Integrand size = 28, antiderivative size = 380 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=-\frac {c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}+\frac {x \left (d+2 e x+3 f x^2\right )}{96 a b \left (a+b x^4\right )^2}+\frac {x \left (7 d+12 e x+15 f x^2\right )}{384 a^2 b \left (a+b x^4\right )}+\frac {e \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{5/2} b^{3/2}}-\frac {\left (7 \sqrt {b} d+5 \sqrt {a} f\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{11/4} b^{7/4}}+\frac {\left (7 \sqrt {b} d+5 \sqrt {a} f\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 \sqrt {2} a^{11/4} b^{7/4}}-\frac {\left (7 \sqrt {b} d-5 \sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{11/4} b^{7/4}}+\frac {\left (7 \sqrt {b} d-5 \sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{512 \sqrt {2} a^{11/4} b^{7/4}} \]
1/12*(-f*x^3-e*x^2-d*x-c)/b/(b*x^4+a)^3+1/96*x*(3*f*x^2+2*e*x+d)/a/b/(b*x^ 4+a)^2+1/384*x*(15*f*x^2+12*e*x+7*d)/a^2/b/(b*x^4+a)+1/32*e*arctan(x^2*b^( 1/2)/a^(1/2))/a^(5/2)/b^(3/2)-1/1024*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2) +x^2*b^(1/2))*(-5*f*a^(1/2)+7*d*b^(1/2))/a^(11/4)/b^(7/4)*2^(1/2)+1/1024*l n(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))*(-5*f*a^(1/2)+7*d*b^(1/2) )/a^(11/4)/b^(7/4)*2^(1/2)+1/512*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))*(5*f *a^(1/2)+7*d*b^(1/2))/a^(11/4)/b^(7/4)*2^(1/2)+1/512*arctan(1+b^(1/4)*x*2^ (1/2)/a^(1/4))*(5*f*a^(1/2)+7*d*b^(1/2))/a^(11/4)/b^(7/4)*2^(1/2)
Time = 0.44 (sec) , antiderivative size = 366, normalized size of antiderivative = 0.96 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=\frac {\frac {32 b^{3/4} x (d+x (2 e+3 f x))}{a \left (a+b x^4\right )^2}+\frac {8 b^{3/4} x (7 d+3 x (4 e+5 f x))}{a^2 \left (a+b x^4\right )}-\frac {256 b^{3/4} (c+x (d+x (e+f x)))}{\left (a+b x^4\right )^3}-\frac {6 \left (7 \sqrt {2} \sqrt {b} d+16 \sqrt [4]{a} \sqrt [4]{b} e+5 \sqrt {2} \sqrt {a} f\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+\frac {6 \left (7 \sqrt {2} \sqrt {b} d-16 \sqrt [4]{a} \sqrt [4]{b} e+5 \sqrt {2} \sqrt {a} f\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{11/4}}+\frac {3 \sqrt {2} \left (-7 \sqrt {b} d+5 \sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{11/4}}+\frac {3 \sqrt {2} \left (7 \sqrt {b} d-5 \sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{11/4}}}{3072 b^{7/4}} \]
((32*b^(3/4)*x*(d + x*(2*e + 3*f*x)))/(a*(a + b*x^4)^2) + (8*b^(3/4)*x*(7* d + 3*x*(4*e + 5*f*x)))/(a^2*(a + b*x^4)) - (256*b^(3/4)*(c + x*(d + x*(e + f*x))))/(a + b*x^4)^3 - (6*(7*Sqrt[2]*Sqrt[b]*d + 16*a^(1/4)*b^(1/4)*e + 5*Sqrt[2]*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(11/4) + (6*(7*Sqrt[2]*Sqrt[b]*d - 16*a^(1/4)*b^(1/4)*e + 5*Sqrt[2]*Sqrt[a]*f)*ArcT an[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(11/4) + (3*Sqrt[2]*(-7*Sqrt[b]*d + 5*Sqrt[a]*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(1 1/4) + (3*Sqrt[2]*(7*Sqrt[b]*d - 5*Sqrt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4 )*b^(1/4)*x + Sqrt[b]*x^2])/a^(11/4))/(3072*b^(7/4))
Time = 0.66 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.04, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {2363, 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.
| \(\displaystyle \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx\) |
| \(\Big \downarrow \) 2363 |
| \(\displaystyle \frac {\int \frac {3 f x^2+2 e x+d}{\left (b x^4+a\right )^3}dx}{12 b}-\frac {c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {x \left (d+2 e x+3 f x^2\right )}{8 a \left (a+b x^4\right )^2}-\frac {\int -\frac {15 f x^2+12 e x+7 d}{\left (b x^4+a\right )^2}dx}{8 a}}{12 b}-\frac {c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {15 f x^2+12 e x+7 d}{\left (b x^4+a\right )^2}dx}{8 a}+\frac {x \left (d+2 e x+3 f x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 b}-\frac {c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {\frac {x \left (7 d+12 e x+15 f x^2\right )}{4 a \left (a+b x^4\right )}-\frac {\int -\frac {3 \left (5 f x^2+8 e x+7 d\right )}{b x^4+a}dx}{4 a}}{8 a}+\frac {x \left (d+2 e x+3 f x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 b}-\frac {c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {3 \int \frac {5 f x^2+8 e x+7 d}{b x^4+a}dx}{4 a}+\frac {x \left (7 d+12 e x+15 f x^2\right )}{4 a \left (a+b x^4\right )}}{8 a}+\frac {x \left (d+2 e x+3 f x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 b}-\frac {c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\frac {\frac {3 \int \left (\frac {8 e x}{b x^4+a}+\frac {5 f x^2+7 d}{b x^4+a}\right )dx}{4 a}+\frac {x \left (7 d+12 e x+15 f x^2\right )}{4 a \left (a+b x^4\right )}}{8 a}+\frac {x \left (d+2 e x+3 f x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 b}-\frac {c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {\frac {3 \left (-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} f+7 \sqrt {b} d\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt {a} f+7 \sqrt {b} d\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (7 \sqrt {b} d-5 \sqrt {a} f\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (7 \sqrt {b} d-5 \sqrt {a} f\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {4 e \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}\right )}{4 a}+\frac {x \left (7 d+12 e x+15 f x^2\right )}{4 a \left (a+b x^4\right )}}{8 a}+\frac {x \left (d+2 e x+3 f x^2\right )}{8 a \left (a+b x^4\right )^2}}{12 b}-\frac {c+d x+e x^2+f x^3}{12 b \left (a+b x^4\right )^3}\) |
-1/12*(c + d*x + e*x^2 + f*x^3)/(b*(a + b*x^4)^3) + ((x*(d + 2*e*x + 3*f*x ^2))/(8*a*(a + b*x^4)^2) + ((x*(7*d + 12*e*x + 15*f*x^2))/(4*a*(a + b*x^4) ) + (3*((4*e*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[b]) - ((7*Sqrt[b ]*d + 5*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^( 3/4)*b^(3/4)) + ((7*Sqrt[b]*d + 5*Sqrt[a]*f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x )/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)) - ((7*Sqrt[b]*d - 5*Sqrt[a]*f)*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)) + ((7*Sqrt[b]*d - 5*Sqrt[a]*f)*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))))/(4*a))/(8*a))/(12*b)
3.5.94.3.1 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_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Pq*(( a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[1/(b*n*(p + 1)) Int[D[Pq, x] *(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, m, n}, x] && PolyQ[Pq, x] && E qQ[m - n + 1, 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 = 1.56 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.38
| method | result | size |
| risch | \(\frac {\frac {5 b f \,x^{11}}{128 a^{2}}+\frac {b e \,x^{10}}{32 a^{2}}+\frac {7 b d \,x^{9}}{384 a^{2}}+\frac {7 f \,x^{7}}{64 a}+\frac {e \,x^{6}}{12 a}+\frac {3 d \,x^{5}}{64 a}-\frac {5 f \,x^{3}}{384 b}-\frac {e \,x^{2}}{32 b}-\frac {7 d x}{128 b}-\frac {c}{12 b}}{\left (b \,x^{4}+a \right )^{3}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (5 f \,\textit {\_R}^{2}+8 e \textit {\_R} +7 d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{512 a^{2} b^{2}}\) | \(144\) |
| default | \(\frac {\frac {5 b f \,x^{11}}{128 a^{2}}+\frac {b e \,x^{10}}{32 a^{2}}+\frac {7 b d \,x^{9}}{384 a^{2}}+\frac {7 f \,x^{7}}{64 a}+\frac {e \,x^{6}}{12 a}+\frac {3 d \,x^{5}}{64 a}-\frac {5 f \,x^{3}}{384 b}-\frac {e \,x^{2}}{32 b}-\frac {7 d x}{128 b}-\frac {c}{12 b}}{\left (b \,x^{4}+a \right )^{3}}+\frac {\frac {7 d \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {4 e \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{\sqrt {a b}}+\frac {5 f \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{128 a^{2} b}\) | \(334\) |
(5/128/a^2*b*f*x^11+1/32*b*e/a^2*x^10+7/384*b*d/a^2*x^9+7/64*f/a*x^7+1/12/ a*e*x^6+3/64*d/a*x^5-5/384*f*x^3/b-1/32*e*x^2/b-7/128*d*x/b-1/12*c/b)/(b*x ^4+a)^3+1/512/a^2/b^2*sum((5*_R^2*f+8*_R*e+7*d)/_R^3*ln(x-_R),_R=RootOf(_Z ^4*b+a))
Result contains complex when optimal does not.
Time = 21.92 (sec) , antiderivative size = 125996, normalized size of antiderivative = 331.57 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=\text {Too large to display} \]
Timed out. \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 396, normalized size of antiderivative = 1.04 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=\frac {15 \, b^{2} f x^{11} + 12 \, b^{2} e x^{10} + 7 \, b^{2} d x^{9} + 42 \, a b f x^{7} + 32 \, a b e x^{6} + 18 \, a b d x^{5} - 5 \, a^{2} f x^{3} - 12 \, a^{2} e x^{2} - 21 \, a^{2} d x - 32 \, a^{2} c}{384 \, {\left (a^{2} b^{4} x^{12} + 3 \, a^{3} b^{3} x^{8} + 3 \, a^{4} b^{2} x^{4} + a^{5} b\right )}} + \frac {\frac {\sqrt {2} {\left (7 \, \sqrt {b} d - 5 \, \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (7 \, \sqrt {b} d - 5 \, \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} d + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} f - 16 \, \sqrt {a} \sqrt {b} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (7 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} d + 5 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} f + 16 \, \sqrt {a} \sqrt {b} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{1024 \, a^{2} b} \]
1/384*(15*b^2*f*x^11 + 12*b^2*e*x^10 + 7*b^2*d*x^9 + 42*a*b*f*x^7 + 32*a*b *e*x^6 + 18*a*b*d*x^5 - 5*a^2*f*x^3 - 12*a^2*e*x^2 - 21*a^2*d*x - 32*a^2*c )/(a^2*b^4*x^12 + 3*a^3*b^3*x^8 + 3*a^4*b^2*x^4 + a^5*b) + 1/1024*(sqrt(2) *(7*sqrt(b)*d - 5*sqrt(a)*f)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) - sqrt(2)*(7*sqrt(b)*d - 5*sqrt(a)*f)*log(sqrt (b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) + 2*(7*sq rt(2)*a^(1/4)*b^(3/4)*d + 5*sqrt(2)*a^(3/4)*b^(1/4)*f - 16*sqrt(a)*sqrt(b) *e)*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*(7*sqrt(2)*a^(1/4) *b^(3/4)*d + 5*sqrt(2)*a^(3/4)*b^(1/4)*f + 16*sqrt(a)*sqrt(b)*e)*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^2*b)
Time = 0.27 (sec) , antiderivative size = 375, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=\frac {\sqrt {2} {\left (8 \, \sqrt {2} \sqrt {a b} b^{2} e + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{3} b^{4}} + \frac {\sqrt {2} {\left (8 \, \sqrt {2} \sqrt {a b} b^{2} e + 7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d + 5 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, a^{3} b^{4}} + \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{3} b^{4}} - \frac {\sqrt {2} {\left (7 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - 5 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{1024 \, a^{3} b^{4}} + \frac {15 \, b^{2} f x^{11} + 12 \, b^{2} e x^{10} + 7 \, b^{2} d x^{9} + 42 \, a b f x^{7} + 32 \, a b e x^{6} + 18 \, a b d x^{5} - 5 \, a^{2} f x^{3} - 12 \, a^{2} e x^{2} - 21 \, a^{2} d x - 32 \, a^{2} c}{384 \, {\left (b x^{4} + a\right )}^{3} a^{2} b} \]
1/512*sqrt(2)*(8*sqrt(2)*sqrt(a*b)*b^2*e + 7*(a*b^3)^(1/4)*b^2*d + 5*(a*b^ 3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a ^3*b^4) + 1/512*sqrt(2)*(8*sqrt(2)*sqrt(a*b)*b^2*e + 7*(a*b^3)^(1/4)*b^2*d + 5*(a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b) ^(1/4))/(a^3*b^4) + 1/1024*sqrt(2)*(7*(a*b^3)^(1/4)*b^2*d - 5*(a*b^3)^(3/4 )*f)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^4) - 1/1024*sqrt( 2)*(7*(a*b^3)^(1/4)*b^2*d - 5*(a*b^3)^(3/4)*f)*log(x^2 - sqrt(2)*x*(a/b)^( 1/4) + sqrt(a/b))/(a^3*b^4) + 1/384*(15*b^2*f*x^11 + 12*b^2*e*x^10 + 7*b^2 *d*x^9 + 42*a*b*f*x^7 + 32*a*b*e*x^6 + 18*a*b*d*x^5 - 5*a^2*f*x^3 - 12*a^2 *e*x^2 - 21*a^2*d*x - 32*a^2*c)/((b*x^4 + a)^3*a^2*b)
Time = 0.50 (sec) , antiderivative size = 888, normalized size of antiderivative = 2.34 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^4} \, dx=\frac {\frac {3\,d\,x^5}{64\,a}-\frac {c}{12\,b}+\frac {e\,x^6}{12\,a}-\frac {e\,x^2}{32\,b}+\frac {7\,f\,x^7}{64\,a}-\frac {5\,f\,x^3}{384\,b}-\frac {7\,d\,x}{128\,b}+\frac {7\,b\,d\,x^9}{384\,a^2}+\frac {b\,e\,x^{10}}{32\,a^2}+\frac {5\,b\,f\,x^{11}}{128\,a^2}}{a^3+3\,a^2\,b\,x^4+3\,a\,b^2\,x^8+b^3\,x^{12}}+\left (\sum _{k=1}^4\ln \left (-\frac {125\,a\,f^3-448\,b\,d\,e^2+245\,b\,d^2\,f-512\,b\,e^3\,x+{\mathrm {root}\left (68719476736\,a^{11}\,b^7\,z^4+36700160\,a^6\,b^4\,d\,f\,z^2+33554432\,a^6\,b^4\,e^2\,z^2+409600\,a^4\,b^2\,e\,f^2\,z-802816\,a^3\,b^3\,d^2\,e\,z-8960\,a\,b\,d\,e^2\,f+2450\,a\,b\,d^2\,f^2+4096\,a\,b\,e^4+625\,a^2\,f^4+2401\,b^2\,d^4,z,k\right )}^2\,a^5\,b^4\,d\,1835008+560\,b\,d\,e\,f\,x+\mathrm {root}\left (68719476736\,a^{11}\,b^7\,z^4+36700160\,a^6\,b^4\,d\,f\,z^2+33554432\,a^6\,b^4\,e^2\,z^2+409600\,a^4\,b^2\,e\,f^2\,z-802816\,a^3\,b^3\,d^2\,e\,z-8960\,a\,b\,d\,e^2\,f+2450\,a\,b\,d^2\,f^2+4096\,a\,b\,e^4+625\,a^2\,f^4+2401\,b^2\,d^4,z,k\right )\,a^2\,b^3\,d^2\,x\,25088-{\mathrm {root}\left (68719476736\,a^{11}\,b^7\,z^4+36700160\,a^6\,b^4\,d\,f\,z^2+33554432\,a^6\,b^4\,e^2\,z^2+409600\,a^4\,b^2\,e\,f^2\,z-802816\,a^3\,b^3\,d^2\,e\,z-8960\,a\,b\,d\,e^2\,f+2450\,a\,b\,d^2\,f^2+4096\,a\,b\,e^4+625\,a^2\,f^4+2401\,b^2\,d^4,z,k\right )}^2\,a^5\,b^4\,e\,x\,2097152-\mathrm {root}\left (68719476736\,a^{11}\,b^7\,z^4+36700160\,a^6\,b^4\,d\,f\,z^2+33554432\,a^6\,b^4\,e^2\,z^2+409600\,a^4\,b^2\,e\,f^2\,z-802816\,a^3\,b^3\,d^2\,e\,z-8960\,a\,b\,d\,e^2\,f+2450\,a\,b\,d^2\,f^2+4096\,a\,b\,e^4+625\,a^2\,f^4+2401\,b^2\,d^4,z,k\right )\,a^3\,b^2\,f^2\,x\,12800+\mathrm {root}\left (68719476736\,a^{11}\,b^7\,z^4+36700160\,a^6\,b^4\,d\,f\,z^2+33554432\,a^6\,b^4\,e^2\,z^2+409600\,a^4\,b^2\,e\,f^2\,z-802816\,a^3\,b^3\,d^2\,e\,z-8960\,a\,b\,d\,e^2\,f+2450\,a\,b\,d^2\,f^2+4096\,a\,b\,e^4+625\,a^2\,f^4+2401\,b^2\,d^4,z,k\right )\,a^3\,b^2\,e\,f\,40960}{a^6\,b^2\,2097152}\right )\,\mathrm {root}\left (68719476736\,a^{11}\,b^7\,z^4+36700160\,a^6\,b^4\,d\,f\,z^2+33554432\,a^6\,b^4\,e^2\,z^2+409600\,a^4\,b^2\,e\,f^2\,z-802816\,a^3\,b^3\,d^2\,e\,z-8960\,a\,b\,d\,e^2\,f+2450\,a\,b\,d^2\,f^2+4096\,a\,b\,e^4+625\,a^2\,f^4+2401\,b^2\,d^4,z,k\right )\right ) \]
((3*d*x^5)/(64*a) - c/(12*b) + (e*x^6)/(12*a) - (e*x^2)/(32*b) + (7*f*x^7) /(64*a) - (5*f*x^3)/(384*b) - (7*d*x)/(128*b) + (7*b*d*x^9)/(384*a^2) + (b *e*x^10)/(32*a^2) + (5*b*f*x^11)/(128*a^2))/(a^3 + b^3*x^12 + 3*a^2*b*x^4 + 3*a*b^2*x^8) + symsum(log(-(125*a*f^3 - 448*b*d*e^2 + 245*b*d^2*f - 512* b*e^3*x + 1835008*root(68719476736*a^11*b^7*z^4 + 36700160*a^6*b^4*d*f*z^2 + 33554432*a^6*b^4*e^2*z^2 + 409600*a^4*b^2*e*f^2*z - 802816*a^3*b^3*d^2* e*z - 8960*a*b*d*e^2*f + 2450*a*b*d^2*f^2 + 4096*a*b*e^4 + 625*a^2*f^4 + 2 401*b^2*d^4, z, k)^2*a^5*b^4*d + 560*b*d*e*f*x + 25088*root(68719476736*a^ 11*b^7*z^4 + 36700160*a^6*b^4*d*f*z^2 + 33554432*a^6*b^4*e^2*z^2 + 409600* a^4*b^2*e*f^2*z - 802816*a^3*b^3*d^2*e*z - 8960*a*b*d*e^2*f + 2450*a*b*d^2 *f^2 + 4096*a*b*e^4 + 625*a^2*f^4 + 2401*b^2*d^4, z, k)*a^2*b^3*d^2*x - 20 97152*root(68719476736*a^11*b^7*z^4 + 36700160*a^6*b^4*d*f*z^2 + 33554432* a^6*b^4*e^2*z^2 + 409600*a^4*b^2*e*f^2*z - 802816*a^3*b^3*d^2*e*z - 8960*a *b*d*e^2*f + 2450*a*b*d^2*f^2 + 4096*a*b*e^4 + 625*a^2*f^4 + 2401*b^2*d^4, z, k)^2*a^5*b^4*e*x - 12800*root(68719476736*a^11*b^7*z^4 + 36700160*a^6* b^4*d*f*z^2 + 33554432*a^6*b^4*e^2*z^2 + 409600*a^4*b^2*e*f^2*z - 802816*a ^3*b^3*d^2*e*z - 8960*a*b*d*e^2*f + 2450*a*b*d^2*f^2 + 4096*a*b*e^4 + 625* a^2*f^4 + 2401*b^2*d^4, z, k)*a^3*b^2*f^2*x + 40960*root(68719476736*a^11* b^7*z^4 + 36700160*a^6*b^4*d*f*z^2 + 33554432*a^6*b^4*e^2*z^2 + 409600*a^4 *b^2*e*f^2*z - 802816*a^3*b^3*d^2*e*z - 8960*a*b*d*e^2*f + 2450*a*b*d^2...