Integrand size = 25, antiderivative size = 382 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^4} \, dx=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a+b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a+b x^4\right )}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}+\frac {5 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}-\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{15/4} b^{3/4}}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{15/4} b^{3/4}} \]
1/96*x*(9*e*x^2+10*d*x+11*c)/a^2/(b*x^4+a)^2+1/384*x*(45*e*x^2+60*d*x+77*c )/a^3/(b*x^4+a)+1/12*(-a*f+b*x*(e*x^2+d*x+c))/a/b/(b*x^4+a)^3+5/32*d*arcta n(x^2*b^(1/2)/a^(1/2))/a^(7/2)/b^(1/2)-1/1024*ln(-a^(1/4)*b^(1/4)*x*2^(1/2 )+a^(1/2)+x^2*b^(1/2))*(-15*e*a^(1/2)+77*c*b^(1/2))/a^(15/4)/b^(3/4)*2^(1/ 2)+1/1024*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-15*e*a^(1/2) +77*c*b^(1/2))/a^(15/4)/b^(3/4)*2^(1/2)+1/512*arctan(-1+b^(1/4)*x*2^(1/2)/ a^(1/4))*(15*e*a^(1/2)+77*c*b^(1/2))/a^(15/4)/b^(3/4)*2^(1/2)+1/512*arctan (1+b^(1/4)*x*2^(1/2)/a^(1/4))*(15*e*a^(1/2)+77*c*b^(1/2))/a^(15/4)/b^(3/4) *2^(1/2)
Time = 0.41 (sec) , antiderivative size = 379, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^4} \, dx=\frac {\frac {8 a x (77 c+15 x (4 d+3 e x))}{a+b x^4}+\frac {32 a^2 x (11 c+x (10 d+9 e x))}{\left (a+b x^4\right )^2}-\frac {256 a^3 (a f-b x (c+x (d+e x)))}{b \left (a+b x^4\right )^3}-\frac {6 \sqrt [4]{a} \left (77 \sqrt {2} \sqrt {b} c+80 \sqrt [4]{a} \sqrt [4]{b} d+15 \sqrt {2} \sqrt {a} e\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {6 \sqrt [4]{a} \left (77 \sqrt {2} \sqrt {b} c-80 \sqrt [4]{a} \sqrt [4]{b} d+15 \sqrt {2} \sqrt {a} e\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {3 \sqrt {2} \left (-77 \sqrt [4]{a} \sqrt {b} c+15 a^{3/4} e\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}+\frac {3 \sqrt {2} \left (77 \sqrt [4]{a} \sqrt {b} c-15 a^{3/4} e\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{b^{3/4}}}{3072 a^4} \]
((8*a*x*(77*c + 15*x*(4*d + 3*e*x)))/(a + b*x^4) + (32*a^2*x*(11*c + x*(10 *d + 9*e*x)))/(a + b*x^4)^2 - (256*a^3*(a*f - b*x*(c + x*(d + e*x))))/(b*( a + b*x^4)^3) - (6*a^(1/4)*(77*Sqrt[2]*Sqrt[b]*c + 80*a^(1/4)*b^(1/4)*d + 15*Sqrt[2]*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + ( 6*a^(1/4)*(77*Sqrt[2]*Sqrt[b]*c - 80*a^(1/4)*b^(1/4)*d + 15*Sqrt[2]*Sqrt[a ]*e)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/b^(3/4) + (3*Sqrt[2]*(-77*a^ (1/4)*Sqrt[b]*c + 15*a^(3/4)*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4) + (3*Sqrt[2]*(77*a^(1/4)*Sqrt[b]*c - 15*a^(3/4)*e)*L og[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(3/4))/(3072*a^4)
Time = 0.66 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.06, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2393, 25, 2394, 25, 2394, 27, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^4} \, dx\) |
| \(\Big \downarrow \) 2393 |
| \(\displaystyle -\frac {\int -\frac {9 e x^2+10 d x+11 c}{\left (b x^4+a\right )^3}dx}{12 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {9 e x^2+10 d x+11 c}{\left (b x^4+a\right )^3}dx}{12 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {x \left (11 c+10 d x+9 e x^2\right )}{8 a \left (a+b x^4\right )^2}-\frac {\int -\frac {45 e x^2+60 d x+77 c}{\left (b x^4+a\right )^2}dx}{8 a}}{12 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {45 e x^2+60 d x+77 c}{\left (b x^4+a\right )^2}dx}{8 a}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {\frac {x \left (77 c+60 d x+45 e x^2\right )}{4 a \left (a+b x^4\right )}-\frac {\int -\frac {3 \left (15 e x^2+40 d x+77 c\right )}{b x^4+a}dx}{4 a}}{8 a}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {15 e x^2+40 d x+77 c}{b x^4+a}dx}{4 a}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{4 a \left (a+b x^4\right )}}{8 a}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\frac {\frac {3 \int \left (\frac {40 d x}{b x^4+a}+\frac {15 e x^2+77 c}{b x^4+a}\right )dx}{4 a}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{4 a \left (a+b x^4\right )}}{8 a}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt {a} e+77 \sqrt {b} c\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (15 \sqrt {a} e+77 \sqrt {b} c\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {20 d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}\right )}{4 a}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{4 a \left (a+b x^4\right )}}{8 a}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 a}-\frac {a f-b x \left (c+d x+e x^2\right )}{12 a b \left (a+b x^4\right )^3}\) |
-1/12*(a*f - b*x*(c + d*x + e*x^2))/(a*b*(a + b*x^4)^3) + ((x*(11*c + 10*d *x + 9*e*x^2))/(8*a*(a + b*x^4)^2) + ((x*(77*c + 60*d*x + 45*e*x^2))/(4*a* (a + b*x^4)) + (3*((20*d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) - ((77*Sqrt[b]*c + 15*Sqrt[a]*e)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/ (2*Sqrt[2]*a^(3/4)*b^(3/4)) + ((77*Sqrt[b]*c + 15*Sqrt[a]*e)*ArcTan[1 + (S qrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)) - ((77*Sqrt[b]*c - 15*Sqrt[a]*e)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4* Sqrt[2]*a^(3/4)*b^(3/4)) + ((77*Sqrt[b]*c - 15*Sqrt[a]*e)*Log[Sqrt[a] + Sq rt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(3/4))))/(4*a ))/(8*a))/(12*a)
3.5.93.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[(a*Coeff[Pq, x, q] - b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q , x])*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) In t[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^( p + 1), x], x] /; q == n - 1] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n , 0] && LtQ[p, -1]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b *x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[ExpandToSum[n *(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x ] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 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 = 1.64 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.40
| method | result | size |
| risch | \(\frac {\frac {15 e \,b^{2} x^{11}}{128 a^{3}}+\frac {5 d \,b^{2} x^{10}}{32 a^{3}}+\frac {77 c \,b^{2} x^{9}}{384 a^{3}}+\frac {21 b e \,x^{7}}{64 a^{2}}+\frac {5 b d \,x^{6}}{12 a^{2}}+\frac {33 b c \,x^{5}}{64 a^{2}}+\frac {113 e \,x^{3}}{384 a}+\frac {11 d \,x^{2}}{32 a}+\frac {51 c x}{128 a}-\frac {f}{12 b}}{\left (b \,x^{4}+a \right )^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (15 \textit {\_R}^{2} e +40 \textit {\_R} d +77 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{512 a^{3} b}\) | \(153\) |
| default | \(\frac {\frac {15 e \,b^{2} x^{11}}{128 a^{3}}+\frac {5 d \,b^{2} x^{10}}{32 a^{3}}+\frac {77 c \,b^{2} x^{9}}{384 a^{3}}+\frac {21 b e \,x^{7}}{64 a^{2}}+\frac {5 b d \,x^{6}}{12 a^{2}}+\frac {33 b c \,x^{5}}{64 a^{2}}+\frac {113 e \,x^{3}}{384 a}+\frac {11 d \,x^{2}}{32 a}+\frac {51 c x}{128 a}-\frac {f}{12 b}}{\left (b \,x^{4}+a \right )^{3}}+\frac {\frac {77 c \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {20 d \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{\sqrt {a b}}+\frac {15 e \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{128 a^{3}}\) | \(340\) |
(15/128*e/a^3*b^2*x^11+5/32*d/a^3*b^2*x^10+77/384*c/a^3*b^2*x^9+21/64*b*e/ a^2*x^7+5/12*b*d/a^2*x^6+33/64*b*c/a^2*x^5+113/384/a*e*x^3+11/32*d/a*x^2+5 1/128*c/a*x-1/12*f/b)/(b*x^4+a)^3+1/512/a^3/b*sum((15*_R^2*e+40*_R*d+77*c) /_R^3*ln(x-_R),_R=RootOf(_Z^4*b+a))
Result contains complex when optimal does not.
Time = 19.78 (sec) , antiderivative size = 125011, normalized size of antiderivative = 327.25 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^4} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^4} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 402, normalized size of antiderivative = 1.05 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^4} \, dx=\frac {45 \, b^{3} e x^{11} + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} + 126 \, a b^{2} e x^{7} + 160 \, a b^{2} d x^{6} + 198 \, a b^{2} c x^{5} + 113 \, a^{2} b e x^{3} + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x - 32 \, a^{3} f}{384 \, {\left (a^{3} b^{4} x^{12} + 3 \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{4} + a^{6} b\right )}} + \frac {\frac {\sqrt {2} {\left (77 \, \sqrt {b} c - 15 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (77 \, \sqrt {b} c - 15 \, \sqrt {a} e\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 15 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e - 80 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (77 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} c + 15 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} e + 80 \, \sqrt {a} \sqrt {b} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{1024 \, a^{3}} \]
1/384*(45*b^3*e*x^11 + 60*b^3*d*x^10 + 77*b^3*c*x^9 + 126*a*b^2*e*x^7 + 16 0*a*b^2*d*x^6 + 198*a*b^2*c*x^5 + 113*a^2*b*e*x^3 + 132*a^2*b*d*x^2 + 153* a^2*b*c*x - 32*a^3*f)/(a^3*b^4*x^12 + 3*a^4*b^3*x^8 + 3*a^5*b^2*x^4 + a^6* b) + 1/1024*(sqrt(2)*(77*sqrt(b)*c - 15*sqrt(a)*e)*log(sqrt(b)*x^2 + sqrt( 2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) - sqrt(2)*(77*sqrt(b)*c - 15*sqrt(a)*e)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^ (3/4)*b^(3/4)) + 2*(77*sqrt(2)*a^(1/4)*b^(3/4)*c + 15*sqrt(2)*a^(3/4)*b^(1 /4)*e - 80*sqrt(a)*sqrt(b)*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^(3/ 4)) + 2*(77*sqrt(2)*a^(1/4)*b^(3/4)*c + 15*sqrt(2)*a^(3/4)*b^(1/4)*e + 80* sqrt(a)*sqrt(b)*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^(3/4)))/a^3
Time = 0.28 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^4} \, dx=\frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac {\sqrt {2} {\left (40 \, \sqrt {2} \sqrt {a b} b^{2} d + 77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c + 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{4} b^{3}} + \frac {\sqrt {2} {\left (77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac {\sqrt {2} {\left (77 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} c - 15 \, \left (a b^{3}\right )^{\frac {3}{4}} e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{4} b^{3}} + \frac {45 \, b^{3} e x^{11} + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} + 126 \, a b^{2} e x^{7} + 160 \, a b^{2} d x^{6} + 198 \, a b^{2} c x^{5} + 113 \, a^{2} b e x^{3} + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x - 32 \, a^{3} f}{384 \, {\left (b x^{4} + a\right )}^{3} a^{3} b} \]
1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b^2*d + 77*(a*b^3)^(1/4)*b^2*c + 15*(a *b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4)) /(a^4*b^3) + 1/512*sqrt(2)*(40*sqrt(2)*sqrt(a*b)*b^2*d + 77*(a*b^3)^(1/4)* b^2*c + 15*(a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4)) /(a/b)^(1/4))/(a^4*b^3) + 1/1024*sqrt(2)*(77*(a*b^3)^(1/4)*b^2*c - 15*(a*b ^3)^(3/4)*e)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^4*b^3) - 1/10 24*sqrt(2)*(77*(a*b^3)^(1/4)*b^2*c - 15*(a*b^3)^(3/4)*e)*log(x^2 - sqrt(2) *x*(a/b)^(1/4) + sqrt(a/b))/(a^4*b^3) + 1/384*(45*b^3*e*x^11 + 60*b^3*d*x^ 10 + 77*b^3*c*x^9 + 126*a*b^2*e*x^7 + 160*a*b^2*d*x^6 + 198*a*b^2*c*x^5 + 113*a^2*b*e*x^3 + 132*a^2*b*d*x^2 + 153*a^2*b*c*x - 32*a^3*f)/((b*x^4 + a) ^3*a^3*b)
Time = 9.45 (sec) , antiderivative size = 879, normalized size of antiderivative = 2.30 \[ \int \frac {c+d x+e x^2+f x^3}{\left (a+b x^4\right )^4} \, dx=\left (\sum _{k=1}^4\ln \left (-\frac {b\,\left (3375\,a\,e^3-123200\,b\,c\,d^2+88935\,b\,c^2\,e-64000\,b\,d^3\,x+{\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )}^2\,a^7\,b^2\,c\,20185088-\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )\,a^4\,b\,e^2\,x\,115200+92400\,b\,c\,d\,e\,x+\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )\,a^3\,b^2\,c^2\,x\,3035648-{\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )}^2\,a^7\,b^2\,d\,x\,10485760+\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )\,a^4\,b\,d\,e\,614400\right )}{a^9\,2097152}\right )\,\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4+1211105280\,a^8\,b^2\,c\,e\,z^2+838860800\,a^8\,b^2\,d^2\,z^2-485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4+35153041\,b^2\,c^4+50625\,a^2\,e^4,z,k\right )\right )+\frac {\frac {11\,d\,x^2}{32\,a}-\frac {f}{12\,b}+\frac {113\,e\,x^3}{384\,a}+\frac {51\,c\,x}{128\,a}+\frac {77\,b^2\,c\,x^9}{384\,a^3}+\frac {5\,b^2\,d\,x^{10}}{32\,a^3}+\frac {15\,b^2\,e\,x^{11}}{128\,a^3}+\frac {33\,b\,c\,x^5}{64\,a^2}+\frac {5\,b\,d\,x^6}{12\,a^2}+\frac {21\,b\,e\,x^7}{64\,a^2}}{a^3+3\,a^2\,b\,x^4+3\,a\,b^2\,x^8+b^3\,x^{12}} \]
symsum(log(-(b*(3375*a*e^3 - 123200*b*c*d^2 + 88935*b*c^2*e - 64000*b*d^3* x + 20185088*root(68719476736*a^15*b^3*z^4 + 1211105280*a^8*b^2*c*e*z^2 + 838860800*a^8*b^2*d^2*z^2 - 485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e ^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 + 35153 041*b^2*c^4 + 50625*a^2*e^4, z, k)^2*a^7*b^2*c - 115200*root(68719476736*a ^15*b^3*z^4 + 1211105280*a^8*b^2*c*e*z^2 + 838860800*a^8*b^2*d^2*z^2 - 485 703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 7392000*a*b*c*d^2*e + 26 68050*a*b*c^2*e^2 + 2560000*a*b*d^4 + 35153041*b^2*c^4 + 50625*a^2*e^4, z, k)*a^4*b*e^2*x + 92400*b*c*d*e*x + 3035648*root(68719476736*a^15*b^3*z^4 + 1211105280*a^8*b^2*c*e*z^2 + 838860800*a^8*b^2*d^2*z^2 - 485703680*a^4*b ^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^ 2*e^2 + 2560000*a*b*d^4 + 35153041*b^2*c^4 + 50625*a^2*e^4, z, k)*a^3*b^2* c^2*x - 10485760*root(68719476736*a^15*b^3*z^4 + 1211105280*a^8*b^2*c*e*z^ 2 + 838860800*a^8*b^2*d^2*z^2 - 485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b *d*e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 + 3 5153041*b^2*c^4 + 50625*a^2*e^4, z, k)^2*a^7*b^2*d*x + 614400*root(6871947 6736*a^15*b^3*z^4 + 1211105280*a^8*b^2*c*e*z^2 + 838860800*a^8*b^2*d^2*z^2 - 485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 7392000*a*b*c*d^2* e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 + 35153041*b^2*c^4 + 50625*a^2*e ^4, z, k)*a^4*b*d*e))/(2097152*a^9))*root(68719476736*a^15*b^3*z^4 + 12...