Integrand size = 22, antiderivative size = 659 \[ \int \frac {1}{x^5 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=-\frac {1}{4 a d^2 x^4}+\frac {2 e}{a d^3 x}+\frac {e^4 x^2}{3 d^3 \left (c d^2+a e^2\right ) \left (d+e x^3\right )}+\frac {2 c^{13/6} d e \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{7/6} \left (c d^2+a e^2\right )^2}-\frac {c^{13/6} d e \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{7/6} \left (c d^2+a e^2\right )^2}+\frac {c^{13/6} d e \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{3 a^{7/6} \left (c d^2+a e^2\right )^2}-\frac {e^{10/3} \left (13 c d^2+7 a e^2\right ) \arctan \left (\frac {\sqrt [3]{d}-2 \sqrt [3]{e} x}{\sqrt {3} \sqrt [3]{d}}\right )}{3 \sqrt {3} d^{10/3} \left (c d^2+a e^2\right )^2}+\frac {c^{5/3} \left (c d^2-a e^2\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{c} x^2}{\sqrt {3} \sqrt [3]{a}}\right )}{2 \sqrt {3} a^{5/3} \left (c d^2+a e^2\right )^2}-\frac {c^{13/6} d e \text {arctanh}\left (\frac {\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x}{\sqrt [3]{a}+\sqrt [3]{c} x^2}\right )}{\sqrt {3} a^{7/6} \left (c d^2+a e^2\right )^2}-\frac {e^{10/3} \left (13 c d^2+7 a e^2\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{9 d^{10/3} \left (c d^2+a e^2\right )^2}-\frac {c^{5/3} \left (c d^2-a e^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{6 a^{5/3} \left (c d^2+a e^2\right )^2}+\frac {e^{10/3} \left (13 c d^2+7 a e^2\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{18 d^{10/3} \left (c d^2+a e^2\right )^2}+\frac {c^{5/3} \left (c d^2-a e^2\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{c} x^2+c^{2/3} x^4\right )}{12 a^{5/3} \left (c d^2+a e^2\right )^2} \] Output:
-1/4/a/d^2/x^4+2*e/a/d^3/x+1/3*e^4*x^2/d^3/(a*e^2+c*d^2)/(e*x^3+d)+2/3*c^( 13/6)*d*e*arctan(c^(1/6)*x/a^(1/6))/a^(7/6)/(a*e^2+c*d^2)^2+1/3*c^(13/6)*d *e*arctan(-3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(7/6)/(a*e^2+c*d^2)^2+1/3*c^(13/ 6)*d*e*arctan(3^(1/2)+2*c^(1/6)*x/a^(1/6))/a^(7/6)/(a*e^2+c*d^2)^2-1/9*e^( 10/3)*(7*a*e^2+13*c*d^2)*arctan(1/3*(d^(1/3)-2*e^(1/3)*x)*3^(1/2)/d^(1/3)) *3^(1/2)/d^(10/3)/(a*e^2+c*d^2)^2+1/6*c^(5/3)*(-a*e^2+c*d^2)*arctan(1/3*(a ^(1/3)-2*c^(1/3)*x^2)*3^(1/2)/a^(1/3))*3^(1/2)/a^(5/3)/(a*e^2+c*d^2)^2-1/3 *c^(13/6)*d*e*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)^2-1/9*e^(10/3)*(7*a*e^2+13*c*d^2)*ln(d^(1/3)+e^ (1/3)*x)/d^(10/3)/(a*e^2+c*d^2)^2-1/6*c^(5/3)*(-a*e^2+c*d^2)*ln(a^(1/3)+c^ (1/3)*x^2)/a^(5/3)/(a*e^2+c*d^2)^2+1/18*e^(10/3)*(7*a*e^2+13*c*d^2)*ln(d^( 2/3)-d^(1/3)*e^(1/3)*x+e^(2/3)*x^2)/d^(10/3)/(a*e^2+c*d^2)^2+1/12*c^(5/3)* (-a*e^2+c*d^2)*ln(a^(2/3)-a^(1/3)*c^(1/3)*x^2+c^(2/3)*x^4)/a^(5/3)/(a*e^2+ c*d^2)^2
Time = 0.88 (sec) , antiderivative size = 679, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^5 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\frac {1}{36} \left (-\frac {9}{a d^2 x^4}+\frac {72 e}{a d^3 x}+\frac {12 e^4 x^2}{d^3 \left (c d^2+a e^2\right ) \left (d+e x^3\right )}+\frac {24 c^{13/6} d e \arctan \left (\frac {\sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{a^{7/6} \left (c d^2+a e^2\right )^2}-\frac {6 c^{5/3} \left (-\sqrt {3} c d^2+2 \sqrt {a} \sqrt {c} d e+\sqrt {3} a e^2\right ) \arctan \left (\sqrt {3}-\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{a^{5/3} \left (c d^2+a e^2\right )^2}+\frac {6 c^{5/3} \left (\sqrt {3} c d^2+2 \sqrt {a} \sqrt {c} d e-\sqrt {3} a e^2\right ) \arctan \left (\sqrt {3}+\frac {2 \sqrt [6]{c} x}{\sqrt [6]{a}}\right )}{a^{5/3} \left (c d^2+a e^2\right )^2}-\frac {4 \sqrt {3} e^{10/3} \left (13 c d^2+7 a e^2\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{e} x}{\sqrt [3]{d}}}{\sqrt {3}}\right )}{d^{10/3} \left (c d^2+a e^2\right )^2}-\frac {4 e^{10/3} \left (13 c d^2+7 a e^2\right ) \log \left (\sqrt [3]{d}+\sqrt [3]{e} x\right )}{d^{10/3} \left (c d^2+a e^2\right )^2}+\frac {6 c^{5/3} \left (-c d^2+a e^2\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{c} x^2\right )}{a^{5/3} \left (c d^2+a e^2\right )^2}+\frac {3 c^{5/3} \left (c d^2+2 \sqrt {3} \sqrt {a} \sqrt {c} d e-a e^2\right ) \log \left (\sqrt [3]{a}-\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{a^{5/3} \left (c d^2+a e^2\right )^2}-\frac {3 c^{5/3} \left (-c d^2+2 \sqrt {3} \sqrt {a} \sqrt {c} d e+a e^2\right ) \log \left (\sqrt [3]{a}+\sqrt {3} \sqrt [6]{a} \sqrt [6]{c} x+\sqrt [3]{c} x^2\right )}{a^{5/3} \left (c d^2+a e^2\right )^2}+\frac {2 e^{10/3} \left (13 c d^2+7 a e^2\right ) \log \left (d^{2/3}-\sqrt [3]{d} \sqrt [3]{e} x+e^{2/3} x^2\right )}{d^{10/3} \left (c d^2+a e^2\right )^2}\right ) \] Input:
Integrate[1/(x^5*(d + e*x^3)^2*(a + c*x^6)),x]
Output:
(-9/(a*d^2*x^4) + (72*e)/(a*d^3*x) + (12*e^4*x^2)/(d^3*(c*d^2 + a*e^2)*(d + e*x^3)) + (24*c^(13/6)*d*e*ArcTan[(c^(1/6)*x)/a^(1/6)])/(a^(7/6)*(c*d^2 + a*e^2)^2) - (6*c^(5/3)*(-(Sqrt[3]*c*d^2) + 2*Sqrt[a]*Sqrt[c]*d*e + Sqrt[ 3]*a*e^2)*ArcTan[Sqrt[3] - (2*c^(1/6)*x)/a^(1/6)])/(a^(5/3)*(c*d^2 + a*e^2 )^2) + (6*c^(5/3)*(Sqrt[3]*c*d^2 + 2*Sqrt[a]*Sqrt[c]*d*e - Sqrt[3]*a*e^2)* ArcTan[Sqrt[3] + (2*c^(1/6)*x)/a^(1/6)])/(a^(5/3)*(c*d^2 + a*e^2)^2) - (4* Sqrt[3]*e^(10/3)*(13*c*d^2 + 7*a*e^2)*ArcTan[(1 - (2*e^(1/3)*x)/d^(1/3))/S qrt[3]])/(d^(10/3)*(c*d^2 + a*e^2)^2) - (4*e^(10/3)*(13*c*d^2 + 7*a*e^2)*L og[d^(1/3) + e^(1/3)*x])/(d^(10/3)*(c*d^2 + a*e^2)^2) + (6*c^(5/3)*(-(c*d^ 2) + a*e^2)*Log[a^(1/3) + c^(1/3)*x^2])/(a^(5/3)*(c*d^2 + a*e^2)^2) + (3*c ^(5/3)*(c*d^2 + 2*Sqrt[3]*Sqrt[a]*Sqrt[c]*d*e - a*e^2)*Log[a^(1/3) - Sqrt[ 3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(a^(5/3)*(c*d^2 + a*e^2)^2) - (3*c^(5 /3)*(-(c*d^2) + 2*Sqrt[3]*Sqrt[a]*Sqrt[c]*d*e + a*e^2)*Log[a^(1/3) + Sqrt[ 3]*a^(1/6)*c^(1/6)*x + c^(1/3)*x^2])/(a^(5/3)*(c*d^2 + a*e^2)^2) + (2*e^(1 0/3)*(13*c*d^2 + 7*a*e^2)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/ (d^(10/3)*(c*d^2 + a*e^2)^2))/36
Time = 1.29 (sec) , antiderivative size = 903, normalized size of antiderivative = 1.37, 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.
| \(\displaystyle \int \frac {1}{x^5 \left (a+c x^6\right ) \left (d+e x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 1837 |
| \(\displaystyle \int \left (\frac {c^2 x \left (a e^2-c d^2+2 c d e x^3\right )}{a \left (a+c x^6\right ) \left (a e^2+c d^2\right )^2}+\frac {e^4 x}{d^2 \left (d+e x^3\right )^2 \left (a e^2+c d^2\right )}+\frac {2 e^4 x \left (a e^2+2 c d^2\right )}{d^3 \left (d+e x^3\right ) \left (a e^2+c d^2\right )^2}-\frac {2 e}{a d^3 x^2}+\frac {1}{a d^2 x^5}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^2 e^4}{3 d^3 \left (c d^2+a e^2\right ) \left (e x^3+d\right )}-\frac {2 \left (2 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^{10/3}}{\sqrt {3} d^{10/3} \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^{10/3}}{3 \sqrt {3} d^{10/3} \left (c d^2+a e^2\right )}-\frac {2 \left (2 c d^2+a e^2\right ) \log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) e^{10/3}}{3 d^{10/3} \left (c d^2+a e^2\right )^2}-\frac {\log \left (\sqrt [3]{e} x+\sqrt [3]{d}\right ) e^{10/3}}{9 d^{10/3} \left (c d^2+a e^2\right )}+\frac {\left (2 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^{10/3}}{3 d^{10/3} \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^{10/3}}{18 d^{10/3} \left (c d^2+a e^2\right )}+\frac {2 e}{a d^3 x}+\frac {c^{5/3} \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)^{5/3} \left (c d^2+a e^2\right )^2}+\frac {c^{5/3} \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)^{5/3} \left (c d^2+a e^2\right )^2}+\frac {c^{5/3} \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)^{5/3} \left (c d^2+a e^2\right )^2}+\frac {c^{5/3} \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)^{5/3} \left (c d^2+a e^2\right )^2}-\frac {c^{5/3} \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)^{5/3} \left (c d^2+a e^2\right )^2}-\frac {c^{5/3} \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)^{5/3} \left (c d^2+a e^2\right )^2}-\frac {1}{4 a d^2 x^4}\) |
Input:
Int[1/(x^5*(d + e*x^3)^2*(a + c*x^6)),x]
Output:
-1/4*1/(a*d^2*x^4) + (2*e)/(a*d^3*x) + (e^4*x^2)/(3*d^3*(c*d^2 + a*e^2)*(d + e*x^3)) + (c^(5/3)*(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)^(5/3)*(c*d^2 + a*e^2) ^2) + (c^(5/3)*(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)^(5/3)*(c*d^2 + a*e^2)^2) - ( e^(10/3)*ArcTan[(d^(1/3) - 2*e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(3*Sqrt[3]*d^( 10/3)*(c*d^2 + a*e^2)) - (2*e^(10/3)*(2*c*d^2 + a*e^2)*ArcTan[(d^(1/3) - 2 *e^(1/3)*x)/(Sqrt[3]*d^(1/3))])/(Sqrt[3]*d^(10/3)*(c*d^2 + a*e^2)^2) + (c^ (5/3)*(c*d^2 - 2*Sqrt[-a]*Sqrt[c]*d*e - a*e^2)*Log[(-a)^(1/6) - c^(1/6)*x] )/(6*(-a)^(5/3)*(c*d^2 + a*e^2)^2) + (c^(5/3)*(c*d^2 + 2*Sqrt[-a]*Sqrt[c]* d*e - a*e^2)*Log[(-a)^(1/6) + c^(1/6)*x])/(6*(-a)^(5/3)*(c*d^2 + a*e^2)^2) - (e^(10/3)*Log[d^(1/3) + e^(1/3)*x])/(9*d^(10/3)*(c*d^2 + a*e^2)) - (2*e ^(10/3)*(2*c*d^2 + a*e^2)*Log[d^(1/3) + e^(1/3)*x])/(3*d^(10/3)*(c*d^2 + a *e^2)^2) - (c^(5/3)*(c*d^2 + 2*Sqrt[-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)^(5/3)*(c*d^2 + a*e^2)^2) - (c^(5/3)*(c*d^2 - 2*Sqrt[-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)^(5/3)*(c*d^2 + a*e^2)^2) + (e^(1 0/3)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e^(2/3)*x^2])/(18*d^(10/3)*(c*d^2 + a*e^2)) + (e^(10/3)*(2*c*d^2 + a*e^2)*Log[d^(2/3) - d^(1/3)*e^(1/3)*x + e ^(2/3)*x^2])/(3*d^(10/3)*(c*d^2 + a*e^2)^2)
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]
Time = 0.30 (sec) , antiderivative size = 695, normalized size of antiderivative = 1.05
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(695\) |
| risch | \(\text {Expression too large to display}\) | \(2449\) |
Input:
int(1/x^5/(e*x^3+d)^2/(c*x^6+a),x,method=_RETURNVERBOSE)
Output:
e^4/d^3/(a*e^2+c*d^2)^2*((1/3*a*e^2+1/3*c*d^2)*x^2/(e*x^3+d)+(7/3*a*e^2+13 /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))))-1/4/a/d^2/x^4+2*e/a/d^3/x+(1/6*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*e-1/12*ln(x^2-3^(1/2)*(a/c)^( 1/6)*x+(a/c)^(1/3))*(a/c)^(1/3)*e^2+1/12*c/a*ln(x^2-3^(1/2)*(a/c)^(1/6)*x+ (a/c)^(1/3))*(a/c)^(1/3)*d^2+1/3/(a/c)^(1/6)*arctan(2*x/(a/c)^(1/6)-3^(1/2 ))*d*e+1/6*(a/c)^(1/3)*arctan(2*x/(a/c)^(1/6)-3^(1/2))*3^(1/2)*e^2-1/6*c/a *(a/c)^(1/3)*arctan(2*x/(a/c)^(1/6)-3^(1/2))*3^(1/2)*d^2-1/6*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*e-1/12*ln(x^2+3^(1/ 2)*(a/c)^(1/6)*x+(a/c)^(1/3))*(a/c)^(1/3)*e^2+1/12*c/a*ln(x^2+3^(1/2)*(a/c )^(1/6)*x+(a/c)^(1/3))*(a/c)^(1/3)*d^2+1/3/(a/c)^(1/6)*arctan(2*x/(a/c)^(1 /6)+3^(1/2))*d*e-1/6*(a/c)^(1/3)*arctan(2*x/(a/c)^(1/6)+3^(1/2))*3^(1/2)*e ^2+1/6*c/a*(a/c)^(1/3)*arctan(2*x/(a/c)^(1/6)+3^(1/2))*3^(1/2)*d^2+1/6*ln( x^2+(a/c)^(1/3))*(a/c)^(1/3)*e^2-1/6*c/a*ln(x^2+(a/c)^(1/3))*(a/c)^(1/3)*d ^2+2/3*d*e/(a/c)^(1/6)*arctan(x/(a/c)^(1/6)))*c^2/(a*e^2+c*d^2)^2/a
Timed out. \[ \int \frac {1}{x^5 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x^5/(e*x^3+d)^2/(c*x^6+a),x, algorithm="fricas")
Output:
Timed out
Timed out. \[ \int \frac {1}{x^5 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x**5/(e*x**3+d)**2/(c*x**6+a),x)
Output:
Timed out
Exception generated. \[ \int \frac {1}{x^5 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(1/x^5/(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
Time = 0.17 (sec) , antiderivative size = 897, normalized size of antiderivative = 1.36 \[ \int \frac {1}{x^5 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\text {Too large to display} \] Input:
integrate(1/x^5/(e*x^3+d)^2/(c*x^6+a),x, algorithm="giac")
Output:
1/3*e^4*x^2/((c*d^5 + a*d^3*e^2)*(e*x^3 + d)) + 2/3*(a*c^5)^(5/6)*d*e*arct an(x/(a/c)^(1/6))/(a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4) - 1/9*(1 3*c*d^2*e^4*(-d/e)^(1/3) + 7*a*e^6*(-d/e)^(1/3))*(-d/e)^(1/3)*log(abs(x - (-d/e)^(1/3)))/(c^2*d^8 + 2*a*c*d^6*e^2 + a^2*d^4*e^4) - 1/3*(13*(-d*e^2)^ (2/3)*c*d^2*e^2 + 7*(-d*e^2)^(2/3)*a*e^4)*arctan(1/3*sqrt(3)*(2*x + (-d/e) ^(1/3))/(-d/e)^(1/3))/(sqrt(3)*c^2*d^8 + 2*sqrt(3)*a*c*d^6*e^2 + sqrt(3)*a ^2*d^4*e^4) + 1/6*(sqrt(3)*(a*c^5)^(1/3)*c^3*d^2 - sqrt(3)*(a*c^5)^(1/3)*a *c^2*e^2 + 2*(a*c^5)^(5/6)*d*e)*arctan((2*x + sqrt(3)*(a/c)^(1/6))/(a/c)^( 1/6))/(a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4) - 1/6*(sqrt(3)*(a*c^ 5)^(1/3)*c^3*d^2 - sqrt(3)*(a*c^5)^(1/3)*a*c^2*e^2 - 2*(a*c^5)^(5/6)*d*e)* arctan((2*x - sqrt(3)*(a/c)^(1/6))/(a/c)^(1/6))/(a^2*c^4*d^4 + 2*a^3*c^3*d ^2*e^2 + a^4*c^2*e^4) + 1/12*((a*c^5)^(1/3)*c^3*d^2 - (a*c^5)^(1/3)*a*c^2* e^2 - 2*sqrt(3)*(a*c^5)^(5/6)*d*e)*log(x^2 + sqrt(3)*x*(a/c)^(1/6) + (a/c) ^(1/3))/(a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4) + 1/12*((a*c^5)^(1 /3)*c^3*d^2 - (a*c^5)^(1/3)*a*c^2*e^2 + 2*sqrt(3)*(a*c^5)^(5/6)*d*e)*log(x ^2 - sqrt(3)*x*(a/c)^(1/6) + (a/c)^(1/3))/(a^2*c^4*d^4 + 2*a^3*c^3*d^2*e^2 + a^4*c^2*e^4) + 1/18*(13*(-d*e^2)^(2/3)*c*d^2*e^2 + 7*(-d*e^2)^(2/3)*a*e ^4)*log(x^2 + x*(-d/e)^(1/3) + (-d/e)^(2/3))/(c^2*d^8 + 2*a*c*d^6*e^2 + a^ 2*d^4*e^4) - 1/6*((a*c^5)^(1/3)*c*d^2 - (a*c^5)^(1/3)*a*e^2)*log(x^2 + (a/ c)^(1/3))/(a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4) + 1/4*(8*e*x^3 - d)...
Timed out. \[ \int \frac {1}{x^5 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\text {Hanged} \] Input:
int(1/(x^5*(a + c*x^6)*(d + e*x^3)^2),x)
Output:
\text{Hanged}
\[ \int \frac {1}{x^5 \left (d+e x^3\right )^2 \left (a+c x^6\right )} \, dx=\int \frac {1}{x^{5} \left (e \,x^{3}+d \right )^{2} \left (c \,x^{6}+a \right )}d x \] Input:
int(1/x^5/(e*x^3+d)^2/(c*x^6+a),x)
Output:
int(1/x^5/(e*x^3+d)^2/(c*x^6+a),x)