3}{88-402 x+855 x^2-837 x^3+324 x^4} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{324 \, x^{4} - 837 \, x^{3} + 855 \, x^{2} - 402 \, x + 88} \,d x } \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(324*x^4-837*x^3+855*x^2-402*x+88),x, algori thm="maxima")
Output:
integrate((D*x^3 + C*x^2 + B*x + A)/(324*x^4 - 837*x^3 + 855*x^2 - 402*x + 88), x)
Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{88-402 x+855 x^2-837 x^3+324 x^4} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((D*x^3+C*x^2+B*x+A)/(324*x^4-837*x^3+855*x^2-402*x+88),x, algori thm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{88-402 x+855 x^2-837 x^3+324 x^4} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{324\,x^4-837\,x^3+855\,x^2-402\,x+88} \,d x \] Input:
int((A + B*x + C*x^2 + x^3*D)/(855*x^2 - 402*x - 837*x^3 + 324*x^4 + 88),x )
Output:
int((A + B*x + C*x^2 + x^3*D)/(855*x^2 - 402*x - 837*x^3 + 324*x^4 + 88), x)
\[ \int \frac {A+B x+C x^2+D x^3}{88-402 x+855 x^2-837 x^3+324 x^4} \, dx=\left (\int \frac {x^{2}}{324 x^{4}-837 x^{3}+855 x^{2}-402 x +88}d x \right ) c +\frac {31 \left (\int \frac {x^{2}}{324 x^{4}-837 x^{3}+855 x^{2}-402 x +88}d x \right ) d}{16}+\left (\int \frac {x}{324 x^{4}-837 x^{3}+855 x^{2}-402 x +88}d x \right ) b -\frac {95 \left (\int \frac {x}{324 x^{4}-837 x^{3}+855 x^{2}-402 x +88}d x \right ) d}{72}+\left (\int \frac {1}{324 x^{4}-837 x^{3}+855 x^{2}-402 x +88}d x \right ) a +\frac {67 \left (\int \frac {1}{324 x^{4}-837 x^{3}+855 x^{2}-402 x +88}d x \right ) d}{216}+\frac {\mathrm {log}\left (324 x^{4}-837 x^{3}+855 x^{2}-402 x +88\right ) d}{1296} \] Input:
int((D*x^3+C*x^2+B*x+A)/(324*x^4-837*x^3+855*x^2-402*x+88),x)
Output:
(1296*int(x**2/(324*x**4 - 837*x**3 + 855*x**2 - 402*x + 88),x)*c + 2511*i nt(x**2/(324*x**4 - 837*x**3 + 855*x**2 - 402*x + 88),x)*d + 1296*int(x/(3 24*x**4 - 837*x**3 + 855*x**2 - 402*x + 88),x)*b - 1710*int(x/(324*x**4 - 837*x**3 + 855*x**2 - 402*x + 88),x)*d + 1296*int(1/(324*x**4 - 837*x**3 + 855*x**2 - 402*x + 88),x)*a + 402*int(1/(324*x**4 - 837*x**3 + 855*x**2 - 402*x + 88),x)*d + log(324*x**4 - 837*x**3 + 855*x**2 - 402*x + 88)*d)/12 96