Integrand size = 30, antiderivative size = 369 \[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x}{c^3}+\frac {(c e-b f) x^3}{3 c^2}+\frac {f x^5}{5 c}+\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)-\frac {b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{7/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\left (b^2 c e-a c^2 e-b^3 f-b c (c d-2 a f)+\frac {b^3 c e-3 a b c^2 e-b^4 f-b^2 c (c d-4 a f)+2 a c^2 (c d-a f)}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{7/2} \sqrt {b+\sqrt {b^2-4 a c}}} \]
(c^2*d+b^2*f-c*(a*f+b*e))*x/c^3+1/3*(-b*f+c*e)*x^3/c^2+1/5*f*x^5/c+1/2*arc tan(x*2^(1/2)*c^(1/2)/(b-(-4*a*c+b^2)^(1/2))^(1/2))*(b^2*c*e-a*c^2*e-b^3*f -b*c*(-2*a*f+c*d)+(-b^3*c*e+3*a*b*c^2*e+b^4*f+b^2*c*(-4*a*f+c*d)-2*a*c^2*( -a*f+c*d))/(-4*a*c+b^2)^(1/2))/c^(7/2)*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))*(b^2*c*e-a*c^ 2*e-b^3*f-b*c*(-2*a*f+c*d)+(b^3*c*e-3*a*b*c^2*e-b^4*f-b^2*c*(-4*a*f+c*d)+2 *a*c^2*(-a*f+c*d))/(-4*a*c+b^2)^(1/2))/c^(7/2)*2^(1/2)/(b+(-4*a*c+b^2)^(1/ 2))^(1/2)
Time = 0.36 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.24 \[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\frac {\left (c^2 d+b^2 f-c (b e+a f)\right ) x}{c^3}+\frac {(c e-b f) x^3}{3 c^2}+\frac {f x^5}{5 c}-\frac {\left (-b^4 f-b^2 c \left (c d+\sqrt {b^2-4 a c} e-4 a f\right )+a c^2 \left (2 c d+\sqrt {b^2-4 a c} e-2 a f\right )+b^3 \left (c e+\sqrt {b^2-4 a c} f\right )+b c \left (c \sqrt {b^2-4 a c} d-3 a c e-2 a \sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{7/2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}}-\frac {\left (b^4 f+b^2 c \left (c d-\sqrt {b^2-4 a c} e-4 a f\right )+a c^2 \left (-2 c d+\sqrt {b^2-4 a c} e+2 a f\right )+b^3 \left (-c e+\sqrt {b^2-4 a c} f\right )+b c \left (c \sqrt {b^2-4 a c} d+3 a c e-2 a \sqrt {b^2-4 a c} f\right )\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{\sqrt {2} c^{7/2} \sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}}} \]
((c^2*d + b^2*f - c*(b*e + a*f))*x)/c^3 + ((c*e - b*f)*x^3)/(3*c^2) + (f*x ^5)/(5*c) - ((-(b^4*f) - b^2*c*(c*d + Sqrt[b^2 - 4*a*c]*e - 4*a*f) + a*c^2 *(2*c*d + Sqrt[b^2 - 4*a*c]*e - 2*a*f) + b^3*(c*e + Sqrt[b^2 - 4*a*c]*f) + b*c*(c*Sqrt[b^2 - 4*a*c]*d - 3*a*c*e - 2*a*Sqrt[b^2 - 4*a*c]*f))*ArcTan[( Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) - ((b^4*f + b^2*c*(c*d - Sqrt[b^2 - 4*a*c]*e - 4*a*f) + a*c^2*(-2*c*d + Sqrt[b^2 - 4*a*c]*e + 2*a*f) + b^3*(- (c*e) + Sqrt[b^2 - 4*a*c]*f) + b*c*(c*Sqrt[b^2 - 4*a*c]*d + 3*a*c*e - 2*a* Sqrt[b^2 - 4*a*c]*f))*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + Sqrt[b^2 - 4*a*c ]]])/(Sqrt[2]*c^(7/2)*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])
Time = 3.02 (sec) , antiderivative size = 369, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {2195, 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.
\(\displaystyle \int \frac {x^4 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx\) |
\(\Big \downarrow \) 2195 |
\(\displaystyle \int \left (\frac {-c (a f+b e)+b^2 f+c^2 d}{c^3}-\frac {a \left (-c (a f+b e)+b^2 f+c^2 d\right )-x^2 \left (-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right )}{c^3 \left (a+b x^2+c x^4\right )}+\frac {x^2 (c e-b f)}{c^2}+\frac {f x^4}{c}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (-\frac {-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^4 (-f)+b^3 c e}{\sqrt {b^2-4 a c}}-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right )}{\sqrt {2} c^{7/2} \sqrt {b-\sqrt {b^2-4 a c}}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt {c} x}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {-b^2 c (c d-4 a f)-3 a b c^2 e+2 a c^2 (c d-a f)+b^4 (-f)+b^3 c e}{\sqrt {b^2-4 a c}}-b c (c d-2 a f)-a c^2 e+b^3 (-f)+b^2 c e\right )}{\sqrt {2} c^{7/2} \sqrt {\sqrt {b^2-4 a c}+b}}+\frac {x \left (-c (a f+b e)+b^2 f+c^2 d\right )}{c^3}+\frac {x^3 (c e-b f)}{3 c^2}+\frac {f x^5}{5 c}\) |
((c^2*d + b^2*f - c*(b*e + a*f))*x)/c^3 + ((c*e - b*f)*x^3)/(3*c^2) + (f*x ^5)/(5*c) + ((b^2*c*e - a*c^2*e - b^3*f - b*c*(c*d - 2*a*f) - (b^3*c*e - 3 *a*b*c^2*e - b^4*f - b^2*c*(c*d - 4*a*f) + 2*a*c^2*(c*d - a*f))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2] *c^(7/2)*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + ((b^2*c*e - a*c^2*e - b^3*f - b*c* (c*d - 2*a*f) + (b^3*c*e - 3*a*b*c^2*e - b^4*f - b^2*c*(c*d - 4*a*f) + 2*a *c^2*(c*d - a*f))/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x)/Sqrt[b + S qrt[b^2 - 4*a*c]]])/(Sqrt[2]*c^(7/2)*Sqrt[b + Sqrt[b^2 - 4*a*c]])
3.1.55.3.1 Defintions of rubi rules used
Int[(Pq_)*((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_ Symbol] :> Int[ExpandIntegrand[(d*x)^m*Pq*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && PolyQ[Pq, x^2] && IGtQ[p, -2]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 164, normalized size of antiderivative = 0.44
method | result | size |
risch | \(\frac {f \,x^{5}}{5 c}-\frac {b f \,x^{3}}{3 c^{2}}+\frac {e \,x^{3}}{3 c}-\frac {a f x}{c^{2}}+\frac {b^{2} f x}{c^{3}}-\frac {b e x}{c^{2}}+\frac {d x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+\textit {\_Z}^{2} b +a \right )}{\sum }\frac {\left (\left (2 a b c f -a \,c^{2} e -b^{3} f +b^{2} c e -b \,c^{2} d \right ) \textit {\_R}^{2}+a^{2} c f -a \,b^{2} f +a b c e -a \,c^{2} d \right ) \ln \left (x -\textit {\_R} \right )}{2 c \,\textit {\_R}^{3}+\textit {\_R} b}}{2 c^{3}}\) | \(164\) |
default | \(-\frac {-\frac {1}{5} f \,x^{5} c^{2}+\frac {1}{3} b c f \,x^{3}-\frac {1}{3} c^{2} e \,x^{3}+a c f x -b^{2} f x +b c e x -c^{2} d x}{c^{3}}+\frac {\frac {\left (2 \sqrt {-4 a c +b^{2}}\, a b c f -\sqrt {-4 a c +b^{2}}\, a \,c^{2} e -b^{3} f \sqrt {-4 a c +b^{2}}+b^{2} c e \sqrt {-4 a c +b^{2}}-b \,c^{2} d \sqrt {-4 a c +b^{2}}-2 a^{2} c^{2} f +4 a \,b^{2} c f -3 a b \,c^{2} e +2 a \,c^{3} d -b^{4} f +b^{3} c e -b^{2} c^{2} d \right ) \sqrt {2}\, \arctan \left (\frac {c x \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}-\frac {\left (2 \sqrt {-4 a c +b^{2}}\, a b c f -\sqrt {-4 a c +b^{2}}\, a \,c^{2} e -b^{3} f \sqrt {-4 a c +b^{2}}+b^{2} c e \sqrt {-4 a c +b^{2}}-b \,c^{2} d \sqrt {-4 a c +b^{2}}+2 a^{2} c^{2} f -4 a \,b^{2} c f +3 a b \,c^{2} e -2 a \,c^{3} d +b^{4} f -b^{3} c e +b^{2} c^{2} d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c x \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{2 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}}{c^{2}}\) | \(453\) |
1/5*f*x^5/c-1/3/c^2*b*f*x^3+1/3*e*x^3/c-1/c^2*a*f*x+1/c^3*b^2*f*x-1/c^2*b* e*x+1/c*d*x+1/2/c^3*sum(((2*a*b*c*f-a*c^2*e-b^3*f+b^2*c*e-b*c^2*d)*_R^2+a^ 2*c*f-a*b^2*f+a*b*c*e-a*c^2*d)/(2*_R^3*c+_R*b)*ln(x-_R),_R=RootOf(_Z^4*c+_ Z^2*b+a))
Leaf count of result is larger than twice the leaf count of optimal. 15467 vs. \(2 (331) = 662\).
Time = 39.65 (sec) , antiderivative size = 15467, normalized size of antiderivative = 41.92 \[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\text {Timed out} \]
\[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\int { \frac {{\left (f x^{4} + e x^{2} + d\right )} x^{4}}{c x^{4} + b x^{2} + a} \,d x } \]
1/15*(3*c^2*f*x^5 + 5*(c^2*e - b*c*f)*x^3 + 15*(c^2*d - b*c*e + (b^2 - a*c )*f)*x)/c^3 + integrate(-(a*c^2*d - a*b*c*e + (b*c^2*d - (b^2*c - a*c^2)*e + (b^3 - 2*a*b*c)*f)*x^2 + (a*b^2 - a^2*c)*f)/(c*x^4 + b*x^2 + a), x)/c^3
Leaf count of result is larger than twice the leaf count of optimal. 7235 vs. \(2 (331) = 662\).
Time = 1.16 (sec) , antiderivative size = 7235, normalized size of antiderivative = 19.61 \[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
-1/8*((2*b^5*c^4 - 16*a*b^3*c^5 + 32*a^2*b*c^6 - sqrt(2)*sqrt(b^2 - 4*a*c) *sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^2 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqr t(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt( b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 - 16*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 - 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqr t(b^2 - 4*a*c)*c)*b^3*c^4 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^ 2 - 4*a*c)*c)*a*b*c^5 - 2*(b^2 - 4*a*c)*b^3*c^4 + 8*(b^2 - 4*a*c)*a*b*c^5) *c^2*d - (2*b^6*c^3 - 18*a*b^4*c^4 + 48*a^2*b^2*c^5 - 32*a^3*c^6 - sqrt(2) *sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^6*c + 9*sqrt(2)*sqrt( b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^4*c^2 + 2*sqrt(2)*sqrt(b^ 2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^5*c^2 - 24*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b^2*c^3 - 10*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^3*c^3 - sqrt(2)*sqrt(b^2 - 4* a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c^3 + 16*sqrt(2)*sqrt(b^2 - 4*a*c )*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^3*c^4 + 8*sqrt(2)*sqrt(b^2 - 4*a*c)*sq rt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*b*c^4 + 5*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt (b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^4 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b *c + sqrt(b^2 - 4*a*c)*c)*a^2*c^5 - 2*(b^2 - 4*a*c)*b^4*c^3 + 10*(b^2 - 4* a*c)*a*b^2*c^4 - 8*(b^2 - 4*a*c)*a^2*c^5)*c^2*e + (2*b^7*c^2 - 20*a*b^5...
Time = 10.50 (sec) , antiderivative size = 23332, normalized size of antiderivative = 63.23 \[ \int \frac {x^4 \left (d+e x^2+f x^4\right )}{a+b x^2+c x^4} \, dx=\text {Too large to display} \]
x^3*(e/(3*c) - (b*f)/(3*c^2)) - x*((b*(e/c - (b*f)/c^2))/c - d/c + (a*f)/c ^2) + atan(((((16*a^3*c^6*f - 16*a^2*c^7*d - 20*a^2*b^2*c^5*f + 4*a*b^2*c^ 6*d - 4*a*b^3*c^5*e + 16*a^2*b*c^6*e + 4*a*b^4*c^4*f)/c^5 - (2*x*(4*b^3*c^ 7 - 16*a*b*c^8)*(-(b^9*f^2 + b^5*c^4*d^2 + b^7*c^2*e^2 + b^6*f^2*(-(4*a*c - b^2)^3)^(1/2) - 7*a*b^3*c^5*d^2 + 12*a^2*b*c^6*d^2 - a*c^5*d^2*(-(4*a*c - b^2)^3)^(1/2) - 9*a*b^5*c^3*e^2 - 20*a^3*b*c^5*e^2 + 28*a^4*b*c^4*f^2 - 2*b^8*c*e*f + 25*a^2*b^3*c^4*e^2 + a^2*c^4*e^2*(-(4*a*c - b^2)^3)^(1/2) + b^2*c^4*d^2*(-(4*a*c - b^2)^3)^(1/2) + 42*a^2*b^5*c^2*f^2 - 63*a^3*b^3*c^3 *f^2 - a^3*c^3*f^2*(-(4*a*c - b^2)^3)^(1/2) + b^4*c^2*e^2*(-(4*a*c - b^2)^ 3)^(1/2) - 11*a*b^7*c*f^2 + 16*a^3*c^6*d*e - 2*b^6*c^3*d*e - 16*a^4*c^5*e* f + 2*b^7*c^2*d*f + 16*a*b^4*c^4*d*e - 18*a*b^5*c^3*d*f - 40*a^3*b*c^5*d*f + 20*a*b^6*c^2*e*f - 2*b^5*c*e*f*(-(4*a*c - b^2)^3)^(1/2) + 6*a^2*b^2*c^2 *f^2*(-(4*a*c - b^2)^3)^(1/2) - 5*a*b^4*c*f^2*(-(4*a*c - b^2)^3)^(1/2) - 3 6*a^2*b^2*c^5*d*e + 50*a^2*b^3*c^4*d*f + 2*a^2*c^4*d*f*(-(4*a*c - b^2)^3)^ (1/2) - 2*b^3*c^3*d*e*(-(4*a*c - b^2)^3)^(1/2) - 66*a^2*b^4*c^3*e*f + 76*a ^3*b^2*c^4*e*f + 2*b^4*c^2*d*f*(-(4*a*c - b^2)^3)^(1/2) - 3*a*b^2*c^3*e^2* (-(4*a*c - b^2)^3)^(1/2) + 4*a*b*c^4*d*e*(-(4*a*c - b^2)^3)^(1/2) - 6*a*b^ 2*c^3*d*f*(-(4*a*c - b^2)^3)^(1/2) + 8*a*b^3*c^2*e*f*(-(4*a*c - b^2)^3)^(1 /2) - 6*a^2*b*c^3*e*f*(-(4*a*c - b^2)^3)^(1/2))/(8*(16*a^2*c^9 + b^4*c^7 - 8*a*b^2*c^8)))^(1/2))/c^5)*(-(b^9*f^2 + b^5*c^4*d^2 + b^7*c^2*e^2 + b^...