Integrand size = 30, antiderivative size = 319 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{a+b x^4} \, dx=\frac {g x}{b}+\frac {d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}-\frac {\left (b c+\sqrt {a} \sqrt {b} e-a g\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{5/4}}+\frac {\left (b c+\sqrt {a} \sqrt {b} e-a g\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{5/4}}-\frac {\left (b c-\sqrt {a} \sqrt {b} e-a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{5/4}}+\frac {\left (b c-\sqrt {a} \sqrt {b} e-a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{5/4}}+\frac {f \log \left (a+b x^4\right )}{4 b} \]
g*x/b+1/4*f*ln(b*x^4+a)/b+1/2*d*arctan(x^2*b^(1/2)/a^(1/2))/a^(1/2)/b^(1/2 )-1/8*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(b*c-a*g-e*a^(1/2 )*b^(1/2))/a^(3/4)/b^(5/4)*2^(1/2)+1/8*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2 )+x^2*b^(1/2))*(b*c-a*g-e*a^(1/2)*b^(1/2))/a^(3/4)/b^(5/4)*2^(1/2)+1/4*arc tan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))*(b*c-a*g+e*a^(1/2)*b^(1/2))/a^(3/4)/b^(5 /4)*2^(1/2)+1/4*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))*(b*c-a*g+e*a^(1/2)*b^( 1/2))/a^(3/4)/b^(5/4)*2^(1/2)
Time = 0.34 (sec) , antiderivative size = 311, normalized size of antiderivative = 0.97 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{a+b x^4} \, dx=\frac {8 a^{3/4} \sqrt [4]{b} g x-2 \left (\sqrt {2} b c+2 \sqrt [4]{a} b^{3/4} d+\sqrt {2} \sqrt {a} \sqrt {b} e-\sqrt {2} a g\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (\sqrt {2} b c-2 \sqrt [4]{a} b^{3/4} d+\sqrt {2} \sqrt {a} \sqrt {b} e-\sqrt {2} a g\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt {2} \left (-b c+\sqrt {a} \sqrt {b} e+a g\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+\sqrt {2} \left (b c-\sqrt {a} \sqrt {b} e-a g\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+2 a^{3/4} \sqrt [4]{b} f \log \left (a+b x^4\right )}{8 a^{3/4} b^{5/4}} \]
(8*a^(3/4)*b^(1/4)*g*x - 2*(Sqrt[2]*b*c + 2*a^(1/4)*b^(3/4)*d + Sqrt[2]*Sq rt[a]*Sqrt[b]*e - Sqrt[2]*a*g)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2 *(Sqrt[2]*b*c - 2*a^(1/4)*b^(3/4)*d + Sqrt[2]*Sqrt[a]*Sqrt[b]*e - Sqrt[2]* a*g)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*(-(b*c) + Sqrt[a]*S qrt[b]*e + a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + S qrt[2]*(b*c - Sqrt[a]*Sqrt[b]*e - a*g)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/ 4)*x + Sqrt[b]*x^2] + 2*a^(3/4)*b^(1/4)*f*Log[a + b*x^4])/(8*a^(3/4)*b^(5/ 4))
Time = 0.52 (sec) , antiderivative size = 319, 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 = {2424, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4}{a+b x^4} \, dx\) |
| \(\Big \downarrow \) 2424 |
| \(\displaystyle \int \left (\frac {c+e x^2+g x^4}{a+b x^4}+\frac {x \left (d+f x^2\right )}{a+b x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} \sqrt {b} e-a g+b c\right )}{2 \sqrt {2} a^{3/4} b^{5/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} \sqrt {b} e-a g+b c\right )}{2 \sqrt {2} a^{3/4} b^{5/4}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-\sqrt {a} \sqrt {b} e-a g+b c\right )}{4 \sqrt {2} a^{3/4} b^{5/4}}+\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (-\sqrt {a} \sqrt {b} e-a g+b c\right )}{4 \sqrt {2} a^{3/4} b^{5/4}}+\frac {d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \sqrt {b}}+\frac {f \log \left (a+b x^4\right )}{4 b}+\frac {g x}{b}\) |
(g*x)/b + (d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(2*Sqrt[a]*Sqrt[b]) - ((b*c + Sqrt[a]*Sqrt[b]*e - a*g)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[ 2]*a^(3/4)*b^(5/4)) + ((b*c + Sqrt[a]*Sqrt[b]*e - a*g)*ArcTan[1 + (Sqrt[2] *b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(5/4)) - ((b*c - Sqrt[a]*Sqrt[b ]*e - a*g)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt [2]*a^(3/4)*b^(5/4)) + ((b*c - Sqrt[a]*Sqrt[b]*e - a*g)*Log[Sqrt[a] + Sqrt [2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(5/4)) + (f*Log [a + b*x^4])/(4*b)
3.2.75.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2 *((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && !PolyQ[Pq, x^(n/2)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.50 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.18
| method | result | size |
| risch | \(\frac {g x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{3} b f +\textit {\_R}^{2} b e +\textit {\_R} b d -a g +b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b^{2}}\) | \(58\) |
| default | \(\frac {g x}{b}+\frac {\frac {\left (-a g +b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {b d \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}+\frac {e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}}}+\frac {f \ln \left (b \,x^{4}+a \right )}{4}}{b}\) | \(253\) |
Result contains complex when optimal does not.
Time = 43.66 (sec) , antiderivative size = 622377, normalized size of antiderivative = 1951.03 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{a+b x^4} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{a+b x^4} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.03 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{a+b x^4} \, dx=\frac {g x}{b} + \frac {\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} f + b^{2} c - \sqrt {a} b^{\frac {3}{2}} e - a b g\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} f - b^{2} c + \sqrt {a} b^{\frac {3}{2}} e + a b g\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g - 2 \, \sqrt {a} b^{2} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}} + \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} b^{\frac {9}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {7}{4}} e - \sqrt {2} a^{\frac {5}{4}} b^{\frac {5}{4}} g + 2 \, \sqrt {a} b^{2} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {5}{4}}}}{8 \, b} \]
g*x/b + 1/8*(sqrt(2)*(sqrt(2)*a^(3/4)*b^(5/4)*f + b^2*c - sqrt(a)*b^(3/2)* e - a*b*g)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4) *b^(5/4)) + sqrt(2)*(sqrt(2)*a^(3/4)*b^(5/4)*f - b^2*c + sqrt(a)*b^(3/2)*e + a*b*g)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)* b^(5/4)) + 2*(sqrt(2)*a^(1/4)*b^(9/4)*c + sqrt(2)*a^(3/4)*b^(7/4)*e - sqrt (2)*a^(5/4)*b^(5/4)*g - 2*sqrt(a)*b^2*d)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqr t(b))*b^(5/4)) + 2*(sqrt(2)*a^(1/4)*b^(9/4)*c + sqrt(2)*a^(3/4)*b^(7/4)*e - sqrt(2)*a^(5/4)*b^(5/4)*g + 2*sqrt(a)*b^2*d)*arctan(1/2*sqrt(2)*(2*sqrt( b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt( a)*sqrt(b))*b^(5/4)))/b
Time = 0.48 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.06 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{a+b x^4} \, dx=\frac {g x}{b} + \frac {f \log \left ({\left | b x^{4} + a \right |}\right )}{4 \, b} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{2} d - \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac {1}{4}} a b g - \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{3}} - \frac {\sqrt {2} {\left (\sqrt {2} \sqrt {a b} b^{2} d - \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + \left (a b^{3}\right )^{\frac {1}{4}} a b g - \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, a b^{3}} + \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac {1}{4}} a b g - \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{3}} - \frac {\sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - \left (a b^{3}\right )^{\frac {1}{4}} a b g - \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{8 \, a b^{3}} \]
g*x/b + 1/4*f*log(abs(b*x^4 + a))/b - 1/4*sqrt(2)*(sqrt(2)*sqrt(a*b)*b^2*d - (a*b^3)^(1/4)*b^2*c + (a*b^3)^(1/4)*a*b*g - (a*b^3)^(3/4)*e)*arctan(1/2 *sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b^3) - 1/4*sqrt(2)*(s qrt(2)*sqrt(a*b)*b^2*d - (a*b^3)^(1/4)*b^2*c + (a*b^3)^(1/4)*a*b*g - (a*b^ 3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a *b^3) + 1/8*sqrt(2)*((a*b^3)^(1/4)*b^2*c - (a*b^3)^(1/4)*a*b*g - (a*b^3)^( 3/4)*e)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^3) - 1/8*sqrt(2) *((a*b^3)^(1/4)*b^2*c - (a*b^3)^(1/4)*a*b*g - (a*b^3)^(3/4)*e)*log(x^2 - s qrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a*b^3)
Time = 9.95 (sec) , antiderivative size = 5042, normalized size of antiderivative = 15.81 \[ \int \frac {c+d x+e x^2+f x^3+g x^4}{a+b x^4} \, dx=\text {Too large to display} \]
symsum(log(b^2*c*d^2 - b^2*c^2*e - a^2*e*g^2 + a^2*f^2*g + b^2*d^3*x - a*b *e^3 - a*b*c*f^2 - a*b*d^2*g - 16*root(256*a^3*b^5*z^4 - 256*a^3*b^4*f*z^3 - 64*a^3*b^3*e*g*z^2 + 64*a^2*b^4*c*e*z^2 + 96*a^3*b^3*f^2*z^2 + 32*a^2*b ^4*d^2*z^2 + 32*a^3*b^2*e*f*g*z - 32*a^2*b^3*c*e*f*z + 32*a^2*b^3*c*d*g*z - 16*a^3*b^2*d*g^2*z - 16*a^2*b^3*d^2*f*z + 16*a^2*b^3*d*e^2*z - 16*a*b^4* c^2*d*z - 16*a^3*b^2*f^3*z - 8*a^2*b^2*c*d*f*g + 4*a^2*b^2*d^2*e*g - 4*a^2 *b^2*d*e^2*f - 4*a^2*b^2*c*e^2*g + 4*a^2*b^2*c*e*f^2 - 4*a^3*b*e*f^2*g + 4 *a^3*b*d*f*g^2 + 4*a*b^3*c^2*d*f - 4*a*b^3*c*d^2*e - 4*a^3*b*c*g^3 - 4*a*b ^3*c^3*g + 6*a^2*b^2*c^2*g^2 + 2*a^2*b^2*d^2*f^2 + 2*a^3*b*e^2*g^2 + 2*a*b ^3*c^2*e^2 + a^2*b^2*e^4 + a^3*b*f^4 + a*b^3*d^4 + a^4*g^4 + b^4*c^4, z, k )^2*a*b^3*c - 4*root(256*a^3*b^5*z^4 - 256*a^3*b^4*f*z^3 - 64*a^3*b^3*e*g* z^2 + 64*a^2*b^4*c*e*z^2 + 96*a^3*b^3*f^2*z^2 + 32*a^2*b^4*d^2*z^2 + 32*a^ 3*b^2*e*f*g*z - 32*a^2*b^3*c*e*f*z + 32*a^2*b^3*c*d*g*z - 16*a^3*b^2*d*g^2 *z - 16*a^2*b^3*d^2*f*z + 16*a^2*b^3*d*e^2*z - 16*a*b^4*c^2*d*z - 16*a^3*b ^2*f^3*z - 8*a^2*b^2*c*d*f*g + 4*a^2*b^2*d^2*e*g - 4*a^2*b^2*d*e^2*f - 4*a ^2*b^2*c*e^2*g + 4*a^2*b^2*c*e*f^2 - 4*a^3*b*e*f^2*g + 4*a^3*b*d*f*g^2 + 4 *a*b^3*c^2*d*f - 4*a*b^3*c*d^2*e - 4*a^3*b*c*g^3 - 4*a*b^3*c^3*g + 6*a^2*b ^2*c^2*g^2 + 2*a^2*b^2*d^2*f^2 + 2*a^3*b*e^2*g^2 + 2*a*b^3*c^2*e^2 + a^2*b ^2*e^4 + a^3*b*f^4 + a*b^3*d^4 + a^4*g^4 + b^4*c^4, z, k)*b^3*c^2*x + b^2* c^2*f*x + a^2*f*g^2*x + 16*root(256*a^3*b^5*z^4 - 256*a^3*b^4*f*z^3 - 6...