Integrand size = 17, antiderivative size = 443 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=-\frac {e^3}{\left (c d^4+a e^4\right ) (d+e x)}-\frac {\sqrt {c} d e \left (c d^4-a e^4\right ) \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \left (c d^4+a e^4\right )^2}-\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )+\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )+\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 \sqrt {2} a^{3/4} \left (c d^4+a e^4\right )^2}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (c d^4+a e^4\right )^2}-\frac {c d^3 e^3 \log \left (a+c x^4\right )}{\left (c d^4+a e^4\right )^2} \] Output:
-e^3/(a*e^4+c*d^4)/(e*x+d)-c^(1/2)*d*e*(-a*e^4+c*d^4)*arctan(c^(1/2)*x^2/a ^(1/2))/a^(1/2)/(a*e^4+c*d^4)^2+1/4*c^(1/4)*(c^(1/2)*d^2*(-3*a*e^4+c*d^4)+ a^(1/2)*e^2*(-a*e^4+3*c*d^4))*arctan(-1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2) /a^(3/4)/(a*e^4+c*d^4)^2+1/4*c^(1/4)*(c^(1/2)*d^2*(-3*a*e^4+c*d^4)+a^(1/2) *e^2*(-a*e^4+3*c*d^4))*arctan(1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(3/4) /(a*e^4+c*d^4)^2+1/4*c^(1/4)*(c^(1/2)*d^2*(-3*a*e^4+c*d^4)-a^(1/2)*e^2*(-a *e^4+3*c*d^4))*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)^2+4*c*d^3*e^3*ln(e*x+d)/(a*e^4+c*d^4)^2-c*d^3* e^3*ln(c*x^4+a)/(a*e^4+c*d^4)^2
Time = 0.64 (sec) , antiderivative size = 524, normalized size of antiderivative = 1.18 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=\frac {-\frac {8 e^3 \left (c d^4+a e^4\right )}{d+e x}+\frac {2 \sqrt [4]{c} \left (-\sqrt {c} d^2+\sqrt {a} e^2\right ) \left (\sqrt {2} c d^4-4 \sqrt [4]{a} c^{3/4} d^3 e+4 \sqrt {2} \sqrt {a} \sqrt {c} d^2 e^2-4 a^{3/4} \sqrt [4]{c} d e^3+\sqrt {2} a e^4\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac {2 \sqrt [4]{c} \left (\sqrt {c} d^2-\sqrt {a} e^2\right ) \left (\sqrt {2} c d^4+4 \sqrt [4]{a} c^{3/4} d^3 e+4 \sqrt {2} \sqrt {a} \sqrt {c} d^2 e^2+4 a^{3/4} \sqrt [4]{c} d e^3+\sqrt {2} a e^4\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+32 c d^3 e^3 \log (d+e x)-\frac {\sqrt {2} \sqrt [4]{c} \left (c^{3/2} d^6-3 \sqrt {a} c d^4 e^2-3 a \sqrt {c} d^2 e^4+a^{3/2} e^6\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{3/4}}+\frac {\sqrt {2} \sqrt [4]{c} \left (c^{3/2} d^6-3 \sqrt {a} c d^4 e^2-3 a \sqrt {c} d^2 e^4+a^{3/2} e^6\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{a^{3/4}}-8 c d^3 e^3 \log \left (a+c x^4\right )}{8 \left (c d^4+a e^4\right )^2} \] Input:
Integrate[1/((d + e*x)^2*(a + c*x^4)),x]
Output:
((-8*e^3*(c*d^4 + a*e^4))/(d + e*x) + (2*c^(1/4)*(-(Sqrt[c]*d^2) + Sqrt[a] *e^2)*(Sqrt[2]*c*d^4 - 4*a^(1/4)*c^(3/4)*d^3*e + 4*Sqrt[2]*Sqrt[a]*Sqrt[c] *d^2*e^2 - 4*a^(3/4)*c^(1/4)*d*e^3 + Sqrt[2]*a*e^4)*ArcTan[1 - (Sqrt[2]*c^ (1/4)*x)/a^(1/4)])/a^(3/4) + (2*c^(1/4)*(Sqrt[c]*d^2 - Sqrt[a]*e^2)*(Sqrt[ 2]*c*d^4 + 4*a^(1/4)*c^(3/4)*d^3*e + 4*Sqrt[2]*Sqrt[a]*Sqrt[c]*d^2*e^2 + 4 *a^(3/4)*c^(1/4)*d*e^3 + Sqrt[2]*a*e^4)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^( 1/4)])/a^(3/4) + 32*c*d^3*e^3*Log[d + e*x] - (Sqrt[2]*c^(1/4)*(c^(3/2)*d^6 - 3*Sqrt[a]*c*d^4*e^2 - 3*a*Sqrt[c]*d^2*e^4 + a^(3/2)*e^6)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(3/4) + (Sqrt[2]*c^(1/4)*(c^(3 /2)*d^6 - 3*Sqrt[a]*c*d^4*e^2 - 3*a*Sqrt[c]*d^2*e^4 + a^(3/2)*e^6)*Log[Sqr t[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(3/4) - 8*c*d^3*e^3*Log [a + c*x^4])/(8*(c*d^4 + a*e^4)^2)
Time = 1.58 (sec) , antiderivative size = 552, normalized size of antiderivative = 1.25, 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.
| \(\displaystyle \int \frac {1}{\left (a+c x^4\right ) (d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {e^4}{(d+e x)^2 \left (a e^4+c d^4\right )}+\frac {4 c d^3 e^4}{(d+e x) \left (a e^4+c d^4\right )^2}+\frac {c \left (-2 d e x \left (c d^4-a e^4\right )+e^2 x^2 \left (3 c d^4-a e^4\right )+d^2 \left (c d^4-3 a e^4\right )-4 c d^3 e^3 x^3\right )}{\left (a+c x^4\right ) \left (a e^4+c d^4\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [4]{c} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} e^2 \left (3 c d^4-a e^4\right )+\sqrt {c} d^2 \left (c d^4-3 a e^4\right )\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac {\sqrt [4]{c} \arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} e^2 \left (3 c d^4-a e^4\right )+\sqrt {c} d^2 \left (c d^4-3 a e^4\right )\right )}{2 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}+\frac {\sqrt [4]{c} \left (\sqrt {c} d^2 \left (c d^4-3 a e^4\right )-\sqrt {a} e^2 \left (3 c d^4-a e^4\right )\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} \left (a e^4+c d^4\right )^2}-\frac {\sqrt {c} d e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right ) \left (c d^4-a e^4\right )}{\sqrt {a} \left (a e^4+c d^4\right )^2}-\frac {e^3}{(d+e x) \left (a e^4+c d^4\right )}-\frac {c d^3 e^3 \log \left (a+c x^4\right )}{\left (a e^4+c d^4\right )^2}+\frac {4 c d^3 e^3 \log (d+e x)}{\left (a e^4+c d^4\right )^2}\) |
Input:
Int[1/((d + e*x)^2*(a + c*x^4)),x]
Output:
-(e^3/((c*d^4 + a*e^4)*(d + e*x))) - (Sqrt[c]*d*e*(c*d^4 - a*e^4)*ArcTan[( Sqrt[c]*x^2)/Sqrt[a]])/(Sqrt[a]*(c*d^4 + a*e^4)^2) - (c^(1/4)*(Sqrt[c]*d^2 *(c*d^4 - 3*a*e^4) + Sqrt[a]*e^2*(3*c*d^4 - 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)^2) + (c^(1/4)*(Sqrt[ c]*d^2*(c*d^4 - 3*a*e^4) + Sqrt[a]*e^2*(3*c*d^4 - 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)^2) + (4*c*d^3* e^3*Log[d + e*x])/(c*d^4 + a*e^4)^2 - (c^(1/4)*(Sqrt[c]*d^2*(c*d^4 - 3*a*e ^4) - Sqrt[a]*e^2*(3*c*d^4 - a*e^4))*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)*(Sqrt[ c]*d^2*(c*d^4 - 3*a*e^4) - Sqrt[a]*e^2*(3*c*d^4 - a*e^4))*Log[Sqrt[a] + Sq rt[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*d^3*e^3*Log[a + c*x^4])/(c*d^4 + a*e^4)^2
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 = 0.24 (sec) , antiderivative size = 354, normalized size of antiderivative = 0.80
| method | result | size |
| default | \(-\frac {c \left (\frac {\left (3 a \,d^{2} e^{4}-c \,d^{6}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-2 a d \,e^{5}+2 c \,d^{5} e \right ) \arctan \left (\sqrt {\frac {c}{a}}\, x^{2}\right )}{2 \sqrt {a c}}+\frac {\left (a \,e^{6}-3 c \,d^{4} e^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}+d^{3} e^{3} \ln \left (c \,x^{4}+a \right )\right )}{\left (e^{4} a +c \,d^{4}\right )^{2}}-\frac {e^{3}}{\left (e^{4} a +c \,d^{4}\right ) \left (e x +d \right )}+\frac {4 c \,d^{3} e^{3} \ln \left (e x +d \right )}{\left (e^{4} a +c \,d^{4}\right )^{2}}\) | \(354\) |
| risch | \(-\frac {e^{3}}{\left (e^{4} a +c \,d^{4}\right ) \left (e x +d \right )}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{5} e^{8}+2 c \,d^{4} a^{4} e^{4}+d^{8} c^{2} a^{3}\right ) \textit {\_Z}^{4}+16 a^{3} c \,d^{3} e^{3} \textit {\_Z}^{3}+20 a^{2} c \,d^{2} e^{2} \textit {\_Z}^{2}+8 a c d e \textit {\_Z} +c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (5 a^{5} e^{14}+7 a^{4} c \,d^{4} e^{10}-a^{3} c^{2} d^{8} e^{6}-3 a^{2} c^{3} d^{12} e^{2}\right ) \textit {\_R}^{4}+\left (44 a^{3} c \,d^{3} e^{9}+40 a^{2} c^{2} d^{7} e^{5}-4 a \,c^{3} d^{11} e \right ) \textit {\_R}^{3}+\left (79 a^{2} c \,d^{2} e^{8}+62 a \,c^{2} d^{6} e^{4}-c^{3} d^{10}\right ) \textit {\_R}^{2}+\left (32 a c d \,e^{7}+16 c^{2} d^{5} e^{3}\right ) \textit {\_R} +4 c \,e^{6}\right ) x +\left (6 a^{5} d \,e^{13}+10 a^{4} c \,d^{5} e^{9}+2 a^{3} c^{2} d^{9} e^{5}-2 a^{2} c^{3} d^{13} e \right ) \textit {\_R}^{4}+\left (a^{4} e^{12}+33 a^{3} c \,d^{4} e^{8}+31 a^{2} c^{2} d^{8} e^{4}-a \,c^{3} d^{12}\right ) \textit {\_R}^{3}+\left (64 a^{2} c \,d^{3} e^{7}+16 a \,c^{2} d^{7} e^{3}\right ) \textit {\_R}^{2}+32 a c \,d^{2} e^{6} \textit {\_R} +4 c d \,e^{5}\right )\right )}{4}+\frac {4 c \,d^{3} e^{3} \ln \left (e x +d \right )}{a^{2} e^{8}+2 a c \,d^{4} e^{4}+d^{8} c^{2}}\) | \(460\) |
Input:
int(1/(e*x+d)^2/(c*x^4+a),x,method=_RETURNVERBOSE)
Output:
-c/(a*e^4+c*d^4)^2*(1/8*(3*a*d^2*e^4-c*d^6)*(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*(-2*a*d*e^5+2*c*d^5*e)/(a*c)^(1/2)*arctan((c/a)^(1/2)*x^2)+1/8*(a*e^6-3* c*d^4*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))+d^3*e^3*ln(c*x^4+a))-e^3/(a*e^4+c*d ^4)/(e*x+d)+4*c*d^3*e^3*ln(e*x+d)/(a*e^4+c*d^4)^2
Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x+d)^2/(c*x^4+a),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=\text {Timed out} \] Input:
integrate(1/(e*x+d)**2/(c*x**4+a),x)
Output:
Timed out
Time = 0.13 (sec) , antiderivative size = 561, normalized size of antiderivative = 1.27 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=\frac {4 \, c d^{3} e^{3} \log \left (e x + d\right )}{c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}} - \frac {e^{3}}{c d^{5} + a d e^{4} + {\left (c d^{4} e + a e^{5}\right )} x} - \frac {c {\left (\frac {\sqrt {2} {\left (4 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {5}{4}} d^{3} e^{3} - c^{2} d^{6} + 3 \, \sqrt {a} c^{\frac {3}{2}} d^{4} e^{2} + 3 \, a c d^{2} e^{4} - a^{\frac {3}{2}} \sqrt {c} 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 {3}{4}} c^{\frac {5}{4}} d^{3} e^{3} + c^{2} d^{6} - 3 \, \sqrt {a} c^{\frac {3}{2}} d^{4} e^{2} - 3 \, a c d^{2} e^{4} + a^{\frac {3}{2}} \sqrt {c} 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 (\sqrt {2} a^{\frac {1}{4}} c^{\frac {9}{4}} d^{6} + 3 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {7}{4}} d^{4} e^{2} - 3 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {5}{4}} d^{2} e^{4} - \sqrt {2} a^{\frac {7}{4}} c^{\frac {3}{4}} e^{6} + 4 \, \sqrt {a} c^{2} d^{5} e - 4 \, a^{\frac {3}{2}} c d 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 (\sqrt {2} a^{\frac {1}{4}} c^{\frac {9}{4}} d^{6} + 3 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {7}{4}} d^{4} e^{2} - 3 \, \sqrt {2} a^{\frac {5}{4}} c^{\frac {5}{4}} d^{2} e^{4} - \sqrt {2} a^{\frac {7}{4}} c^{\frac {3}{4}} e^{6} - 4 \, \sqrt {a} c^{2} d^{5} e + 4 \, a^{\frac {3}{2}} c d 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 )}}{8 \, {\left (c^{2} d^{8} + 2 \, a c d^{4} e^{4} + a^{2} e^{8}\right )}} \] Input:
integrate(1/(e*x+d)^2/(c*x^4+a),x, algorithm="maxima")
Output:
4*c*d^3*e^3*log(e*x + d)/(c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8) - e^3/(c*d^5 + a*d*e^4 + (c*d^4*e + a*e^5)*x) - 1/8*c*(sqrt(2)*(4*sqrt(2)*a^(3/4)*c^(5/ 4)*d^3*e^3 - c^2*d^6 + 3*sqrt(a)*c^(3/2)*d^4*e^2 + 3*a*c*d^2*e^4 - a^(3/2) *sqrt(c)*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^(3/4)*c^(5/4)*d^3*e^3 + c^2*d^6 - 3*sq rt(a)*c^(3/2)*d^4*e^2 - 3*a*c*d^2*e^4 + a^(3/2)*sqrt(c)*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*(sqrt(2)*a ^(1/4)*c^(9/4)*d^6 + 3*sqrt(2)*a^(3/4)*c^(7/4)*d^4*e^2 - 3*sqrt(2)*a^(5/4) *c^(5/4)*d^2*e^4 - sqrt(2)*a^(7/4)*c^(3/4)*e^6 + 4*sqrt(a)*c^2*d^5*e - 4*a ^(3/2)*c*d*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*(sqrt( 2)*a^(1/4)*c^(9/4)*d^6 + 3*sqrt(2)*a^(3/4)*c^(7/4)*d^4*e^2 - 3*sqrt(2)*a^( 5/4)*c^(5/4)*d^2*e^4 - sqrt(2)*a^(7/4)*c^(3/4)*e^6 - 4*sqrt(a)*c^2*d^5*e + 4*a^(3/2)*c*d*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)))/(c^2* d^8 + 2*a*c*d^4*e^4 + a^2*e^8)
Time = 0.30 (sec) , antiderivative size = 668, normalized size of antiderivative = 1.51 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx =\text {Too large to display} \] Input:
integrate(1/(e*x+d)^2/(c*x^4+a),x, algorithm="giac")
Output:
4*c*d^3*e^4*log(abs(e*x + d))/(c^2*d^8*e + 2*a*c*d^4*e^5 + a^2*e^9) - c*d^ 3*e^3*log(abs(c*x^4 + a))/(c^2*d^8 + 2*a*c*d^4*e^4 + a^2*e^8) + 1/2*((a*c^ 3)^(1/4)*c^2*d^2 - (a*c^3)^(3/4)*e^2)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a /c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a*c^3*d^4 + sqrt(2)*a^2*c^2*e^4 + 4*sqrt( 2)*sqrt(a*c)*a*c^2*d^2*e^2 - 4*(a*c^3)^(1/4)*a*c^2*d^3*e - 4*(a*c^3)^(3/4) *a*d*e^3) + 1/2*((a*c^3)^(1/4)*c^2*d^2 - (a*c^3)^(3/4)*e^2)*arctan(1/2*sqr t(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(sqrt(2)*a*c^3*d^4 + sqrt(2) *a^2*c^2*e^4 + 4*sqrt(2)*sqrt(a*c)*a*c^2*d^2*e^2 + 4*(a*c^3)^(1/4)*a*c^2*d ^3*e + 4*(a*c^3)^(3/4)*a*d*e^3) + 1/8*(sqrt(2)*(a*c^3)^(1/4)*c^3*d^6 - 3*s qrt(2)*(a*c^3)^(1/4)*a*c^2*d^2*e^4 - 3*sqrt(2)*(a*c^3)^(3/4)*c*d^4*e^2 + s qrt(2)*(a*c^3)^(3/4)*a*e^6)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/( a*c^4*d^8 + 2*a^2*c^3*d^4*e^4 + a^3*c^2*e^8) - 1/8*(sqrt(2)*(a*c^3)^(1/4)* c^3*d^6 - 3*sqrt(2)*(a*c^3)^(1/4)*a*c^2*d^2*e^4 - 3*sqrt(2)*(a*c^3)^(3/4)* c*d^4*e^2 + sqrt(2)*(a*c^3)^(3/4)*a*e^6)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^4*d^8 + 2*a^2*c^3*d^4*e^4 + a^3*c^2*e^8) - (c*d^4*e^3 + a *e^7)/((c*d^4 + a*e^4)^2*(e*x + d))
Time = 22.42 (sec) , antiderivative size = 2436, normalized size of antiderivative = 5.50 \[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=\text {Too large to display} \] Input:
int(1/((a + c*x^4)*(d + e*x)^2),x)
Output:
symsum(log((c^5*d*e^6 + c^5*e^7*x + 16*root(512*a^4*c*d^4*e^4*z^4 + 256*a^ 3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e ^2*z^2 + 32*a*c*d*e*z + c, z, k)^3*a^4*c^4*e^13 + 256*root(512*a^4*c*d^4*e ^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^2*a^2*c^5*d^3*e^8 + 496*ro ot(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^ 3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^3*a^2*c^ 6*d^8*e^5 + 528*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5 *e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^3*a^3*c^5*d^4*e^9 - 128*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2 *d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^ 2 + 32*a*c*d*e*z + c, z, k)^4*a^2*c^7*d^13*e^2 + 128*root(512*a^4*c*d^4*e^ 4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 3 20*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^4*a^3*c^6*d^9*e^6 + 640*roo t(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5*e^8*z^4 + 1024*a^3 *c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)^4*a^4*c^5 *d^5*e^10 + 32*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z^4 + 256*a^5* e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 + 32*a*c*d*e*z + c, z, k)*a*c^5*d^2*e^7 - 16*root(512*a^4*c*d^4*e^4*z^4 + 256*a^3*c^2*d^8*z ^4 + 256*a^5*e^8*z^4 + 1024*a^3*c*d^3*e^3*z^3 + 320*a^2*c*d^2*e^2*z^2 +...
\[ \int \frac {1}{(d+e x)^2 \left (a+c x^4\right )} \, dx=\int \frac {1}{\left (e x +d \right )^{2} \left (c \,x^{4}+a \right )}d x \] Input:
int(1/(e*x+d)^2/(c*x^4+a),x)
Output:
int(1/(e*x+d)^2/(c*x^4+a),x)