Integrand size = 19, antiderivative size = 220 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\frac {e^2 x}{c}-\frac {\left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 \sqrt {2} a^{3/4} c^{5/4}} \] Output:
e^2*x/c+1/4*(c*d^2+2*a^(1/2)*c^(1/2)*d*e-a*e^2)*arctan(-1+2^(1/2)*c^(1/4)* x/a^(1/4))*2^(1/2)/a^(3/4)/c^(5/4)+1/4*(c*d^2+2*a^(1/2)*c^(1/2)*d*e-a*e^2) *arctan(1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(3/4)/c^(5/4)+1/4*(c*d^2-2* a^(1/2)*c^(1/2)*d*e-a*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)/c^(5/4)
Time = 0.17 (sec) , antiderivative size = 269, normalized size of antiderivative = 1.22 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\frac {8 a^{3/4} \sqrt [4]{c} e^2 x-2 \sqrt {2} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+2 \sqrt {2} \left (c d^2+2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+\sqrt {2} \left (-c d^2+2 \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+\sqrt {2} \left (c d^2-2 \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{8 a^{3/4} c^{5/4}} \] Input:
Integrate[(d + e*x^2)^2/(a + c*x^4),x]
Output:
(8*a^(3/4)*c^(1/4)*e^2*x - 2*Sqrt[2]*(c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^ 2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 2*Sqrt[2]*(c*d^2 + 2*Sqrt[a]* Sqrt[c]*d*e - a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + Sqrt[2]*(-( c*d^2) + 2*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1 /4)*x + Sqrt[c]*x^2] + Sqrt[2]*(c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log [Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(8*a^(3/4)*c^(5/4))
Time = 0.75 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.35, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1485, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx\) |
| \(\Big \downarrow \) 1485 |
| \(\displaystyle \int \left (\frac {-a e^2+c d^2+2 c d e x^2}{c \left (a+c x^4\right )}+\frac {e^2}{c}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (2 \sqrt {a} \sqrt {c} d e-a e^2+c d^2\right )}{2 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (2 \sqrt {a} \sqrt {c} d e-a e^2+c d^2\right )}{2 \sqrt {2} a^{3/4} c^{5/4}}-\frac {\left (-2 \sqrt {a} \sqrt {c} d e-a e^2+c d^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{5/4}}+\frac {\left (-2 \sqrt {a} \sqrt {c} d e-a e^2+c d^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{5/4}}+\frac {e^2 x}{c}\) |
Input:
Int[(d + e*x^2)^2/(a + c*x^4),x]
Output:
(e^2*x)/c - ((c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*ArcTan[1 - (Sqrt[2]*c ^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(5/4)) + ((c*d^2 + 2*Sqrt[a]*Sqrt [c]*d*e - a*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/ 4)*c^(5/4)) - ((c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 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^(5/4)) + ((c*d^2 - 2*Sqrt[a]*Sqrt[c]*d*e - 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^(5/4))
Int[((d_) + (e_.)*(x_)^2)^(q_)/((a_) + (c_.)*(x_)^4), x_Symbol] :> Int[Expa ndIntegrand[(d + e*x^2)^q/(a + c*x^4), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IntegerQ[q]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.11 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.25
| method | result | size |
| risch | \(\frac {e^{2} x}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (2 \textit {\_R}^{2} c d e -a \,e^{2}+c \,d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 c^{2}}\) | \(56\) |
| default | \(\frac {e^{2} x}{c}+\frac {\frac {\left (-a \,e^{2}+c \,d^{2}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {e d \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{4 \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{c}\) | \(228\) |
Input:
int((e*x^2+d)^2/(c*x^4+a),x,method=_RETURNVERBOSE)
Output:
e^2*x/c+1/4/c^2*sum((2*_R^2*c*d*e-a*e^2+c*d^2)/_R^3*ln(x-_R),_R=RootOf(_Z^ 4*c+a))
Leaf count of result is larger than twice the leaf count of optimal. 1480 vs. \(2 (161) = 322\).
Time = 0.46 (sec) , antiderivative size = 1480, normalized size of antiderivative = 6.73 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)^2/(c*x^4+a),x, algorithm="fricas")
Output:
1/4*(4*e^2*x + c*sqrt(-(4*c*d^3*e - 4*a*d*e^3 + a*c^2*sqrt(-(c^4*d^8 - 12* a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^3*c^5) ))/(a*c^2))*log((c^4*d^8 - 4*a*c^3*d^6*e^2 - 10*a^2*c^2*d^4*e^4 - 4*a^3*c* d^2*e^6 + a^4*e^8)*x + (a*c^4*d^6 - 7*a^2*c^3*d^4*e^2 + 7*a^3*c^2*d^2*e^4 - a^4*c*e^6 + 2*a^3*c^4*d*e*sqrt(-(c^4*d^8 - 12*a*c^3*d^6*e^2 + 38*a^2*c^2 *d^4*e^4 - 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^3*c^5)))*sqrt(-(4*c*d^3*e - 4*a* d*e^3 + a*c^2*sqrt(-(c^4*d^8 - 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12* a^3*c*d^2*e^6 + a^4*e^8)/(a^3*c^5)))/(a*c^2))) - c*sqrt(-(4*c*d^3*e - 4*a* d*e^3 + a*c^2*sqrt(-(c^4*d^8 - 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12* a^3*c*d^2*e^6 + a^4*e^8)/(a^3*c^5)))/(a*c^2))*log((c^4*d^8 - 4*a*c^3*d^6*e ^2 - 10*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*x - (a*c^4*d^6 - 7*a^ 2*c^3*d^4*e^2 + 7*a^3*c^2*d^2*e^4 - a^4*c*e^6 + 2*a^3*c^4*d*e*sqrt(-(c^4*d ^8 - 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a^3*c*d^2*e^6 + a^4*e^8)/( a^3*c^5)))*sqrt(-(4*c*d^3*e - 4*a*d*e^3 + a*c^2*sqrt(-(c^4*d^8 - 12*a*c^3* d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^3*c^5)))/(a* c^2))) + c*sqrt(-(4*c*d^3*e - 4*a*d*e^3 - a*c^2*sqrt(-(c^4*d^8 - 12*a*c^3* d^6*e^2 + 38*a^2*c^2*d^4*e^4 - 12*a^3*c*d^2*e^6 + a^4*e^8)/(a^3*c^5)))/(a* c^2))*log((c^4*d^8 - 4*a*c^3*d^6*e^2 - 10*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^ 6 + a^4*e^8)*x + (a*c^4*d^6 - 7*a^2*c^3*d^4*e^2 + 7*a^3*c^2*d^2*e^4 - a^4* c*e^6 - 2*a^3*c^4*d*e*sqrt(-(c^4*d^8 - 12*a*c^3*d^6*e^2 + 38*a^2*c^2*d^...
Time = 0.72 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.08 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{3} c^{5} + t^{2} \left (- 128 a^{3} c^{3} d e^{3} + 128 a^{2} c^{4} d^{3} e\right ) + a^{4} e^{8} + 4 a^{3} c d^{2} e^{6} + 6 a^{2} c^{2} d^{4} e^{4} + 4 a c^{3} d^{6} e^{2} + c^{4} d^{8}, \left ( t \mapsto t \log {\left (x + \frac {- 128 t^{3} a^{3} c^{4} d e - 4 t a^{4} c e^{6} + 60 t a^{3} c^{2} d^{2} e^{4} - 60 t a^{2} c^{3} d^{4} e^{2} + 4 t a c^{4} d^{6}}{a^{4} e^{8} - 4 a^{3} c d^{2} e^{6} - 10 a^{2} c^{2} d^{4} e^{4} - 4 a c^{3} d^{6} e^{2} + c^{4} d^{8}} \right )} \right )\right )} + \frac {e^{2} x}{c} \] Input:
integrate((e*x**2+d)**2/(c*x**4+a),x)
Output:
RootSum(256*_t**4*a**3*c**5 + _t**2*(-128*a**3*c**3*d*e**3 + 128*a**2*c**4 *d**3*e) + a**4*e**8 + 4*a**3*c*d**2*e**6 + 6*a**2*c**2*d**4*e**4 + 4*a*c* *3*d**6*e**2 + c**4*d**8, Lambda(_t, _t*log(x + (-128*_t**3*a**3*c**4*d*e - 4*_t*a**4*c*e**6 + 60*_t*a**3*c**2*d**2*e**4 - 60*_t*a**2*c**3*d**4*e**2 + 4*_t*a*c**4*d**6)/(a**4*e**8 - 4*a**3*c*d**2*e**6 - 10*a**2*c**2*d**4*e **4 - 4*a*c**3*d**6*e**2 + c**4*d**8)))) + e**2*x/c
Time = 0.11 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.31 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\frac {e^{2} x}{c} + \frac {\frac {2 \, \sqrt {2} {\left (c^{\frac {3}{2}} d^{2} + 2 \, \sqrt {a} c d e - a \sqrt {c} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (c^{\frac {3}{2}} d^{2} + 2 \, \sqrt {a} c d e - a \sqrt {c} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (c^{\frac {3}{2}} d^{2} - 2 \, \sqrt {a} c d e - a \sqrt {c} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (c^{\frac {3}{2}} d^{2} - 2 \, \sqrt {a} c d e - a \sqrt {c} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{8 \, c} \] Input:
integrate((e*x^2+d)^2/(c*x^4+a),x, algorithm="maxima")
Output:
e^2*x/c + 1/8*(2*sqrt(2)*(c^(3/2)*d^2 + 2*sqrt(a)*c*d*e - a*sqrt(c)*e^2)*a rctan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqr t(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sqrt(c)) + 2*sqrt(2)*(c^(3/2)*d^2 + 2*sqrt(a)*c*d*e - a*sqrt(c)*e^2)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2) *a^(1/4)*c^(1/4))/sqrt(sqrt(a)*sqrt(c)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(c))*sq rt(c)) + sqrt(2)*(c^(3/2)*d^2 - 2*sqrt(a)*c*d*e - a*sqrt(c)*e^2)*log(sqrt( c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)) - sqrt(2)* (c^(3/2)*d^2 - 2*sqrt(a)*c*d*e - a*sqrt(c)*e^2)*log(sqrt(c)*x^2 - sqrt(2)* a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)))/c
Time = 0.17 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.45 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx=\frac {e^{2} x}{c} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} + 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{4 \, a c^{3}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - \left (a c^{3}\right )^{\frac {1}{4}} a c e^{2} - 2 \, \left (a c^{3}\right )^{\frac {3}{4}} d e\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{3}} \] Input:
integrate((e*x^2+d)^2/(c*x^4+a),x, algorithm="giac")
Output:
e^2*x/c + 1/4*sqrt(2)*((a*c^3)^(1/4)*c^2*d^2 - (a*c^3)^(1/4)*a*c*e^2 + 2*( a*c^3)^(3/4)*d*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/ 4))/(a*c^3) + 1/4*sqrt(2)*((a*c^3)^(1/4)*c^2*d^2 - (a*c^3)^(1/4)*a*c*e^2 + 2*(a*c^3)^(3/4)*d*e)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c) ^(1/4))/(a*c^3) + 1/8*sqrt(2)*((a*c^3)^(1/4)*c^2*d^2 - (a*c^3)^(1/4)*a*c*e ^2 - 2*(a*c^3)^(3/4)*d*e)*log(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a* c^3) - 1/8*sqrt(2)*((a*c^3)^(1/4)*c^2*d^2 - (a*c^3)^(1/4)*a*c*e^2 - 2*(a*c ^3)^(3/4)*d*e)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^3)
Time = 17.37 (sec) , antiderivative size = 1479, normalized size of antiderivative = 6.72 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx =\text {Too large to display} \] Input:
int((d + e*x^2)^2/(a + c*x^4),x)
Output:
(e^2*x)/c - 2*atanh((8*c^3*d^4*x*((d*e^3)/(4*c^2) - (d^3*e)/(4*a*c) + (d^4 *(-a^3*c^5)^(1/2))/(16*a^3*c^3) + (e^4*(-a^3*c^5)^(1/2))/(16*a*c^5) - (3*d ^2*e^2*(-a^3*c^5)^(1/2))/(8*a^2*c^4))^(1/2))/(4*a^2*d*e^5 - (2*d^6*(-a^3*c ^5)^(1/2))/a^2 + 4*c^2*d^5*e + (2*a*e^6*(-a^3*c^5)^(1/2))/c^3 - 24*a*c*d^3 *e^3 - (14*d^2*e^4*(-a^3*c^5)^(1/2))/c^2 + (14*d^4*e^2*(-a^3*c^5)^(1/2))/( a*c)) + (8*a^2*c*e^4*x*((d*e^3)/(4*c^2) - (d^3*e)/(4*a*c) + (d^4*(-a^3*c^5 )^(1/2))/(16*a^3*c^3) + (e^4*(-a^3*c^5)^(1/2))/(16*a*c^5) - (3*d^2*e^2*(-a ^3*c^5)^(1/2))/(8*a^2*c^4))^(1/2))/(4*a^2*d*e^5 - (2*d^6*(-a^3*c^5)^(1/2)) /a^2 + 4*c^2*d^5*e + (2*a*e^6*(-a^3*c^5)^(1/2))/c^3 - 24*a*c*d^3*e^3 - (14 *d^2*e^4*(-a^3*c^5)^(1/2))/c^2 + (14*d^4*e^2*(-a^3*c^5)^(1/2))/(a*c)) - (4 8*a*c^2*d^2*e^2*x*((d*e^3)/(4*c^2) - (d^3*e)/(4*a*c) + (d^4*(-a^3*c^5)^(1/ 2))/(16*a^3*c^3) + (e^4*(-a^3*c^5)^(1/2))/(16*a*c^5) - (3*d^2*e^2*(-a^3*c^ 5)^(1/2))/(8*a^2*c^4))^(1/2))/(4*a^2*d*e^5 - (2*d^6*(-a^3*c^5)^(1/2))/a^2 + 4*c^2*d^5*e + (2*a*e^6*(-a^3*c^5)^(1/2))/c^3 - 24*a*c*d^3*e^3 - (14*d^2* e^4*(-a^3*c^5)^(1/2))/c^2 + (14*d^4*e^2*(-a^3*c^5)^(1/2))/(a*c)))*((a^2*e^ 4*(-a^3*c^5)^(1/2) + c^2*d^4*(-a^3*c^5)^(1/2) - 4*a^2*c^4*d^3*e + 4*a^3*c^ 3*d*e^3 - 6*a*c*d^2*e^2*(-a^3*c^5)^(1/2))/(16*a^3*c^5))^(1/2) - 2*atanh((8 *c^3*d^4*x*((d*e^3)/(4*c^2) - (d^3*e)/(4*a*c) - (d^4*(-a^3*c^5)^(1/2))/(16 *a^3*c^3) - (e^4*(-a^3*c^5)^(1/2))/(16*a*c^5) + (3*d^2*e^2*(-a^3*c^5)^(1/2 ))/(8*a^2*c^4))^(1/2))/((2*d^6*(-a^3*c^5)^(1/2))/a^2 + 4*a^2*d*e^5 + 4*...
Time = 0.18 (sec) , antiderivative size = 452, normalized size of antiderivative = 2.05 \[ \int \frac {\left (d+e x^2\right )^2}{a+c x^4} \, dx =\text {Too large to display} \] Input:
int((e*x^2+d)^2/(c*x^4+a),x)
Output:
( - 4*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)))*c*d*e + 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)))*a* e**2 - 2*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqr t(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c*d**2 + 4*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)) )*c*d*e - 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)))*a*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)))*c*d**2 + 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)*c*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)*c*d*e + c**(3/4)*a**(1/4) *sqrt(2)*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*a*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)*c*d**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)*a*e**2 + c**(3/4)*a**(1/4)*sqrt(2)*lo g(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c)*x**2)*c*d**2 + 8*a*c*e** 2*x)/(8*a*c**2)