Integrand size = 17, antiderivative size = 680 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )} \, dx=-\frac {e^3}{2 \left (c d^4+a e^4\right ) (d+e x)^2}-\frac {4 c d^3 e^3}{\left (c d^4+a e^4\right )^2 (d+e x)}-\frac {\sqrt {c} e \left (3 c^2 d^8-12 a c d^4 e^4+a^2 e^8\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (c d^4+a e^4\right )^3}-\frac {c^{3/4} d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}+\frac {c^{3/4} d \left (c^2 d^8-12 a c d^4 e^4+3 a^2 e^8-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^3}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (c d^4+a e^4\right )^3} \]
-1/2*e^3/(a*e^4+c*d^4)/(e*x+d)^2-4*c*d^3*e^3/(a*e^4+c*d^4)^2/(e*x+d)+2*c*d ^2*e^3*(-3*a*e^4+5*c*d^4)*ln(e*x+d)/(a*e^4+c*d^4)^3-1/2*c*d^2*e^3*(-3*a*e^ 4+5*c*d^4)*ln(c*x^4+a)/(a*e^4+c*d^4)^3-1/2*e*(a^2*e^8-12*a*c*d^4*e^4+3*c^2 *d^8)*arctan(x^2*c^(1/2)/a^(1/2))*c^(1/2)/(a*e^4+c*d^4)^3/a^(1/2)-1/8*c^(3 /4)*d*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(c^2*d^8-12*a*c*d ^4*e^4+3*a^2*e^8-2*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a* e^4+c*d^4)^3*2^(1/2)+1/8*c^(3/4)*d*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^ 2*c^(1/2))*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8-2*d^2*e^2*(-5*a*e^4+3*c*d^4)* a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/4*c^(3/4)*d*arctan(-1+c ^(1/4)*x*2^(1/2)/a^(1/4))*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8+2*d^2*e^2*(-5* a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1/2)+1/4*c^(3/4 )*d*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(c^2*d^8-12*a*c*d^4*e^4+3*a^2*e^8+ 2*d^2*e^2*(-5*a*e^4+3*c*d^4)*a^(1/2)*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^3*2^(1 /2)
Time = 0.58 (sec) , antiderivative size = 738, normalized size of antiderivative = 1.09 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )} \, dx=\frac {-4 a^{3/4} e^3 \left (c d^4+a e^4\right )^2-32 a^{3/4} c d^3 e^3 \left (c d^4+a e^4\right ) (d+e x)-2 \sqrt {c} \left (\sqrt {2} c^{9/4} d^9-6 \sqrt [4]{a} c^2 d^8 e+6 \sqrt {2} \sqrt {a} c^{7/4} d^7 e^2-12 \sqrt {2} a c^{5/4} d^5 e^4+24 a^{5/4} c d^4 e^5-10 \sqrt {2} a^{3/2} c^{3/4} d^3 e^6+3 \sqrt {2} a^2 \sqrt [4]{c} d e^8-2 a^{9/4} e^9\right ) (d+e x)^2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt {c} \left (\sqrt {2} c^{9/4} d^9+6 \sqrt [4]{a} c^2 d^8 e+6 \sqrt {2} \sqrt {a} c^{7/4} d^7 e^2-12 \sqrt {2} a c^{5/4} d^5 e^4-24 a^{5/4} c d^4 e^5-10 \sqrt {2} a^{3/2} c^{3/4} d^3 e^6+3 \sqrt {2} a^2 \sqrt [4]{c} d e^8+2 a^{9/4} e^9\right ) (d+e x)^2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+16 a^{3/4} c d^2 e^3 \left (5 c d^4-3 a e^4\right ) (d+e x)^2 \log (d+e x)-\sqrt {2} c^{3/4} d \left (c^2 d^8-6 \sqrt {a} c^{3/2} d^6 e^2-12 a c d^4 e^4+10 a^{3/2} \sqrt {c} d^2 e^6+3 a^2 e^8\right ) (d+e x)^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} c^{3/4} d \left (c^2 d^8-6 \sqrt {a} c^{3/2} d^6 e^2-12 a c d^4 e^4+10 a^{3/2} \sqrt {c} d^2 e^6+3 a^2 e^8\right ) (d+e x)^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+4 a^{3/4} c d^2 e^3 \left (-5 c d^4+3 a e^4\right ) (d+e x)^2 \log \left (a+c x^4\right )}{8 a^{3/4} \left (c d^4+a e^4\right )^3 (d+e x)^2} \]
(-4*a^(3/4)*e^3*(c*d^4 + a*e^4)^2 - 32*a^(3/4)*c*d^3*e^3*(c*d^4 + a*e^4)*( d + e*x) - 2*Sqrt[c]*(Sqrt[2]*c^(9/4)*d^9 - 6*a^(1/4)*c^2*d^8*e + 6*Sqrt[2 ]*Sqrt[a]*c^(7/4)*d^7*e^2 - 12*Sqrt[2]*a*c^(5/4)*d^5*e^4 + 24*a^(5/4)*c*d^ 4*e^5 - 10*Sqrt[2]*a^(3/2)*c^(3/4)*d^3*e^6 + 3*Sqrt[2]*a^2*c^(1/4)*d*e^8 - 2*a^(9/4)*e^9)*(d + e*x)^2*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*Sq rt[c]*(Sqrt[2]*c^(9/4)*d^9 + 6*a^(1/4)*c^2*d^8*e + 6*Sqrt[2]*Sqrt[a]*c^(7/ 4)*d^7*e^2 - 12*Sqrt[2]*a*c^(5/4)*d^5*e^4 - 24*a^(5/4)*c*d^4*e^5 - 10*Sqrt [2]*a^(3/2)*c^(3/4)*d^3*e^6 + 3*Sqrt[2]*a^2*c^(1/4)*d*e^8 + 2*a^(9/4)*e^9) *(d + e*x)^2*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 16*a^(3/4)*c*d^2*e^ 3*(5*c*d^4 - 3*a*e^4)*(d + e*x)^2*Log[d + e*x] - Sqrt[2]*c^(3/4)*d*(c^2*d^ 8 - 6*Sqrt[a]*c^(3/2)*d^6*e^2 - 12*a*c*d^4*e^4 + 10*a^(3/2)*Sqrt[c]*d^2*e^ 6 + 3*a^2*e^8)*(d + e*x)^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[ c]*x^2] + Sqrt[2]*c^(3/4)*d*(c^2*d^8 - 6*Sqrt[a]*c^(3/2)*d^6*e^2 - 12*a*c* d^4*e^4 + 10*a^(3/2)*Sqrt[c]*d^2*e^6 + 3*a^2*e^8)*(d + e*x)^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + 4*a^(3/4)*c*d^2*e^3*(-5*c*d^4 + 3*a*e^4)*(d + e*x)^2*Log[a + c*x^4])/(8*a^(3/4)*(c*d^4 + a*e^4)^3*(d + e*x)^2)
Time = 1.22 (sec) , antiderivative size = 680, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {7276, 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.
| \(\displaystyle \int \frac {1}{\left (a+c x^4\right ) (d+e x)^3} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {c \left (-e x \left (a^2 e^8-12 a c d^4 e^4+3 c^2 d^8\right )+d \left (3 a^2 e^8-12 a c d^4 e^4+c^2 d^8\right )+2 c d^3 e^2 x^2 \left (3 c d^4-5 a e^4\right )-2 c d^2 e^3 x^3 \left (5 c d^4-3 a e^4\right )\right )}{\left (a+c x^4\right ) \left (a e^4+c d^4\right )^3}+\frac {e^4}{(d+e x)^3 \left (a e^4+c d^4\right )}+\frac {4 c d^3 e^4}{(d+e x)^2 \left (a e^4+c d^4\right )^2}+\frac {2 c d^2 e^4 \left (5 c d^4-3 a e^4\right )}{(d+e x) \left (a e^4+c d^4\right )^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {c} e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) \left (a^2 e^8-12 a c d^4 e^4+3 c^2 d^8\right )}{2 \sqrt {a} \left (a e^4+c d^4\right )^3}-\frac {c^{3/4} d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (3 a^2 e^8-12 a c d^4 e^4+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}+\frac {c^{3/4} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (3 a^2 e^8-12 a c d^4 e^4+2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}-\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}+\frac {c^{3/4} d \left (3 a^2 e^8-12 a c d^4 e^4-2 \sqrt {a} \sqrt {c} d^2 e^2 \left (3 c d^4-5 a e^4\right )+c^2 d^8\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^3}-\frac {e^3}{2 (d+e x)^2 \left (a e^4+c d^4\right )}-\frac {4 c d^3 e^3}{(d+e x) \left (a e^4+c d^4\right )^2}-\frac {c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log \left (a+c x^4\right )}{2 \left (a e^4+c d^4\right )^3}+\frac {2 c d^2 e^3 \left (5 c d^4-3 a e^4\right ) \log (d+e x)}{\left (a e^4+c d^4\right )^3}\) |
-1/2*e^3/((c*d^4 + a*e^4)*(d + e*x)^2) - (4*c*d^3*e^3)/((c*d^4 + a*e^4)^2* (d + e*x)) - (Sqrt[c]*e*(3*c^2*d^8 - 12*a*c*d^4*e^4 + a^2*e^8)*ArcTan[(Sqr t[c]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^4 + a*e^4)^3) - (c^(3/4)*d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8 + 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4) )*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e ^4)^3) + (c^(3/4)*d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8 + 2*Sqrt[a]*Sqrt [c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/ (2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) + (2*c*d^2*e^3*(5*c*d^4 - 3*a*e^4)*L og[d + e*x])/(c*d^4 + a*e^4)^3 - (c^(3/4)*d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3* a^2*e^8 - 2*Sqrt[a]*Sqrt[c]*d^2*e^2*(3*c*d^4 - 5*a*e^4))*Log[Sqrt[a] - Sqr t[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^ 3) + (c^(3/4)*d*(c^2*d^8 - 12*a*c*d^4*e^4 + 3*a^2*e^8 - 2*Sqrt[a]*Sqrt[c]* d^2*e^2*(3*c*d^4 - 5*a*e^4))*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqr t[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^3) - (c*d^2*e^3*(5*c*d^4 - 3 *a*e^4)*Log[a + c*x^4])/(2*(c*d^4 + a*e^4)^3)
3.4.100.3.1 Defintions of rubi rules used
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Time = 0.97 (sec) , antiderivative size = 446, normalized size of antiderivative = 0.66
| method | result | size |
| default | \(\frac {c \left (\frac {\left (3 a^{2} d \,e^{8}-12 a c \,d^{5} e^{4}+c^{2} d^{9}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-a^{2} e^{9}+12 a c \,d^{4} e^{5}-3 c^{2} d^{8} e \right ) \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{2 \sqrt {a c}}+\frac {\left (-10 a c \,d^{3} e^{6}+6 c^{2} d^{7} e^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}+\frac {\left (6 a c \,d^{2} e^{7}-10 c^{2} d^{6} e^{3}\right ) \ln \left (c \,x^{4}+a \right )}{4 c}\right )}{\left (e^{4} a +d^{4} c \right )^{3}}-\frac {e^{3}}{2 \left (e^{4} a +d^{4} c \right ) \left (e x +d \right )^{2}}-\frac {4 c \,d^{3} e^{3}}{\left (e^{4} a +d^{4} c \right )^{2} \left (e x +d \right )}-\frac {2 c \,d^{2} e^{3} \left (3 e^{4} a -5 d^{4} c \right ) \ln \left (e x +d \right )}{\left (e^{4} a +d^{4} c \right )^{3}}\) | \(446\) |
| risch | \(\frac {-\frac {4 d^{3} c \,e^{4} x}{a^{2} e^{8}+2 a c \,d^{4} e^{4}+c^{2} d^{8}}-\frac {\left (e^{4} a +9 d^{4} c \right ) e^{3}}{2 \left (a^{2} e^{8}+2 a c \,d^{4} e^{4}+c^{2} d^{8}\right )}}{\left (e x +d \right )^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{6} e^{12}+3 a^{5} c \,d^{4} e^{8}+3 a^{4} c^{2} d^{8} e^{4}+a^{3} c^{3} d^{12}\right ) \textit {\_Z}^{4}+\left (-24 a^{4} c \,d^{2} e^{7}+40 a^{3} c^{2} d^{6} e^{3}\right ) \textit {\_Z}^{3}+\left (2 a^{3} c \,e^{6}+42 a^{2} c^{2} d^{4} e^{2}\right ) \textit {\_Z}^{2}+12 a \,c^{2} d^{2} e \textit {\_Z} +c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 a^{7} e^{22}+17 a^{6} c \,d^{4} e^{18}+18 a^{5} c^{2} d^{8} e^{14}+2 a^{4} c^{3} d^{12} e^{10}-7 a^{3} c^{4} d^{16} e^{6}-3 a^{2} c^{5} d^{20} e^{2}\right ) \textit {\_R}^{4}+\left (-81 a^{5} c \,d^{2} e^{17}-56 a^{4} c^{2} d^{6} e^{13}+126 a^{3} c^{3} d^{10} e^{9}+96 a^{2} c^{4} d^{14} e^{5}-5 a \,c^{5} d^{18} e \right ) \textit {\_R}^{3}+\left (9 a^{4} c \,e^{16}-38 a^{3} c^{2} d^{4} e^{12}+408 a^{2} c^{3} d^{8} e^{8}+198 a \,c^{4} d^{12} e^{4}-c^{5} d^{16}\right ) \textit {\_R}^{2}+\left (72 a^{2} c^{2} d^{2} e^{11}-16 a \,c^{3} d^{6} e^{7}+40 c^{4} d^{10} e^{3}\right ) \textit {\_R} +4 a \,c^{2} e^{10}+4 c^{3} d^{4} e^{6}\right ) x +\left (6 a^{7} d \,e^{21}+22 a^{6} c \,d^{5} e^{17}+28 a^{5} c^{2} d^{9} e^{13}+12 a^{4} c^{3} d^{13} e^{9}-2 a^{3} c^{4} d^{17} e^{5}-2 a^{2} c^{5} d^{21} e \right ) \textit {\_R}^{4}+\left (-61 a^{5} c \,d^{3} e^{16}-56 a^{4} c^{2} d^{7} e^{12}+70 a^{3} c^{3} d^{11} e^{8}+64 a^{2} c^{4} d^{15} e^{4}-a \,c^{5} d^{19}\right ) \textit {\_R}^{3}+\left (4 a^{4} c d \,e^{15}-148 a^{3} c^{2} d^{5} e^{11}+652 a^{2} c^{3} d^{9} e^{7}+36 a \,c^{4} d^{13} e^{3}\right ) \textit {\_R}^{2}+\left (-48 a^{2} c^{2} d^{3} e^{10}+208 a \,c^{3} d^{7} e^{6}\right ) \textit {\_R} +4 a \,c^{2} d \,e^{9}+4 c^{3} d^{5} e^{5}\right )\right )}{4}-\frac {6 c \,d^{2} e^{7} \ln \left (e x +d \right ) a}{a^{3} e^{12}+3 a^{2} c \,d^{4} e^{8}+3 a \,c^{2} d^{8} e^{4}+c^{3} d^{12}}+\frac {10 c^{2} d^{6} e^{3} \ln \left (e x +d \right )}{a^{3} e^{12}+3 a^{2} c \,d^{4} e^{8}+3 a \,c^{2} d^{8} e^{4}+c^{3} d^{12}}\) | \(867\) |
c/(a*e^4+c*d^4)^3*(1/8*(3*a^2*d*e^8-12*a*c*d^5*e^4+c^2*d^9)*(a/c)^(1/4)/a* 2^(1/2)*(ln((x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2-(a/c)^(1/4)*x*2^( 1/2)+(a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c )^(1/4)*x-1))+1/2*(-a^2*e^9+12*a*c*d^4*e^5-3*c^2*d^8*e)/(a*c)^(1/2)*arctan (x^2*(c/a)^(1/2))+1/8*(-10*a*c*d^3*e^6+6*c^2*d^7*e^2)/c/(a/c)^(1/4)*2^(1/2 )*(ln((x^2-(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+( a/c)^(1/2)))+2*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+2*arctan(2^(1/2)/(a/c)^(1/4 )*x-1))+1/4*(6*a*c*d^2*e^7-10*c^2*d^6*e^3)/c*ln(c*x^4+a))-1/2*e^3/(a*e^4+c *d^4)/(e*x+d)^2-4*c*d^3*e^3/(a*e^4+c*d^4)^2/(e*x+d)-2*c*d^2*e^3*(3*a*e^4-5 *c*d^4)/(a*e^4+c*d^4)^3*ln(e*x+d)
Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 817, normalized size of antiderivative = 1.20 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )} \, dx=-\frac {c {\left (\frac {\sqrt {2} {\left (10 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {9}{4}} d^{6} e^{3} - 6 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {5}{4}} d^{2} e^{7} - c^{3} d^{9} + 6 \, \sqrt {a} c^{\frac {5}{2}} d^{7} e^{2} + 12 \, a c^{2} d^{5} e^{4} - 10 \, a^{\frac {3}{2}} c^{\frac {3}{2}} d^{3} e^{6} - 3 \, a^{2} c d e^{8}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (10 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {9}{4}} d^{6} e^{3} - 6 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {5}{4}} d^{2} e^{7} + c^{3} d^{9} - 6 \, \sqrt {a} c^{\frac {5}{2}} d^{7} e^{2} - 12 \, a c^{2} d^{5} e^{4} + 10 \, a^{\frac {3}{2}} c^{\frac {3}{2}} d^{3} e^{6} + 3 \, a^{2} c d e^{8}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {13}{4}} d^{9} + 6 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {11}{4}} d^{7} e^{2} - 12 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {9}{4}} d^{5} e^{4} - 10 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {7}{4}} d^{3} e^{6} + 3 \, \sqrt {2} a^{\frac {9}{4}} c^{\frac {5}{4}} d e^{8} + 6 \, \sqrt {a} c^{3} d^{8} e - 24 \, a^{\frac {3}{2}} c^{2} d^{4} e^{5} + 2 \, a^{\frac {5}{2}} c e^{9}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {13}{4}} d^{9} + 6 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {11}{4}} d^{7} e^{2} - 12 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {9}{4}} d^{5} e^{4} - 10 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {7}{4}} d^{3} e^{6} + 3 \, \sqrt {2} a^{\frac {9}{4}} c^{\frac {5}{4}} d e^{8} - 6 \, \sqrt {a} c^{3} d^{8} e + 24 \, a^{\frac {3}{2}} c^{2} d^{4} e^{5} - 2 \, a^{\frac {5}{2}} c e^{9}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}}\right )}}{8 \, {\left (c^{3} d^{12} + 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{4} e^{8} + a^{3} e^{12}\right )}} + \frac {2 \, {\left (5 \, c^{2} d^{6} e^{3} - 3 \, a c d^{2} e^{7}\right )} \log \left (e x + d\right )}{c^{3} d^{12} + 3 \, a c^{2} d^{8} e^{4} + 3 \, a^{2} c d^{4} e^{8} + a^{3} e^{12}} - \frac {8 \, c d^{3} e^{4} x + 9 \, c d^{4} e^{3} + a e^{7}}{2 \, {\left (c^{2} d^{10} + 2 \, a c d^{6} e^{4} + a^{2} d^{2} e^{8} + {\left (c^{2} d^{8} e^{2} + 2 \, a c d^{4} e^{6} + a^{2} e^{10}\right )} x^{2} + 2 \, {\left (c^{2} d^{9} e + 2 \, a c d^{5} e^{5} + a^{2} d e^{9}\right )} x\right )}} \]
-1/8*c*(sqrt(2)*(10*sqrt(2)*a^(3/4)*c^(9/4)*d^6*e^3 - 6*sqrt(2)*a^(7/4)*c^ (5/4)*d^2*e^7 - c^3*d^9 + 6*sqrt(a)*c^(5/2)*d^7*e^2 + 12*a*c^2*d^5*e^4 - 1 0*a^(3/2)*c^(3/2)*d^3*e^6 - 3*a^2*c*d*e^8)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/ 4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) + sqrt(2)*(10*sqrt(2)*a^(3/4)*c^ (9/4)*d^6*e^3 - 6*sqrt(2)*a^(7/4)*c^(5/4)*d^2*e^7 + c^3*d^9 - 6*sqrt(a)*c^ (5/2)*d^7*e^2 - 12*a*c^2*d^5*e^4 + 10*a^(3/2)*c^(3/2)*d^3*e^6 + 3*a^2*c*d* e^8)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(5/ 4)) - 2*(sqrt(2)*a^(1/4)*c^(13/4)*d^9 + 6*sqrt(2)*a^(3/4)*c^(11/4)*d^7*e^2 - 12*sqrt(2)*a^(5/4)*c^(9/4)*d^5*e^4 - 10*sqrt(2)*a^(7/4)*c^(7/4)*d^3*e^6 + 3*sqrt(2)*a^(9/4)*c^(5/4)*d*e^8 + 6*sqrt(a)*c^3*d^8*e - 24*a^(3/2)*c^2* d^4*e^5 + 2*a^(5/2)*c*e^9)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/ 4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(5/4)) - 2*(sqrt(2)*a^(1/4)*c^(13/4)*d^9 + 6*sqrt(2)*a^(3/4)*c^(11/4)*d^7*e^2 - 12*sqrt(2)*a^(5/4)*c^(9/4)*d^5*e^4 - 10*sqrt(2)*a^(7/4)*c^(7/4)*d^3*e^6 + 3*sqrt(2)*a^(9/4)*c^(5/4)*d*e^8 - 6*sqrt(a)*c^3*d^8*e + 24*a^(3/2)*c^2*d^4 *e^5 - 2*a^(5/2)*c*e^9)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)* c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(5/4)))/( c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8 + a^3*e^12) + 2*(5*c^2*d^6*e^ 3 - 3*a*c*d^2*e^7)*log(e*x + d)/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4* e^8 + a^3*e^12) - 1/2*(8*c*d^3*e^4*x + 9*c*d^4*e^3 + a*e^7)/(c^2*d^10 +...
Time = 0.37 (sec) , antiderivative size = 941, normalized size of antiderivative = 1.38 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )} \, dx=\text {Too large to display} \]
1/4*(2*a*c^2*e^3 + sqrt(2)*(a*c^3)^(1/4)*c^2*d^3 - 3*sqrt(2)*(a*c^3)^(3/4) *d*e^2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^3 *d^6 + 9*a^2*c^2*d^2*e^4 - 3*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^5*e - 3*sqrt(2) *(a*c^3)^(1/4)*a^2*c*d*e^5 + 9*sqrt(a*c)*a*c^2*d^4*e^2 + sqrt(a*c)*a^2*c*e ^6 - 8*sqrt(2)*(a*c^3)^(3/4)*a*d^3*e^3) - 1/4*(2*a*c^2*e^3 - sqrt(2)*(a*c^ 3)^(1/4)*c^2*d^3 + 3*sqrt(2)*(a*c^3)^(3/4)*d*e^2)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^3*d^6 + 9*a^2*c^2*d^2*e^4 + 3*sqr t(2)*(a*c^3)^(1/4)*a*c^2*d^5*e + 3*sqrt(2)*(a*c^3)^(1/4)*a^2*c*d*e^5 - 9*s qrt(a*c)*a*c^2*d^4*e^2 + sqrt(a*c)*a^2*c*e^6 + 8*sqrt(2)*(a*c^3)^(3/4)*a*d ^3*e^3) + 1/8*(sqrt(2)*(a*c^3)^(1/4)*c^3*d^9 - 12*sqrt(2)*(a*c^3)^(1/4)*a* c^2*d^5*e^4 + 3*sqrt(2)*(a*c^3)^(1/4)*a^2*c*d*e^8 - 6*sqrt(2)*(a*c^3)^(3/4 )*c*d^7*e^2 + 10*sqrt(2)*(a*c^3)^(3/4)*a*d^3*e^6)*log(x^2 + sqrt(2)*x*(a/c )^(1/4) + sqrt(a/c))/(a*c^4*d^12 + 3*a^2*c^3*d^8*e^4 + 3*a^3*c^2*d^4*e^8 + a^4*c*e^12) - 1/8*(sqrt(2)*(a*c^3)^(1/4)*c^3*d^9 - 12*sqrt(2)*(a*c^3)^(1/ 4)*a*c^2*d^5*e^4 + 3*sqrt(2)*(a*c^3)^(1/4)*a^2*c*d*e^8 - 6*sqrt(2)*(a*c^3) ^(3/4)*c*d^7*e^2 + 10*sqrt(2)*(a*c^3)^(3/4)*a*d^3*e^6)*log(x^2 - sqrt(2)*x *(a/c)^(1/4) + sqrt(a/c))/(a*c^4*d^12 + 3*a^2*c^3*d^8*e^4 + 3*a^3*c^2*d^4* e^8 + a^4*c*e^12) - 1/2*(5*c^2*d^6*e^3 - 3*a*c*d^2*e^7)*log(abs(c*x^4 + a) )/(c^3*d^12 + 3*a*c^2*d^8*e^4 + 3*a^2*c*d^4*e^8 + a^3*e^12) + 2*(5*c^2*d^6 *e^4 - 3*a*c*d^2*e^8)*log(abs(e*x + d))/(c^3*d^12*e + 3*a*c^2*d^8*e^5 +...
Time = 9.89 (sec) , antiderivative size = 1955, normalized size of antiderivative = 2.88 \[ \int \frac {1}{(d+e x)^3 \left (a+c x^4\right )} \, dx=\text {Too large to display} \]
symsum(log((c^7*d^5*e^6 + a*c^6*d*e^10)/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^1 2*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8) + root(768*a^5*c*d^4*e^8*z^4 + 768*a^4*c^2*d^8*e^4*z^4 + 256*a^3*c^3*d^12*z^4 + 256*a^6*e^12*z^4 - 153 6*a^4*c*d^2*e^7*z^3 + 2560*a^3*c^2*d^6*e^3*z^3 + 672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z^2 + 48*a*c^2*d^2*e*z + c^2, z, k)*((208*a*c^7*d^7*e^7 - 48 *a^2*c^6*d^3*e^11)/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e ^12 + 6*a^2*c^2*d^8*e^8) + root(768*a^5*c*d^4*e^8*z^4 + 768*a^4*c^2*d^8*e^ 4*z^4 + 256*a^3*c^3*d^12*z^4 + 256*a^6*e^12*z^4 - 1536*a^4*c*d^2*e^7*z^3 + 2560*a^3*c^2*d^6*e^3*z^3 + 672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z^2 + 4 8*a*c^2*d^2*e*z + c^2, z, k)*((144*a*c^8*d^13*e^4 + 16*a^4*c^5*d*e^16 + 26 08*a^2*c^7*d^9*e^8 - 592*a^3*c^6*d^5*e^12)/(a^4*e^16 + c^4*d^16 + 4*a*c^3* d^12*e^4 + 4*a^3*c*d^4*e^12 + 6*a^2*c^2*d^8*e^8) - root(768*a^5*c*d^4*e^8* z^4 + 768*a^4*c^2*d^8*e^4*z^4 + 256*a^3*c^3*d^12*z^4 + 256*a^6*e^12*z^4 - 1536*a^4*c*d^2*e^7*z^3 + 2560*a^3*c^2*d^6*e^3*z^3 + 672*a^2*c^2*d^4*e^2*z^ 2 + 32*a^3*c*e^6*z^2 + 48*a*c^2*d^2*e*z + c^2, z, k)*((896*a^4*c^6*d^7*e^1 3 - 1120*a^3*c^7*d^11*e^9 - 1024*a^2*c^8*d^15*e^5 + 976*a^5*c^5*d^3*e^17 + 16*a*c^9*d^19*e)/(a^4*e^16 + c^4*d^16 + 4*a*c^3*d^12*e^4 + 4*a^3*c*d^4*e^ 12 + 6*a^2*c^2*d^8*e^8) - root(768*a^5*c*d^4*e^8*z^4 + 768*a^4*c^2*d^8*e^4 *z^4 + 256*a^3*c^3*d^12*z^4 + 256*a^6*e^12*z^4 - 1536*a^4*c*d^2*e^7*z^3 + 2560*a^3*c^2*d^6*e^3*z^3 + 672*a^2*c^2*d^4*e^2*z^2 + 32*a^3*c*e^6*z^2 +...