Integrand size = 17, antiderivative size = 855 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {a e^3+c x \left (d^3-d^2 e x+d e^2 x^2\right )}{4 a \left (c d^4+a e^4\right ) \left (a+c x^4\right )}-\frac {\sqrt {c} d^2 e^5 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\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 )^2}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2+\sqrt {a} e^2\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 )^2}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {e^7 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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 )^2}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d e^4 \left (\sqrt {c} d^2-\sqrt {a} e^2\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 )^2}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}-\frac {e^7 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )^2} \]
1/4*(a*e^3+c*x*(d*e^2*x^2-d^2*e*x+d^3))/a/(a*e^4+c*d^4)/(c*x^4+a)+e^7*ln(e *x+d)/(a*e^4+c*d^4)^2-1/4*e^7*ln(c*x^4+a)/(a*e^4+c*d^4)^2-1/4*d^2*e*arctan (x^2*c^(1/2)/a^(1/2))*c^(1/2)/a^(3/2)/(a*e^4+c*d^4)-1/2*d^2*e^5*arctan(x^2 *c^(1/2)/a^(1/2))*c^(1/2)/(a*e^4+c*d^4)^2/a^(1/2)-1/8*c^(1/4)*d*e^4*ln(-a^ (1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-e^2*a^(1/2)+d^2*c^(1/2))/a^ (3/4)/(a*e^4+c*d^4)^2*2^(1/2)+1/8*c^(1/4)*d*e^4*ln(a^(1/4)*c^(1/4)*x*2^(1/ 2)+a^(1/2)+x^2*c^(1/2))*(-e^2*a^(1/2)+d^2*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^2 *2^(1/2)+1/4*c^(1/4)*d*e^4*arctan(-1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*a^(1/ 2)+d^2*c^(1/2))/a^(3/4)/(a*e^4+c*d^4)^2*2^(1/2)+1/4*c^(1/4)*d*e^4*arctan(1 +c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*a^(1/2)+d^2*c^(1/2))/a^(3/4)/(a*e^4+c*d^4 )^2*2^(1/2)-1/32*c^(1/4)*d*ln(-a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/ 2))*(-e^2*a^(1/2)+3*d^2*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)*2^(1/2)+1/32*c^(1/4 )*d*ln(a^(1/4)*c^(1/4)*x*2^(1/2)+a^(1/2)+x^2*c^(1/2))*(-e^2*a^(1/2)+3*d^2* c^(1/2))/a^(7/4)/(a*e^4+c*d^4)*2^(1/2)+1/16*c^(1/4)*d*arctan(-1+c^(1/4)*x* 2^(1/2)/a^(1/4))*(e^2*a^(1/2)+3*d^2*c^(1/2))/a^(7/4)/(a*e^4+c*d^4)*2^(1/2) +1/16*c^(1/4)*d*arctan(1+c^(1/4)*x*2^(1/2)/a^(1/4))*(e^2*a^(1/2)+3*d^2*c^( 1/2))/a^(7/4)/(a*e^4+c*d^4)*2^(1/2)
Time = 0.26 (sec) , antiderivative size = 558, normalized size of antiderivative = 0.65 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {\frac {8 \left (c d^4+a e^4\right ) \left (a e^3+c d x \left (d^2-d e x+e^2 x^2\right )\right )}{a \left (a+c x^4\right )}-\frac {2 \sqrt [4]{c} d \left (3 \sqrt {2} c^{3/2} d^6-4 \sqrt [4]{a} c^{5/4} d^5 e+\sqrt {2} \sqrt {a} c d^4 e^2+7 \sqrt {2} a \sqrt {c} d^2 e^4-12 a^{5/4} \sqrt [4]{c} d e^5+5 \sqrt {2} a^{3/2} e^6\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {2 \sqrt [4]{c} d \left (3 \sqrt {2} c^{3/2} d^6+4 \sqrt [4]{a} c^{5/4} d^5 e+\sqrt {2} \sqrt {a} c d^4 e^2+7 \sqrt {2} a \sqrt {c} d^2 e^4+12 a^{5/4} \sqrt [4]{c} d e^5+5 \sqrt {2} a^{3/2} e^6\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+32 e^7 \log (d+e x)+\frac {\sqrt {2} \sqrt [4]{c} \left (-3 c^{3/2} d^7+\sqrt {a} c d^5 e^2-7 a \sqrt {c} d^3 e^4+5 a^{3/2} d e^6\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}+\frac {\sqrt {2} \sqrt [4]{c} \left (3 c^{3/2} d^7-\sqrt {a} c d^5 e^2+7 a \sqrt {c} d^3 e^4-5 a^{3/2} d e^6\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{7/4}}-8 e^7 \log \left (a+c x^4\right )}{32 \left (c d^4+a e^4\right )^2} \]
((8*(c*d^4 + a*e^4)*(a*e^3 + c*d*x*(d^2 - d*e*x + e^2*x^2)))/(a*(a + c*x^4 )) - (2*c^(1/4)*d*(3*Sqrt[2]*c^(3/2)*d^6 - 4*a^(1/4)*c^(5/4)*d^5*e + Sqrt[ 2]*Sqrt[a]*c*d^4*e^2 + 7*Sqrt[2]*a*Sqrt[c]*d^2*e^4 - 12*a^(5/4)*c^(1/4)*d* e^5 + 5*Sqrt[2]*a^(3/2)*e^6)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7 /4) + (2*c^(1/4)*d*(3*Sqrt[2]*c^(3/2)*d^6 + 4*a^(1/4)*c^(5/4)*d^5*e + Sqrt [2]*Sqrt[a]*c*d^4*e^2 + 7*Sqrt[2]*a*Sqrt[c]*d^2*e^4 + 12*a^(5/4)*c^(1/4)*d *e^5 + 5*Sqrt[2]*a^(3/2)*e^6)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^( 7/4) + 32*e^7*Log[d + e*x] + (Sqrt[2]*c^(1/4)*(-3*c^(3/2)*d^7 + Sqrt[a]*c* d^5*e^2 - 7*a*Sqrt[c]*d^3*e^4 + 5*a^(3/2)*d*e^6)*Log[Sqrt[a] - Sqrt[2]*a^( 1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) + (Sqrt[2]*c^(1/4)*(3*c^(3/2)*d^7 - Sqrt[a]*c*d^5*e^2 + 7*a*Sqrt[c]*d^3*e^4 - 5*a^(3/2)*d*e^6)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(7/4) - 8*e^7*Log[a + c*x^4])/ (32*(c*d^4 + a*e^4)^2)
Time = 1.23 (sec) , antiderivative size = 855, 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 = {7293, 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.
| \(\displaystyle \int \frac {1}{\left (a+c x^4\right )^2 (d+e x)} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (\frac {e^8}{(d+e x) \left (a e^4+c d^4\right )^2}-\frac {c e^4 \left (-d^3+d^2 e x-d e^2 x^2+e^3 x^3\right )}{\left (a+c x^4\right ) \left (a e^4+c d^4\right )^2}+\frac {c \left (d^3-d^2 e x+d e^2 x^2-e^3 x^3\right )}{\left (a+c x^4\right )^2 \left (a e^4+c d^4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\log (d+e x) e^7}{\left (c d^4+a e^4\right )^2}-\frac {\log \left (c x^4+a\right ) e^7}{4 \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e^5}{2 \sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) e^4}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) e^4}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right ) e^4}{4 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}-\frac {\sqrt {c} d^2 \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) e}{4 a^{3/2} \left (c d^4+a e^4\right )}+\frac {a e^3+c x \left (d^3-e x d^2+e^2 x^2 d\right )}{4 a \left (c d^4+a e^4\right ) \left (c x^4+a\right )}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (3 \sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {c} x^2+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}\right )}{16 \sqrt {2} a^{7/4} \left (c d^4+a e^4\right )}\) |
(a*e^3 + c*x*(d^3 - d^2*e*x + d*e^2*x^2))/(4*a*(c*d^4 + a*e^4)*(a + c*x^4) ) - (Sqrt[c]*d^2*e^5*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*Sqrt[a]*(c*d^4 + a* e^4)^2) - (Sqrt[c]*d^2*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(4*a^(3/2)*(c*d^4 + a*e^4)) - (c^(1/4)*d*e^4*(Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2] *c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^2) - (c^(1/4)*d*( 3*Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(8*S qrt[2]*a^(7/4)*(c*d^4 + a*e^4)) + (c^(1/4)*d*e^4*(Sqrt[c]*d^2 + Sqrt[a]*e^ 2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*(c*d^4 + a* e^4)^2) + (c^(1/4)*d*(3*Sqrt[c]*d^2 + Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2]*c^( 1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)) + (e^7*Log[d + e*x]) /(c*d^4 + a*e^4)^2 - (c^(1/4)*d*e^4*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log[Sqrt[a ] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^2) - (c^(1/4)*d*(3*Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log[Sqrt[a] - Sqrt[2] *a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)) + (c^(1/4)*d*e^4*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c ^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4)*(c*d^4 + a*e^4)^2) + (c^(1/4)* d*(3*Sqrt[c]*d^2 - Sqrt[a]*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(16*Sqrt[2]*a^(7/4)*(c*d^4 + a*e^4)) - (e^7*Log[a + c*x^4])/ (4*(c*d^4 + a*e^4)^2)
3.5.5.3.1 Defintions of rubi rules used
Time = 0.94 (sec) , antiderivative size = 430, normalized size of antiderivative = 0.50
| method | result | size |
| default | \(\frac {c \left (\frac {\frac {d \,e^{2} \left (e^{4} a +d^{4} c \right ) x^{3}}{4 a}-\frac {d^{2} e \left (e^{4} a +d^{4} c \right ) x^{2}}{4 a}+\frac {d^{3} \left (e^{4} a +d^{4} c \right ) x}{4 a}+\frac {e^{3} \left (e^{4} a +d^{4} c \right )}{4 c}}{c \,x^{4}+a}+\frac {\frac {\left (7 a \,d^{3} e^{4}+3 c \,d^{7}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-6 a \,d^{2} e^{5}-2 c \,d^{6} e \right ) \arctan \left (x^{2} \sqrt {\frac {c}{a}}\right )}{2 \sqrt {a c}}+\frac {\left (5 a d \,e^{6}+c \,d^{5} e^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}-\frac {a \,e^{7} \ln \left (c \,x^{4}+a \right )}{c}}{4 a}\right )}{\left (e^{4} a +d^{4} c \right )^{2}}+\frac {e^{7} \ln \left (e x +d \right )}{\left (e^{4} a +d^{4} c \right )^{2}}\) | \(430\) |
| risch | \(\frac {\frac {c d \,e^{2} x^{3}}{4 a \left (e^{4} a +d^{4} c \right )}-\frac {d^{2} c e \,x^{2}}{4 a \left (e^{4} a +d^{4} c \right )}+\frac {d^{3} c x}{4 a \left (e^{4} a +d^{4} c \right )}+\frac {e^{3}}{4 e^{4} a +4 d^{4} c}}{c \,x^{4}+a}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{9} e^{8}+2 a^{8} c \,d^{4} e^{4}+a^{7} c^{2} d^{8}\right ) \textit {\_Z}^{4}+16 a^{7} e^{7} \textit {\_Z}^{3}+\left (96 a^{5} e^{6}+20 a^{4} c \,d^{4} e^{2}\right ) \textit {\_Z}^{2}+\left (256 a^{3} e^{5}+72 a^{2} c \,d^{4} e \right ) \textit {\_Z} +256 e^{4} a +81 d^{4} c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 a^{9} e^{14}+7 a^{8} c \,d^{4} e^{10}-a^{7} c^{2} d^{8} e^{6}-3 a^{6} c^{3} d^{12} e^{2}\right ) \textit {\_R}^{4}+\left (60 a^{7} e^{13}+64 a^{6} c \,d^{4} e^{9}-4 a^{5} c^{2} d^{8} e^{5}-8 a^{4} c^{3} d^{12} e \right ) \textit {\_R}^{3}+\left (240 a^{5} e^{12}+295 a^{4} c \,d^{4} e^{8}+30 a^{3} c^{2} d^{8} e^{4}-9 a^{2} c^{3} d^{12}\right ) \textit {\_R}^{2}+\left (320 a^{3} e^{11}+560 a^{2} c \,d^{4} e^{7}+64 a \,c^{2} d^{8} e^{3}\right ) \textit {\_R} +324 c \,d^{4} e^{6}\right ) x +\left (6 a^{9} d \,e^{13}+10 a^{8} c \,d^{5} e^{9}+2 a^{7} c^{2} d^{9} e^{5}-2 a^{6} c^{3} d^{13} e \right ) \textit {\_R}^{4}+\left (51 a^{7} d \,e^{12}+67 a^{6} c \,d^{5} e^{8}+13 a^{5} c^{2} d^{9} e^{4}-3 a^{4} c^{3} d^{13}\right ) \textit {\_R}^{3}+\left (136 a^{5} d \,e^{11}+240 a^{4} c \,d^{5} e^{7}+56 a^{3} c^{2} d^{9} e^{3}\right ) \textit {\_R}^{2}+\left (176 a^{3} d \,e^{10}+384 a^{2} c \,d^{5} e^{6}+48 a \,c^{2} d^{9} e^{2}\right ) \textit {\_R} +256 a d \,e^{9}+324 c \,d^{5} e^{5}\right )\right )}{16}+\frac {e^{7} \ln \left (e x +d \right )}{a^{2} e^{8}+2 a c \,d^{4} e^{4}+c^{2} d^{8}}\) | \(653\) |
c/(a*e^4+c*d^4)^2*((1/4*d*e^2*(a*e^4+c*d^4)/a*x^3-1/4*d^2*e*(a*e^4+c*d^4)/ a*x^2+1/4*d^3*(a*e^4+c*d^4)/a*x+1/4*e^3*(a*e^4+c*d^4)/c)/(c*x^4+a)+1/4/a*( 1/8*(7*a*d^3*e^4+3*c*d^7)*(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*(-6*a*d^2*e^5-2* c*d^6*e)/(a*c)^(1/2)*arctan(x^2*(c/a)^(1/2))+1/8*(5*a*d*e^6+c*d^5*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))-a*e^7/c*ln(c*x^4+a)))+e^7*ln(e*x+d)/(a*e^4+c*d^4) ^2
Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 601, normalized size of antiderivative = 0.70 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {e^{7} \log \left (e x + d\right )}{c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}} - \frac {c {\left (\frac {\sqrt {2} {\left (4 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {1}{4}} e^{7} - 3 \, c^{2} d^{7} + \sqrt {a} c^{\frac {3}{2}} d^{5} e^{2} - 7 \, a c d^{3} e^{4} + 5 \, a^{\frac {3}{2}} \sqrt {c} d e^{6}\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 (4 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {1}{4}} e^{7} + 3 \, c^{2} d^{7} - \sqrt {a} c^{\frac {3}{2}} d^{5} e^{2} + 7 \, a c d^{3} e^{4} - 5 \, a^{\frac {3}{2}} \sqrt {c} d e^{6}\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 (3 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {9}{4}} d^{7} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {7}{4}} d^{5} e^{2} + 7 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {5}{4}} d^{3} e^{4} + 5 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {3}{4}} d e^{6} + 4 \, \sqrt {a} c^{2} d^{6} e + 12 \, a^{\frac {3}{2}} c d^{2} e^{5}\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 (3 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {9}{4}} d^{7} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {7}{4}} d^{5} e^{2} + 7 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {5}{4}} d^{3} e^{4} + 5 \, \sqrt {2} a^{\frac {7}{4}} c^{\frac {3}{4}} d e^{6} - 4 \, \sqrt {a} c^{2} d^{6} e - 12 \, a^{\frac {3}{2}} c d^{2} e^{5}\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 )}}{32 \, {\left (a c^{2} d^{8} + 2 \, a^{2} c d^{4} e^{4} + a^{3} e^{8}\right )}} + \frac {c d e^{2} x^{3} - c d^{2} e x^{2} + c d^{3} x + a e^{3}}{4 \, {\left (a^{2} c d^{4} + a^{3} e^{4} + {\left (a c^{2} d^{4} + a^{2} c e^{4}\right )} x^{4}\right )}} \]
e^7*log(e*x + d)/(c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8) - 1/32*c*(sqrt(2)*(4* sqrt(2)*a^(7/4)*c^(1/4)*e^7 - 3*c^2*d^7 + sqrt(a)*c^(3/2)*d^5*e^2 - 7*a*c* d^3*e^4 + 5*a^(3/2)*sqrt(c)*d*e^6)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/ 4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) + sqrt(2)*(4*sqrt(2)*a^(7/4)*c^(1/4)*e^7 + 3*c^2*d^7 - sqrt(a)*c^(3/2)*d^5*e^2 + 7*a*c*d^3*e^4 - 5*a^(3/2)*sqrt(c) *d*e^6)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^ (5/4)) - 2*(3*sqrt(2)*a^(1/4)*c^(9/4)*d^7 + sqrt(2)*a^(3/4)*c^(7/4)*d^5*e^ 2 + 7*sqrt(2)*a^(5/4)*c^(5/4)*d^3*e^4 + 5*sqrt(2)*a^(7/4)*c^(3/4)*d*e^6 + 4*sqrt(a)*c^2*d^6*e + 12*a^(3/2)*c*d^2*e^5)*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*(3*sqrt(2)*a^(1/4)*c^(9/4)*d^7 + sqrt(2)*a^(3/4)*c^( 7/4)*d^5*e^2 + 7*sqrt(2)*a^(5/4)*c^(5/4)*d^3*e^4 + 5*sqrt(2)*a^(7/4)*c^(3/ 4)*d*e^6 - 4*sqrt(a)*c^2*d^6*e - 12*a^(3/2)*c*d^2*e^5)*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)*sq rt(sqrt(a)*sqrt(c))*c^(5/4)))/(a*c^2*d^8 + 2*a^2*c*d^4*e^4 + a^3*e^8) + 1/ 4*(c*d*e^2*x^3 - c*d^2*e*x^2 + c*d^3*x + a*e^3)/(a^2*c*d^4 + a^3*e^4 + (a* c^2*d^4 + a^2*c*e^4)*x^4)
Time = 0.46 (sec) , antiderivative size = 795, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\frac {e^{8} \log \left ({\left | e x + d \right |}\right )}{c^{2} d^{8} e + 2 \, a c d^{4} e^{5} + a^{2} e^{9}} - \frac {e^{7} \log \left ({\left | c x^{4} + a \right |}\right )}{4 \, {\left (c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}\right )}} + \frac {{\left (4 \, \sqrt {2} \sqrt {a c} c^{2} d^{2} e + 3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} a^{2} c^{3} d^{4} + \sqrt {2} a^{3} c^{2} e^{4} + 4 \, \sqrt {2} \sqrt {a c} a^{2} c^{2} d^{2} e^{2} - 4 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c^{2} d^{3} e - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a^{2} d e^{3}\right )}} + \frac {{\left (4 \, \sqrt {2} \sqrt {a c} c^{2} d^{2} e + 3 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{8 \, {\left (\sqrt {2} a^{2} c^{3} d^{4} + \sqrt {2} a^{3} c^{2} e^{4} + 4 \, \sqrt {2} \sqrt {a c} a^{2} c^{2} d^{2} e^{2} + 4 \, \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c^{2} d^{3} e + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a^{2} d e^{3}\right )}} + \frac {{\left (3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{7} + 7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{3} e^{4} - \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{5} e^{2} - 5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d e^{6}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, {\left (a^{2} c^{4} d^{8} + 2 \, a^{3} c^{3} d^{4} e^{4} + a^{4} c^{2} e^{8}\right )}} - \frac {{\left (3 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{7} + 7 \, \sqrt {2} \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{3} e^{4} - \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} c d^{5} e^{2} - 5 \, \sqrt {2} \left (a c^{3}\right )^{\frac {3}{4}} a d e^{6}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{32 \, {\left (a^{2} c^{4} d^{8} + 2 \, a^{3} c^{3} d^{4} e^{4} + a^{4} c^{2} e^{8}\right )}} + \frac {a c d^{4} e^{3} + a^{2} e^{7} + {\left (c^{2} d^{5} e^{2} + a c d e^{6}\right )} x^{3} - {\left (c^{2} d^{6} e + a c d^{2} e^{5}\right )} x^{2} + {\left (c^{2} d^{7} + a c d^{3} e^{4}\right )} x}{4 \, {\left (c d^{4} + a e^{4}\right )}^{2} {\left (c x^{4} + a\right )} a} \]
e^8*log(abs(e*x + d))/(c^2*d^8*e + 2*a*c*d^4*e^5 + a^2*e^9) - 1/4*e^7*log( abs(c*x^4 + a))/(c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8) + 1/8*(4*sqrt(2)*sqrt( a*c)*c^2*d^2*e + 3*(a*c^3)^(1/4)*c^2*d^3 + 5*(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))/(sqrt(2)*a^2*c^3*d^4 + sqrt(2)*a^3*c^2*e^4 + 4*sqrt(2)*sqrt(a*c)*a^2*c^2*d^2*e^2 - 4*(a*c^3)^(1/ 4)*a^2*c^2*d^3*e - 4*(a*c^3)^(3/4)*a^2*d*e^3) + 1/8*(4*sqrt(2)*sqrt(a*c)*c ^2*d^2*e + 3*(a*c^3)^(1/4)*c^2*d^3 + 5*(a*c^3)^(3/4)*d*e^2)*arctan(1/2*sqr t(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a^2*c^3*d^4 + sqrt( 2)*a^3*c^2*e^4 + 4*sqrt(2)*sqrt(a*c)*a^2*c^2*d^2*e^2 + 4*(a*c^3)^(1/4)*a^2 *c^2*d^3*e + 4*(a*c^3)^(3/4)*a^2*d*e^3) + 1/32*(3*sqrt(2)*(a*c^3)^(1/4)*c^ 3*d^7 + 7*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^3*e^4 - sqrt(2)*(a*c^3)^(3/4)*c*d^ 5*e^2 - 5*sqrt(2)*(a*c^3)^(3/4)*a*d*e^6)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^4*d^8 + 2*a^3*c^3*d^4*e^4 + a^4*c^2*e^8) - 1/32*(3*sqrt (2)*(a*c^3)^(1/4)*c^3*d^7 + 7*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^3*e^4 - sqrt(2 )*(a*c^3)^(3/4)*c*d^5*e^2 - 5*sqrt(2)*(a*c^3)^(3/4)*a*d*e^6)*log(x^2 - sqr t(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^4*d^8 + 2*a^3*c^3*d^4*e^4 + a^4*c^2 *e^8) + 1/4*(a*c*d^4*e^3 + a^2*e^7 + (c^2*d^5*e^2 + a*c*d*e^6)*x^3 - (c^2* d^6*e + a*c*d^2*e^5)*x^2 + (c^2*d^7 + a*c*d^3*e^4)*x)/((c*d^4 + a*e^4)^2*( c*x^4 + a)*a)
Time = 9.66 (sec) , antiderivative size = 1591, normalized size of antiderivative = 1.86 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )^2} \, dx=\text {Too large to display} \]
e^3/(4*(a^2*e^4 + c^2*d^4*x^4 + a*c*d^4 + a*c*e^4*x^4)) + symsum(log((81*c ^5*d^5*e^6 + 64*a*c^4*d*e^10)/(256*(a^6*e^8 + a^4*c^2*d^8 + 2*a^5*c*d^4*e^ 4)) + root(131072*a^8*c*d^4*e^4*z^4 + 65536*a^7*c^2*d^8*z^4 + 65536*a^9*e^ 8*z^4 + 65536*a^7*e^7*z^3 + 5120*a^4*c*d^4*e^2*z^2 + 24576*a^5*e^6*z^2 + 1 152*a^2*c*d^4*e*z + 4096*a^3*e^5*z + 81*c*d^4 + 256*a*e^4, z, k)*(root(131 072*a^8*c*d^4*e^4*z^4 + 65536*a^7*c^2*d^8*z^4 + 65536*a^9*e^8*z^4 + 65536* a^7*e^7*z^3 + 5120*a^4*c*d^4*e^2*z^2 + 24576*a^5*e^6*z^2 + 1152*a^2*c*d^4* e*z + 4096*a^3*e^5*z + 81*c*d^4 + 256*a*e^4, z, k)*(root(131072*a^8*c*d^4* e^4*z^4 + 65536*a^7*c^2*d^8*z^4 + 65536*a^9*e^8*z^4 + 65536*a^7*e^7*z^3 + 5120*a^4*c*d^4*e^2*z^2 + 24576*a^5*e^6*z^2 + 1152*a^2*c*d^4*e*z + 4096*a^3 *e^5*z + 81*c*d^4 + 256*a*e^4, z, k)*(root(131072*a^8*c*d^4*e^4*z^4 + 6553 6*a^7*c^2*d^8*z^4 + 65536*a^9*e^8*z^4 + 65536*a^7*e^7*z^3 + 5120*a^4*c*d^4 *e^2*z^2 + 24576*a^5*e^6*z^2 + 1152*a^2*c*d^4*e*z + 4096*a^3*e^5*z + 81*c* d^4 + 256*a*e^4, z, k)*((98304*a^9*c^4*d*e^14 - 32768*a^6*c^7*d^13*e^2 + 3 2768*a^7*c^6*d^9*e^6 + 163840*a^8*c^5*d^5*e^10)/(256*(a^6*e^8 + a^4*c^2*d^ 8 + 2*a^5*c*d^4*e^4)) + (x*(81920*a^9*c^4*e^15 - 49152*a^6*c^7*d^12*e^3 - 16384*a^7*c^6*d^8*e^7 + 114688*a^8*c^5*d^4*e^11))/(256*(a^6*e^8 + a^4*c^2* d^8 + 2*a^5*c*d^4*e^4))) + (52224*a^7*c^4*d*e^13 - 3072*a^4*c^7*d^13*e + 1 3312*a^5*c^6*d^9*e^5 + 68608*a^6*c^5*d^5*e^9)/(256*(a^6*e^8 + a^4*c^2*d^8 + 2*a^5*c*d^4*e^4)) + (x*(61440*a^7*c^4*e^14 - 8192*a^4*c^7*d^12*e^2 - ...