Integrand size = 38, antiderivative size = 605 \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\frac {\left (4 a^2 B+b \left (b-\sqrt {8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b-\sqrt {8 a^2+b^2-4 a c}\right )+b C-\sqrt {8 a^2+b^2-4 a c} C+2 c D\right )\right ) \arctan \left (\frac {b-\sqrt {8 a^2+b^2-4 a c}+4 a x}{\sqrt {2} \sqrt {4 a^2+2 a c-b \left (b-\sqrt {8 a^2+b^2-4 a c}\right )}}\right )}{\sqrt {2} a \sqrt {8 a^2+b^2-4 a c} \sqrt {4 a^2+2 a c-b \left (b-\sqrt {8 a^2+b^2-4 a c}\right )}}-\frac {\left (4 a^2 B+b \left (b+\sqrt {8 a^2+b^2-4 a c}\right ) D-a \left (A \left (b+\sqrt {8 a^2+b^2-4 a c}\right )+b C+\sqrt {8 a^2+b^2-4 a c} C+2 c D\right )\right ) \arctan \left (\frac {b+\sqrt {8 a^2+b^2-4 a c}+4 a x}{\sqrt {2} \sqrt {4 a^2+2 a c-b \left (b+\sqrt {8 a^2+b^2-4 a c}\right )}}\right )}{\sqrt {2} a \sqrt {8 a^2+b^2-4 a c} \sqrt {4 a^2+2 a c-b \left (b+\sqrt {8 a^2+b^2-4 a c}\right )}}-\frac {\left (2 a (A-C)+\left (b-\sqrt {8 a^2+b^2-4 a c}\right ) D\right ) \log \left (2 a+\left (b-\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2\right )}{4 a \sqrt {8 a^2+b^2-4 a c}}+\frac {\left (2 a (A-C)+\left (b+\sqrt {8 a^2+b^2-4 a c}\right ) D\right ) \log \left (2 a+\left (b+\sqrt {8 a^2+b^2-4 a c}\right ) x+2 a x^2\right )}{4 a \sqrt {8 a^2+b^2-4 a c}} \]
-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*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+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)
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 \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\text {RootSum}\left [a+b \text {$\#$1}+c \text {$\#$1}^2+b \text {$\#$1}^3+a \text {$\#$1}^4\&,\frac {A \log (x-\text {$\#$1})+B \log (x-\text {$\#$1}) \text {$\#$1}+C \log (x-\text {$\#$1}) \text {$\#$1}^2+D \log (x-\text {$\#$1}) \text {$\#$1}^3}{b+2 c \text {$\#$1}+3 b \text {$\#$1}^2+4 a \text {$\#$1}^3}\&\right ] \]
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) & ]
Time = 3.85 (sec) , antiderivative size = 597, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {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.
| \(\displaystyle \int \frac {A+B x+C x^2+D x^3}{a x^4+a+b x^3+b x+c x^2} \, dx\) |
| \(\Big \downarrow \) 2492 |
| \(\displaystyle \frac {\int \left (\frac {a \left (A \left (b+\sqrt {8 a^2-4 c a+b^2}\right )-2 a (B-D)+\left (2 a (A-C)+\left (b+\sqrt {8 a^2-4 c a+b^2}\right ) D\right ) x\right )}{\sqrt {8 a^2-4 c a+b^2} \left (2 a x^2+\left (b+\sqrt {8 a^2-4 c a+b^2}\right ) x+2 a\right )}-\frac {a \left (A \left (b-\sqrt {8 a^2-4 c a+b^2}\right )-2 a (B-D)+\left (2 a A-2 a C+b D-\sqrt {8 a^2-4 c a+b^2} D\right ) x\right )}{\sqrt {8 a^2-4 c a+b^2} \left (2 a x^2+\left (b-\sqrt {8 a^2-4 c a+b^2}\right ) x+2 a\right )}\right )dx}{a}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\arctan \left (\frac {-\sqrt {8 a^2-4 a c+b^2}+4 a x+b}{\sqrt {2} \sqrt {-b \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+4 a^2+2 a c}}\right ) \left (-a \left (A \left (b-\sqrt {8 a^2-4 a c+b^2}\right )-C \sqrt {8 a^2-4 a c+b^2}+b C+2 c D\right )+b D \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+4 a^2 B\right )}{\sqrt {2} \sqrt {8 a^2-4 a c+b^2} \sqrt {-b \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+4 a^2+2 a c}}-\frac {\arctan \left (\frac {\sqrt {8 a^2-4 a c+b^2}+4 a x+b}{\sqrt {2} \sqrt {-b \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+4 a^2+2 a c}}\right ) \left (-a \left (A \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+C \sqrt {8 a^2-4 a c+b^2}+b C+2 c D\right )+b D \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+4 a^2 B\right )}{\sqrt {2} \sqrt {8 a^2-4 a c+b^2} \sqrt {-b \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+4 a^2+2 a c}}-\frac {\log \left (x \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+2 a x^2+2 a\right ) \left (D \left (b-\sqrt {8 a^2-4 a c+b^2}\right )+2 a (A-C)\right )}{4 \sqrt {8 a^2-4 a c+b^2}}+\frac {\log \left (x \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+2 a x^2+2 a\right ) \left (D \left (\sqrt {8 a^2-4 a c+b^2}+b\right )+2 a (A-C)\right )}{4 \sqrt {8 a^2-4 a c+b^2}}}{a}\) |
(((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 - Sq rt[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]*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 + 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 + Sqrt[8*a^2 + b^2 - 4*a*c])])])/( Sqrt[2]*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])]) - ((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*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*Sqrt[8*a^2 + b^2 - 4*a*c]))/ a
3.3.27.3.1 Defintions of rubi rules used
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]
Time = 0.16 (sec) , antiderivative size = 535, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(4 a \left (\frac {-\frac {\left (2 A a -2 C a -\sqrt {8 a^{2}-4 a c +b^{2}}\, D+D b \right ) \ln \left (-2 a \,x^{2}+\sqrt {8 a^{2}-4 a c +b^{2}}\, x -b x -2 a \right )}{4 a}+\frac {2 \left (\frac {\left (2 A a -2 C a -\sqrt {8 a^{2}-4 a c +b^{2}}\, D+D b \right ) \left (\sqrt {8 a^{2}-4 a c +b^{2}}-b \right )}{4 a}-\sqrt {8 a^{2}-4 a c +b^{2}}\, A +A b -2 B a +2 D a \right ) \arctan \left (\frac {-4 a x +\sqrt {8 a^{2}-4 a c +b^{2}}-b}{\sqrt {8 a^{2}+4 a c -2 b^{2}+2 \sqrt {8 a^{2}-4 a c +b^{2}}\, b}}\right )}{\sqrt {8 a^{2}+4 a c -2 b^{2}+2 \sqrt {8 a^{2}-4 a c +b^{2}}\, b}}}{4 a \sqrt {8 a^{2}-4 a c +b^{2}}}+\frac {\frac {\left (2 A a -2 C a +\sqrt {8 a^{2}-4 a c +b^{2}}\, D+D b \right ) \ln \left (2 a \,x^{2}+\sqrt {8 a^{2}-4 a c +b^{2}}\, x +b x +2 a \right )}{4 a}+\frac {2 \left (-\frac {\left (2 A a -2 C a +\sqrt {8 a^{2}-4 a c +b^{2}}\, D+D b \right ) \left (b +\sqrt {8 a^{2}-4 a c +b^{2}}\right )}{4 a}+\sqrt {8 a^{2}-4 a c +b^{2}}\, A +A b -2 B a +2 D a \right ) \arctan \left (\frac {b +4 a x +\sqrt {8 a^{2}-4 a c +b^{2}}}{\sqrt {8 a^{2}+4 a c -2 b^{2}-2 \sqrt {8 a^{2}-4 a c +b^{2}}\, b}}\right )}{\sqrt {8 a^{2}+4 a c -2 b^{2}-2 \sqrt {8 a^{2}-4 a c +b^{2}}\, b}}}{4 a \sqrt {8 a^{2}-4 a c +b^{2}}}\right )\) | \(535\) |
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))))
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\text {Timed out} \]
\[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{a x^{4} + b x^{3} + c x^{2} + b x + a} \,d x } \]
Exception generated. \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Not invertible Error: Bad Argument Value
Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{a+b x+c x^2+b x^3+a x^4} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{a\,x^4+b\,x^3+c\,x^2+b\,x+a} \,d x \]