\(\int \frac {1}{x^2 (d+e x^3)^2 (a+c x^6)} \, dx\) [30]

Optimal result

Integrand size = 22, antiderivative size = 654 \[ \int \frac {1}{x^2 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=-\frac {1}{a d^2 x}-\frac {e^3 x^2}{3 d^2 \left (c d^2+a e^2\right ) \left (d+e x^3\right )}-\frac {c^{7/6} \left (c d^2-a e^2\right ) \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{7/6} \left (c d^2+a e^2\right )^2}+\frac {c^{7/6} \left (c d^2-a e^2\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{7/6} \left (c d^2+a e^2\right )^2}-\frac {c^{7/6} \left (c d^2-a e^2\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{6 a^{7/6} \left (c d^2+a e^2\right )^2}+\frac {2 e^{7/3} \left (5 c d^2+2 a e^2\right ) \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{3 \sqrt {3} d^{7/3} \left (c d^2+a e^2\right )^2}+\frac {c^{5/3} d e \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{c} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} \left (c d^2+a e^2\right )^2}+\frac {c^{7/6} \left (c d^2-a e^2\right ) \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x}{\sqrt [3]{a}+\sqrt [3]{c} x^2}\right )}{2 \sqrt {3} a^{7/6} \left (c d^2+a e^2\right )^2}+\frac {2 e^{7/3} \left (5 c d^2+2 a e^2\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{7/3} \left (c d^2+a e^2\right )^2}-\frac {c^{5/3} d e \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{3 a^{2/3} \left (c d^2+a e^2\right )^2}-\frac {e^{7/3} \left (5 c d^2+2 a e^2\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{9 d^{7/3} \left (c d^2+a e^2\right )^2}+\frac {c^{5/3} d e \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{c} x^2+c^{2/3} x^4\right )}{6 a^{2/3} \left (c d^2+a e^2\right )^2} \] Output:

-1/a/d^2/x-1/3*e^3*x^2/d^2/(a*e^2+c*d^2)/(e*x^3+d)-1/3*c^(7/6)*(-a*e^2+c*d 
^2)*arctan(c^(1/6)*x/a^(1/6))/a^(7/6)/(a*e^2+c*d^2)^2-1/6*c^(7/6)*(-a*e^2+ 
c*d^2)*arctan(-3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(7/6)/(a*e^2+c*d^2)^2-1/6*c^ 
(7/6)*(-a*e^2+c*d^2)*arctan(3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(7/6)/(a*e^2+c* 
d^2)^2+2/9*e^(7/3)*(2*a*e^2+5*c*d^2)*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+c*d^2)^2+1/3*c^(5/3)*d*e*arctan(1/3*(a 
^(1/3)-2*c^(1/3)*x^2)*3^(1/2)/a^(1/3))*3^(1/2)/a^(2/3)/(a*e^2+c*d^2)^2+1/6 
*c^(7/6)*(-a*e^2+c*d^2)*arctanh(3^(1/2)*a^(1/6)*c^(1/6)*x/(a^(1/3)+c^(1/3) 
*x^2))*3^(1/2)/a^(7/6)/(a*e^2+c*d^2)^2+2/9*e^(7/3)*(2*a*e^2+5*c*d^2)*ln(d^ 
(1/3)+e^(1/3)*x)/d^(7/3)/(a*e^2+c*d^2)^2-1/3*c^(5/3)*d*e*ln(a^(1/3)+c^(1/3 
)*x^2)/a^(2/3)/(a*e^2+c*d^2)^2-1/9*e^(7/3)*(2*a*e^2+5*c*d^2)*ln(d^(2/3)-d^ 
(1/3)*e^(1/3)*x+e^(2/3)*x^2)/d^(7/3)/(a*e^2+c*d^2)^2+1/6*c^(5/3)*d*e*ln(a^ 
(2/3)-a^(1/3)*c^(1/3)*x^2+c^(2/3)*x^4)/a^(2/3)/(a*e^2+c*d^2)^2
 

Mathematica [A] (verified)

Time = 0.63 (sec) , antiderivative size = 678, normalized size of antiderivative = 1.04 \[ \int \frac {1}{x^2 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\frac {-12 a^{7/6} \sqrt [3]{d} e^3 \left (c d^2+a e^2\right ) x^3-36 \sqrt [6]{a} \sqrt [3]{d} \left (c d^2+a e^2\right )^2 \left (d+e x^3\right )+12 c^{7/6} d^{7/3} \left (-c d^2+a e^2\right ) x \left (d+e x^3\right ) \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )+6 c^{7/6} d^{7/3} \left (c d^2+2 \sqrt {3} \sqrt {a} \sqrt {c} d e-a e^2\right ) x \left (d+e x^3\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )+6 c^{7/6} d^{7/3} \left (-c d^2+2 \sqrt {3} \sqrt {a} \sqrt {c} d e+a e^2\right ) x \left (d+e x^3\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )+8 \sqrt {3} a^{7/6} e^{7/3} \left (5 c d^2+2 a e^2\right ) x \left (d+e x^3\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )+8 a^{7/6} e^{7/3} \left (5 c d^2+2 a e^2\right ) x \left (d+e x^3\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )-12 \sqrt {a} c^{5/3} d^{10/3} e x \left (d+e x^3\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )+3 c^{7/6} d^{7/3} \left (-\sqrt {3} c d^2+2 \sqrt {a} \sqrt {c} d e+\sqrt {3} a e^2\right ) x \left (d+e x^3\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )+3 c^{7/6} d^{7/3} \left (\sqrt {3} c d^2+2 \sqrt {a} \sqrt {c} d e-\sqrt {3} a e^2\right ) x \left (d+e x^3\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )-4 a^{7/6} e^{7/3} \left (5 c d^2+2 a e^2\right ) x \left (d+e x^3\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{36 a^{7/6} d^{7/3} \left (c d^2+a e^2\right )^2 x \left (d+e x^3\right )} \] Input:

Integrate[1/(x^2*(d + e*x^3)^2*(a + c*x^6)),x]
 

Output:

(-12*a^(7/6)*d^(1/3)*e^3*(c*d^2 + a*e^2)*x^3 - 36*a^(1/6)*d^(1/3)*(c*d^2 + 
 a*e^2)^2*(d + e*x^3) + 12*c^(7/6)*d^(7/3)*(-(c*d^2) + a*e^2)*x*(d + e*x^3 
)*ArcTan[(c^(1/6)*x)/a^(1/6)] + 6*c^(7/6)*d^(7/3)*(c*d^2 + 2*Sqrt[3]*Sqrt[ 
a]*Sqrt[c]*d*e - a*e^2)*x*(d + e*x^3)*ArcTan[Sqrt[3] - (2*c^(1/6)*x)/a^(1/ 
6)] + 6*c^(7/6)*d^(7/3)*(-(c*d^2) + 2*Sqrt[3]*Sqrt[a]*Sqrt[c]*d*e + a*e^2) 
*x*(d + e*x^3)*ArcTan[Sqrt[3] + (2*c^(1/6)*x)/a^(1/6)] + 8*Sqrt[3]*a^(7/6) 
*e^(7/3)*(5*c*d^2 + 2*a*e^2)*x*(d + e*x^3)*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/ 
3))/Sqrt[3]] + 8*a^(7/6)*e^(7/3)*(5*c*d^2 + 2*a*e^2)*x*(d + e*x^3)*Log[d^( 
1/3) + e^(1/3)*x] - 12*Sqrt[a]*c^(5/3)*d^(10/3)*e*x*(d + e*x^3)*Log[a^(1/3 
) + c^(1/3)*x^2] + 3*c^(7/6)*d^(7/3)*(-(Sqrt[3]*c*d^2) + 2*Sqrt[a]*Sqrt[c] 
*d*e + Sqrt[3]*a*e^2)*x*(d + e*x^3)*Log[a^(1/3) - Sqrt[3]*a^(1/6)*c^(1/6)* 
x + c^(1/3)*x^2] + 3*c^(7/6)*d^(7/3)*(Sqrt[3]*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d* 
e - Sqrt[3]*a*e^2)*x*(d + e*x^3)*Log[a^(1/3) + Sqrt[3]*a^(1/6)*c^(1/6)*x + 
 c^(1/3)*x^2] - 4*a^(7/6)*e^(7/3)*(5*c*d^2 + 2*a*e^2)*x*(d + e*x^3)*Log[d^ 
(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(36*a^(7/6)*d^(7/3)*(c*d^2 + a*e 
^2)^2*x*(d + e*x^3))
 

Rubi [A] (verified)

Time = 1.23 (sec) , antiderivative size = 888, normalized size of antiderivative = 1.36, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {1837, 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.

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

\(\Big \downarrow \) 1837

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

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {x^2 e^3}{3 d^2 \left (c d^2+a e^2\right ) \left (e x^3+d\right )}+\frac {\left (3 c d^2+a e^2\right ) \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) e^{7/3}}{\sqrt {3} d^{7/3} \left (c d^2+a e^2\right )^2}+\frac {\arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right ) e^{7/3}}{3 \sqrt {3} d^{7/3} \left (c d^2+a e^2\right )}+\frac {\left (3 c d^2+a e^2\right ) \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) e^{7/3}}{3 d^{7/3} \left (c d^2+a e^2\right )^2}+\frac {\log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) e^{7/3}}{9 d^{7/3} \left (c d^2+a e^2\right )}-\frac {\left (3 c d^2+a e^2\right ) \log \left (e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}\right ) e^{7/3}}{6 d^{7/3} \left (c d^2+a e^2\right )^2}-\frac {\log \left (e^{2/3} x^2-\sqrt [3]{d} \sqrt [3]{e} x+d^{2/3}\right ) e^{7/3}}{18 d^{7/3} \left (c d^2+a e^2\right )}-\frac {c^{7/6} \left (c d^2+2 \sqrt {-a} \sqrt {c} e d-a e^2\right ) \arctan \left (\frac {1-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}}{\sqrt {3}}\right )}{2 \sqrt {3} (-a)^{7/6} \left (c d^2+a e^2\right )^2}+\frac {c^{7/6} \left (c d^2-2 \sqrt {-a} \sqrt {c} e d-a e^2\right ) \arctan \left (\frac {\frac {2 \sqrt [6]{c} x}{\sqrt [6]{-a}}+1}{\sqrt {3}}\right )}{2 \sqrt {3} (-a)^{7/6} \left (c d^2+a e^2\right )^2}+\frac {c^{7/6} \left (c d^2-2 \sqrt {-a} \sqrt {c} e d-a e^2\right ) \log \left (\sqrt [6]{-a}-\sqrt [6]{c} x\right )}{6 (-a)^{7/6} \left (c d^2+a e^2\right )^2}-\frac {c^{7/6} \left (c d^2+2 \sqrt {-a} \sqrt {c} e d-a e^2\right ) \log \left (\sqrt [6]{c} x+\sqrt [6]{-a}\right )}{6 (-a)^{7/6} \left (c d^2+a e^2\right )^2}+\frac {c^{7/6} \left (c d^2+2 \sqrt {-a} \sqrt {c} e d-a e^2\right ) \log \left (\sqrt [3]{c} x^2-\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}\right )}{12 (-a)^{7/6} \left (c d^2+a e^2\right )^2}-\frac {c^{7/6} \left (c d^2-2 \sqrt {-a} \sqrt {c} e d-a e^2\right ) \log \left (\sqrt [3]{c} x^2+\sqrt [6]{-a} \sqrt [6]{c} x+\sqrt [3]{-a}\right )}{12 (-a)^{7/6} \left (c d^2+a e^2\right )^2}-\frac {1}{a d^2 x}\)

Input:

Int[1/(x^2*(d + e*x^3)^2*(a + c*x^6)),x]
 

Output:

-(1/(a*d^2*x)) - (e^3*x^2)/(3*d^2*(c*d^2 + a*e^2)*(d + e*x^3)) - (c^(7/6)* 
(c*d^2 + 2*Sqrt[-a]*Sqrt[c]*d*e - a*e^2)*ArcTan[(1 - (2*c^(1/6)*x)/(-a)^(1 
/6))/Sqrt[3]])/(2*Sqrt[3]*(-a)^(7/6)*(c*d^2 + a*e^2)^2) + (c^(7/6)*(c*d^2 
- 2*Sqrt[-a]*Sqrt[c]*d*e - a*e^2)*ArcTan[(1 + (2*c^(1/6)*x)/(-a)^(1/6))/Sq 
rt[3]])/(2*Sqrt[3]*(-a)^(7/6)*(c*d^2 + a*e^2)^2) + (e^(7/3)*ArcTan[(d^(1/3 
) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(3*Sqrt[3]*d^(7/3)*(c*d^2 + a*e^2)) + 
 (e^(7/3)*(3*c*d^2 + a*e^2)*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3 
))])/(Sqrt[3]*d^(7/3)*(c*d^2 + a*e^2)^2) + (c^(7/6)*(c*d^2 - 2*Sqrt[-a]*Sq 
rt[c]*d*e - a*e^2)*Log[(-a)^(1/6) - c^(1/6)*x])/(6*(-a)^(7/6)*(c*d^2 + a*e 
^2)^2) - (c^(7/6)*(c*d^2 + 2*Sqrt[-a]*Sqrt[c]*d*e - a*e^2)*Log[(-a)^(1/6) 
+ c^(1/6)*x])/(6*(-a)^(7/6)*(c*d^2 + a*e^2)^2) + (e^(7/3)*Log[d^(1/3) + e^ 
(1/3)*x])/(9*d^(7/3)*(c*d^2 + a*e^2)) + (e^(7/3)*(3*c*d^2 + a*e^2)*Log[d^( 
1/3) + e^(1/3)*x])/(3*d^(7/3)*(c*d^2 + a*e^2)^2) + (c^(7/6)*(c*d^2 + 2*Sqr 
t[-a]*Sqrt[c]*d*e - a*e^2)*Log[(-a)^(1/3) - (-a)^(1/6)*c^(1/6)*x + c^(1/3) 
*x^2])/(12*(-a)^(7/6)*(c*d^2 + a*e^2)^2) - (c^(7/6)*(c*d^2 - 2*Sqrt[-a]*Sq 
rt[c]*d*e - a*e^2)*Log[(-a)^(1/3) + (-a)^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/( 
12*(-a)^(7/6)*(c*d^2 + a*e^2)^2) - (e^(7/3)*Log[d^(2/3) - d^(1/3)*e^(1/3)* 
x + e^(2/3)*x^2])/(18*d^(7/3)*(c*d^2 + a*e^2)) - (e^(7/3)*(3*c*d^2 + a*e^2 
)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(6*d^(7/3)*(c*d^2 + a*e^ 
2)^2)
 

Defintions of rubi rules used

rule 1837

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

rule 2009

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

Maple [A] (verified)

Time = 0.29 (sec) , antiderivative size = 751, normalized size of antiderivative = 1.15

method	result	size
default	\(\text {Expression too large to display}\)	\(751\)
risch	\(\text {Expression too large to display}\)	\(2257\)

Input:

int(1/x^2/(e*x^3+d)^2/(c*x^6+a),x,method=_RETURNVERBOSE)
 

Output:

-e^3/d^2/(a*e^2+c*d^2)^2*((1/3*a*e^2+1/3*c*d^2)*x^2/(e*x^3+d)+(4/3*a*e^2+1 
0/3*c*d^2)*(-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))))-1/a/d^2/x-(-1/12*ln(x^2-3^(1/2)*(a/c)^(1/6)*x+(a/c)^( 
1/3))*3^(1/2)*(a/c)^(5/6)*e^2+1/12*c/a*ln(x^2-3^(1/2)*(a/c)^(1/6)*x+(a/c)^ 
(1/3))*3^(1/2)*(a/c)^(5/6)*d^2-1/6*c/a*ln(x^2-3^(1/2)*(a/c)^(1/6)*x+(a/c)^ 
(1/3))*(a/c)^(4/3)*d*e-1/6/c*a/(a/c)^(1/6)*arctan(2*x/(a/c)^(1/6)-3^(1/2)) 
*e^2+1/6/(a/c)^(1/6)*arctan(2*x/(a/c)^(1/6)-3^(1/2))*d^2-1/3*c/a*(a/c)^(4/ 
3)*arctan(2*x/(a/c)^(1/6)-3^(1/2))*3^(1/2)*d*e+2/3*(a/c)^(1/3)*arctan(2*x/ 
(a/c)^(1/6)-3^(1/2))*3^(1/2)*d*e+1/12*ln(x^2+3^(1/2)*(a/c)^(1/6)*x+(a/c)^( 
1/3))*3^(1/2)*(a/c)^(5/6)*e^2-1/12*c/a*ln(x^2+3^(1/2)*(a/c)^(1/6)*x+(a/c)^ 
(1/3))*3^(1/2)*(a/c)^(5/6)*d^2-1/6*c/a*ln(x^2+3^(1/2)*(a/c)^(1/6)*x+(a/c)^ 
(1/3))*(a/c)^(4/3)*d*e-1/6/c*a/(a/c)^(1/6)*arctan(2*x/(a/c)^(1/6)+3^(1/2)) 
*e^2+1/6/(a/c)^(1/6)*arctan(2*x/(a/c)^(1/6)+3^(1/2))*d^2+1/3*c/a*(a/c)^(4/ 
3)*arctan(2*x/(a/c)^(1/6)+3^(1/2))*3^(1/2)*d*e-2/3*(a/c)^(1/3)*arctan(2*x/ 
(a/c)^(1/6)+3^(1/2))*3^(1/2)*d*e+1/3*(a/c)^(1/3)*d*e*ln(x^2+(a/c)^(1/3))-1 
/3/c/(a/c)^(1/6)*arctan(x/(a/c)^(1/6))*a*e^2+1/3/(a/c)^(1/6)*arctan(x/(a/c 
)^(1/6))*d^2)/(a*e^2+c*d^2)^2*c^2/a
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\text {Timed out} \] Input:

integrate(1/x^2/(e*x^3+d)^2/(c*x^6+a),x, algorithm="fricas")
 

Output:

Timed out
 

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\text {Timed out} \] Input:

integrate(1/x**2/(e*x**3+d)**2/(c*x**6+a),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {1}{x^2 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\text {Exception raised: ValueError} \] Input:

integrate(1/x^2/(e*x^3+d)^2/(c*x^6+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 [A] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 898, normalized size of antiderivative = 1.37 \[ \int \frac {1}{x^2 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\text {Too large to display} \] Input:

integrate(1/x^2/(e*x^3+d)^2/(c*x^6+a),x, algorithm="giac")
 

Output:

-1/3*(a*c^5)^(1/3)*d*e*log(x^2 + (a/c)^(1/3))/(a*c^2*d^4 + 2*a^2*c*d^2*e^2 
 + a^3*e^4) + 2/9*(5*c*d^2*e^3*(-d/e)^(1/3) + 2*a*e^5*(-d/e)^(1/3))*(-d/e) 
^(1/3)*log(abs(x - (-d/e)^(1/3)))/(c^2*d^7 + 2*a*c*d^5*e^2 + a^2*d^3*e^4) 
+ 2/3*(5*(-d*e^2)^(2/3)*c*d^2*e + 2*(-d*e^2)^(2/3)*a*e^3)*arctan(1/3*sqrt( 
3)*(2*x + (-d/e)^(1/3))/(-d/e)^(1/3))/(sqrt(3)*c^2*d^7 + 2*sqrt(3)*a*c*d^5 
*e^2 + sqrt(3)*a^2*d^3*e^4) + 1/6*(2*sqrt(3)*(a*c^5)^(1/3)*a*c^3*d*e - (a* 
c^5)^(5/6)*c*d^2 + (a*c^5)^(5/6)*a*e^2)*arctan((2*x + sqrt(3)*(a/c)^(1/6)) 
/(a/c)^(1/6))/(a^2*c^5*d^4 + 2*a^3*c^4*d^2*e^2 + a^4*c^3*e^4) - 1/6*(2*sqr 
t(3)*(a*c^5)^(1/3)*a*c^3*d*e + (a*c^5)^(5/6)*c*d^2 - (a*c^5)^(5/6)*a*e^2)* 
arctan((2*x - sqrt(3)*(a/c)^(1/6))/(a/c)^(1/6))/(a^2*c^5*d^4 + 2*a^3*c^4*d 
^2*e^2 + a^4*c^3*e^4) - 1/3*((a*c^5)^(5/6)*c*d^2 - (a*c^5)^(5/6)*a*e^2)*ar 
ctan(x/(a/c)^(1/6))/(a^2*c^5*d^4 + 2*a^3*c^4*d^2*e^2 + a^4*c^3*e^4) + 1/12 
*(2*(a*c^5)^(1/3)*a*c^3*d*e + sqrt(3)*(a*c^5)^(5/6)*c*d^2 - sqrt(3)*(a*c^5 
)^(5/6)*a*e^2)*log(x^2 + sqrt(3)*x*(a/c)^(1/6) + (a/c)^(1/3))/(a^2*c^5*d^4 
 + 2*a^3*c^4*d^2*e^2 + a^4*c^3*e^4) + 1/12*(2*(a*c^5)^(1/3)*a*c^3*d*e - sq 
rt(3)*(a*c^5)^(5/6)*c*d^2 + sqrt(3)*(a*c^5)^(5/6)*a*e^2)*log(x^2 - sqrt(3) 
*x*(a/c)^(1/6) + (a/c)^(1/3))/(a^2*c^5*d^4 + 2*a^3*c^4*d^2*e^2 + a^4*c^3*e 
^4) - 1/9*(5*(-d*e^2)^(2/3)*c*d^2*e + 2*(-d*e^2)^(2/3)*a*e^3)*log(x^2 + x* 
(-d/e)^(1/3) + (-d/e)^(2/3))/(c^2*d^7 + 2*a*c*d^5*e^2 + a^2*d^3*e^4) - 1/3 
*(3*c*d^2*e*x^3 + 4*a*e^3*x^3 + 3*c*d^3 + 3*a*d*e^2)/((a*c*d^4 + a^2*d^...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{x^2 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\text {Hanged} \] Input:

int(1/(x^2*(a + c*x^6)*(d + e*x^3)^2),x)
 

Output:

\text{Hanged}
 

Reduce [F]

\[ \int \frac {1}{x^2 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\int \frac {1}{x^{2} \left (e \,x^{3}+d \right )^{2} \left (c \,x^{6}+a \right )}d x \] Input:

int(1/x^2/(e*x^3+d)^2/(c*x^6+a),x)
 

Output:

int(1/x^2/(e*x^3+d)^2/(c*x^6+a),x)