Integrand size = 17, antiderivative size = 331 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=-\frac {\sqrt {c} d^2 e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (c d^4+a e^4\right )}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2+\sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )}+\frac {e^3 \log (d+e x)}{c d^4+a e^4}-\frac {e^3 \log \left (a+c x^4\right )}{4 \left (c d^4+a e^4\right )} \] Output:
-1/2*c^(1/2)*d^2*e*arctan(c^(1/2)*x^2/a^(1/2))/a^(1/2)/(a*e^4+c*d^4)+1/4*c ^(1/4)*d*(c^(1/2)*d^2+a^(1/2)*e^2)*arctan(-1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^ (1/2)/a^(3/4)/(a*e^4+c*d^4)+1/4*c^(1/4)*d*(c^(1/2)*d^2+a^(1/2)*e^2)*arctan (1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(3/4)/(a*e^4+c*d^4)+1/4*c^(1/4)*d* (c^(1/2)*d^2-a^(1/2)*e^2)*arctanh(2^(1/2)*a^(1/4)*c^(1/4)*x/(a^(1/2)+c^(1/ 2)*x^2))*2^(1/2)/a^(3/4)/(a*e^4+c*d^4)+e^3*ln(e*x+d)/(a*e^4+c*d^4)-e^3*ln( c*x^4+a)/(4*a*e^4+4*c*d^4)
Time = 0.16 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.22 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\frac {-2 \sqrt [4]{c} d \left (\sqrt {2} \sqrt {c} d^2-2 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt [4]{c} d \left (\sqrt {2} \sqrt {c} d^2+2 \sqrt [4]{a} \sqrt [4]{c} d e+\sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+8 a^{3/4} e^3 \log (d+e x)-\sqrt {2} c^{3/4} d^3 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} \sqrt {a} \sqrt [4]{c} d e^2 \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} c^{3/4} d^3 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )-\sqrt {2} \sqrt {a} \sqrt [4]{c} d e^2 \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )-2 a^{3/4} e^3 \log \left (a+c x^4\right )}{8 a^{3/4} \left (c d^4+a e^4\right )} \] Input:
Integrate[1/((d + e*x)*(a + c*x^4)),x]
Output:
(-2*c^(1/4)*d*(Sqrt[2]*Sqrt[c]*d^2 - 2*a^(1/4)*c^(1/4)*d*e + Sqrt[2]*Sqrt[ a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*c^(1/4)*d*(Sqrt[2]*Sqr t[c]*d^2 + 2*a^(1/4)*c^(1/4)*d*e + Sqrt[2]*Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2 ]*c^(1/4)*x)/a^(1/4)] + 8*a^(3/4)*e^3*Log[d + e*x] - Sqrt[2]*c^(3/4)*d^3*L og[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + Sqrt[2]*Sqrt[a]*c^ (1/4)*d*e^2*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] + Sqrt[ 2]*c^(3/4)*d^3*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2] - Sq rt[2]*Sqrt[a]*c^(1/4)*d*e^2*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt [c]*x^2] - 2*a^(3/4)*e^3*Log[a + c*x^4])/(8*a^(3/4)*(c*d^4 + a*e^4))
Time = 1.08 (sec) , antiderivative size = 416, normalized size of antiderivative = 1.26, 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.
| \(\displaystyle \int \frac {1}{\left (a+c x^4\right ) (d+e x)} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {e^4}{(d+e x) \left (a e^4+c d^4\right )}+\frac {c \left (d^3-d^2 e x+d e^2 x^2-e^3 x^3\right )}{\left (a+c x^4\right ) \left (a e^4+c d^4\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [4]{c} d \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )}+\frac {\sqrt [4]{c} d \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e^2+\sqrt {c} d^2\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )}-\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )}+\frac {\sqrt [4]{c} d \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )}-\frac {\sqrt {c} d^2 e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} \left (a e^4+c d^4\right )}-\frac {e^3 \log \left (a+c x^4\right )}{4 \left (a e^4+c d^4\right )}+\frac {e^3 \log (d+e x)}{a e^4+c d^4}\) |
Input:
Int[1/((d + e*x)*(a + c*x^4)),x]
Output:
-1/2*(Sqrt[c]*d^2*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(Sqrt[a]*(c*d^4 + a*e^4 )) - (c^(1/4)*d*(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)) + (c^(1/4)*d*(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)) + (e^3*Log[d + e*x])/(c*d^4 + a*e^4) - (c^(1/4)*d*(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)) + (c^(1/4)*d*(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)) - (e^3*Log[a + c*x^4])/(4*(c*d^4 + a*e^4))
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]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.30 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.64
| method | result | size |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (a^{4} e^{4}+a^{3} c \,d^{4}\right ) \textit {\_Z}^{4}+4 a^{3} e^{3} \textit {\_Z}^{3}+6 a^{2} e^{2} \textit {\_Z}^{2}+4 a e \textit {\_Z} \right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 a^{3} e^{6}-3 a^{2} d^{4} e^{2} c \right ) \textit {\_R}^{3}+\left (15 a^{2} e^{5}-3 a \,d^{4} e c \right ) \textit {\_R}^{2}+\left (15 e^{4} a -c \,d^{4}\right ) \textit {\_R} +5 e^{3}\right ) x +\left (6 a^{3} d \,e^{5}-2 a^{2} c \,d^{5} e \right ) \textit {\_R}^{3}+\left (13 a^{2} d \,e^{4}-a c \,d^{5}\right ) \textit {\_R}^{2}+8 a d \,e^{3} \textit {\_R} +d \,e^{2}\right )\right )}{4}+\frac {e^{3} \ln \left (e x +d \right )}{e^{4} a +c \,d^{4}}\) | \(211\) |
| default | \(\frac {c \left (\frac {d^{3} \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}-\frac {d^{2} e \arctan \left (\sqrt {\frac {c}{a}}\, x^{2}\right )}{2 \sqrt {a c}}+\frac {d \,e^{2} \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}-\frac {e^{3} \ln \left (c \,x^{4}+a \right )}{4 c}\right )}{e^{4} a +c \,d^{4}}+\frac {e^{3} \ln \left (e x +d \right )}{e^{4} a +c \,d^{4}}\) | \(289\) |
Input:
int(1/(e*x+d)/(c*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/4*sum(_R*ln(((5*a^3*e^6-3*a^2*c*d^4*e^2)*_R^3+(15*a^2*e^5-3*a*c*d^4*e)*_ R^2+(15*a*e^4-c*d^4)*_R+5*e^3)*x+(6*a^3*d*e^5-2*a^2*c*d^5*e)*_R^3+(13*a^2* d*e^4-a*c*d^5)*_R^2+8*a*d*e^3*_R+d*e^2),_R=RootOf(1+(a^4*e^4+a^3*c*d^4)*_Z ^4+4*a^3*e^3*_Z^3+6*a^2*e^2*_Z^2+4*a*e*_Z))+e^3*ln(e*x+d)/(a*e^4+c*d^4)
Result contains complex when optimal does not.
Time = 93.92 (sec) , antiderivative size = 352864, normalized size of antiderivative = 1066.05 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/(e*x+d)/(c*x^4+a),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x+d)/(c*x**4+a),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 345, normalized size of antiderivative = 1.04 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\frac {e^{3} \log \left (e x + d\right )}{c d^{4} + a e^{4}} - \frac {c {\left (\frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{3} - c d^{3} + \sqrt {a} \sqrt {c} d e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} + \frac {\sqrt {2} {\left (\sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{3} + c d^{3} - \sqrt {a} \sqrt {c} d e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {5}{4}} d^{3} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {3}{4}} d e^{2} + 2 \, \sqrt {a} c d^{2} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}} - \frac {2 \, {\left (\sqrt {2} a^{\frac {1}{4}} c^{\frac {5}{4}} d^{3} + \sqrt {2} a^{\frac {3}{4}} c^{\frac {3}{4}} d e^{2} - 2 \, \sqrt {a} c d^{2} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {5}{4}}}\right )}}{8 \, {\left (c d^{4} + a e^{4}\right )}} \] Input:
integrate(1/(e*x+d)/(c*x^4+a),x, algorithm="maxima")
Output:
e^3*log(e*x + d)/(c*d^4 + a*e^4) - 1/8*c*(sqrt(2)*(sqrt(2)*a^(3/4)*c^(1/4) *e^3 - c*d^3 + sqrt(a)*sqrt(c)*d*e^2)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^ (1/4)*x + sqrt(a))/(a^(3/4)*c^(5/4)) + sqrt(2)*(sqrt(2)*a^(3/4)*c^(1/4)*e^ 3 + c*d^3 - sqrt(a)*sqrt(c)*d*e^2)*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^(5/4)*d^3 + sqrt( 2)*a^(3/4)*c^(3/4)*d*e^2 + 2*sqrt(a)*c*d^2*e)*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^(5/4)*d^3 + sqrt(2)*a^(3/4)*c^( 3/4)*d*e^2 - 2*sqrt(a)*c*d^2*e)*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*d^4 + a*e^4)
Time = 0.15 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.14 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\frac {e^{4} \log \left ({\left | e x + d \right |}\right )}{c d^{4} e + a e^{5}} - \frac {e^{3} \log \left ({\left | c x^{4} + a \right |}\right )}{4 \, {\left (c d^{4} + a e^{4}\right )}} + \frac {\left (a c^{3}\right )^{\frac {1}{4}} c d \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a c^{2} d^{2} + \sqrt {2} \sqrt {a c} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {1}{4}} a c d e\right )}} + \frac {\left (a c^{3}\right )^{\frac {1}{4}} c d \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{2 \, {\left (\sqrt {2} a c^{2} d^{2} + \sqrt {2} \sqrt {a c} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {1}{4}} a c d e\right )}} + \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{4 \, {\left (\sqrt {2} a c^{3} d^{4} + \sqrt {2} a^{2} c^{2} e^{4}\right )}} - \frac {{\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{3} - \left (a c^{3}\right )^{\frac {3}{4}} d e^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{4 \, {\left (\sqrt {2} a c^{3} d^{4} + \sqrt {2} a^{2} c^{2} e^{4}\right )}} \] Input:
integrate(1/(e*x+d)/(c*x^4+a),x, algorithm="giac")
Output:
e^4*log(abs(e*x + d))/(c*d^4*e + a*e^5) - 1/4*e^3*log(abs(c*x^4 + a))/(c*d ^4 + a*e^4) + 1/2*(a*c^3)^(1/4)*c*d*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c )^(1/4))/(a/c)^(1/4))/(sqrt(2)*a*c^2*d^2 + sqrt(2)*sqrt(a*c)*a*c*e^2 - 2*( a*c^3)^(1/4)*a*c*d*e) + 1/2*(a*c^3)^(1/4)*c*d*arctan(1/2*sqrt(2)*(2*x - sq rt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a*c^2*d^2 + sqrt(2)*sqrt(a*c)*a*c *e^2 + 2*(a*c^3)^(1/4)*a*c*d*e) + 1/4*((a*c^3)^(1/4)*c^2*d^3 - (a*c^3)^(3/ 4)*d*e^2)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a*c^3*d^4 + sqrt(2)*a^2*c^2*e^4) - 1/4*((a*c^3)^(1/4)*c^2*d^3 - (a*c^3)^(3/4)*d*e^2) *log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(sqrt(2)*a*c^3*d^4 + sqrt(2) *a^2*c^2*e^4)
Time = 22.05 (sec) , antiderivative size = 874, normalized size of antiderivative = 2.64 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\left (\sum _{k=1}^4\ln \left (\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )\,c^4\,e\,\left (d\,e^2+5\,e^3\,x+{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^2\,a^2\,e^5\,x\,240+{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^3\,a^3\,e^6\,x\,320+\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )\,a\,d\,e^3\,32+\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )\,a\,e^4\,x\,60-\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )\,c\,d^4\,x\,4-{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^2\,a\,c\,d^5\,16+{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^2\,a^2\,d\,e^4\,208+{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^3\,a^3\,d\,e^5\,384-{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^3\,a^2\,c\,d^5\,e\,128-{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^3\,a^2\,c\,d^4\,e^2\,x\,192-{\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )}^2\,a\,c\,d^4\,e\,x\,48\right )\right )\,\mathrm {root}\left (256\,a^3\,c\,d^4\,z^4+256\,a^4\,e^4\,z^4+256\,a^3\,e^3\,z^3+96\,a^2\,e^2\,z^2+16\,a\,e\,z+1,z,k\right )\right )+\frac {e^3\,\ln \left (d+e\,x\right )}{c\,d^4+a\,e^4} \] Input:
int(1/((a + c*x^4)*(d + e*x)),x)
Output:
symsum(log(root(256*a^3*c*d^4*z^4 + 256*a^4*e^4*z^4 + 256*a^3*e^3*z^3 + 96 *a^2*e^2*z^2 + 16*a*e*z + 1, z, k)*c^4*e*(d*e^2 + 5*e^3*x + 240*root(256*a ^3*c*d^4*z^4 + 256*a^4*e^4*z^4 + 256*a^3*e^3*z^3 + 96*a^2*e^2*z^2 + 16*a*e *z + 1, z, k)^2*a^2*e^5*x + 320*root(256*a^3*c*d^4*z^4 + 256*a^4*e^4*z^4 + 256*a^3*e^3*z^3 + 96*a^2*e^2*z^2 + 16*a*e*z + 1, z, k)^3*a^3*e^6*x + 32*r oot(256*a^3*c*d^4*z^4 + 256*a^4*e^4*z^4 + 256*a^3*e^3*z^3 + 96*a^2*e^2*z^2 + 16*a*e*z + 1, z, k)*a*d*e^3 + 60*root(256*a^3*c*d^4*z^4 + 256*a^4*e^4*z ^4 + 256*a^3*e^3*z^3 + 96*a^2*e^2*z^2 + 16*a*e*z + 1, z, k)*a*e^4*x - 4*ro ot(256*a^3*c*d^4*z^4 + 256*a^4*e^4*z^4 + 256*a^3*e^3*z^3 + 96*a^2*e^2*z^2 + 16*a*e*z + 1, z, k)*c*d^4*x - 16*root(256*a^3*c*d^4*z^4 + 256*a^4*e^4*z^ 4 + 256*a^3*e^3*z^3 + 96*a^2*e^2*z^2 + 16*a*e*z + 1, z, k)^2*a*c*d^5 + 208 *root(256*a^3*c*d^4*z^4 + 256*a^4*e^4*z^4 + 256*a^3*e^3*z^3 + 96*a^2*e^2*z ^2 + 16*a*e*z + 1, z, k)^2*a^2*d*e^4 + 384*root(256*a^3*c*d^4*z^4 + 256*a^ 4*e^4*z^4 + 256*a^3*e^3*z^3 + 96*a^2*e^2*z^2 + 16*a*e*z + 1, z, k)^3*a^3*d *e^5 - 128*root(256*a^3*c*d^4*z^4 + 256*a^4*e^4*z^4 + 256*a^3*e^3*z^3 + 96 *a^2*e^2*z^2 + 16*a*e*z + 1, z, k)^3*a^2*c*d^5*e - 192*root(256*a^3*c*d^4* z^4 + 256*a^4*e^4*z^4 + 256*a^3*e^3*z^3 + 96*a^2*e^2*z^2 + 16*a*e*z + 1, z , k)^3*a^2*c*d^4*e^2*x - 48*root(256*a^3*c*d^4*z^4 + 256*a^4*e^4*z^4 + 256 *a^3*e^3*z^3 + 96*a^2*e^2*z^2 + 16*a*e*z + 1, z, k)^2*a*c*d^4*e*x))*root(2 56*a^3*c*d^4*z^4 + 256*a^4*e^4*z^4 + 256*a^3*e^3*z^3 + 96*a^2*e^2*z^2 +...
Time = 0.16 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.37 \[ \int \frac {1}{(d+e x) \left (a+c x^4\right )} \, dx=\frac {-2 c^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d \,e^{2}-2 c^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d^{3}+4 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}-2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d^{2} e +2 c^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d \,e^{2}+2 c^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d^{3}+4 \sqrt {c}\, \sqrt {a}\, \mathit {atan} \left (\frac {c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}+2 \sqrt {c}\, x}{c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}}\right ) d^{2} e +c^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) d \,e^{2}-c^{\frac {1}{4}} a^{\frac {3}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) d \,e^{2}-c^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) d^{3}+c^{\frac {3}{4}} a^{\frac {1}{4}} \sqrt {2}\, \mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) d^{3}-2 \,\mathrm {log}\left (-c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a \,e^{3}-2 \,\mathrm {log}\left (c^{\frac {1}{4}} a^{\frac {1}{4}} \sqrt {2}\, x +\sqrt {a}+\sqrt {c}\, x^{2}\right ) a \,e^{3}+8 \,\mathrm {log}\left (e x +d \right ) a \,e^{3}}{8 a \left (a \,e^{4}+c \,d^{4}\right )} \] Input:
int(1/(e*x+d)/(c*x^4+a),x)
Output:
( - 2*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c )*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*d*e**2 - 2*c**(3/4)*a**(1/4)*sqrt(2)*ata n((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*d **3 + 4*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c* *(1/4)*a**(1/4)*sqrt(2)))*d**2*e + 2*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1 /4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*d*e**2 + 2*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x) /(c**(1/4)*a**(1/4)*sqrt(2)))*d**3 + 4*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**( 1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*d**2*e + c**(1/4) *a**(3/4)*sqrt(2)*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x **2)*d*e**2 - c**(1/4)*a**(3/4)*sqrt(2)*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*d*e**2 - c**(3/4)*a**(1/4)*sqrt(2)*log( - c**(1/4) *a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*d**3 + c**(3/4)*a**(1/4)*sqr t(2)*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*d**3 - 2*lo g( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a*e**3 - 2*log( c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a*e**3 + 8*log(d + e *x)*a*e**3)/(8*a*(a*e**4 + c*d**4))