\(\int \frac {x^4}{(d+e x^4) (a+b x^4+c x^8)} \, dx\) [84]

Optimal result

Integrand size = 27, antiderivative size = 630 \[ \int \frac {x^4}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=-\frac {c^{3/4} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [4]{2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right )}-\frac {c^{3/4} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [4]{2} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right )}+\frac {\sqrt [4]{d} e^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )}{2 \sqrt {2} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt [4]{d} e^{3/4} \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )}{2 \sqrt {2} \left (c d^2-b d e+a e^2\right )}-\frac {c^{3/4} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [4]{2} \left (-b-\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right )}-\frac {c^{3/4} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt [4]{2} \left (-b+\sqrt {b^2-4 a c}\right )^{3/4} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt [4]{d} e^{3/4} \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x}{\sqrt {d}+\sqrt {e} x^2}\right )}{2 \sqrt {2} \left (c d^2-b d e+a e^2\right )} \] Output:

-1/4*c^(3/4)*(d+(-2*a*e+b*d)/(-4*a*c+b^2)^(1/2))*arctan(2^(1/4)*c^(1/4)*x/ 
(-b-(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)/(a*e^ 
2-b*d*e+c*d^2)-1/4*c^(3/4)*(d-(-2*a*e+b*d)/(-4*a*c+b^2)^(1/2))*arctan(2^(1 
/4)*c^(1/4)*x/(-b+(-4*a*c+b^2)^(1/2))^(1/4))*2^(3/4)/(-b+(-4*a*c+b^2)^(1/2 
))^(3/4)/(a*e^2-b*d*e+c*d^2)-1/4*d^(1/4)*e^(3/4)*arctan(-1+2^(1/2)*e^(1/4) 
*x/d^(1/4))*2^(1/2)/(a*e^2-b*d*e+c*d^2)-1/4*d^(1/4)*e^(3/4)*arctan(1+2^(1/ 
2)*e^(1/4)*x/d^(1/4))*2^(1/2)/(a*e^2-b*d*e+c*d^2)-1/4*c^(3/4)*(d+(-2*a*e+b 
*d)/(-4*a*c+b^2)^(1/2))*arctanh(2^(1/4)*c^(1/4)*x/(-b-(-4*a*c+b^2)^(1/2))^ 
(1/4))*2^(3/4)/(-b-(-4*a*c+b^2)^(1/2))^(3/4)/(a*e^2-b*d*e+c*d^2)-1/4*c^(3/ 
4)*(d-(-2*a*e+b*d)/(-4*a*c+b^2)^(1/2))*arctanh(2^(1/4)*c^(1/4)*x/(-b+(-4*a 
*c+b^2)^(1/2))^(1/4))*2^(3/4)/(-b+(-4*a*c+b^2)^(1/2))^(3/4)/(a*e^2-b*d*e+c 
*d^2)-1/4*d^(1/4)*e^(3/4)*arctanh(2^(1/2)*d^(1/4)*e^(1/4)*x/(d^(1/2)+e^(1/ 
2)*x^2))*2^(1/2)/(a*e^2-b*d*e+c*d^2)
 

Mathematica [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.

Time = 0.24 (sec) , antiderivative size = 215, normalized size of antiderivative = 0.34 \[ \int \frac {x^4}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\frac {\sqrt {2} \sqrt [4]{d} e^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )+\log \left (\sqrt {d}-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {e} x^2\right )-\log \left (\sqrt {d}+\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {e} x^2\right )\right )+2 \text {RootSum}\left [a+b \text {$\#$1}^4+c \text {$\#$1}^8\&,\frac {a e \log (x-\text {$\#$1})+c d \log (x-\text {$\#$1}) \text {$\#$1}^4}{b \text {$\#$1}^3+2 c \text {$\#$1}^7}\&\right ]}{8 \left (c d^2+e (-b d+a e)\right )} \] Input:

Integrate[x^4/((d + e*x^4)*(a + b*x^4 + c*x^8)),x]
 

Output:

(Sqrt[2]*d^(1/4)*e^(3/4)*(2*ArcTan[1 - (Sqrt[2]*e^(1/4)*x)/d^(1/4)] - 2*Ar 
cTan[1 + (Sqrt[2]*e^(1/4)*x)/d^(1/4)] + Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1 
/4)*x + Sqrt[e]*x^2] - Log[Sqrt[d] + Sqrt[2]*d^(1/4)*e^(1/4)*x + Sqrt[e]*x 
^2]) + 2*RootSum[a + b*#1^4 + c*#1^8 & , (a*e*Log[x - #1] + c*d*Log[x - #1 
]*#1^4)/(b*#1^3 + 2*c*#1^7) & ])/(8*(c*d^2 + e*(-(b*d) + a*e)))
 

Rubi [A] (verified)

Time = 1.31 (sec) , antiderivative size = 699, normalized size of antiderivative = 1.11, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1836, 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.

\(\displaystyle \int \frac {x^4}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx\)

\(\Big \downarrow \) 1836

\(\displaystyle \int \left (\frac {a e+c d x^4}{\left (a+b x^4+c x^8\right ) \left (a e^2-b d e+c d^2\right )}-\frac {d e}{\left (d+e x^4\right ) \left (a e^2-b d e+c d^2\right )}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {c^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right ) \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right )}{2 \sqrt [4]{2} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4} \left (a e^2-b d e+c d^2\right )}-\frac {c^{3/4} \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right ) \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt [4]{2} \left (\sqrt {b^2-4 a c}-b\right )^{3/4} \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt [4]{d} e^{3/4} \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}\right )}{2 \sqrt {2} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt [4]{d} e^{3/4} \arctan \left (\frac {\sqrt {2} \sqrt [4]{e} x}{\sqrt [4]{d}}+1\right )}{2 \sqrt {2} \left (a e^2-b d e+c d^2\right )}-\frac {c^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{-\sqrt {b^2-4 a c}-b}}\right ) \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right )}{2 \sqrt [4]{2} \left (-\sqrt {b^2-4 a c}-b\right )^{3/4} \left (a e^2-b d e+c d^2\right )}-\frac {c^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{c} x}{\sqrt [4]{\sqrt {b^2-4 a c}-b}}\right ) \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )}{2 \sqrt [4]{2} \left (\sqrt {b^2-4 a c}-b\right )^{3/4} \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt [4]{d} e^{3/4} \log \left (-\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {d}+\sqrt {e} x^2\right )}{4 \sqrt {2} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt [4]{d} e^{3/4} \log \left (\sqrt {2} \sqrt [4]{d} \sqrt [4]{e} x+\sqrt {d}+\sqrt {e} x^2\right )}{4 \sqrt {2} \left (a e^2-b d e+c d^2\right )}\)

Input:

Int[x^4/((d + e*x^4)*(a + b*x^4 + c*x^8)),x]
 

Output:

-1/2*(c^(3/4)*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(2^(1/4)*c^(1/4 
)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2^(1/4)*(-b - Sqrt[b^2 - 4*a*c])^(3 
/4)*(c*d^2 - b*d*e + a*e^2)) - (c^(3/4)*(d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a* 
c])*ArcTan[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4) 
*(-b + Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)) + (d^(1/4)*e^(3/4 
)*ArcTan[1 - (Sqrt[2]*e^(1/4)*x)/d^(1/4)])/(2*Sqrt[2]*(c*d^2 - b*d*e + a*e 
^2)) - (d^(1/4)*e^(3/4)*ArcTan[1 + (Sqrt[2]*e^(1/4)*x)/d^(1/4)])/(2*Sqrt[2 
]*(c*d^2 - b*d*e + a*e^2)) - (c^(3/4)*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c] 
)*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b - Sqrt[b^2 - 4*a*c])^(1/4)])/(2*2^(1/4)* 
(-b - Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e + a*e^2)) - (c^(3/4)*(d - (b 
*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTanh[(2^(1/4)*c^(1/4)*x)/(-b + Sqrt[b^2 
- 4*a*c])^(1/4)])/(2*2^(1/4)*(-b + Sqrt[b^2 - 4*a*c])^(3/4)*(c*d^2 - b*d*e 
 + a*e^2)) + (d^(1/4)*e^(3/4)*Log[Sqrt[d] - Sqrt[2]*d^(1/4)*e^(1/4)*x + Sq 
rt[e]*x^2])/(4*Sqrt[2]*(c*d^2 - b*d*e + a*e^2)) - (d^(1/4)*e^(3/4)*Log[Sqr 
t[d] + Sqrt[2]*d^(1/4)*e^(1/4)*x + Sqrt[e]*x^2])/(4*Sqrt[2]*(c*d^2 - b*d*e 
 + a*e^2))
 

Defintions of rubi rules used

rule 1836

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(n_))^(q_.))/((a_) + (c_.)*(x_)^ 
(n2_.) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*((d + e 
*x^n)^q/(a + b*x^n + c*x^(2*n))), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] 
 && EqQ[n2, 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[q] && Int 
egerQ[m]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [C] (verified)

Result contains higher order function than in optimal. Order 9 vs. order 3.

Time = 0.29 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.30

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8} c +\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (\textit {\_R}^{4} c d +a e \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{7} c +\textit {\_R}^{3} b}}{4 a \,e^{2}-4 b d e +4 c \,d^{2}}-\frac {e \left (\frac {d}{e}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {d}{e}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d}{e}}}{x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {d}{e}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d}{e}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {d}{e}\right )^{\frac {1}{4}}}-1\right )\right )}{8 \left (a \,e^{2}-b d e +c \,d^{2}\right )}\)	\(186\)
risch	\(\text {Expression too large to display}\)	\(8669\)

Input:

int(x^4/(e*x^4+d)/(c*x^8+b*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

1/4/(a*e^2-b*d*e+c*d^2)*sum((_R^4*c*d+a*e)/(2*_R^7*c+_R^3*b)*ln(x-_R),_R=R 
ootOf(_Z^8*c+_Z^4*b+a))-1/8*e/(a*e^2-b*d*e+c*d^2)*(d/e)^(1/4)*2^(1/2)*(ln( 
(x^2+(d/e)^(1/4)*x*2^(1/2)+(d/e)^(1/2))/(x^2-(d/e)^(1/4)*x*2^(1/2)+(d/e)^( 
1/2)))+2*arctan(2^(1/2)/(d/e)^(1/4)*x+1)+2*arctan(2^(1/2)/(d/e)^(1/4)*x-1) 
)
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 153.05 (sec) , antiderivative size = 47212, normalized size of antiderivative = 74.94 \[ \int \frac {x^4}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Too large to display} \] Input:

integrate(x^4/(e*x^4+d)/(c*x^8+b*x^4+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^4}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Timed out} \] Input:

integrate(x**4/(e*x**4+d)/(c*x**8+b*x**4+a),x)
                                                                                    
                                                                                    
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {x^4}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\int { \frac {x^{4}}{{\left (c x^{8} + b x^{4} + a\right )} {\left (e x^{4} + d\right )}} \,d x } \] Input:

integrate(x^4/(e*x^4+d)/(c*x^8+b*x^4+a),x, algorithm="maxima")
 

Output:

-1/8*(sqrt(2)*e^(3/4)*log(sqrt(e)*x^2 + sqrt(2)*d^(1/4)*e^(1/4)*x + sqrt(d 
))/d^(3/4) - sqrt(2)*e^(3/4)*log(sqrt(e)*x^2 - sqrt(2)*d^(1/4)*e^(1/4)*x + 
 sqrt(d))/d^(3/4) + sqrt(2)*e*log((2*sqrt(e)*x - sqrt(2)*sqrt(-sqrt(d)*sqr 
t(e)) + sqrt(2)*d^(1/4)*e^(1/4))/(2*sqrt(e)*x + sqrt(2)*sqrt(-sqrt(d)*sqrt 
(e)) + sqrt(2)*d^(1/4)*e^(1/4)))/(sqrt(d)*sqrt(-sqrt(d)*sqrt(e))) + sqrt(2 
)*e*log((2*sqrt(e)*x - sqrt(2)*sqrt(-sqrt(d)*sqrt(e)) - sqrt(2)*d^(1/4)*e^ 
(1/4))/(2*sqrt(e)*x + sqrt(2)*sqrt(-sqrt(d)*sqrt(e)) - sqrt(2)*d^(1/4)*e^( 
1/4)))/(sqrt(d)*sqrt(-sqrt(d)*sqrt(e))))*d/(c*d^2 - b*d*e + a*e^2) + integ 
rate((c*d*x^4 + a*e)/(c*x^8 + b*x^4 + a), x)/(c*d^2 - b*d*e + a*e^2)
 

Giac [F(-1)]

Timed out. \[ \int \frac {x^4}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Timed out} \] Input:

integrate(x^4/(e*x^4+d)/(c*x^8+b*x^4+a),x, algorithm="giac")
 

Output:

Timed out
 

Mupad [B] (verification not implemented)

Time = 51.64 (sec) , antiderivative size = 359899, normalized size of antiderivative = 571.27 \[ \int \frac {x^4}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Too large to display} \] Input:

int(x^4/((d + e*x^4)*(a + b*x^4 + c*x^8)),x)
 

Output:

(2^(1/2)*atan(((2^(1/2)*(x*(4*a^4*c^7*d^2*e^9 - 8*a^3*c^8*d^4*e^7 + 4*a^2* 
b^2*c^7*d^4*e^7) - (2^(1/2)*(-(-(d*e^3)/64)^(1/2))^(1/2)*((2^(1/2)*(x*(204 
8*a^3*c^12*d^11*e^4 + 8192*a^4*c^11*d^9*e^6 - 118784*a^5*c^10*d^7*e^8 + 27 
0336*a^6*c^9*d^5*e^10 - 129024*a^7*c^8*d^3*e^12 - 1024*a^2*b^2*c^11*d^11*e 
^4 + 4096*a^2*b^3*c^10*d^10*e^5 - 6144*a^2*b^4*c^9*d^9*e^6 + 3072*a^2*b^5* 
c^8*d^8*e^7 + 3072*a^2*b^6*c^7*d^7*e^8 - 6144*a^2*b^7*c^6*d^6*e^9 + 4096*a 
^2*b^8*c^5*d^5*e^10 - 1024*a^2*b^9*c^4*d^4*e^11 + 8192*a^3*b^2*c^10*d^9*e^ 
6 + 12288*a^3*b^3*c^9*d^8*e^7 - 51200*a^3*b^4*c^8*d^7*e^8 + 70656*a^3*b^5* 
c^7*d^6*e^9 - 49152*a^3*b^6*c^6*d^5*e^10 + 13312*a^3*b^7*c^5*d^4*e^11 + 14 
9504*a^4*b^2*c^9*d^7*e^8 - 233472*a^4*b^3*c^8*d^6*e^9 + 206848*a^4*b^4*c^7 
*d^5*e^10 - 68608*a^4*b^5*c^6*d^4*e^11 + 4096*a^4*b^6*c^5*d^3*e^12 - 1024* 
a^4*b^7*c^4*d^2*e^13 - 385024*a^5*b^2*c^8*d^5*e^10 + 176128*a^5*b^3*c^7*d^ 
4*e^11 - 40960*a^5*b^4*c^6*d^3*e^12 + 11264*a^5*b^5*c^5*d^2*e^13 + 130048* 
a^6*b^2*c^7*d^3*e^12 - 40960*a^6*b^3*c^6*d^2*e^13 - 8192*a^3*b*c^11*d^10*e 
^5 - 40960*a^4*b*c^10*d^8*e^7 + 253952*a^5*b*c^9*d^6*e^9 - 188416*a^6*b*c^ 
8*d^4*e^11 + 49152*a^7*b*c^7*d^2*e^13) - (2^(1/2)*(-(-(d*e^3)/64)^(1/2))^( 
1/2)*((2^(1/2)*(x*(33554432*a^5*c^14*d^14*e^5 + 100663296*a^6*c^13*d^12*e^ 
7 + 134217728*a^7*c^12*d^10*e^9 + 201326592*a^8*c^11*d^8*e^11 + 301989888* 
a^9*c^10*d^6*e^13 + 234881024*a^10*c^9*d^4*e^15 + 67108864*a^11*c^8*d^2*e^ 
17 + 262144*a^2*b^5*c^12*d^15*e^4 - 2097152*a^2*b^6*c^11*d^14*e^5 + 734...
 

Reduce [F]

\[ \int \frac {x^4}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\int \frac {x^{4}}{\left (e \,x^{4}+d \right ) \left (c \,x^{8}+b \,x^{4}+a \right )}d x \] Input:

int(x^4/(e*x^4+d)/(c*x^8+b*x^4+a),x)
 

Output:

int(x^4/(e*x^4+d)/(c*x^8+b*x^4+a),x)