Integrand size = 23, antiderivative size = 242 \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )^2} \, dx=-\frac {c}{2 a^2 x^2}-\frac {d}{a^2 x}-\frac {x \left (b c+b d x+b e x^2\right )}{3 a^2 \left (a+b x^3\right )}+\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} c+4 \sqrt [3]{a} d\right ) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{8/3}}+\frac {e \log (x)}{a^2}-\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} c-4 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{8/3}}+\frac {\sqrt [3]{b} \left (5 \sqrt [3]{b} c-4 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{18 a^{8/3}}-\frac {e \log \left (a+b x^3\right )}{3 a^2} \]
-1/2*c/a^2/x^2-d/a^2/x-1/3*x*(b*e*x^2+b*d*x+b*c)/a^2/(b*x^3+a)+e*ln(x)/a^2 -1/9*b^(1/3)*(5*b^(1/3)*c-4*a^(1/3)*d)*ln(a^(1/3)+b^(1/3)*x)/a^(8/3)+1/18* b^(1/3)*(5*b^(1/3)*c-4*a^(1/3)*d)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2 )/a^(8/3)-1/3*e*ln(b*x^3+a)/a^2+1/9*b^(1/3)*(5*b^(1/3)*c+4*a^(1/3)*d)*arct an(1/3*(a^(1/3)-2*b^(1/3)*x)/a^(1/3)*3^(1/2))/a^(8/3)*3^(1/2)
Time = 0.17 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.91 \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )^2} \, dx=\frac {-\frac {9 a c}{x^2}-\frac {18 a d}{x}+\frac {6 a (a e-b x (c+d x))}{a+b x^3}+2 \sqrt {3} \sqrt [3]{a} \sqrt [3]{b} \left (5 \sqrt [3]{b} c+4 \sqrt [3]{a} d\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+18 a e \log (x)+2 \sqrt [3]{b} \left (-5 \sqrt [3]{a} \sqrt [3]{b} c+4 a^{2/3} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )+\sqrt [3]{b} \left (5 \sqrt [3]{a} \sqrt [3]{b} c-4 a^{2/3} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-6 a e \log \left (a+b x^3\right )}{18 a^3} \]
((-9*a*c)/x^2 - (18*a*d)/x + (6*a*(a*e - b*x*(c + d*x)))/(a + b*x^3) + 2*S qrt[3]*a^(1/3)*b^(1/3)*(5*b^(1/3)*c + 4*a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)* x)/a^(1/3))/Sqrt[3]] + 18*a*e*Log[x] + 2*b^(1/3)*(-5*a^(1/3)*b^(1/3)*c + 4 *a^(2/3)*d)*Log[a^(1/3) + b^(1/3)*x] + b^(1/3)*(5*a^(1/3)*b^(1/3)*c - 4*a^ (2/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2] - 6*a*e*Log[a + b* x^3])/(18*a^3)
Time = 0.61 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2368, 25, 2373, 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.
| \(\displaystyle \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )^2} \, dx\) |
| \(\Big \downarrow \) 2368 |
| \(\displaystyle -\frac {\int -\frac {-\frac {b^2 d x^4}{a}-\frac {2 b^2 c x^3}{a}+3 b e x^2+3 b d x+3 b c}{x^3 \left (b x^3+a\right )}dx}{3 a b}-\frac {x \left (b c+b d x+b e x^2\right )}{3 a^2 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {-\frac {b^2 d x^4}{a}-\frac {2 b^2 c x^3}{a}+3 b e x^2+3 b d x+3 b c}{x^3 \left (b x^3+a\right )}dx}{3 a b}-\frac {x \left (b c+b d x+b e x^2\right )}{3 a^2 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2373 |
| \(\displaystyle \frac {\int \left (-\frac {\left (3 e x^2+4 d x+5 c\right ) b^2}{a \left (b x^3+a\right )}+\frac {3 e b}{a x}+\frac {3 d b}{a x^2}+\frac {3 c b}{a x^3}\right )dx}{3 a b}-\frac {x \left (b c+b d x+b e x^2\right )}{3 a^2 \left (a+b x^3\right )}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b^{4/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) \left (4 \sqrt [3]{a} d+5 \sqrt [3]{b} c\right )}{\sqrt {3} a^{5/3}}+\frac {b^{4/3} \left (5 \sqrt [3]{b} c-4 \sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3}}-\frac {b^{4/3} \left (5 \sqrt [3]{b} c-4 \sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3}}-\frac {3 b c}{2 a x^2}-\frac {3 b d}{a x}-\frac {b e \log \left (a+b x^3\right )}{a}+\frac {3 b e \log (x)}{a}}{3 a b}-\frac {x \left (b c+b d x+b e x^2\right )}{3 a^2 \left (a+b x^3\right )}\) |
-1/3*(x*(b*c + b*d*x + b*e*x^2))/(a^2*(a + b*x^3)) + ((-3*b*c)/(2*a*x^2) - (3*b*d)/(a*x) + (b^(4/3)*(5*b^(1/3)*c + 4*a^(1/3)*d)*ArcTan[(a^(1/3) - 2* b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(5/3)) + (3*b*e*Log[x])/a - (b^( 4/3)*(5*b^(1/3)*c - 4*a^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x])/(3*a^(5/3)) + ( b^(4/3)*(5*b^(1/3)*c - 4*a^(1/3)*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2 /3)*x^2])/(6*a^(5/3)) - (b*e*Log[a + b*x^3])/a)/(3*a*b)
3.4.49.3.1 Defintions of rubi rules used
Int[(Pq_)*(x_)^(m_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient[a*b^(Floor[(q - 1)/n] + 1)*x^ m*Pq, a + b*x^n, x], R = PolynomialRemainder[a*b^(Floor[(q - 1)/n] + 1)*x^m *Pq, a + b*x^n, x], i}, Simp[(-x)*R*((a + b*x^n)^(p + 1)/(a^2*n*(p + 1)*b^( Floor[(q - 1)/n] + 1))), x] + Simp[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)) Int[x^m*(a + b*x^n)^(p + 1)*ExpandToSum[(n*(p + 1)*Q)/x^m + Sum[((n*(p + 1) + i + 1)/a)*Coeff[R, x, i]*x^(i - m), {i, 0, n - 1}], x], x], x]]] /; F reeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && ILtQ[m, 0]
Int[((Pq_)*((c_.)*(x_))^(m_.))/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[E xpandIntegrand[(c*x)^m*(Pq/(a + b*x^n)), x], x] /; FreeQ[{a, b, c, m}, x] & & PolyQ[Pq, x] && IntegerQ[n] && !IGtQ[m, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.61 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.03
| method | result | size |
| risch | \(\frac {-\frac {4 b d \,x^{4}}{3 a^{2}}-\frac {5 b c \,x^{3}}{6 a^{2}}+\frac {e \,x^{2}}{3 a}-\frac {x d}{a}-\frac {c}{2 a}}{x^{2} \left (b \,x^{3}+a \right )}+\frac {e \ln \left (-x \right )}{a^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (a^{8} \textit {\_Z}^{3}+9 a^{6} e \,\textit {\_Z}^{2}+\left (27 a^{4} e^{2}+60 a^{3} b c d \right ) \textit {\_Z} +27 a^{2} e^{3}+180 a b c d e -64 a b \,d^{3}+125 b^{2} c^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (-4 \textit {\_R}^{3} a^{8}-24 \textit {\_R}^{2} a^{6} e +\left (-36 a^{4} e^{2}-200 a^{3} b c d \right ) \textit {\_R} -360 a b c d e +192 a b \,d^{3}-375 b^{2} c^{3}\right ) x -4 a^{6} d \,\textit {\_R}^{2}+\left (24 a^{4} d e -25 a^{3} b \,c^{2}\right ) \textit {\_R} +108 a^{2} d \,e^{2}+225 a b \,c^{2} e \right )\right )}{9}\) | \(249\) |
| default | \(-\frac {c}{2 a^{2} x^{2}}-\frac {d}{a^{2} x}+\frac {e \ln \left (x \right )}{a^{2}}-\frac {b \left (\frac {\frac {d \,x^{2}}{3}+\frac {c x}{3}-\frac {a e}{3 b}}{b \,x^{3}+a}+\frac {5 c \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3}+\frac {4 d \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3}+\frac {e \ln \left (b \,x^{3}+a \right )}{3 b}\right )}{a^{2}}\) | \(262\) |
(-4/3*b*d/a^2*x^4-5/6*b*c/a^2*x^3+1/3/a*e*x^2-1/a*x*d-1/2*c/a)/x^2/(b*x^3+ a)+1/a^2*e*ln(-x)+1/9*sum(_R*ln((-4*_R^3*a^8-24*_R^2*a^6*e+(-36*a^4*e^2-20 0*a^3*b*c*d)*_R-360*a*b*c*d*e+192*a*b*d^3-375*b^2*c^3)*x-4*a^6*d*_R^2+(24* a^4*d*e-25*a^3*b*c^2)*_R+108*a^2*d*e^2+225*a*b*c^2*e),_R=RootOf(a^8*_Z^3+9 *a^6*e*_Z^2+(27*a^4*e^2+60*a^3*b*c*d)*_Z+27*a^2*e^3+180*a*b*c*d*e-64*a*b*d ^3+125*b^2*c^3))
Result contains complex when optimal does not.
Time = 1.09 (sec) , antiderivative size = 4774, normalized size of antiderivative = 19.73 \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )^2} \, dx=\text {Too large to display} \]
-1/324*(432*b*d*x^4 + 270*b*c*x^3 - 108*a*e*x^2 + 324*a*d*x + 2*(a^2*b*x^5 + a^3*x^2)*((-I*sqrt(3) + 1)*(9*e^2/a^4 - (20*b*c*d + 9*a*e^2)/a^5)/(-1/2 7*e^3/a^6 + 1/162*(20*b*c*d + 9*a*e^2)*e/a^7 + 1/1458*(125*b*c^3 + 64*a*d^ 3)*b/a^8 - 1/1458*(125*b^2*c^3 + 27*a^2*e^3 - 4*(16*d^3 - 45*c*d*e)*a*b)/a ^8)^(1/3) + 81*(I*sqrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b*c*d + 9*a*e^2) *e/a^7 + 1/1458*(125*b*c^3 + 64*a*d^3)*b/a^8 - 1/1458*(125*b^2*c^3 + 27*a^ 2*e^3 - 4*(16*d^3 - 45*c*d*e)*a*b)/a^8)^(1/3) + 54*e/a^2)*log(1/81*((-I*sq rt(3) + 1)*(9*e^2/a^4 - (20*b*c*d + 9*a*e^2)/a^5)/(-1/27*e^3/a^6 + 1/162*( 20*b*c*d + 9*a*e^2)*e/a^7 + 1/1458*(125*b*c^3 + 64*a*d^3)*b/a^8 - 1/1458*( 125*b^2*c^3 + 27*a^2*e^3 - 4*(16*d^3 - 45*c*d*e)*a*b)/a^8)^(1/3) + 81*(I*s qrt(3) + 1)*(-1/27*e^3/a^6 + 1/162*(20*b*c*d + 9*a*e^2)*e/a^7 + 1/1458*(12 5*b*c^3 + 64*a*d^3)*b/a^8 - 1/1458*(125*b^2*c^3 + 27*a^2*e^3 - 4*(16*d^3 - 45*c*d*e)*a*b)/a^8)^(1/3) + 54*e/a^2)^2*a^6*d + 160*a*b*c*d^2 - 75*a*b*c^ 2*e + 36*a^2*d*e^2 + 1/18*(25*a^3*b*c^2 - 24*a^4*d*e)*((-I*sqrt(3) + 1)*(9 *e^2/a^4 - (20*b*c*d + 9*a*e^2)/a^5)/(-1/27*e^3/a^6 + 1/162*(20*b*c*d + 9* a*e^2)*e/a^7 + 1/1458*(125*b*c^3 + 64*a*d^3)*b/a^8 - 1/1458*(125*b^2*c^3 + 27*a^2*e^3 - 4*(16*d^3 - 45*c*d*e)*a*b)/a^8)^(1/3) + 81*(I*sqrt(3) + 1)*( -1/27*e^3/a^6 + 1/162*(20*b*c*d + 9*a*e^2)*e/a^7 + 1/1458*(125*b*c^3 + 64* a*d^3)*b/a^8 - 1/1458*(125*b^2*c^3 + 27*a^2*e^3 - 4*(16*d^3 - 45*c*d*e)*a* b)/a^8)^(1/3) + 54*e/a^2) + (125*b^2*c^3 + 64*a*b*d^3)*x) + 162*a*c + (...
Timed out. \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )^2} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 220, normalized size of antiderivative = 0.91 \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )^2} \, dx=-\frac {8 \, b d x^{4} + 5 \, b c x^{3} - 2 \, a e x^{2} + 6 \, a d x + 3 \, a c}{6 \, {\left (a^{2} b x^{5} + a^{3} x^{2}\right )}} + \frac {e \log \left (x\right )}{a^{2}} - \frac {\sqrt {3} {\left (4 \, b d \left (\frac {a}{b}\right )^{\frac {2}{3}} + 5 \, b c \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3}} - \frac {{\left (6 \, e \left (\frac {a}{b}\right )^{\frac {2}{3}} + 4 \, d \left (\frac {a}{b}\right )^{\frac {1}{3}} - 5 \, c\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (3 \, e \left (\frac {a}{b}\right )^{\frac {2}{3}} - 4 \, d \left (\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, c\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{9 \, a^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \]
-1/6*(8*b*d*x^4 + 5*b*c*x^3 - 2*a*e*x^2 + 6*a*d*x + 3*a*c)/(a^2*b*x^5 + a^ 3*x^2) + e*log(x)/a^2 - 1/9*sqrt(3)*(4*b*d*(a/b)^(2/3) + 5*b*c*(a/b)^(1/3) )*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/a^3 - 1/18*(6*e*(a/b )^(2/3) + 4*d*(a/b)^(1/3) - 5*c)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a ^2*(a/b)^(2/3)) - 1/9*(3*e*(a/b)^(2/3) - 4*d*(a/b)^(1/3) + 5*c)*log(x + (a /b)^(1/3))/(a^2*(a/b)^(2/3))
Time = 0.28 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.01 \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )^2} \, dx=-\frac {e \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, a^{2}} + \frac {e \log \left ({\left | x \right |}\right )}{a^{2}} - \frac {\sqrt {3} {\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c - 4 \, \left (-a b^{2}\right )^{\frac {2}{3}} d\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{9 \, a^{3} b} - \frac {{\left (5 \, \left (-a b^{2}\right )^{\frac {1}{3}} b c + 4 \, \left (-a b^{2}\right )^{\frac {2}{3}} d\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{18 \, a^{3} b} + \frac {{\left (4 \, a^{2} b^{2} d \left (-\frac {a}{b}\right )^{\frac {1}{3}} + 5 \, a^{2} b^{2} c\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{9 \, a^{5} b} - \frac {8 \, b d x^{4} + 5 \, b c x^{3} - 2 \, a e x^{2} + 6 \, a d x + 3 \, a c}{6 \, {\left (b x^{3} + a\right )} a^{2} x^{2}} \]
-1/3*e*log(abs(b*x^3 + a))/a^2 + e*log(abs(x))/a^2 - 1/9*sqrt(3)*(5*(-a*b^ 2)^(1/3)*b*c - 4*(-a*b^2)^(2/3)*d)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3)) /(-a/b)^(1/3))/(a^3*b) - 1/18*(5*(-a*b^2)^(1/3)*b*c + 4*(-a*b^2)^(2/3)*d)* log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^3*b) + 1/9*(4*a^2*b^2*d*(-a/b) ^(1/3) + 5*a^2*b^2*c)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^5*b) - 1/ 6*(8*b*d*x^4 + 5*b*c*x^3 - 2*a*e*x^2 + 6*a*d*x + 3*a*c)/((b*x^3 + a)*a^2*x ^2)
Time = 9.20 (sec) , antiderivative size = 733, normalized size of antiderivative = 3.03 \[ \int \frac {c+d x+e x^2}{x^3 \left (a+b x^3\right )^2} \, dx=\left (\sum _{k=1}^3\ln \left (-\frac {b^3\,\left ({\mathrm {root}\left (729\,a^8\,z^3+729\,a^6\,e\,z^2+540\,a^3\,b\,c\,d\,z+243\,a^4\,e^2\,z+180\,a\,b\,c\,d\,e-64\,a\,b\,d^3+27\,a^2\,e^3+125\,b^2\,c^3,z,k\right )}^2\,a^6\,d\,108-36\,a^2\,d\,e^2+{\mathrm {root}\left (729\,a^8\,z^3+729\,a^6\,e\,z^2+540\,a^3\,b\,c\,d\,z+243\,a^4\,e^2\,z+180\,a\,b\,c\,d\,e-64\,a\,b\,d^3+27\,a^2\,e^3+125\,b^2\,c^3,z,k\right )}^3\,a^8\,x\,972+125\,b^2\,c^3\,x-\mathrm {root}\left (729\,a^8\,z^3+729\,a^6\,e\,z^2+540\,a^3\,b\,c\,d\,z+243\,a^4\,e^2\,z+180\,a\,b\,c\,d\,e-64\,a\,b\,d^3+27\,a^2\,e^3+125\,b^2\,c^3,z,k\right )\,a^4\,d\,e\,72-75\,a\,b\,c^2\,e-64\,a\,b\,d^3\,x+\mathrm {root}\left (729\,a^8\,z^3+729\,a^6\,e\,z^2+540\,a^3\,b\,c\,d\,z+243\,a^4\,e^2\,z+180\,a\,b\,c\,d\,e-64\,a\,b\,d^3+27\,a^2\,e^3+125\,b^2\,c^3,z,k\right )\,a^3\,b\,c^2\,75+\mathrm {root}\left (729\,a^8\,z^3+729\,a^6\,e\,z^2+540\,a^3\,b\,c\,d\,z+243\,a^4\,e^2\,z+180\,a\,b\,c\,d\,e-64\,a\,b\,d^3+27\,a^2\,e^3+125\,b^2\,c^3,z,k\right )\,a^4\,e^2\,x\,108+{\mathrm {root}\left (729\,a^8\,z^3+729\,a^6\,e\,z^2+540\,a^3\,b\,c\,d\,z+243\,a^4\,e^2\,z+180\,a\,b\,c\,d\,e-64\,a\,b\,d^3+27\,a^2\,e^3+125\,b^2\,c^3,z,k\right )}^2\,a^6\,e\,x\,648+\mathrm {root}\left (729\,a^8\,z^3+729\,a^6\,e\,z^2+540\,a^3\,b\,c\,d\,z+243\,a^4\,e^2\,z+180\,a\,b\,c\,d\,e-64\,a\,b\,d^3+27\,a^2\,e^3+125\,b^2\,c^3,z,k\right )\,a^3\,b\,c\,d\,x\,600+120\,a\,b\,c\,d\,e\,x\right )}{a^6\,27}\right )\,\mathrm {root}\left (729\,a^8\,z^3+729\,a^6\,e\,z^2+540\,a^3\,b\,c\,d\,z+243\,a^4\,e^2\,z+180\,a\,b\,c\,d\,e-64\,a\,b\,d^3+27\,a^2\,e^3+125\,b^2\,c^3,z,k\right )\right )-\frac {\frac {c}{2\,a}-\frac {e\,x^2}{3\,a}+\frac {d\,x}{a}+\frac {5\,b\,c\,x^3}{6\,a^2}+\frac {4\,b\,d\,x^4}{3\,a^2}}{b\,x^5+a\,x^2}+\frac {e\,\ln \left (x\right )}{a^2} \]
symsum(log(-(b^3*(108*root(729*a^8*z^3 + 729*a^6*e*z^2 + 540*a^3*b*c*d*z + 243*a^4*e^2*z + 180*a*b*c*d*e - 64*a*b*d^3 + 27*a^2*e^3 + 125*b^2*c^3, z, k)^2*a^6*d - 36*a^2*d*e^2 + 972*root(729*a^8*z^3 + 729*a^6*e*z^2 + 540*a^ 3*b*c*d*z + 243*a^4*e^2*z + 180*a*b*c*d*e - 64*a*b*d^3 + 27*a^2*e^3 + 125* b^2*c^3, z, k)^3*a^8*x + 125*b^2*c^3*x - 72*root(729*a^8*z^3 + 729*a^6*e*z ^2 + 540*a^3*b*c*d*z + 243*a^4*e^2*z + 180*a*b*c*d*e - 64*a*b*d^3 + 27*a^2 *e^3 + 125*b^2*c^3, z, k)*a^4*d*e - 75*a*b*c^2*e - 64*a*b*d^3*x + 75*root( 729*a^8*z^3 + 729*a^6*e*z^2 + 540*a^3*b*c*d*z + 243*a^4*e^2*z + 180*a*b*c* d*e - 64*a*b*d^3 + 27*a^2*e^3 + 125*b^2*c^3, z, k)*a^3*b*c^2 + 108*root(72 9*a^8*z^3 + 729*a^6*e*z^2 + 540*a^3*b*c*d*z + 243*a^4*e^2*z + 180*a*b*c*d* e - 64*a*b*d^3 + 27*a^2*e^3 + 125*b^2*c^3, z, k)*a^4*e^2*x + 648*root(729* a^8*z^3 + 729*a^6*e*z^2 + 540*a^3*b*c*d*z + 243*a^4*e^2*z + 180*a*b*c*d*e - 64*a*b*d^3 + 27*a^2*e^3 + 125*b^2*c^3, z, k)^2*a^6*e*x + 600*root(729*a^ 8*z^3 + 729*a^6*e*z^2 + 540*a^3*b*c*d*z + 243*a^4*e^2*z + 180*a*b*c*d*e - 64*a*b*d^3 + 27*a^2*e^3 + 125*b^2*c^3, z, k)*a^3*b*c*d*x + 120*a*b*c*d*e*x ))/(27*a^6))*root(729*a^8*z^3 + 729*a^6*e*z^2 + 540*a^3*b*c*d*z + 243*a^4* e^2*z + 180*a*b*c*d*e - 64*a*b*d^3 + 27*a^2*e^3 + 125*b^2*c^3, z, k), k, 1 , 3) - (c/(2*a) - (e*x^2)/(3*a) + (d*x)/a + (5*b*c*x^3)/(6*a^2) + (4*b*d*x ^4)/(3*a^2))/(a*x^2 + b*x^5) + (e*log(x))/a^2