Integrand size = 40, antiderivative size = 395 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^2} \, dx=\frac {x \left (b c-a g+(b d-a h) x+(b e-a i) x^2+b f x^3\right )}{4 a b \left (a+b x^4\right )}+\frac {(b d+a h) \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} b^{3/2}}-\frac {\left (\sqrt {b} (3 b c+a g)+\sqrt {a} (b e+3 a i)\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\left (\sqrt {b} (3 b c+a g)+\sqrt {a} (b e+3 a i)\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{7/4}}-\frac {\left (\sqrt {b} (3 b c+a g)-\sqrt {a} (b e+3 a i)\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{7/4}}+\frac {\left (\sqrt {b} (3 b c+a g)-\sqrt {a} (b e+3 a i)\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{7/4}} \]
1/4*x*(b*c-a*g+(-a*h+b*d)*x+(-a*i+b*e)*x^2+b*f*x^3)/a/b/(b*x^4+a)+1/4*(a*h +b*d)*arctan(x^2*b^(1/2)/a^(1/2))/a^(3/2)/b^(3/2)-1/32*ln(-a^(1/4)*b^(1/4) *x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-(3*a*i+b*e)*a^(1/2)+(a*g+3*b*c)*b^(1/2)) /a^(7/4)/b^(7/4)*2^(1/2)+1/32*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^( 1/2))*(-(3*a*i+b*e)*a^(1/2)+(a*g+3*b*c)*b^(1/2))/a^(7/4)/b^(7/4)*2^(1/2)+1 /16*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))*((3*a*i+b*e)*a^(1/2)+(a*g+3*b*c)* b^(1/2))/a^(7/4)/b^(7/4)*2^(1/2)+1/16*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))* ((3*a*i+b*e)*a^(1/2)+(a*g+3*b*c)*b^(1/2))/a^(7/4)/b^(7/4)*2^(1/2)
Time = 0.36 (sec) , antiderivative size = 415, normalized size of antiderivative = 1.05 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^2} \, dx=\frac {-\frac {8 a^{3/4} b^{3/4} (-b x (c+x (d+e x))+a (f+x (g+x (h+i x))))}{a+b x^4}-2 \left (3 \sqrt {2} b^{3/2} c+4 \sqrt [4]{a} b^{5/4} d+\sqrt {2} \sqrt {a} b e+\sqrt {2} a \sqrt {b} g+4 a^{5/4} \sqrt [4]{b} h+3 \sqrt {2} a^{3/2} i\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \left (3 \sqrt {2} b^{3/2} c-4 \sqrt [4]{a} b^{5/4} d+\sqrt {2} \sqrt {a} b e+\sqrt {2} a \sqrt {b} g-4 a^{5/4} \sqrt [4]{b} h+3 \sqrt {2} a^{3/2} i\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+\sqrt {2} \left (-3 b^{3/2} c+\sqrt {a} b e-a \sqrt {b} g+3 a^{3/2} i\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+\sqrt {2} \left (3 b^{3/2} c-\sqrt {a} b e+a \sqrt {b} g-3 a^{3/2} i\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{32 a^{7/4} b^{7/4}} \]
((-8*a^(3/4)*b^(3/4)*(-(b*x*(c + x*(d + e*x))) + a*(f + x*(g + x*(h + i*x) ))))/(a + b*x^4) - 2*(3*Sqrt[2]*b^(3/2)*c + 4*a^(1/4)*b^(5/4)*d + Sqrt[2]* Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g + 4*a^(5/4)*b^(1/4)*h + 3*Sqrt[2]*a^(3/2 )*i)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*(3*Sqrt[2]*b^(3/2)*c - 4* a^(1/4)*b^(5/4)*d + Sqrt[2]*Sqrt[a]*b*e + Sqrt[2]*a*Sqrt[b]*g - 4*a^(5/4)* b^(1/4)*h + 3*Sqrt[2]*a^(3/2)*i)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + Sqrt[2]*(-3*b^(3/2)*c + Sqrt[a]*b*e - a*Sqrt[b]*g + 3*a^(3/2)*i)*Log[Sqrt [a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(3*b^(3/2)*c - Sq rt[a]*b*e + a*Sqrt[b]*g - 3*a^(3/2)*i)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/ 4)*x + Sqrt[b]*x^2])/(32*a^(7/4)*b^(7/4))
Time = 0.66 (sec) , antiderivative size = 403, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2397, 25, 2415, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 2397 |
| \(\displaystyle \frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{4 a b \left (a+b x^4\right )}-\frac {\int -\frac {b (b e+3 a i) x^2+2 b (b d+a h) x+b (3 b c+a g)}{b x^4+a}dx}{4 a b^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {b (b e+3 a i) x^2+2 b (b d+a h) x+b (3 b c+a g)}{b x^4+a}dx}{4 a b^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{4 a b \left (a+b x^4\right )}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\int \left (\frac {2 b (b d+a h) x}{b x^4+a}+\frac {b (b e+3 a i) x^2+b (3 b c+a g)}{b x^4+a}\right )dx}{4 a b^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{4 a b \left (a+b x^4\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\sqrt [4]{b} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {b} (a g+3 b c)+\sqrt {a} (3 a i+b e)\right )}{2 \sqrt {2} a^{3/4}}+\frac {\sqrt [4]{b} \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {b} (a g+3 b c)+\sqrt {a} (3 a i+b e)\right )}{2 \sqrt {2} a^{3/4}}-\frac {\sqrt [4]{b} \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (a g+3 b c)-\sqrt {a} (3 a i+b e)\right )}{4 \sqrt {2} a^{3/4}}+\frac {\sqrt [4]{b} \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right ) \left (\sqrt {b} (a g+3 b c)-\sqrt {a} (3 a i+b e)\right )}{4 \sqrt {2} a^{3/4}}+\frac {\sqrt {b} \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (a h+b d)}{\sqrt {a}}}{4 a b^2}+\frac {x \left (x (b d-a h)+x^2 (b e-a i)-a g+b c+b f x^3\right )}{4 a b \left (a+b x^4\right )}\) |
(x*(b*c - a*g + (b*d - a*h)*x + (b*e - a*i)*x^2 + b*f*x^3))/(4*a*b*(a + b* x^4)) + ((Sqrt[b]*(b*d + a*h)*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/Sqrt[a] - (b^ (1/4)*(Sqrt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*ArcTan[1 - (Sqrt[2]* b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)) + (b^(1/4)*(Sqrt[b]*(3*b*c + a*g) + Sqrt[a]*(b*e + 3*a*i))*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt [2]*a^(3/4)) - (b^(1/4)*(Sqrt[b]*(3*b*c + a*g) - Sqrt[a]*(b*e + 3*a*i))*Lo g[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)) + (b^(1/4)*(Sqrt[b]*(3*b*c + a*g) - Sqrt[a]*(b*e + 3*a*i))*Log[Sqrt[a] + S qrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)))/(4*a*b^2)
3.2.96.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x]}, S imp[(-x)*R*((a + b*x^n)^(p + 1)/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[(a + b*x^n)^(p + 1)* ExpandToSum[a*n*(p + 1)*Q + n*(p + 1)*R + D[x*R, x], x], x], x]] /; GeQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff [Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1 }]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.52 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.34
| method | result | size |
| risch | \(\frac {-\frac {\left (a i -b e \right ) x^{3}}{4 a b}-\frac {\left (a h -b d \right ) x^{2}}{4 a b}-\frac {\left (a g -b c \right ) x}{4 a b}-\frac {f}{4 b}}{b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\left (3 a i +b e \right ) \textit {\_R}^{2}+2 \left (a h +b d \right ) \textit {\_R} +a g +3 b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 a \,b^{2}}\) | \(133\) |
| default | \(\frac {-\frac {\left (a i -b e \right ) x^{3}}{4 a b}-\frac {\left (a h -b d \right ) x^{2}}{4 a b}-\frac {\left (a g -b c \right ) x}{4 a b}-\frac {f}{4 b}}{b \,x^{4}+a}+\frac {\frac {\left (a g +3 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (2 a h +2 b d \right ) \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{2 \sqrt {a b}}+\frac {\left (3 a i +b e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{4 b a}\) | \(329\) |
(-1/4*(a*i-b*e)/a/b*x^3-1/4*(a*h-b*d)/a/b*x^2-1/4*(a*g-b*c)/a/b*x-1/4*f/b) /(b*x^4+a)+1/16/a/b^2*sum(((3*a*i+b*e)*_R^2+2*(a*h+b*d)*_R+a*g+3*b*c)/_R^3 *ln(x-_R),_R=RootOf(_Z^4*b+a))
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^2} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.05 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^2} \, dx=\frac {{\left (b e - a i\right )} x^{3} + {\left (b d - a h\right )} x^{2} - a f + {\left (b c - a g\right )} x}{4 \, {\left (a b^{2} x^{4} + a^{2} b\right )}} + \frac {\frac {\sqrt {2} {\left (3 \, b^{\frac {3}{2}} c - \sqrt {a} b e + a \sqrt {b} g - 3 \, a^{\frac {3}{2}} i\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (3 \, b^{\frac {3}{2}} c - \sqrt {a} b e + a \sqrt {b} g - 3 \, a^{\frac {3}{2}} i\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g + 3 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} i - 4 \, \sqrt {a} b^{\frac {3}{2}} d - 4 \, a^{\frac {3}{2}} \sqrt {b} h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {7}{4}} c + \sqrt {2} a^{\frac {3}{4}} b^{\frac {5}{4}} e + \sqrt {2} a^{\frac {5}{4}} b^{\frac {3}{4}} g + 3 \, \sqrt {2} a^{\frac {7}{4}} b^{\frac {1}{4}} i + 4 \, \sqrt {a} b^{\frac {3}{2}} d + 4 \, a^{\frac {3}{2}} \sqrt {b} h\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{32 \, a b} \]
1/4*((b*e - a*i)*x^3 + (b*d - a*h)*x^2 - a*f + (b*c - a*g)*x)/(a*b^2*x^4 + a^2*b) + 1/32*(sqrt(2)*(3*b^(3/2)*c - sqrt(a)*b*e + a*sqrt(b)*g - 3*a^(3/ 2)*i)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3 /4)) - sqrt(2)*(3*b^(3/2)*c - sqrt(a)*b*e + a*sqrt(b)*g - 3*a^(3/2)*i)*log (sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) + 2* (3*sqrt(2)*a^(1/4)*b^(7/4)*c + sqrt(2)*a^(3/4)*b^(5/4)*e + sqrt(2)*a^(5/4) *b^(3/4)*g + 3*sqrt(2)*a^(7/4)*b^(1/4)*i - 4*sqrt(a)*b^(3/2)*d - 4*a^(3/2) *sqrt(b)*h)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqr t(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(b))*b^(3/4)) + 2*(3*sqrt(2) *a^(1/4)*b^(7/4)*c + sqrt(2)*a^(3/4)*b^(5/4)*e + sqrt(2)*a^(5/4)*b^(3/4)*g + 3*sqrt(2)*a^(7/4)*b^(1/4)*i + 4*sqrt(a)*b^(3/2)*d + 4*a^(3/2)*sqrt(b)*h )*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^(3/4)))/(a*b)
Time = 0.28 (sec) , antiderivative size = 460, normalized size of antiderivative = 1.16 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^2} \, dx=\frac {b e x^{3} - a i x^{3} + b d x^{2} - a h x^{2} + b c x - a g x - a f}{4 \, {\left (b x^{4} + a\right )} a b} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b^{3} d + 2 \, \sqrt {2} \sqrt {a b} a b^{2} h + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g + \left (a b^{3}\right )^{\frac {3}{4}} b e + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b^{3} d + 2 \, \sqrt {2} \sqrt {a b} a b^{2} h + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g + \left (a b^{3}\right )^{\frac {3}{4}} b e + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b^{4}} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{3} c + \left (a b^{3}\right )^{\frac {1}{4}} a b^{2} g - \left (a b^{3}\right )^{\frac {3}{4}} b e - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a i\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b^{4}} \]
1/4*(b*e*x^3 - a*i*x^3 + b*d*x^2 - a*h*x^2 + b*c*x - a*g*x - a*f)/((b*x^4 + a)*a*b) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(a*b)*b^3*d + 2*sqrt(2)*sqrt(a*b)* a*b^2*h + 3*(a*b^3)^(1/4)*b^3*c + (a*b^3)^(1/4)*a*b^2*g + (a*b^3)^(3/4)*b* e + 3*(a*b^3)^(3/4)*a*i)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a /b)^(1/4))/(a^2*b^4) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(a*b)*b^3*d + 2*sqrt(2) *sqrt(a*b)*a*b^2*h + 3*(a*b^3)^(1/4)*b^3*c + (a*b^3)^(1/4)*a*b^2*g + (a*b^ 3)^(3/4)*b*e + 3*(a*b^3)^(3/4)*a*i)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b )^(1/4))/(a/b)^(1/4))/(a^2*b^4) + 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b^3*c + (a *b^3)^(1/4)*a*b^2*g - (a*b^3)^(3/4)*b*e - 3*(a*b^3)^(3/4)*a*i)*log(x^2 + s qrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^4) - 1/32*sqrt(2)*(3*(a*b^3)^(1/4 )*b^3*c + (a*b^3)^(1/4)*a*b^2*g - (a*b^3)^(3/4)*b*e - 3*(a*b^3)^(3/4)*a*i) *log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^4)
Time = 10.17 (sec) , antiderivative size = 2605, normalized size of antiderivative = 6.59 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{\left (a+b x^4\right )^2} \, dx=\text {Too large to display} \]
symsum(log(- root(65536*a^7*b^7*z^4 + 3072*a^6*b^4*g*i*z^2 + 9216*a^5*b^5* c*i*z^2 + 4096*a^5*b^5*d*h*z^2 + 1024*a^5*b^5*e*g*z^2 + 3072*a^4*b^6*c*e*z ^2 + 2048*a^6*b^4*h^2*z^2 + 2048*a^4*b^6*d^2*z^2 + 768*a^5*b^3*e*h*i*z + 7 68*a^4*b^4*d*e*i*z - 768*a^4*b^4*c*g*h*z - 768*a^3*b^5*c*d*g*z + 1152*a^6* b^2*h*i^2*z - 128*a^5*b^3*g^2*h*z + 1152*a^5*b^3*d*i^2*z + 128*a^4*b^4*e^2 *h*z - 1152*a^3*b^5*c^2*h*z - 128*a^4*b^4*d*g^2*z + 128*a^3*b^5*d*e^2*z - 1152*a^2*b^6*c^2*d*z - 96*a^4*b^2*d*g*h*i - 288*a^3*b^3*c*d*h*i + 72*a^3*b ^3*c*e*g*i - 32*a^3*b^3*d*e*g*h - 96*a^2*b^4*c*d*e*h + 12*a^4*b^2*e*g^2*i - 144*a^4*b^2*c*h^2*i - 48*a^3*b^3*d^2*g*i - 16*a^4*b^2*e*g*h^2 + 108*a^4* b^2*c*g*i^2 + 108*a^2*b^4*c^2*e*i - 144*a^2*b^4*c*d^2*i - 48*a^3*b^3*c*e*h ^2 - 16*a^2*b^4*d^2*e*g + 12*a^2*b^4*c*e^2*g - 48*a^5*b*g*h^2*i - 48*a*b^5 *c*d^2*e + 108*a^5*b*e*i^3 + 108*a*b^5*c^3*g + 54*a^4*b^2*e^2*i^2 + 162*a^ 3*b^3*c^2*i^2 + 96*a^3*b^3*d^2*h^2 + 2*a^3*b^3*e^2*g^2 + 54*a^2*b^4*c^2*g^ 2 + 18*a^5*b*g^2*i^2 + 12*a^3*b^3*e^3*i + 64*a^4*b^2*d*h^3 + 64*a^2*b^4*d^ 3*h + 12*a^3*b^3*c*g^3 + 18*a*b^5*c^2*e^2 + 16*a^5*b*h^4 + 16*a*b^5*d^4 + 81*a^6*i^4 + 81*b^6*c^4 + a^4*b^2*g^4 + a^2*b^4*e^4, z, l)*(root(65536*a^7 *b^7*z^4 + 3072*a^6*b^4*g*i*z^2 + 9216*a^5*b^5*c*i*z^2 + 4096*a^5*b^5*d*h* z^2 + 1024*a^5*b^5*e*g*z^2 + 3072*a^4*b^6*c*e*z^2 + 2048*a^6*b^4*h^2*z^2 + 2048*a^4*b^6*d^2*z^2 + 768*a^5*b^3*e*h*i*z + 768*a^4*b^4*d*e*i*z - 768*a^ 4*b^4*c*g*h*z - 768*a^3*b^5*c*d*g*z + 1152*a^6*b^2*h*i^2*z - 128*a^5*b^...