Integrand size = 17, antiderivative size = 245 \[ \int \frac {c+d x^4}{\left (a+b x^4\right )^2} \, dx=\frac {(b c-a d) x}{4 a b \left (a+b x^4\right )}-\frac {(3 b c+a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(3 b c+a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{5/4}}-\frac {(3 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{5/4}}+\frac {(3 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{5/4}} \]
1/4*(-a*d+b*c)*x/a/b/(b*x^4+a)+1/16*(a*d+3*b*c)*arctan(-1+b^(1/4)*x*2^(1/2 )/a^(1/4))/a^(7/4)/b^(5/4)*2^(1/2)+1/16*(a*d+3*b*c)*arctan(1+b^(1/4)*x*2^( 1/2)/a^(1/4))/a^(7/4)/b^(5/4)*2^(1/2)-1/32*(a*d+3*b*c)*ln(-a^(1/4)*b^(1/4) *x*2^(1/2)+a^(1/2)+x^2*b^(1/2))/a^(7/4)/b^(5/4)*2^(1/2)+1/32*(a*d+3*b*c)*l n(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))/a^(7/4)/b^(5/4)*2^(1/2)
Time = 0.17 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.87 \[ \int \frac {c+d x^4}{\left (a+b x^4\right )^2} \, dx=\frac {-\frac {8 a^{3/4} \sqrt [4]{b} (-b c+a d) x}{a+b x^4}-2 \sqrt {2} (3 b c+a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )+2 \sqrt {2} (3 b c+a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\sqrt {2} (3 b c+a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )+\sqrt {2} (3 b c+a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{32 a^{7/4} b^{5/4}} \]
((-8*a^(3/4)*b^(1/4)*(-(b*c) + a*d)*x)/(a + b*x^4) - 2*Sqrt[2]*(3*b*c + a* d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)] + 2*Sqrt[2]*(3*b*c + a*d)*ArcTa n[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)] - Sqrt[2]*(3*b*c + a*d)*Log[Sqrt[a] - S qrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2] + Sqrt[2]*(3*b*c + a*d)*Log[Sqrt[a ] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(32*a^(7/4)*b^(5/4))
Time = 0.43 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.01, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.529, Rules used = {910, 755, 1476, 1082, 217, 1479, 25, 27, 1103}
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.
| \(\displaystyle \int \frac {c+d x^4}{\left (a+b x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 910 |
| \(\displaystyle \frac {(a d+3 b c) \int \frac {1}{b x^4+a}dx}{4 a b}+\frac {x (b c-a d)}{4 a b \left (a+b x^4\right )}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {(a d+3 b c) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {b} x^2+\sqrt {a}}{b x^4+a}dx}{2 \sqrt {a}}\right )}{4 a b}+\frac {x (b c-a d)}{4 a b \left (a+b x^4\right )}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {(a d+3 b c) \left (\frac {\frac {\int \frac {1}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {b}}+\frac {\int \frac {1}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {b}}}{2 \sqrt {a}}+\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}\right )}{4 a b}+\frac {x (b c-a d)}{4 a b \left (a+b x^4\right )}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {(a d+3 b c) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\int \frac {1}{-\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )^2-1}d\left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int \frac {1}{-\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )^2-1}d\left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{4 a b}+\frac {x (b c-a d)}{4 a b \left (a+b x^4\right )}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {(a d+3 b c) \left (\frac {\int \frac {\sqrt {a}-\sqrt {b} x^2}{b x^4+a}dx}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{4 a b}+\frac {x (b c-a d)}{4 a b \left (a+b x^4\right )}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {(a d+3 b c) \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{4 a b}+\frac {x (b c-a d)}{4 a b \left (a+b x^4\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {(a d+3 b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{\sqrt [4]{b} \left (x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}\right )}{\sqrt [4]{b} \left (x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}\right )}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{4 a b}+\frac {x (b c-a d)}{4 a b \left (a+b x^4\right )}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(a d+3 b c) \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt [4]{a}-2 \sqrt [4]{b} x}{x^2-\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt {2} \sqrt [4]{a} \sqrt {b}}+\frac {\int \frac {\sqrt {2} \sqrt [4]{b} x+\sqrt [4]{a}}{x^2+\frac {\sqrt {2} \sqrt [4]{a} x}{\sqrt [4]{b}}+\frac {\sqrt {a}}{\sqrt {b}}}dx}{2 \sqrt [4]{a} \sqrt {b}}}{2 \sqrt {a}}+\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{4 a b}+\frac {x (b c-a d)}{4 a b \left (a+b x^4\right )}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {(a d+3 b c) \left (\frac {\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}+\frac {\frac {\log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}-\frac {\log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{2 \sqrt {2} \sqrt [4]{a} \sqrt [4]{b}}}{2 \sqrt {a}}\right )}{4 a b}+\frac {x (b c-a d)}{4 a b \left (a+b x^4\right )}\) |
((b*c - a*d)*x)/(4*a*b*(a + b*x^4)) + ((3*b*c + a*d)*((-(ArcTan[1 - (Sqrt[ 2]*b^(1/4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4))) + ArcTan[1 + (Sqrt[2]*b^ (1/4)*x)/a^(1/4)]/(Sqrt[2]*a^(1/4)*b^(1/4)))/(2*Sqrt[a]) + (-1/2*Log[Sqrt[ a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2]/(Sqrt[2]*a^(1/4)*b^(1/4)) + Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2]/(2*Sqrt[2]*a^(1/4)* b^(1/4)))/(2*Sqrt[a])))/(4*a*b)
3.2.70.3.1 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[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) Int[(r + s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] & & AtomQ[SplitProduct[SumBaseQ, b]]))
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Si mp[(-(b*c - a*d))*x*((a + b*x^n)^(p + 1)/(a*b*n*(p + 1))), x] - Simp[(a*d - b*c*(n*(p + 1) + 1))/(a*b*n*(p + 1)) Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && (LtQ[p, -1] || ILtQ[1/ n + p, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b )], x] /; RationalQ[q] && (EqQ[q^2, 1] || !RationalQ[b^2 - 4*a*c])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ e/(2*c) Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 4.00 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.27
| method | result | size |
| risch | \(-\frac {\left (a d -b c \right ) x}{4 b a \left (b \,x^{4}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (a d +3 b c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 a \,b^{2}}\) | \(65\) |
| default | \(-\frac {\left (a d -b c \right ) x}{4 b a \left (b \,x^{4}+a \right )}+\frac {\left (a d +3 b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2} b}\) | \(140\) |
-1/4*(a*d-b*c)/b/a*x/(b*x^4+a)+1/16/a/b^2*sum((a*d+3*b*c)/_R^3*ln(x-_R),_R =RootOf(_Z^4*b+a))
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 648, normalized size of antiderivative = 2.64 \[ \int \frac {c+d x^4}{\left (a+b x^4\right )^2} \, dx=\frac {{\left (a b^{2} x^{4} + a^{2} b\right )} \left (-\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (a^{2} b \left (-\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} + {\left (3 \, b c + a d\right )} x\right ) - {\left (-i \, a b^{2} x^{4} - i \, a^{2} b\right )} \left (-\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (i \, a^{2} b \left (-\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} + {\left (3 \, b c + a d\right )} x\right ) - {\left (i \, a b^{2} x^{4} + i \, a^{2} b\right )} \left (-\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (-i \, a^{2} b \left (-\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} + {\left (3 \, b c + a d\right )} x\right ) - {\left (a b^{2} x^{4} + a^{2} b\right )} \left (-\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} \log \left (-a^{2} b \left (-\frac {81 \, b^{4} c^{4} + 108 \, a b^{3} c^{3} d + 54 \, a^{2} b^{2} c^{2} d^{2} + 12 \, a^{3} b c d^{3} + a^{4} d^{4}}{a^{7} b^{5}}\right )^{\frac {1}{4}} + {\left (3 \, b c + a d\right )} x\right ) + 4 \, {\left (b c - a d\right )} x}{16 \, {\left (a b^{2} x^{4} + a^{2} b\right )}} \]
1/16*((a*b^2*x^4 + a^2*b)*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2 *d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*b^5))^(1/4)*log(a^2*b*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*b^ 5))^(1/4) + (3*b*c + a*d)*x) - (-I*a*b^2*x^4 - I*a^2*b)*(-(81*b^4*c^4 + 10 8*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*b^5))^ (1/4)*log(I*a^2*b*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 1 2*a^3*b*c*d^3 + a^4*d^4)/(a^7*b^5))^(1/4) + (3*b*c + a*d)*x) - (I*a*b^2*x^ 4 + I*a^2*b)*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3 *b*c*d^3 + a^4*d^4)/(a^7*b^5))^(1/4)*log(-I*a^2*b*(-(81*b^4*c^4 + 108*a*b^ 3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*b^5))^(1/4) + (3*b*c + a*d)*x) - (a*b^2*x^4 + a^2*b)*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*b^5))^(1/4)*log(-a^2* b*(-(81*b^4*c^4 + 108*a*b^3*c^3*d + 54*a^2*b^2*c^2*d^2 + 12*a^3*b*c*d^3 + a^4*d^4)/(a^7*b^5))^(1/4) + (3*b*c + a*d)*x) + 4*(b*c - a*d)*x)/(a*b^2*x^4 + a^2*b)
Time = 0.46 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.46 \[ \int \frac {c+d x^4}{\left (a+b x^4\right )^2} \, dx=\frac {x \left (- a d + b c\right )}{4 a^{2} b + 4 a b^{2} x^{4}} + \operatorname {RootSum} {\left (65536 t^{4} a^{7} b^{5} + a^{4} d^{4} + 12 a^{3} b c d^{3} + 54 a^{2} b^{2} c^{2} d^{2} + 108 a b^{3} c^{3} d + 81 b^{4} c^{4}, \left ( t \mapsto t \log {\left (\frac {16 t a^{2} b}{a d + 3 b c} + x \right )} \right )\right )} \]
x*(-a*d + b*c)/(4*a**2*b + 4*a*b**2*x**4) + RootSum(65536*_t**4*a**7*b**5 + a**4*d**4 + 12*a**3*b*c*d**3 + 54*a**2*b**2*c**2*d**2 + 108*a*b**3*c**3* d + 81*b**4*c**4, Lambda(_t, _t*log(16*_t*a**2*b/(a*d + 3*b*c) + x)))
Time = 0.29 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.96 \[ \int \frac {c+d x^4}{\left (a+b x^4\right )^2} \, dx=\frac {{\left (b c - a d\right )} x}{4 \, {\left (a b^{2} x^{4} + a^{2} b\right )}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, b c + a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (3 \, b c + a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (3 \, b c + a d\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {\sqrt {2} {\left (3 \, b c + a d\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}}}{32 \, a b} \]
1/4*(b*c - a*d)*x/(a*b^2*x^4 + a^2*b) + 1/32*(2*sqrt(2)*(3*b*c + a*d)*arct an(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b )))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(3*b*c + a*d)*arctan(1/2*s qrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqr t(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(3*b*c + a*d)*log(sqrt(b)*x^2 + sqrt (2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(3*b*c + a*d) *log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4))) /(a*b)
Time = 0.30 (sec) , antiderivative size = 266, normalized size of antiderivative = 1.09 \[ \int \frac {c+d x^4}{\left (a+b x^4\right )^2} \, dx=\frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b c + \left (a b^{3}\right )^{\frac {1}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{2}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b c + \left (a b^{3}\right )^{\frac {1}{4}} a d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{2}} + \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b c + \left (a b^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b^{2}} - \frac {\sqrt {2} {\left (3 \, \left (a b^{3}\right )^{\frac {1}{4}} b c + \left (a b^{3}\right )^{\frac {1}{4}} a d\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b^{2}} + \frac {b c x - a d x}{4 \, {\left (b x^{4} + a\right )} a b} \]
1/16*sqrt(2)*(3*(a*b^3)^(1/4)*b*c + (a*b^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)* (2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^2) + 1/16*sqrt(2)*(3*(a*b^ 3)^(1/4)*b*c + (a*b^3)^(1/4)*a*d)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^ (1/4))/(a/b)^(1/4))/(a^2*b^2) + 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b*c + (a*b^3 )^(1/4)*a*d)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^2) - 1/32 *sqrt(2)*(3*(a*b^3)^(1/4)*b*c + (a*b^3)^(1/4)*a*d)*log(x^2 - sqrt(2)*x*(a/ b)^(1/4) + sqrt(a/b))/(a^2*b^2) + 1/4*(b*c*x - a*d*x)/((b*x^4 + a)*a*b)
Time = 5.70 (sec) , antiderivative size = 740, normalized size of antiderivative = 3.02 \[ \int \frac {c+d x^4}{\left (a+b x^4\right )^2} \, dx=\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {x\,\left (a^2\,b\,d^2+6\,a\,b^2\,c\,d+9\,b^3\,c^2\right )}{4\,a^2}-\frac {\left (a\,d+3\,b\,c\right )\,\left (12\,c\,b^3+4\,a\,d\,b^2\right )\,1{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,\left (a\,d+3\,b\,c\right )}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}+\frac {\left (\frac {x\,\left (a^2\,b\,d^2+6\,a\,b^2\,c\,d+9\,b^3\,c^2\right )}{4\,a^2}+\frac {\left (a\,d+3\,b\,c\right )\,\left (12\,c\,b^3+4\,a\,d\,b^2\right )\,1{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,\left (a\,d+3\,b\,c\right )}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}}{\frac {\left (\frac {x\,\left (a^2\,b\,d^2+6\,a\,b^2\,c\,d+9\,b^3\,c^2\right )}{4\,a^2}-\frac {\left (a\,d+3\,b\,c\right )\,\left (12\,c\,b^3+4\,a\,d\,b^2\right )\,1{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,\left (a\,d+3\,b\,c\right )\,1{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}-\frac {\left (\frac {x\,\left (a^2\,b\,d^2+6\,a\,b^2\,c\,d+9\,b^3\,c^2\right )}{4\,a^2}+\frac {\left (a\,d+3\,b\,c\right )\,\left (12\,c\,b^3+4\,a\,d\,b^2\right )\,1{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,\left (a\,d+3\,b\,c\right )\,1{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}}\right )\,\left (a\,d+3\,b\,c\right )}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}}-\frac {x\,\left (a\,d-b\,c\right )}{4\,a\,b\,\left (b\,x^4+a\right )}+\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {x\,\left (a^2\,b\,d^2+6\,a\,b^2\,c\,d+9\,b^3\,c^2\right )}{4\,a^2}-\frac {\left (a\,d+3\,b\,c\right )\,\left (12\,c\,b^3+4\,a\,d\,b^2\right )}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,\left (a\,d+3\,b\,c\right )\,1{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}+\frac {\left (\frac {x\,\left (a^2\,b\,d^2+6\,a\,b^2\,c\,d+9\,b^3\,c^2\right )}{4\,a^2}+\frac {\left (a\,d+3\,b\,c\right )\,\left (12\,c\,b^3+4\,a\,d\,b^2\right )}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,\left (a\,d+3\,b\,c\right )\,1{}\mathrm {i}}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}}{\frac {\left (\frac {x\,\left (a^2\,b\,d^2+6\,a\,b^2\,c\,d+9\,b^3\,c^2\right )}{4\,a^2}-\frac {\left (a\,d+3\,b\,c\right )\,\left (12\,c\,b^3+4\,a\,d\,b^2\right )}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,\left (a\,d+3\,b\,c\right )}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}-\frac {\left (\frac {x\,\left (a^2\,b\,d^2+6\,a\,b^2\,c\,d+9\,b^3\,c^2\right )}{4\,a^2}+\frac {\left (a\,d+3\,b\,c\right )\,\left (12\,c\,b^3+4\,a\,d\,b^2\right )}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}\right )\,\left (a\,d+3\,b\,c\right )}{16\,{\left (-a\right )}^{7/4}\,b^{5/4}}}\right )\,\left (a\,d+3\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{7/4}\,b^{5/4}} \]
(atan(((((x*(9*b^3*c^2 + a^2*b*d^2 + 6*a*b^2*c*d))/(4*a^2) - ((a*d + 3*b*c )*(12*b^3*c + 4*a*b^2*d))/(16*(-a)^(7/4)*b^(5/4)))*(a*d + 3*b*c)*1i)/(16*( -a)^(7/4)*b^(5/4)) + (((x*(9*b^3*c^2 + a^2*b*d^2 + 6*a*b^2*c*d))/(4*a^2) + ((a*d + 3*b*c)*(12*b^3*c + 4*a*b^2*d))/(16*(-a)^(7/4)*b^(5/4)))*(a*d + 3* b*c)*1i)/(16*(-a)^(7/4)*b^(5/4)))/((((x*(9*b^3*c^2 + a^2*b*d^2 + 6*a*b^2*c *d))/(4*a^2) - ((a*d + 3*b*c)*(12*b^3*c + 4*a*b^2*d))/(16*(-a)^(7/4)*b^(5/ 4)))*(a*d + 3*b*c))/(16*(-a)^(7/4)*b^(5/4)) - (((x*(9*b^3*c^2 + a^2*b*d^2 + 6*a*b^2*c*d))/(4*a^2) + ((a*d + 3*b*c)*(12*b^3*c + 4*a*b^2*d))/(16*(-a)^ (7/4)*b^(5/4)))*(a*d + 3*b*c))/(16*(-a)^(7/4)*b^(5/4))))*(a*d + 3*b*c)*1i) /(8*(-a)^(7/4)*b^(5/4)) + (atan(((((x*(9*b^3*c^2 + a^2*b*d^2 + 6*a*b^2*c*d ))/(4*a^2) - ((a*d + 3*b*c)*(12*b^3*c + 4*a*b^2*d)*1i)/(16*(-a)^(7/4)*b^(5 /4)))*(a*d + 3*b*c))/(16*(-a)^(7/4)*b^(5/4)) + (((x*(9*b^3*c^2 + a^2*b*d^2 + 6*a*b^2*c*d))/(4*a^2) + ((a*d + 3*b*c)*(12*b^3*c + 4*a*b^2*d)*1i)/(16*( -a)^(7/4)*b^(5/4)))*(a*d + 3*b*c))/(16*(-a)^(7/4)*b^(5/4)))/((((x*(9*b^3*c ^2 + a^2*b*d^2 + 6*a*b^2*c*d))/(4*a^2) - ((a*d + 3*b*c)*(12*b^3*c + 4*a*b^ 2*d)*1i)/(16*(-a)^(7/4)*b^(5/4)))*(a*d + 3*b*c)*1i)/(16*(-a)^(7/4)*b^(5/4) ) - (((x*(9*b^3*c^2 + a^2*b*d^2 + 6*a*b^2*c*d))/(4*a^2) + ((a*d + 3*b*c)*( 12*b^3*c + 4*a*b^2*d)*1i)/(16*(-a)^(7/4)*b^(5/4)))*(a*d + 3*b*c)*1i)/(16*( -a)^(7/4)*b^(5/4))))*(a*d + 3*b*c))/(8*(-a)^(7/4)*b^(5/4)) - (x*(a*d - b*c ))/(4*a*b*(a + b*x^4))