Integrand size = 40, antiderivative size = 321 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{a+b x^2+c x^4} \, dx=\frac {h x}{c}+\frac {i x^2}{2 c}+\frac {\left (c f-b h+\frac {2 c^2 d+b^2 h-c (b f+2 a h)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (c f-b h-\frac {2 c^2 d-b c f+b^2 h-2 a c h}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{3/2} \sqrt {b+\sqrt {b^2-4 a c}}}-\frac {\left (2 c^2 e-b c g+b^2 i-2 a c i\right ) \text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right )}{2 c^2 \sqrt {b^2-4 a c}}+\frac {(c g-b i) \log \left (a+b x^2+c x^4\right )}{4 c^2} \]
h*x/c+1/2*i*x^2/c+1/4*(-b*i+c*g)*ln(c*x^4+b*x^2+a)/c^2-1/2*(-2*a*c*i+b^2*i -b*c*g+2*c^2*e)*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))*(c*f-b*h+( 2*c^2*d+b^2*h-c*(2*a*h+b*f))/(-4*a*c+b^2)^(1/2))/c^(3/2)*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))*(c*f-b*h+(2*a*c*h-b^2*h+b*c*f-2*c^2*d)/(-4*a*c+b^2)^(1/2))/c^(3/2)*2^ (1/2)/(b+(-4*a*c+b^2)^(1/2))^(1/2)
Time = 0.39 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.37 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{a+b x^2+c x^4} \, dx=\frac {4 c h x+2 c i x^2+\frac {2 \sqrt {2} \sqrt {c} \left (2 c^2 d+b \left (b-\sqrt {b^2-4 a c}\right ) h+c \left (-b f+\sqrt {b^2-4 a c} f-2 a h\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {2 \sqrt {2} \sqrt {c} \left (2 c^2 d+b \left (b+\sqrt {b^2-4 a c}\right ) h-c \left (b f+\sqrt {b^2-4 a c} f+2 a h\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}}+\frac {\left (2 c^2 e+b \left (b-\sqrt {b^2-4 a c}\right ) i+c \left (-b g+\sqrt {b^2-4 a c} g-2 a i\right )\right ) \log \left (-b+\sqrt {b^2-4 a c}-2 c x^2\right )}{\sqrt {b^2-4 a c}}-\frac {\left (2 c^2 e+b \left (b+\sqrt {b^2-4 a c}\right ) i-c \left (b g+\sqrt {b^2-4 a c} g+2 a i\right )\right ) \log \left (b+\sqrt {b^2-4 a c}+2 c x^2\right )}{\sqrt {b^2-4 a c}}}{4 c^2} \]
(4*c*h*x + 2*c*i*x^2 + (2*Sqrt[2]*Sqrt[c]*(2*c^2*d + b*(b - Sqrt[b^2 - 4*a *c])*h + c*(-(b*f) + Sqrt[b^2 - 4*a*c]*f - 2*a*h))*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]]) - (2*Sqrt[2]*Sqrt[c]*(2*c^2*d + b*(b + Sqrt[b^2 - 4*a*c])*h - c*(b *f + Sqrt[b^2 - 4*a*c]*f + 2*a*h))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqr t[b^2 - 4*a*c]]])/(Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((2*c^ 2*e + b*(b - Sqrt[b^2 - 4*a*c])*i + c*(-(b*g) + Sqrt[b^2 - 4*a*c]*g - 2*a* i))*Log[-b + Sqrt[b^2 - 4*a*c] - 2*c*x^2])/Sqrt[b^2 - 4*a*c] - ((2*c^2*e + b*(b + Sqrt[b^2 - 4*a*c])*i - c*(b*g + Sqrt[b^2 - 4*a*c]*g + 2*a*i))*Log[ b + Sqrt[b^2 - 4*a*c] + 2*c*x^2])/Sqrt[b^2 - 4*a*c])/(4*c^2)
Time = 0.79 (sec) , antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2202, 2194, 2188, 2009, 2205, 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.
| \(\displaystyle \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{a+b x^2+c x^4} \, dx\) |
| \(\Big \downarrow \) 2202 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{c x^4+b x^2+a}dx+\int \frac {x \left (i x^4+g x^2+e\right )}{c x^4+b x^2+a}dx\) |
| \(\Big \downarrow \) 2194 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{c x^4+b x^2+a}dx+\frac {1}{2} \int \frac {i x^4+g x^2+e}{c x^4+b x^2+a}dx^2\) |
| \(\Big \downarrow \) 2188 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{c x^4+b x^2+a}dx+\frac {1}{2} \int \left (\frac {i}{c}+\frac {(c g-b i) x^2+c e-a i}{c \left (c x^4+b x^2+a\right )}\right )dx^2\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \int \frac {h x^4+f x^2+d}{c x^4+b x^2+a}dx+\frac {1}{2} \left (-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {(c g-b i) \log \left (a+b x^2+c x^4\right )}{2 c^2}+\frac {i x^2}{c}\right )\) |
| \(\Big \downarrow \) 2205 |
| \(\displaystyle \int \left (\frac {h}{c}+\frac {(c f-b h) x^2+c d-a h}{c \left (c x^4+b x^2+a\right )}\right )dx+\frac {1}{2} \left (-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {(c g-b i) \log \left (a+b x^2+c x^4\right )}{2 c^2}+\frac {i x^2}{c}\right )\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (\frac {-c (2 a h+b f)+b^2 h+2 c^2 d}{\sqrt {b^2-4 a c}}-b h+c f\right )}{\sqrt {2} c^{3/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (-\frac {-2 a c h+b^2 h-b c f+2 c^2 d}{\sqrt {b^2-4 a c}}-b h+c f\right )}{\sqrt {2} c^{3/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {1}{2} \left (-\frac {\text {arctanh}\left (\frac {b+2 c x^2}{\sqrt {b^2-4 a c}}\right ) \left (-2 a c i+b^2 i-b c g+2 c^2 e\right )}{c^2 \sqrt {b^2-4 a c}}+\frac {(c g-b i) \log \left (a+b x^2+c x^4\right )}{2 c^2}+\frac {i x^2}{c}\right )+\frac {h x}{c}\) |
(h*x)/c + ((c*f - b*h + (2*c^2*d + b^2*h - c*(b*f + 2*a*h))/Sqrt[b^2 - 4*a *c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^( 3/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((c*f - b*h - (2*c^2*d - b*c*f + b^2*h - 2*a*c*h)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^ 2 - 4*a*c]]])/(Sqrt[2]*c^(3/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]]) + ((i*x^2)/c - ((2*c^2*e - b*c*g + b^2*i - 2*a*c*i)*ArcTanh[(b + 2*c*x^2)/Sqrt[b^2 - 4*a *c]])/(c^2*Sqrt[b^2 - 4*a*c]) + ((c*g - b*i)*Log[a + b*x^2 + c*x^4])/(2*c^ 2))/2
3.1.24.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[Expand Integrand[Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Pq , x] && IGtQ[p, -2]
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] : > Simp[1/2 Subst[Int[x^((m - 1)/2)*SubstFor[x^2, Pq, x]*(a + b*x + c*x^2) ^p, x], x, x^2], x] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pq, x^2] && IntegerQ [(m - 1)/2]
Int[(Pn_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Module[{n = Expon[Pn, x], k}, Int[Sum[Coeff[Pn, x, 2*k]*x^(2*k), {k, 0, n/2}]*(a + b *x^2 + c*x^4)^p, x] + Int[x*Sum[Coeff[Pn, x, 2*k + 1]*x^(2*k), {k, 0, (n - 1)/2}]*(a + b*x^2 + c*x^4)^p, x]] /; FreeQ[{a, b, c, p}, x] && PolyQ[Pn, x] && !PolyQ[Pn, x^2]
Int[(Px_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[ExpandInte grand[Px/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c}, x] && PolyQ[Px, x^ 2] && Expon[Px, x^2] > 1
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.10 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.31
| method | result | size |
| risch | \(\frac {h x}{c}+\frac {i \,x^{2}}{2 c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\left (-b i +g c \right ) \textit {\_R}^{3}+\left (-b h +c f \right ) \textit {\_R}^{2}+\left (-a i +e c \right ) \textit {\_R} -a h +c d \right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{2 c}\) | \(99\) |
| default | \(\frac {h x +\frac {1}{2} i \,x^{2}}{c}+\frac {\sqrt {-4 a c +b^{2}}\, \left (\frac {\left (\sqrt {-4 a c +b^{2}}\, b i -\sqrt {-4 a c +b^{2}}\, c g -2 a c i +b^{2} i -g b c +2 e \,c^{2}\right ) \ln \left (2 c \,x^{2}+\sqrt {-4 a c +b^{2}}+b \right )}{4 c}+\frac {\left (\sqrt {-4 a c +b^{2}}\, b h -\sqrt {-4 a c +b^{2}}\, f c -2 a c h +b^{2} h -f b c +2 c^{2} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \left (4 a c -b^{2}\right )}+\frac {\sqrt {-4 a c +b^{2}}\, \left (-\frac {\left (-\sqrt {-4 a c +b^{2}}\, b i +\sqrt {-4 a c +b^{2}}\, c g -2 a c i +b^{2} i -g b c +2 e \,c^{2}\right ) \ln \left (-2 c \,x^{2}+\sqrt {-4 a c +b^{2}}-b \right )}{4 c}+\frac {\left (-\sqrt {-4 a c +b^{2}}\, b h +\sqrt {-4 a c +b^{2}}\, f c -2 a c h +b^{2} h -f b c +2 c^{2} d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{c \left (4 a c -b^{2}\right )}\) | \(408\) |
h*x/c+1/2*i*x^2/c+1/2/c*sum(((-b*i+c*g)*_R^3+(-b*h+c*f)*_R^2+(-a*i+c*e)*_R -a*h+c*d)/(2*_R^3*c+_R*b)*ln(x-_R),_R=RootOf(_Z^4*c+_Z^2*b+a))
Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{a+b x^2+c x^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{a+b x^2+c x^4} \, dx=\text {Timed out} \]
\[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{a+b x^2+c x^4} \, dx=\int { \frac {i x^{5} + h x^{4} + g x^{3} + f x^{2} + e x + d}{c x^{4} + b x^{2} + a} \,d x } \]
1/2*(i*x^2 + 2*h*x)/c - integrate(-((c*g - b*i)*x^3 + (c*f - b*h)*x^2 + c* d - a*h + (c*e - a*i)*x)/(c*x^4 + b*x^2 + a), x)/c
Leaf count of result is larger than twice the leaf count of optimal. 5941 vs. \(2 (277) = 554\).
Time = 1.55 (sec) , antiderivative size = 5941, normalized size of antiderivative = 18.51 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
1/4*(c*g - b*i)*log(abs(c*x^4 + b*x^2 + a))/c^2 + 1/2*(c*i*x^2 + 2*c*h*x)/ c^2 + 1/8*((2*b^4*c^3 - 16*a*b^2*c^4 + 32*a^2*c^5 - sqrt(2)*sqrt(b^2 - 4*a *c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sq rt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt (b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b* c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b ^2 - 4*a*c)*c)*b^2*c^3 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^4 - 2*(b^2 - 4*a*c)*b^2*c^3 + 8*(b^2 - 4*a*c)*a*c^4)*c^2*f - (2*b^5*c^2 - 16*a*b^3*c^3 + 32*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*b^5 + 8*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)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt (b^2 - 4*a*c)*c)*b^4*c - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^2 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a* c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c )*a*b*c^3 - 2*(b^2 - 4*a*c)*b^3*c^2 + 8*(b^2 - 4*a*c)*a*b*c^3)*c^2*h + 2*( sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 - 8*sqrt(2)*sqrt(b*c + sqr t(b^2 - 4*a*c)*c)*a*b^2*c^4 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^ 3*c^4 - 2*b^4*c^4 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^5 ...
Time = 8.74 (sec) , antiderivative size = 11383, normalized size of antiderivative = 35.46 \[ \int \frac {d+e x+f x^2+g x^3+h x^4+i x^5}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
symsum(log((x*(c^4*e^3 - a^3*c*i^3 + c^4*d^2*g + b^4*e*i^2 + a^2*b^2*i^3 + b^2*c^2*e*g^2 + 3*a^2*c^2*e*i^2 + a^2*c^2*g*h^2 + 2*b^2*c^2*e^2*i - a^2*c ^2*g^2*i - 2*c^4*d*e*f - a*b*c^2*g^3 + a*c^3*e*g^2 + b*c^3*e*f^2 - a*c^3*f ^2*g - 2*b*c^3*e^2*g - 3*a*c^3*e^2*i - b*c^3*d^2*i + b^3*c*e*h^2 - a*b^3*g *i^2 - 2*a*b*c^2*e*h^2 - 3*a*b^2*c*e*i^2 - a*b^2*c*g*h^2 + 2*a*b^2*c*g^2*i + a^2*b*c*h^2*i - 2*b^2*c^2*e*f*h - 2*a^2*c^2*f*h*i + 2*b*c^3*d*e*h + 2*a *c^3*d*f*i - 2*a*c^3*d*g*h + 2*a*c^3*e*f*h - 2*b^3*c*e*g*i + 2*a*b*c^2*e*g *i + 2*a*b*c^2*f*g*h))/c^2 - (a*c^3*f^3 - c^4*d*e^2 + c^4*d^2*f - b^4*d*i^ 2 - b^2*c^2*d*g^2 - a^2*c^2*d*i^2 + a^2*c^2*f*h^2 - a^2*c^2*g^2*h - a^2*b^ 2*h*i^2 - a^2*b*c*h^3 + a*c^3*d*g^2 - b*c^3*d*f^2 + a*c^3*e^2*h - b*c^3*d^ 2*h - b^3*c*d*h^2 + a*b^3*f*i^2 + a^3*c*h*i^2 + 2*a*b*c^2*d*h^2 + a*b*c^2* f*g^2 + 3*a*b^2*c*d*i^2 - 2*a*b*c^2*f^2*h + a*b^2*c*f*h^2 - 2*a^2*b*c*f*i^ 2 - 2*b^2*c^2*d*e*i + 2*b^2*c^2*d*f*h - 2*a^2*c^2*e*h*i + 2*a^2*c^2*f*g*i + 2*b*c^3*d*e*g + 2*a*c^3*d*e*i - 2*a*c^3*d*f*h - 2*a*c^3*e*f*g + 2*b^3*c* d*g*i - 4*a*b*c^2*d*g*i + 2*a*b*c^2*e*f*i - 2*a*b^2*c*f*g*i + 2*a^2*b*c*g* h*i)/c^2 - root(128*a^2*b^2*c^5*z^4 - 16*a*b^4*c^4*z^4 - 256*a^3*c^6*z^4 + 128*a^2*b^3*c^3*i*z^3 - 128*a^2*b^2*c^4*g*z^3 - 256*a^3*b*c^4*i*z^3 - 16* a*b^5*c^2*i*z^3 + 16*a*b^4*c^3*g*z^3 + 256*a^3*c^5*g*z^3 + 160*a^3*b*c^3*g *i*z^2 + 8*a*b^4*c^2*f*h*z^2 + 8*a*b^4*c^2*e*i*z^2 + 32*a^2*b*c^4*e*g*z^2 + 32*a^2*b*c^4*d*h*z^2 - 8*a*b^3*c^3*e*g*z^2 - 8*a*b^3*c^3*d*h*z^2 + 16...