\(\int \frac {1}{x^5 (d+e x^3) (a+b x^3+c x^6)} \, dx\) [63]

Optimal result

Integrand size = 27, antiderivative size = 1138 \[ \int \frac {1}{x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )} \, dx =\text {Too large to display} \] Output:

-1/4/a/d/x^4+(a*e+b*d)/a^2/d^2/x-1/6*c^(1/3)*(b*c*d-b^2*e+a*c*e+(3*a*b*c*e 
-2*a*c^2*d-b^3*e+b^2*c*d)/(-4*a*c+b^2)^(1/2))*arctan(1/3*(1-2*2^(1/3)*c^(1 
/3)*x/(b-(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)/a^2/(b-(-4*a* 
c+b^2)^(1/2))^(1/3)/(a*e^2-b*d*e+c*d^2)-1/6*c^(1/3)*(b*c*d-b^2*e+a*c*e-(3* 
a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)/(-4*a*c+b^2)^(1/2))*arctan(1/3*(1-2*2^(1/ 
3)*c^(1/3)*x/(b+(-4*a*c+b^2)^(1/2))^(1/3))*3^(1/2))*2^(1/3)*3^(1/2)/a^2/(b 
+(-4*a*c+b^2)^(1/2))^(1/3)/(a*e^2-b*d*e+c*d^2)-1/3*e^(10/3)*arctan(1/3*(d^ 
(1/3)-2*e^(1/3)*x)*3^(1/2)/d^(1/3))*3^(1/2)/d^(7/3)/(a*e^2-b*d*e+c*d^2)-1/ 
6*c^(1/3)*(b*c*d-b^2*e+a*c*e+(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)/(-4*a*c+b 
^2)^(1/2))*ln((b-(-4*a*c+b^2)^(1/2))^(1/3)+2^(1/3)*c^(1/3)*x)*2^(1/3)/a^2/ 
(b-(-4*a*c+b^2)^(1/2))^(1/3)/(a*e^2-b*d*e+c*d^2)-1/6*c^(1/3)*(b*c*d-b^2*e+ 
a*c*e-(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)/(-4*a*c+b^2)^(1/2))*ln((b+(-4*a* 
c+b^2)^(1/2))^(1/3)+2^(1/3)*c^(1/3)*x)*2^(1/3)/a^2/(b+(-4*a*c+b^2)^(1/2))^ 
(1/3)/(a*e^2-b*d*e+c*d^2)-1/3*e^(10/3)*ln(d^(1/3)+e^(1/3)*x)/d^(7/3)/(a*e^ 
2-b*d*e+c*d^2)+1/12*c^(1/3)*(b*c*d-b^2*e+a*c*e+(3*a*b*c*e-2*a*c^2*d-b^3*e+ 
b^2*c*d)/(-4*a*c+b^2)^(1/2))*ln((b-(-4*a*c+b^2)^(1/2))^(2/3)-2^(1/3)*c^(1/ 
3)*(b-(-4*a*c+b^2)^(1/2))^(1/3)*x+2^(2/3)*c^(2/3)*x^2)*2^(1/3)/a^2/(b-(-4* 
a*c+b^2)^(1/2))^(1/3)/(a*e^2-b*d*e+c*d^2)+1/12*c^(1/3)*(b*c*d-b^2*e+a*c*e- 
(3*a*b*c*e-2*a*c^2*d-b^3*e+b^2*c*d)/(-4*a*c+b^2)^(1/2))*ln((b+(-4*a*c+b^2) 
^(1/2))^(2/3)-2^(1/3)*c^(1/3)*(b+(-4*a*c+b^2)^(1/2))^(1/3)*x+2^(2/3)*c^...
 

Mathematica [C] (verified)

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

Time = 0.28 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.31 \[ \int \frac {1}{x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )} \, dx=\frac {-3 a d^{4/3} \left (c d^2+e (-b d+a e)\right )+12 \sqrt [3]{d} (b d+a e) \left (c d^2+e (-b d+a e)\right ) x^3-4 \sqrt {3} a^2 e^{10/3} x^4 \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )-4 a^2 e^{10/3} x^4 \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )+2 a^2 e^{10/3} x^4 \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )-4 d^{7/3} x^4 \text {RootSum}\left [a+b \text {$\#$1}^3+c \text {$\#$1}^6\&,\frac {-b^2 c d \log (x-\text {$\#$1})+a c^2 d \log (x-\text {$\#$1})+b^3 e \log (x-\text {$\#$1})-2 a b c e \log (x-\text {$\#$1})-b c^2 d \log (x-\text {$\#$1}) \text {$\#$1}^3+b^2 c e \log (x-\text {$\#$1}) \text {$\#$1}^3-a c^2 e \log (x-\text {$\#$1}) \text {$\#$1}^3}{b \text {$\#$1}+2 c \text {$\#$1}^4}\&\right ]}{12 a^2 d^{7/3} \left (c d^2+e (-b d+a e)\right ) x^4} \] Input:

Integrate[1/(x^5*(d + e*x^3)*(a + b*x^3 + c*x^6)),x]
 

Output:

(-3*a*d^(4/3)*(c*d^2 + e*(-(b*d) + a*e)) + 12*d^(1/3)*(b*d + a*e)*(c*d^2 + 
 e*(-(b*d) + a*e))*x^3 - 4*Sqrt[3]*a^2*e^(10/3)*x^4*ArcTan[(1 - (2*e^(1/3) 
*x)/d^(1/3))/Sqrt[3]] - 4*a^2*e^(10/3)*x^4*Log[d^(1/3) + e^(1/3)*x] + 2*a^ 
2*e^(10/3)*x^4*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2] - 4*d^(7/3)* 
x^4*RootSum[a + b*#1^3 + c*#1^6 & , (-(b^2*c*d*Log[x - #1]) + a*c^2*d*Log[ 
x - #1] + b^3*e*Log[x - #1] - 2*a*b*c*e*Log[x - #1] - b*c^2*d*Log[x - #1]* 
#1^3 + b^2*c*e*Log[x - #1]*#1^3 - a*c^2*e*Log[x - #1]*#1^3)/(b*#1 + 2*c*#1 
^4) & ])/(12*a^2*d^(7/3)*(c*d^2 + e*(-(b*d) + a*e))*x^4)
 

Rubi [A] (verified)

Time = 2.76 (sec) , antiderivative size = 1138, normalized size of antiderivative = 1.00, 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.

\(\displaystyle \int \frac {1}{x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )} \, dx\)

\(\Big \downarrow \) 1836

\(\displaystyle \int \left (\frac {x \left (c x^3 \left (a c e+b^2 (-e)+b c d\right )+2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right )}{a^2 \left (a+b x^3+c x^6\right ) \left (a e^2-b d e+c d^2\right )}+\frac {-a e-b d}{a^2 d^2 x^2}+\frac {e^4 x}{d^2 \left (d+e x^3\right ) \left (a e^2-b d e+c d^2\right )}+\frac {1}{a d x^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {\arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) e^{10/3}}{\sqrt {3} d^{7/3} \left (c d^2-b e d+a e^2\right )}-\frac {\log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) e^{10/3}}{3 d^{7/3} \left (c d^2-b e d+a e^2\right )}+\frac {\log \left (e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}\right ) e^{10/3}}{6 d^{7/3} \left (c d^2-b e d+a e^2\right )}-\frac {\sqrt [3]{c} \left (-e b^2+c d b+a c e+\frac {-e b^3+c d b^2+3 a c e b-2 a c^2 d}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b-\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a^2 \sqrt [3]{b-\sqrt {b^2-4 a c}} \left (c d^2-b e d+a e^2\right )}-\frac {\sqrt [3]{c} \left (-e b^2+c d b+a c e-\frac {-e b^3+c d b^2+3 a c e b-2 a c^2 d}{\sqrt {b^2-4 a c}}\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{2} \sqrt [3]{c} x}{\sqrt [3]{b+\sqrt {b^2-4 a c}}}}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} a^2 \sqrt [3]{b+\sqrt {b^2-4 a c}} \left (c d^2-b e d+a e^2\right )}-\frac {\sqrt [3]{c} \left (-e b^2+c d b+a c e+\frac {-e b^3+c d b^2+3 a c e b-2 a c^2 d}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b-\sqrt {b^2-4 a c}}\right )}{3\ 2^{2/3} a^2 \sqrt [3]{b-\sqrt {b^2-4 a c}} \left (c d^2-b e d+a e^2\right )}-\frac {\sqrt [3]{c} \left (-e b^2+c d b+a c e-\frac {-e b^3+c d b^2+3 a c e b-2 a c^2 d}{\sqrt {b^2-4 a c}}\right ) \log \left (\sqrt [3]{2} \sqrt [3]{c} x+\sqrt [3]{b+\sqrt {b^2-4 a c}}\right )}{3\ 2^{2/3} a^2 \sqrt [3]{b+\sqrt {b^2-4 a c}} \left (c d^2-b e d+a e^2\right )}+\frac {\sqrt [3]{c} \left (-e b^2+c d b+a c e+\frac {-e b^3+c d b^2+3 a c e b-2 a c^2 d}{\sqrt {b^2-4 a c}}\right ) \log \left (2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b-\sqrt {b^2-4 a c}} x+\left (b-\sqrt {b^2-4 a c}\right )^{2/3}\right )}{6\ 2^{2/3} a^2 \sqrt [3]{b-\sqrt {b^2-4 a c}} \left (c d^2-b e d+a e^2\right )}+\frac {\sqrt [3]{c} \left (-e b^2+c d b+a c e-\frac {-e b^3+c d b^2+3 a c e b-2 a c^2 d}{\sqrt {b^2-4 a c}}\right ) \log \left (2^{2/3} c^{2/3} x^2-\sqrt [3]{2} \sqrt [3]{c} \sqrt [3]{b+\sqrt {b^2-4 a c}} x+\left (b+\sqrt {b^2-4 a c}\right )^{2/3}\right )}{6\ 2^{2/3} a^2 \sqrt [3]{b+\sqrt {b^2-4 a c}} \left (c d^2-b e d+a e^2\right )}+\frac {b d+a e}{a^2 d^2 x}-\frac {1}{4 a d x^4}\)

Input:

Int[1/(x^5*(d + e*x^3)*(a + b*x^3 + c*x^6)),x]
 

Output:

-1/4*1/(a*d*x^4) + (b*d + a*e)/(a^2*d^2*x) - (c^(1/3)*(b*c*d - b^2*e + a*c 
*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c])*ArcTan[( 
1 - (2*2^(1/3)*c^(1/3)*x)/(b - Sqrt[b^2 - 4*a*c])^(1/3))/Sqrt[3]])/(2^(2/3 
)*Sqrt[3]*a^2*(b - Sqrt[b^2 - 4*a*c])^(1/3)*(c*d^2 - b*d*e + a*e^2)) - (c^ 
(1/3)*(b*c*d - b^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/S 
qrt[b^2 - 4*a*c])*ArcTan[(1 - (2*2^(1/3)*c^(1/3)*x)/(b + Sqrt[b^2 - 4*a*c] 
)^(1/3))/Sqrt[3]])/(2^(2/3)*Sqrt[3]*a^2*(b + Sqrt[b^2 - 4*a*c])^(1/3)*(c*d 
^2 - b*d*e + a*e^2)) - (e^(10/3)*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d 
^(1/3))])/(Sqrt[3]*d^(7/3)*(c*d^2 - b*d*e + a*e^2)) - (c^(1/3)*(b*c*d - b^ 
2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c]) 
*Log[(b - Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(2/3)*a^2*(b 
 - Sqrt[b^2 - 4*a*c])^(1/3)*(c*d^2 - b*d*e + a*e^2)) - (c^(1/3)*(b*c*d - b 
^2*e + a*c*e - (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c] 
)*Log[(b + Sqrt[b^2 - 4*a*c])^(1/3) + 2^(1/3)*c^(1/3)*x])/(3*2^(2/3)*a^2*( 
b + Sqrt[b^2 - 4*a*c])^(1/3)*(c*d^2 - b*d*e + a*e^2)) - (e^(10/3)*Log[d^(1 
/3) + e^(1/3)*x])/(3*d^(7/3)*(c*d^2 - b*d*e + a*e^2)) + (c^(1/3)*(b*c*d - 
b^2*e + a*c*e + (b^2*c*d - 2*a*c^2*d - b^3*e + 3*a*b*c*e)/Sqrt[b^2 - 4*a*c 
])*Log[(b - Sqrt[b^2 - 4*a*c])^(2/3) - 2^(1/3)*c^(1/3)*(b - Sqrt[b^2 - 4*a 
*c])^(1/3)*x + 2^(2/3)*c^(2/3)*x^2])/(6*2^(2/3)*a^2*(b - Sqrt[b^2 - 4*a*c] 
)^(1/3)*(c*d^2 - b*d*e + a*e^2)) + (c^(1/3)*(b*c*d - b^2*e + a*c*e - (b...
 

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 = 0.32 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.23

method	result	size
default	\(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{6} c +\textit {\_Z}^{3} b +a \right )}{\sum }\frac {\left (c \left (a c e -b^{2} e +c b d \right ) \textit {\_R}^{4}+\left (2 a b c e -a \,c^{2} d -b^{3} e +b^{2} c d \right ) \textit {\_R} \right ) \ln \left (x -\textit {\_R} \right )}{2 \textit {\_R}^{5} c +\textit {\_R}^{2} b}}{3 \left (a \,e^{2}-b d e +c \,d^{2}\right ) a^{2}}+\frac {\left (-\frac {\ln \left (x +\left (\frac {d}{e}\right )^{\frac {1}{3}}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {d}{e}\right )^{\frac {1}{3}} x +\left (\frac {d}{e}\right )^{\frac {2}{3}}\right )}{6 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {d}{e}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 e \left (\frac {d}{e}\right )^{\frac {1}{3}}}\right ) e^{4}}{d^{2} \left (a \,e^{2}-b d e +c \,d^{2}\right )}-\frac {1}{4 a d \,x^{4}}-\frac {-a e -b d}{d^{2} a^{2} x}\)	\(257\)
risch	\(\text {Expression too large to display}\)	\(11771\)

Input:

int(1/x^5/(e*x^3+d)/(c*x^6+b*x^3+a),x,method=_RETURNVERBOSE)
 

Output:

1/3/(a*e^2-b*d*e+c*d^2)/a^2*sum((c*(a*c*e-b^2*e+b*c*d)*_R^4+(2*a*b*c*e-a*c 
^2*d-b^3*e+b^2*c*d)*_R)/(2*_R^5*c+_R^2*b)*ln(x-_R),_R=RootOf(_Z^6*c+_Z^3*b 
+a))+(-1/3/e/(d/e)^(1/3)*ln(x+(d/e)^(1/3))+1/6/e/(d/e)^(1/3)*ln(x^2-(d/e)^ 
(1/3)*x+(d/e)^(2/3))+1/3*3^(1/2)/e/(d/e)^(1/3)*arctan(1/3*3^(1/2)*(2/(d/e) 
^(1/3)*x-1)))*e^4/d^2/(a*e^2-b*d*e+c*d^2)-1/4/a/d/x^4-1/d^2/a^2*(-a*e-b*d) 
/x
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )} \, dx=\text {Timed out} \] Input:

integrate(1/x^5/(e*x^3+d)/(c*x^6+b*x^3+a),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )} \, dx=\text {Timed out} \] Input:

integrate(1/x**5/(e*x**3+d)/(c*x**6+b*x**3+a),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/x^5/(e*x^3+d)/(c*x^6+b*x^3+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 [F]

\[ \int \frac {1}{x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )} \, dx=\int { \frac {1}{{\left (c x^{6} + b x^{3} + a\right )} {\left (e x^{3} + d\right )} x^{5}} \,d x } \] Input:

integrate(1/x^5/(e*x^3+d)/(c*x^6+b*x^3+a),x, algorithm="giac")
 

Output:

integrate(1/((c*x^6 + b*x^3 + a)*(e*x^3 + d)*x^5), x)
                                                                                    
                                                                                    
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )} \, dx=\text {Hanged} \] Input:

int(1/(x^5*(d + e*x^3)*(a + b*x^3 + c*x^6)),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {1}{x^5 \left (d+e x^3\right ) \left (a+b x^3+c x^6\right )} \, dx=\int \frac {1}{c e \,x^{14}+b e \,x^{11}+c d \,x^{11}+a e \,x^{8}+b d \,x^{8}+a d \,x^{5}}d x \] Input:

int(1/x^5/(e*x^3+d)/(c*x^6+b*x^3+a),x)
 

Output:

int(1/(a*d*x**5 + a*e*x**8 + b*d*x**8 + b*e*x**11 + c*d*x**11 + c*e*x**14) 
,x)