\(\int \frac {x^5}{a+b x^4+c x^8} \, dx\) [48]

Optimal result

Integrand size = 18, antiderivative size = 159 \[ \int \frac {x^5}{a+b x^4+c x^8} \, dx=-\frac {\sqrt {b-\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}}+\frac {\sqrt {b+\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {b^2-4 a c}} \] Output:

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.08 \[ \int \frac {x^5}{a+b x^4+c x^8} \, dx=\frac {\left (-b+\sqrt {b^2-4 a c}\right ) \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b-\sqrt {b^2-4 a c}}}\right )+\sqrt {b-\sqrt {b^2-4 a c}} \sqrt {b+\sqrt {b^2-4 a c}} \arctan \left (\frac {\sqrt {2} \sqrt {c} x^2}{\sqrt {b+\sqrt {b^2-4 a c}}}\right )}{2 \sqrt {2} \sqrt {c} \sqrt {b^2-4 a c} \sqrt {b-\sqrt {b^2-4 a c}}} \] Input:

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

Output:

((-b + Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b - Sqrt[b^2 - 
 4*a*c]]] + Sqrt[b - Sqrt[b^2 - 4*a*c]]*Sqrt[b + Sqrt[b^2 - 4*a*c]]*ArcTan 
[(Sqrt[2]*Sqrt[c]*x^2)/Sqrt[b + Sqrt[b^2 - 4*a*c]]])/(2*Sqrt[2]*Sqrt[c]*Sq 
rt[b^2 - 4*a*c]*Sqrt[b - Sqrt[b^2 - 4*a*c]])
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {1695, 1450, 218}

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.

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

\(\Big \downarrow \) 1695

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

\(\Big \downarrow \) 1450

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

\(\Big \downarrow \) 218

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 218

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R 
t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
 

rule 1450

Int[((d_.)*(x_))^(m_)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Wi 
th[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(d^2/2)*(b/q + 1)   Int[(d*x)^(m - 2)/(b/ 
2 + q/2 + c*x^2), x], x] - Simp[(d^2/2)*(b/q - 1)   Int[(d*x)^(m - 2)/(b/2 
- q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d}, x] && NeQ[b^2 - 4*a*c, 0] && 
 GeQ[m, 2]
 

rule 1695

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

Maple [C] (verified)

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

Time = 0.06 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.66

method	result	size
risch	\(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (16 a^{2} c^{3}-8 a \,b^{2} c^{2}+b^{4} c \right ) \textit {\_Z}^{4}+\left (-4 a b c +b^{3}\right ) \textit {\_Z}^{2}+a \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a \,c^{2}+b^{2} c \right ) \textit {\_R}^{2}+b \right ) x^{2}+\left (4 a b \,c^{2}-b^{3} c \right ) \textit {\_R}^{3}+\left (2 a c -b^{2}\right ) \textit {\_R} \right )\right )}{4}\)	\(105\)
default	\(2 c \left (-\frac {\left (-b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \operatorname {arctanh}\left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (-b +\sqrt {-4 a c +b^{2}}\right ) c}}+\frac {\left (b +\sqrt {-4 a c +b^{2}}\right ) \sqrt {2}\, \arctan \left (\frac {c \,x^{2} \sqrt {2}}{\sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )}{8 c \sqrt {-4 a c +b^{2}}\, \sqrt {\left (b +\sqrt {-4 a c +b^{2}}\right ) c}}\right )\)	\(153\)

Input:

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

Output:

1/4*sum(_R*ln(((-4*a*c^2+b^2*c)*_R^2+b)*x^2+(4*a*b*c^2-b^3*c)*_R^3+(2*a*c- 
b^2)*_R),_R=RootOf((16*a^2*c^3-8*a*b^2*c^2+b^4*c)*_Z^4+(-4*a*b*c+b^3)*_Z^2 
+a))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 567 vs. \(2 (119) = 238\).

Time = 0.08 (sec) , antiderivative size = 567, normalized size of antiderivative = 3.57 \[ \int \frac {x^5}{a+b x^4+c x^8} \, dx=\frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b + \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (x^{2} + \frac {\sqrt {\frac {1}{2}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {-\frac {b + \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b + \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (x^{2} - \frac {\sqrt {\frac {1}{2}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {-\frac {b + \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right ) - \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b - \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (x^{2} + \frac {\sqrt {\frac {1}{2}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {-\frac {b - \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right ) + \frac {1}{4} \, \sqrt {\frac {1}{2}} \sqrt {-\frac {b - \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}} \log \left (x^{2} - \frac {\sqrt {\frac {1}{2}} {\left (b^{2} c - 4 \, a c^{2}\right )} \sqrt {-\frac {b - \frac {b^{2} c - 4 \, a c^{2}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}}{b^{2} c - 4 \, a c^{2}}}}{\sqrt {b^{2} c^{2} - 4 \, a c^{3}}}\right ) \] Input:

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

Output:

1/4*sqrt(1/2)*sqrt(-(b + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c 
 - 4*a*c^2))*log(x^2 + sqrt(1/2)*(b^2*c - 4*a*c^2)*sqrt(-(b + (b^2*c - 4*a 
*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))/sqrt(b^2*c^2 - 4*a*c^3)) 
 - 1/4*sqrt(1/2)*sqrt(-(b + (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^ 
2*c - 4*a*c^2))*log(x^2 - sqrt(1/2)*(b^2*c - 4*a*c^2)*sqrt(-(b + (b^2*c - 
4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))/sqrt(b^2*c^2 - 4*a*c^ 
3)) - 1/4*sqrt(1/2)*sqrt(-(b - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/ 
(b^2*c - 4*a*c^2))*log(x^2 + sqrt(1/2)*(b^2*c - 4*a*c^2)*sqrt(-(b - (b^2*c 
 - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))/sqrt(b^2*c^2 - 4*a 
*c^3)) + 1/4*sqrt(1/2)*sqrt(-(b - (b^2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3 
))/(b^2*c - 4*a*c^2))*log(x^2 - sqrt(1/2)*(b^2*c - 4*a*c^2)*sqrt(-(b - (b^ 
2*c - 4*a*c^2)/sqrt(b^2*c^2 - 4*a*c^3))/(b^2*c - 4*a*c^2))/sqrt(b^2*c^2 - 
4*a*c^3))
 

Sympy [A] (verification not implemented)

Time = 1.44 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.48 \[ \int \frac {x^5}{a+b x^4+c x^8} \, dx=\operatorname {RootSum} {\left (t^{4} \cdot \left (4096 a^{2} c^{3} - 2048 a b^{2} c^{2} + 256 b^{4} c\right ) + t^{2} \left (- 64 a b c + 16 b^{3}\right ) + a, \left ( t \mapsto t \log {\left (512 t^{3} a c^{2} - 128 t^{3} b^{2} c - 4 t b + x^{2} \right )} \right )\right )} \] Input:

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

Output:

RootSum(_t**4*(4096*a**2*c**3 - 2048*a*b**2*c**2 + 256*b**4*c) + _t**2*(-6 
4*a*b*c + 16*b**3) + a, Lambda(_t, _t*log(512*_t**3*a*c**2 - 128*_t**3*b** 
2*c - 4*_t*b + x**2)))
 

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1034 vs. \(2 (119) = 238\).

Time = 1.26 (sec) , antiderivative size = 1034, normalized size of antiderivative = 6.50 \[ \int \frac {x^5}{a+b x^4+c x^8} \, dx =\text {Too large to display} \] Input:

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

Output:

1/8*(sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^4 - 8*sqrt(2)*sqrt(b*c + sq 
rt(b^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3 
*c - 2*b^4*c + 16*sqrt(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a^2*c^2 + 8*sqrt 
(2)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqrt(2)*sqrt(b*c + sqrt(b^2 
- 4*a*c)*c)*b^2*c^2 + 16*a*b^2*c^2 - 2*b^3*c^2 - 4*sqrt(2)*sqrt(b*c + sqrt 
(b^2 - 4*a*c)*c)*a*c^3 - 32*a^2*c^3 + 8*a*b*c^3 + sqrt(2)*sqrt(b^2 - 4*a*c 
)*sqrt(b*c + sqrt(b^2 - 4*a*c)*c)*b^3 - 4*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b 
*c + sqrt(b^2 - 4*a*c)*c)*a*b*c - 2*sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + s 
qrt(b^2 - 4*a*c)*c)*b^2*c + sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c + sqrt(b^2 
- 4*a*c)*c)*b*c^2 + 2*(b^2 - 4*a*c)*b^2*c - 8*(b^2 - 4*a*c)*a*c^2 + 2*(b^2 
 - 4*a*c)*b*c^2)*x^4*arctan(2*sqrt(1/2)*x^2/sqrt((b + sqrt(b^2 - 4*a*c))/c 
))/((a*b^4 - 8*a^2*b^2*c - 2*a*b^3*c + 16*a^3*c^2 + 8*a^2*b*c^2 + a*b^2*c^ 
2 - 4*a^2*c^3)*abs(c)) + 1/8*(sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^4 
- 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b^2*c - 2*sqrt(2)*sqrt(b*c - 
 sqrt(b^2 - 4*a*c)*c)*b^3*c + 2*b^4*c + 16*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4 
*a*c)*c)*a^2*c^2 + 8*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c^2 + sqr 
t(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^2*c^2 - 16*a*b^2*c^2 - 2*b^3*c^2 - 
4*sqrt(2)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*c^3 + 32*a^2*c^3 + 8*a*b*c^3 + 
 sqrt(2)*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*b^3 - 4*sqrt(2) 
*sqrt(b^2 - 4*a*c)*sqrt(b*c - sqrt(b^2 - 4*a*c)*c)*a*b*c - 2*sqrt(2)*sq...
 

Mupad [B] (verification not implemented)

Time = 21.03 (sec) , antiderivative size = 1220, normalized size of antiderivative = 7.67 \[ \int \frac {x^5}{a+b x^4+c x^8} \, dx =\text {Too large to display} \] Input:

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

Output:

atan((x^2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2)*1i + b^3* 
x^2*1i - a*b*c*x^2*4i)/(8*b^4*(((b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a* 
b^4*c)^(1/2) - b^3 + 4*a*b*c)/(32*b^4*c + 512*a^2*c^3 - 256*a*b^2*c^2))^(1 
/2) + 128*b^5*c*(((b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) - 
 b^3 + 4*a*b*c)/(32*b^4*c + 512*a^2*c^3 - 256*a*b^2*c^2))^(3/2) + 64*a^2*c 
^2*(((b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) - b^3 + 4*a*b* 
c)/(32*b^4*c + 512*a^2*c^3 - 256*a*b^2*c^2))^(1/2) - 1024*a*b^3*c^2*(((b^6 
 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) - b^3 + 4*a*b*c)/(32*b^ 
4*c + 512*a^2*c^3 - 256*a*b^2*c^2))^(3/2) + 2048*a^2*b*c^3*(((b^6 - 64*a^3 
*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) - b^3 + 4*a*b*c)/(32*b^4*c + 512 
*a^2*c^3 - 256*a*b^2*c^2))^(3/2) - 48*a*b^2*c*(((b^6 - 64*a^3*c^3 + 48*a^2 
*b^2*c^2 - 12*a*b^4*c)^(1/2) - b^3 + 4*a*b*c)/(32*b^4*c + 512*a^2*c^3 - 25 
6*a*b^2*c^2))^(1/2)))*(((b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^( 
1/2) - b^3 + 4*a*b*c)/(32*b^4*c + 512*a^2*c^3 - 256*a*b^2*c^2))^(1/2)*2i - 
 atan((x^2*(b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2)*1i - b^3 
*x^2*1i + a*b*c*x^2*4i)/(8*b^4*(-(b^3 + (b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 
 - 12*a*b^4*c)^(1/2) - 4*a*b*c)/(32*b^4*c + 512*a^2*c^3 - 256*a*b^2*c^2))^ 
(1/2) + 128*b^5*c*(-(b^3 + (b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c 
)^(1/2) - 4*a*b*c)/(32*b^4*c + 512*a^2*c^3 - 256*a*b^2*c^2))^(3/2) + 64*a^ 
2*c^2*(-(b^3 + (b^6 - 64*a^3*c^3 + 48*a^2*b^2*c^2 - 12*a*b^4*c)^(1/2) -...
 

Reduce [F]

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

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

Output:

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