\(\int \frac {x^5}{(d+e x^4) (a+b x^4+c x^8)} \, dx\) [80]

Optimal result

Integrand size = 27, antiderivative size = 265 \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\frac {\sqrt {c} \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}+\frac {\sqrt {c} \left (d+\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {b+\sqrt {b^2-4 a c}} \left (c d^2-b d e+a e^2\right )}-\frac {\sqrt {d} \sqrt {e} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{2 \left (c d^2-b d e+a e^2\right )} \] Output:

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

Mathematica [A] (verified)

Time = 0.69 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.09 \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=-\frac {\sqrt {c} \left (-b d+\sqrt {b^2-4 a c} d+2 a e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}} \left (-c d^2+b d e-a e^2\right )}-\frac {\sqrt {c} \left (b d+\sqrt {b^2-4 a c} d-2 a e\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {b^2-4 a c} \sqrt {b+\sqrt {b^2-4 a c}} \left (-c d^2+b d e-a e^2\right )}-\frac {\sqrt {d} \sqrt {e} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{2 \left (c d^2-b d e+a e^2\right )} \] Input:

Integrate[x^5/((d + e*x^4)*(a + b*x^4 + c*x^8)),x]
 

Output:

-1/2*(Sqrt[c]*(-(b*d) + Sqrt[b^2 - 4*a*c]*d + 2*a*e)*ArcTan[(Sqrt[2]*Sqrt[ 
c]*x^2)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b - 
Sqrt[b^2 - 4*a*c]]*(-(c*d^2) + b*d*e - a*e^2)) - (Sqrt[c]*(b*d + Sqrt[b^2 
- 4*a*c]*d - 2*a*e)*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c 
]]])/(2*Sqrt[2]*Sqrt[b^2 - 4*a*c]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*(-(c*d^2) + 
b*d*e - a*e^2)) - (Sqrt[d]*Sqrt[e]*ArcTan[(Sqrt[e]*x^2)/Sqrt[d]])/(2*(c*d^ 
2 - b*d*e + a*e^2))
 

Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.98, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1814, 1610, 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.

\(\displaystyle \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx\)

\(\Big \downarrow \) 1814

\(\displaystyle \frac {1}{2} \int \frac {x^4}{\left (e x^4+d\right ) \left (c x^8+b x^4+a\right )}dx^2\)

\(\Big \downarrow \) 1610

\(\displaystyle \frac {1}{2} \int \left (\frac {c d x^4+a e}{\left (c d^2-b e d+a e^2\right ) \left (c x^8+b x^4+a\right )}-\frac {d e}{\left (c d^2-b e d+a e^2\right ) \left (e x^4+d\right )}\right )dx^2\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {1}{2} \left (\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right ) \left (d-\frac {b d-2 a e}{\sqrt {b^2-4 a c}}\right )}{\sqrt {2} \sqrt {b-\sqrt {b^2-4 a c}} \left (a e^2-b d e+c d^2\right )}+\frac {\sqrt {c} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {\sqrt {b^2-4 a c}+b}}\right ) \left (\frac {b d-2 a e}{\sqrt {b^2-4 a c}}+d\right )}{\sqrt {2} \sqrt {\sqrt {b^2-4 a c}+b} \left (a e^2-b d e+c d^2\right )}-\frac {\sqrt {d} \sqrt {e} \arctan \left (\frac {\sqrt {e} x^2}{\sqrt {d}}\right )}{a e^2-b d e+c d^2}\right )\)

Input:

Int[x^5/((d + e*x^4)*(a + b*x^4 + c*x^8)),x]
 

Output:

((Sqrt[c]*(d - (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x^ 
2)/Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b - Sqrt[b^2 - 4*a*c]]*(c*d 
^2 - b*d*e + a*e^2)) + (Sqrt[c]*(d + (b*d - 2*a*e)/Sqrt[b^2 - 4*a*c])*ArcT 
an[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(Sqrt[2]*Sqrt[b + S 
qrt[b^2 - 4*a*c]]*(c*d^2 - b*d*e + a*e^2)) - (Sqrt[d]*Sqrt[e]*ArcTan[(Sqrt 
[e]*x^2)/Sqrt[d]])/(c*d^2 - b*d*e + a*e^2))/2
 

Defintions of rubi rules used

rule 1610

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

rule 1814

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_)*((d_) + (e 
_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{k = GCD[m + 1, n]}, Simp[1/k   Sub 
st[Int[x^((m + 1)/k - 1)*(d + e*x^(n/k))^q*(a + b*x^(n/k) + c*x^(2*(n/k)))^ 
p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, c, d, e, p, q}, x] && EqQ[n2, 
 2*n] && NeQ[b^2 - 4*a*c, 0] && IGtQ[n, 0] && IntegerQ[m]
 

rule 2009

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

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 219, normalized size of antiderivative = 0.83

method	result	size
default	\(\frac {2 c \left (-\frac {\left (\sqrt {-4 a c +b^{2}}\, d +2 a e -b d \right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (b d +\sqrt {-4 a c +b^{2}}\, d -2 a e \right ) \sqrt {2}\, \arctan \left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{a \,e^{2}-b d e +c \,d^{2}}-\frac {d e \arctan \left (\frac {e \,x^{2}}{\sqrt {d e}}\right )}{2 \left (a \,e^{2}-b d e +c \,d^{2}\right ) \sqrt {d e}}\)	\(219\)
risch	\(\text {Expression too large to display}\)	\(2715\)

Input:

int(x^5/(e*x^4+d)/(c*x^8+b*x^4+a),x,method=_RETURNVERBOSE)
 

Output:

2/(a*e^2-b*d*e+c*d^2)*c*(-1/8*((-4*a*c+b^2)^(1/2)*d+2*a*e-b*d)/(-4*a*c+b^2 
)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^2*2^(1/2)/(( 
-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))+1/8*(b*d+(-4*a*c+b^2)^(1/2)*d-2*a*e)/(-4* 
a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^2*2^(1/ 
2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)))-1/2*d*e/(a*e^2-b*d*e+c*d^2)/(d*e)^(1 
/2)*arctan(e*x^2/(d*e)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 6137 vs. \(2 (215) = 430\).

Time = 67.20 (sec) , antiderivative size = 12293, normalized size of antiderivative = 46.39 \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(e*x^4+d)/(c*x^8+b*x^4+a),x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Timed out} \] Input:

integrate(x**5/(e*x**4+d)/(c*x**8+b*x**4+a),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(x^5/(e*x^4+d)/(c*x^8+b*x^4+a),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4297 vs. \(2 (215) = 430\).

Time = 1.56 (sec) , antiderivative size = 4297, normalized size of antiderivative = 16.22 \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Too large to display} \] Input:

integrate(x^5/(e*x^4+d)/(c*x^8+b*x^4+a),x, algorithm="giac")
 

Output:

-1/2*d*e*arctan(e*x^2/sqrt(d*e))/((c*d^2 - b*d*e + a*e^2)*sqrt(d*e)) + 1/8 
*((2*b^3*c^4 - 8*a*b*c^5 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 
 4*a*c)*c)*b^3*c^2 + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a 
*c)*c)*a*b*c^3 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)* 
c)*b^2*c^3 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b*c 
^4 - 2*(b^2 - 4*a*c)*b*c^4)*d^3*x^4 - (2*b^4*c^3 - 8*a*b^2*c^4 - sqrt(2)*s 
qrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4*c + 4*sqrt(2)*sqrt(b^ 
2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b^2*c^2 + 2*sqrt(2)*sqrt(b^2 
- 4*a*c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - sqrt(2)*sqrt(b^2 - 4*a* 
c)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^2*c^3 - 2*(b^2 - 4*a*c)*b^2*c^3)*d^2* 
e*x^4 + (2*a*b^3*c^3 - 8*a^2*b*c^4 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + 
sqrt(b^2 - 4*a*c)*c)*a*b^3*c + 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt 
(b^2 - 4*a*c)*c)*a^2*b*c^2 + 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b 
^2 - 4*a*c)*c)*a*b^2*c^2 - sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 - 
 4*a*c)*c)*a*b*c^3 - 2*(b^2 - 4*a*c)*a*b*c^3)*d*e^2*x^4 + (sqrt(2)*sqrt(b* 
c + sqrt(b^2 - 4*a*c)*c)*b^4*c - 8*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c) 
*a*b^2*c^2 - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3*c^2 - 2*b^4*c^2 
 + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^3 + 8*sqrt(2)*sqrt(b*c 
 + sqrt(b^2 - 4*a*c)*c)*a*b*c^3 + sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)* 
b^2*c^3 + 16*a*b^2*c^3 - 4*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*c^...
 

Mupad [B] (verification not implemented)

Time = 75.34 (sec) , antiderivative size = 104563, normalized size of antiderivative = 394.58 \[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\text {Too large to display} \] Input:

int(x^5/((d + e*x^4)*(a + b*x^4 + c*x^8)),x)
 

Output:

atan(((x^2*(4*a^4*c^6*d*e^9 - 8*a^3*c^7*d^3*e^7 + 4*a^2*b^2*c^6*d^3*e^7 + 
4*a*b^3*c^6*d^4*e^6 - 8*a^2*b*c^7*d^4*e^6 + 4*a^3*b*c^6*d^2*e^8) + (-(a*b^ 
3*e^2 + a*e^2*(-(4*a*c - b^2)^3)^(1/2) + b^3*c*d^2 - c*d^2*(-(4*a*c - b^2) 
^3)^(1/2) - 4*a*b*c^2*d^2 - 4*a^2*b*c*e^2 + 16*a^2*c^2*d*e - 4*a*b^2*c*d*e 
)/(32*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d 
^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5* 
d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b* 
c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a^3*b*c^2*d*e^3)))^(1/2)*((x^2*(256*a^ 
3*c^9*d^6*e^6 - 3648*a^4*c^8*d^4*e^8 + 1984*a^5*c^7*d^2*e^10 + 1344*a^2*b^ 
3*c^7*d^5*e^7 - 1280*a^2*b^4*c^6*d^4*e^8 + 128*a^2*b^5*c^5*d^3*e^9 + 3712* 
a^3*b^2*c^7*d^4*e^8 - 1152*a^3*b^3*c^6*d^3*e^9 + 128*a^3*b^4*c^5*d^2*e^10 
- 1024*a^4*b^2*c^6*d^2*e^10 - 512*a^5*b*c^6*d*e^11 + 128*a*b^3*c^8*d^7*e^5 
 - 128*a*b^4*c^7*d^6*e^6 - 128*a*b^5*c^6*d^5*e^7 + 128*a*b^6*c^5*d^4*e^8 - 
 256*a^2*b*c^9*d^7*e^5 - 2240*a^3*b*c^8*d^5*e^7 + 2688*a^4*b*c^7*d^3*e^9 + 
 128*a^4*b^3*c^5*d*e^11) + (-(a*b^3*e^2 + a*e^2*(-(4*a*c - b^2)^3)^(1/2) + 
 b^3*c*d^2 - c*d^2*(-(4*a*c - b^2)^3)^(1/2) - 4*a*b*c^2*d^2 - 4*a^2*b*c*e^ 
2 + 16*a^2*c^2*d*e - 4*a*b^2*c*d*e)/(32*(a^2*b^4*e^4 + 16*a^2*c^4*d^4 + 16 
*a^4*c^2*e^4 + b^4*c^2*d^4 + b^6*d^2*e^2 - 8*a*b^2*c^3*d^4 - 8*a^3*b^2*c*e 
^4 + 32*a^3*c^3*d^2*e^2 - 2*a*b^5*d*e^3 - 2*b^5*c*d^3*e + 16*a*b^3*c^2*d^3 
*e - 6*a*b^4*c*d^2*e^2 - 32*a^2*b*c^3*d^3*e + 16*a^2*b^3*c*d*e^3 - 32*a...
 

Reduce [F]

\[ \int \frac {x^5}{\left (d+e x^4\right ) \left (a+b x^4+c x^8\right )} \, dx=\int \frac {x^{5}}{\left (e \,x^{4}+d \right ) \left (c \,x^{8}+b \,x^{4}+a \right )}d x \] Input:

int(x^5/(e*x^4+d)/(c*x^8+b*x^4+a),x)
 

Output:

int(x^5/(e*x^4+d)/(c*x^8+b*x^4+a),x)