Integrand size = 19, antiderivative size = 353 \[ \int \frac {\left (d+e x^2\right )^4}{a+c x^4} \, dx=\frac {e^2 \left (6 c d^2-a e^2\right ) x}{c^2}+\frac {4 d e^3 x^3}{3 c}+\frac {e^4 x^5}{5 c}-\frac {\left (c^2 d^4+4 \sqrt {a} c^{3/2} d^3 e-6 a c d^2 e^2-4 a^{3/2} \sqrt {c} d e^3+a^2 e^4\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{9/4}}+\frac {\left (c^2 d^4+4 \sqrt {a} c^{3/2} d^3 e-6 a c d^2 e^2-4 a^{3/2} \sqrt {c} d e^3+a^2 e^4\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} c^{9/4}}+\frac {\left (c^2 d^4-4 \sqrt {a} c^{3/2} d^3 e-6 a c d^2 e^2+4 a^{3/2} \sqrt {c} d e^3+a^2 e^4\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{2 \sqrt {2} a^{3/4} c^{9/4}} \] Output:
e^2*(-a*e^2+6*c*d^2)*x/c^2+4/3*d*e^3*x^3/c+1/5*e^4*x^5/c+1/4*(c^2*d^4+4*a^ (1/2)*c^(3/2)*d^3*e-6*a*c*d^2*e^2-4*a^(3/2)*c^(1/2)*d*e^3+a^2*e^4)*arctan( -1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(3/4)/c^(9/4)+1/4*(c^2*d^4+4*a^(1/ 2)*c^(3/2)*d^3*e-6*a*c*d^2*e^2-4*a^(3/2)*c^(1/2)*d*e^3+a^2*e^4)*arctan(1+2 ^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(3/4)/c^(9/4)+1/4*(c^2*d^4-4*a^(1/2)*c ^(3/2)*d^3*e-6*a*c*d^2*e^2+4*a^(3/2)*c^(1/2)*d*e^3+a^2*e^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)/c^(9/4)
Time = 0.23 (sec) , antiderivative size = 444, normalized size of antiderivative = 1.26 \[ \int \frac {\left (d+e x^2\right )^4}{a+c x^4} \, dx=\frac {-120 a^{3/4} \sqrt [4]{c} e^2 \left (-6 c d^2+a e^2\right ) x+160 a^{3/4} c^{5/4} d e^3 x^3+24 a^{3/4} c^{5/4} e^4 x^5-30 \sqrt {2} \left (c^2 d^4+4 \sqrt {a} c^{3/2} d^3 e-6 a c d^2 e^2-4 a^{3/2} \sqrt {c} d e^3+a^2 e^4\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+30 \sqrt {2} \left (c^2 d^4+4 \sqrt {a} c^{3/2} d^3 e-6 a c d^2 e^2-4 a^{3/2} \sqrt {c} d e^3+a^2 e^4\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )-15 \sqrt {2} \left (c^2 d^4-4 \sqrt {a} c^{3/2} d^3 e-6 a c d^2 e^2+4 a^{3/2} \sqrt {c} d e^3+a^2 e^4\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )+15 \sqrt {2} \left (c^2 d^4-4 \sqrt {a} c^{3/2} d^3 e-6 a c d^2 e^2+4 a^{3/2} \sqrt {c} d e^3+a^2 e^4\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{120 a^{3/4} c^{9/4}} \] Input:
Integrate[(d + e*x^2)^4/(a + c*x^4),x]
Output:
(-120*a^(3/4)*c^(1/4)*e^2*(-6*c*d^2 + a*e^2)*x + 160*a^(3/4)*c^(5/4)*d*e^3 *x^3 + 24*a^(3/4)*c^(5/4)*e^4*x^5 - 30*Sqrt[2]*(c^2*d^4 + 4*Sqrt[a]*c^(3/2 )*d^3*e - 6*a*c*d^2*e^2 - 4*a^(3/2)*Sqrt[c]*d*e^3 + a^2*e^4)*ArcTan[1 - (S qrt[2]*c^(1/4)*x)/a^(1/4)] + 30*Sqrt[2]*(c^2*d^4 + 4*Sqrt[a]*c^(3/2)*d^3*e - 6*a*c*d^2*e^2 - 4*a^(3/2)*Sqrt[c]*d*e^3 + a^2*e^4)*ArcTan[1 + (Sqrt[2]* c^(1/4)*x)/a^(1/4)] - 15*Sqrt[2]*(c^2*d^4 - 4*Sqrt[a]*c^(3/2)*d^3*e - 6*a* c*d^2*e^2 + 4*a^(3/2)*Sqrt[c]*d*e^3 + a^2*e^4)*Log[Sqrt[a] - Sqrt[2]*a^(1/ 4)*c^(1/4)*x + Sqrt[c]*x^2] + 15*Sqrt[2]*(c^2*d^4 - 4*Sqrt[a]*c^(3/2)*d^3* e - 6*a*c*d^2*e^2 + 4*a^(3/2)*Sqrt[c]*d*e^3 + a^2*e^4)*Log[Sqrt[a] + Sqrt[ 2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(120*a^(3/4)*c^(9/4))
Time = 1.09 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.24, 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.
| \(\displaystyle \int \frac {\left (d+e x^2\right )^4}{a+c x^4} \, dx\) |
| \(\Big \downarrow \) 1485 |
| \(\displaystyle \int \left (\frac {a^2 e^4+4 c d e x^2 \left (c d^2-a e^2\right )-6 a c d^2 e^2+c^2 d^4}{c^2 \left (a+c x^4\right )}+\frac {e^2 \left (6 c d^2-a e^2\right )}{c^2}+\frac {4 d e^3 x^2}{c}+\frac {e^4 x^4}{c}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (a^2 e^4-6 a c d^2 e^2+4 \sqrt {a} \sqrt {c} d e \left (c d^2-a e^2\right )+c^2 d^4\right )}{2 \sqrt {2} a^{3/4} c^{9/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (a^2 e^4-6 a c d^2 e^2+4 \sqrt {a} \sqrt {c} d e \left (c d^2-a e^2\right )+c^2 d^4\right )}{2 \sqrt {2} a^{3/4} c^{9/4}}-\frac {\left (a^2 e^4-6 a c d^2 e^2-4 \sqrt {a} \sqrt {c} d e \left (c d^2-a e^2\right )+c^2 d^4\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{9/4}}+\frac {\left (a^2 e^4-6 a c d^2 e^2-4 \sqrt {a} \sqrt {c} d e \left (c d^2-a e^2\right )+c^2 d^4\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{9/4}}+\frac {e^2 x \left (6 c d^2-a e^2\right )}{c^2}+\frac {4 d e^3 x^3}{3 c}+\frac {e^4 x^5}{5 c}\) |
Input:
Int[(d + e*x^2)^4/(a + c*x^4),x]
Output:
(e^2*(6*c*d^2 - a*e^2)*x)/c^2 + (4*d*e^3*x^3)/(3*c) + (e^4*x^5)/(5*c) - (( c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 + 4*Sqrt[a]*Sqrt[c]*d*e*(c*d^2 - a*e^2)) *ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(9/4)) + (( c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 + 4*Sqrt[a]*Sqrt[c]*d*e*(c*d^2 - a*e^2)) *ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(9/4)) - (( c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 - 4*Sqrt[a]*Sqrt[c]*d*e*(c*d^2 - 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^(9/4)) + ((c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4 - 4*Sqrt[a]*Sqrt[c]*d*e*( c*d^2 - 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^(9/4))
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 = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.33
| method | result | size |
| risch | \(\frac {e^{4} x^{5}}{5 c}+\frac {4 d \,e^{3} x^{3}}{3 c}-\frac {e^{4} x a}{c^{2}}+\frac {6 e^{2} x \,d^{2}}{c}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (4 e c d \left (-a \,e^{2}+c \,d^{2}\right ) \textit {\_R}^{2}+a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 c^{3}}\) | \(118\) |
| default | \(-\frac {e^{2} \left (-\frac {1}{5} c \,x^{5} e^{2}-\frac {4}{3} c d \,x^{3} e +x a \,e^{2}-6 x c \,d^{2}\right )}{c^{2}}+\frac {\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {\left (-4 a c d \,e^{3}+4 c^{2} d^{3} e \right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{8 c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{c^{2}}\) | \(291\) |
Input:
int((e*x^2+d)^4/(c*x^4+a),x,method=_RETURNVERBOSE)
Output:
1/5*e^4*x^5/c+4/3*d*e^3*x^3/c-e^4/c^2*x*a+6*e^2/c*x*d^2+1/4/c^3*sum((4*e*c *d*(-a*e^2+c*d^2)*_R^2+a^2*e^4-6*a*c*d^2*e^2+c^2*d^4)/_R^3*ln(x-_R),_R=Roo tOf(_Z^4*c+a))
Leaf count of result is larger than twice the leaf count of optimal. 2878 vs. \(2 (278) = 556\).
Time = 9.62 (sec) , antiderivative size = 2878, normalized size of antiderivative = 8.15 \[ \int \frac {\left (d+e x^2\right )^4}{a+c x^4} \, dx=\text {Too large to display} \] Input:
integrate((e*x^2+d)^4/(c*x^4+a),x, algorithm="fricas")
Output:
1/60*(12*c*e^4*x^5 + 80*c*d*e^3*x^3 + 15*c^2*sqrt(-(8*c^3*d^7*e - 56*a*c^2 *d^5*e^3 + 56*a^2*c*d^3*e^5 - 8*a^3*d*e^7 + a*c^4*sqrt(-(c^8*d^16 - 56*a*c ^7*d^14*e^2 + 924*a^2*c^6*d^12*e^4 - 3976*a^3*c^5*d^10*e^6 + 6470*a^4*c^4* d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^12 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))/(a*c^4))*log((c^8*d^16 - 24*a*c^7*d^14*e^2 - 36*a ^2*c^6*d^12*e^4 + 88*a^3*c^5*d^10*e^6 + 198*a^4*c^4*d^8*e^8 + 88*a^5*c^3*d ^6*e^10 - 36*a^6*c^2*d^4*e^12 - 24*a^7*c*d^2*e^14 + a^8*e^16)*x + (a*c^8*d ^12 - 34*a^2*c^7*d^10*e^2 + 239*a^3*c^6*d^8*e^4 - 476*a^4*c^5*d^6*e^6 + 23 9*a^5*c^4*d^4*e^8 - 34*a^6*c^3*d^2*e^10 + a^7*c^2*e^12 + 4*(a^3*c^8*d^3*e - a^4*c^7*d*e^3)*sqrt(-(c^8*d^16 - 56*a*c^7*d^14*e^2 + 924*a^2*c^6*d^12*e^ 4 - 3976*a^3*c^5*d^10*e^6 + 6470*a^4*c^4*d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^12 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))*sqrt(-(8 *c^3*d^7*e - 56*a*c^2*d^5*e^3 + 56*a^2*c*d^3*e^5 - 8*a^3*d*e^7 + a*c^4*sqr t(-(c^8*d^16 - 56*a*c^7*d^14*e^2 + 924*a^2*c^6*d^12*e^4 - 3976*a^3*c^5*d^1 0*e^6 + 6470*a^4*c^4*d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^1 2 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))/(a*c^4))) - 15*c^2*sqrt(-(8* c^3*d^7*e - 56*a*c^2*d^5*e^3 + 56*a^2*c*d^3*e^5 - 8*a^3*d*e^7 + a*c^4*sqrt (-(c^8*d^16 - 56*a*c^7*d^14*e^2 + 924*a^2*c^6*d^12*e^4 - 3976*a^3*c^5*d^10 *e^6 + 6470*a^4*c^4*d^8*e^8 - 3976*a^5*c^3*d^6*e^10 + 924*a^6*c^2*d^4*e^12 - 56*a^7*c*d^2*e^14 + a^8*e^16)/(a^3*c^9)))/(a*c^4))*log((c^8*d^16 - 2...
Time = 1.81 (sec) , antiderivative size = 500, normalized size of antiderivative = 1.42 \[ \int \frac {\left (d+e x^2\right )^4}{a+c x^4} \, dx=x \left (- \frac {a e^{4}}{c^{2}} + \frac {6 d^{2} e^{2}}{c}\right ) + \operatorname {RootSum} {\left (256 t^{4} a^{3} c^{9} + t^{2} \left (- 256 a^{5} c^{5} d e^{7} + 1792 a^{4} c^{6} d^{3} e^{5} - 1792 a^{3} c^{7} d^{5} e^{3} + 256 a^{2} c^{8} d^{7} e\right ) + a^{8} e^{16} + 8 a^{7} c d^{2} e^{14} + 28 a^{6} c^{2} d^{4} e^{12} + 56 a^{5} c^{3} d^{6} e^{10} + 70 a^{4} c^{4} d^{8} e^{8} + 56 a^{3} c^{5} d^{10} e^{6} + 28 a^{2} c^{6} d^{12} e^{4} + 8 a c^{7} d^{14} e^{2} + c^{8} d^{16}, \left ( t \mapsto t \log {\left (x + \frac {256 t^{3} a^{4} c^{7} d e^{3} - 256 t^{3} a^{3} c^{8} d^{3} e + 4 t a^{7} c^{2} e^{12} - 264 t a^{6} c^{3} d^{2} e^{10} + 1980 t a^{5} c^{4} d^{4} e^{8} - 3696 t a^{4} c^{5} d^{6} e^{6} + 1980 t a^{3} c^{6} d^{8} e^{4} - 264 t a^{2} c^{7} d^{10} e^{2} + 4 t a c^{8} d^{12}}{a^{8} e^{16} - 24 a^{7} c d^{2} e^{14} - 36 a^{6} c^{2} d^{4} e^{12} + 88 a^{5} c^{3} d^{6} e^{10} + 198 a^{4} c^{4} d^{8} e^{8} + 88 a^{3} c^{5} d^{10} e^{6} - 36 a^{2} c^{6} d^{12} e^{4} - 24 a c^{7} d^{14} e^{2} + c^{8} d^{16}} \right )} \right )\right )} + \frac {4 d e^{3} x^{3}}{3 c} + \frac {e^{4} x^{5}}{5 c} \] Input:
integrate((e*x**2+d)**4/(c*x**4+a),x)
Output:
x*(-a*e**4/c**2 + 6*d**2*e**2/c) + RootSum(256*_t**4*a**3*c**9 + _t**2*(-2 56*a**5*c**5*d*e**7 + 1792*a**4*c**6*d**3*e**5 - 1792*a**3*c**7*d**5*e**3 + 256*a**2*c**8*d**7*e) + a**8*e**16 + 8*a**7*c*d**2*e**14 + 28*a**6*c**2* d**4*e**12 + 56*a**5*c**3*d**6*e**10 + 70*a**4*c**4*d**8*e**8 + 56*a**3*c* *5*d**10*e**6 + 28*a**2*c**6*d**12*e**4 + 8*a*c**7*d**14*e**2 + c**8*d**16 , Lambda(_t, _t*log(x + (256*_t**3*a**4*c**7*d*e**3 - 256*_t**3*a**3*c**8* d**3*e + 4*_t*a**7*c**2*e**12 - 264*_t*a**6*c**3*d**2*e**10 + 1980*_t*a**5 *c**4*d**4*e**8 - 3696*_t*a**4*c**5*d**6*e**6 + 1980*_t*a**3*c**6*d**8*e** 4 - 264*_t*a**2*c**7*d**10*e**2 + 4*_t*a*c**8*d**12)/(a**8*e**16 - 24*a**7 *c*d**2*e**14 - 36*a**6*c**2*d**4*e**12 + 88*a**5*c**3*d**6*e**10 + 198*a* *4*c**4*d**8*e**8 + 88*a**3*c**5*d**10*e**6 - 36*a**2*c**6*d**12*e**4 - 24 *a*c**7*d**14*e**2 + c**8*d**16)))) + 4*d*e**3*x**3/(3*c) + e**4*x**5/(5*c )
Time = 0.11 (sec) , antiderivative size = 432, normalized size of antiderivative = 1.22 \[ \int \frac {\left (d+e x^2\right )^4}{a+c x^4} \, dx=\frac {3 \, c e^{4} x^{5} + 20 \, c d e^{3} x^{3} + 15 \, {\left (6 \, c d^{2} e^{2} - a e^{4}\right )} x}{15 \, c^{2}} + \frac {\frac {2 \, \sqrt {2} {\left (c^{\frac {5}{2}} d^{4} + 4 \, \sqrt {a} c^{2} d^{3} e - 6 \, a c^{\frac {3}{2}} d^{2} e^{2} - 4 \, a^{\frac {3}{2}} c d e^{3} + a^{2} \sqrt {c} e^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {2 \, \sqrt {2} {\left (c^{\frac {5}{2}} d^{4} + 4 \, \sqrt {a} c^{2} d^{3} e - 6 \, a c^{\frac {3}{2}} d^{2} e^{2} - 4 \, a^{\frac {3}{2}} c d e^{3} + a^{2} \sqrt {c} e^{4}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {c}} \sqrt {c}} + \frac {\sqrt {2} {\left (c^{\frac {5}{2}} d^{4} - 4 \, \sqrt {a} c^{2} d^{3} e - 6 \, a c^{\frac {3}{2}} d^{2} e^{2} + 4 \, a^{\frac {3}{2}} c d e^{3} + a^{2} \sqrt {c} e^{4}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (c^{\frac {5}{2}} d^{4} - 4 \, \sqrt {a} c^{2} d^{3} e - 6 \, a c^{\frac {3}{2}} d^{2} e^{2} + 4 \, a^{\frac {3}{2}} c d e^{3} + a^{2} \sqrt {c} e^{4}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}}}{8 \, c^{2}} \] Input:
integrate((e*x^2+d)^4/(c*x^4+a),x, algorithm="maxima")
Output:
1/15*(3*c*e^4*x^5 + 20*c*d*e^3*x^3 + 15*(6*c*d^2*e^2 - a*e^4)*x)/c^2 + 1/8 *(2*sqrt(2)*(c^(5/2)*d^4 + 4*sqrt(a)*c^2*d^3*e - 6*a*c^(3/2)*d^2*e^2 - 4*a ^(3/2)*c*d*e^3 + a^2*sqrt(c)*e^4)*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))*s qrt(c)) + 2*sqrt(2)*(c^(5/2)*d^4 + 4*sqrt(a)*c^2*d^3*e - 6*a*c^(3/2)*d^2*e ^2 - 4*a^(3/2)*c*d*e^3 + a^2*sqrt(c)*e^4)*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)*sq rt(c))*sqrt(c)) + sqrt(2)*(c^(5/2)*d^4 - 4*sqrt(a)*c^2*d^3*e - 6*a*c^(3/2) *d^2*e^2 + 4*a^(3/2)*c*d*e^3 + a^2*sqrt(c)*e^4)*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^(5/2)*d^4 - 4* sqrt(a)*c^2*d^3*e - 6*a*c^(3/2)*d^2*e^2 + 4*a^(3/2)*c*d*e^3 + a^2*sqrt(c)* e^4)*log(sqrt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/ 4)))/c^2
Time = 0.12 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.44 \[ \int \frac {\left (d+e x^2\right )^4}{a+c x^4} \, dx=\frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} - 6 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\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^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} - 6 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\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^{4}} + \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} - 6 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{4}} - \frac {\sqrt {2} {\left (\left (a c^{3}\right )^{\frac {1}{4}} c^{3} d^{4} - 6 \, \left (a c^{3}\right )^{\frac {1}{4}} a c^{2} d^{2} e^{2} + \left (a c^{3}\right )^{\frac {1}{4}} a^{2} c e^{4} - 4 \, \left (a c^{3}\right )^{\frac {3}{4}} c d^{3} e + 4 \, \left (a c^{3}\right )^{\frac {3}{4}} a d e^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{8 \, a c^{4}} + \frac {3 \, c^{4} e^{4} x^{5} + 20 \, c^{4} d e^{3} x^{3} + 90 \, c^{4} d^{2} e^{2} x - 15 \, a c^{3} e^{4} x}{15 \, c^{5}} \] Input:
integrate((e*x^2+d)^4/(c*x^4+a),x, algorithm="giac")
Output:
1/4*sqrt(2)*((a*c^3)^(1/4)*c^3*d^4 - 6*(a*c^3)^(1/4)*a*c^2*d^2*e^2 + (a*c^ 3)^(1/4)*a^2*c*e^4 + 4*(a*c^3)^(3/4)*c*d^3*e - 4*(a*c^3)^(3/4)*a*d*e^3)*ar ctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^4) + 1/4*sq rt(2)*((a*c^3)^(1/4)*c^3*d^4 - 6*(a*c^3)^(1/4)*a*c^2*d^2*e^2 + (a*c^3)^(1/ 4)*a^2*c*e^4 + 4*(a*c^3)^(3/4)*c*d^3*e - 4*(a*c^3)^(3/4)*a*d*e^3)*arctan(1 /2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^4) + 1/8*sqrt(2)* ((a*c^3)^(1/4)*c^3*d^4 - 6*(a*c^3)^(1/4)*a*c^2*d^2*e^2 + (a*c^3)^(1/4)*a^2 *c*e^4 - 4*(a*c^3)^(3/4)*c*d^3*e + 4*(a*c^3)^(3/4)*a*d*e^3)*log(x^2 + sqrt (2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^4) - 1/8*sqrt(2)*((a*c^3)^(1/4)*c^3*d^ 4 - 6*(a*c^3)^(1/4)*a*c^2*d^2*e^2 + (a*c^3)^(1/4)*a^2*c*e^4 - 4*(a*c^3)^(3 /4)*c*d^3*e + 4*(a*c^3)^(3/4)*a*d*e^3)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + s qrt(a/c))/(a*c^4) + 1/15*(3*c^4*e^4*x^5 + 20*c^4*d*e^3*x^3 + 90*c^4*d^2*e^ 2*x - 15*a*c^3*e^4*x)/c^5
Time = 17.78 (sec) , antiderivative size = 4022, normalized size of antiderivative = 11.39 \[ \int \frac {\left (d+e x^2\right )^4}{a+c x^4} \, dx=\text {Too large to display} \] Input:
int((d + e*x^2)^4/(a + c*x^4),x)
Output:
atan((((4*x*(a^4*e^8 + c^4*d^8 - 28*a*c^3*d^6*e^2 - 28*a^3*c*d^2*e^6 + 70* a^2*c^2*d^4*e^4))/c - (4*(4*a*c^6*d^4 + 4*a^3*c^4*e^4 - 24*a^2*c^5*d^2*e^2 )*((a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) - 8*a^2*c^8*d^7*e + 8*a^5*c^5*d*e^7 + 56*a^3*c^7*d^5*e^3 - 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6 *e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4 *e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2))/c^3)*((a^4*e^8*(-a^3*c^9)^(1/2 ) + c^4*d^8*(-a^3*c^9)^(1/2) - 8*a^2*c^8*d^7*e + 8*a^5*c^5*d*e^7 + 56*a^3* c^7*d^5*e^3 - 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28* a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16* a^3*c^9))^(1/2)*1i + ((4*x*(a^4*e^8 + c^4*d^8 - 28*a*c^3*d^6*e^2 - 28*a^3* c*d^2*e^6 + 70*a^2*c^2*d^4*e^4))/c + (4*(4*a*c^6*d^4 + 4*a^3*c^4*e^4 - 24* a^2*c^5*d^2*e^2)*((a^4*e^8*(-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) - 8 *a^2*c^8*d^7*e + 8*a^5*c^5*d*e^7 + 56*a^3*c^7*d^5*e^3 - 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c^9)^(1/2))/(16*a^3*c^9))^(1/2))/c^3)*((a^4*e^8* (-a^3*c^9)^(1/2) + c^4*d^8*(-a^3*c^9)^(1/2) - 8*a^2*c^8*d^7*e + 8*a^5*c^5* d*e^7 + 56*a^3*c^7*d^5*e^3 - 56*a^4*c^6*d^3*e^5 - 28*a*c^3*d^6*e^2*(-a^3*c ^9)^(1/2) - 28*a^3*c*d^2*e^6*(-a^3*c^9)^(1/2) + 70*a^2*c^2*d^4*e^4*(-a^3*c ^9)^(1/2))/(16*a^3*c^9))^(1/2)*1i)/(((4*x*(a^4*e^8 + c^4*d^8 - 28*a*c^3*d^ 6*e^2 - 28*a^3*c*d^2*e^6 + 70*a^2*c^2*d^4*e^4))/c - (4*(4*a*c^6*d^4 + 4...
Time = 0.20 (sec) , antiderivative size = 812, normalized size of antiderivative = 2.30 \[ \int \frac {\left (d+e x^2\right )^4}{a+c x^4} \, dx =\text {Too large to display} \] Input:
int((e*x^2+d)^4/(c*x^4+a),x)
Output:
(120*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)))*a*c*d*e**3 - 120*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**2*d**3*e - 30*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqr t(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**2*e**4 + 180*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*c*d**2*e**2 - 30*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**2*d** 4 - 120*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)))*a*c*d*e**3 + 120*c**(1/4)*a**(3/4)*sqr t(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqr t(2)))*c**2*d**3*e + 30*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**2*e**4 - 180*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*c*d**2*e**2 + 30*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**2* d**4 - 60*c**(1/4)*a**(3/4)*sqrt(2)*log( - c**(1/4)*a**(1/4)*sqrt(2)*x + s qrt(a) + sqrt(c)*x**2)*a*c*d*e**3 + 60*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**2*d**3*e + 60*c**(1 /4)*a**(3/4)*sqrt(2)*log(c**(1/4)*a**(1/4)*sqrt(2)*x + sqrt(a) + sqrt(c...