Integrand size = 19, antiderivative size = 173 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=\frac {d (2 b c-a d) x}{b^2}+\frac {d^2 x^4}{4 b}-\frac {(b c-a d)^2 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{2/3} b^{7/3}}+\frac {(b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{7/3}}-\frac {(b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{7/3}} \] Output:
d*(-a*d+2*b*c)*x/b^2+1/4*d^2*x^4/b-1/3*(-a*d+b*c)^2*arctan(1/3*(a^(1/3)-2* b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^(2/3)/b^(7/3)+1/3*(-a*d+b*c)^2*ln(a^ (1/3)+b^(1/3)*x)/a^(2/3)/b^(7/3)-1/6*(-a*d+b*c)^2*ln(a^(2/3)-a^(1/3)*b^(1/ 3)*x+b^(2/3)*x^2)/a^(2/3)/b^(7/3)
Time = 0.08 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.97 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=\frac {-12 a^{2/3} \sqrt [3]{b} d (-2 b c+a d) x+3 a^{2/3} b^{4/3} d^2 x^4+4 \sqrt {3} (b c-a d)^2 \arctan \left (\frac {-\sqrt [3]{a}+2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )+4 (b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-2 (b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{12 a^{2/3} b^{7/3}} \] Input:
Integrate[(c + d*x^3)^2/(a + b*x^3),x]
Output:
(-12*a^(2/3)*b^(1/3)*d*(-2*b*c + a*d)*x + 3*a^(2/3)*b^(4/3)*d^2*x^4 + 4*Sq rt[3]*(b*c - a*d)^2*ArcTan[(-a^(1/3) + 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))] + 4 *(b*c - a*d)^2*Log[a^(1/3) + b^(1/3)*x] - 2*(b*c - a*d)^2*Log[a^(2/3) - a^ (1/3)*b^(1/3)*x + b^(2/3)*x^2])/(12*a^(2/3)*b^(7/3))
Time = 0.50 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {915, 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.
| \(\displaystyle \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx\) |
| \(\Big \downarrow \) 915 |
| \(\displaystyle \int \left (\frac {a^2 d^2-2 a b c d+b^2 c^2}{b^2 \left (a+b x^3\right )}+\frac {d (2 b c-a d)}{b^2}+\frac {d^2 x^3}{b}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right ) (b c-a d)^2}{\sqrt {3} a^{2/3} b^{7/3}}-\frac {(b c-a d)^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{2/3} b^{7/3}}+\frac {(b c-a d)^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} b^{7/3}}+\frac {d x (2 b c-a d)}{b^2}+\frac {d^2 x^4}{4 b}\) |
Input:
Int[(c + d*x^3)^2/(a + b*x^3),x]
Output:
(d*(2*b*c - a*d)*x)/b^2 + (d^2*x^4)/(4*b) - ((b*c - a*d)^2*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*a^(2/3)*b^(7/3)) + ((b*c - a* d)^2*Log[a^(1/3) + b^(1/3)*x])/(3*a^(2/3)*b^(7/3)) - ((b*c - a*d)^2*Log[a^ (2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*a^(2/3)*b^(7/3))
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a , b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.89 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.45
| method | result | size |
| risch | \(\frac {d^{2} x^{4}}{4 b}-\frac {d^{2} a x}{b^{2}}+\frac {2 d c x}{b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{3}+a \right )}{\sum }\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{2}}}{3 b^{3}}\) | \(78\) |
| default | \(-\frac {d \left (-\frac {1}{4} b d \,x^{4}+a d x -2 b c x \right )}{b^{2}}+\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}{b^{2}}\) | \(140\) |
Input:
int((d*x^3+c)^2/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
1/4*d^2*x^4/b-d^2/b^2*a*x+2*d/b*c*x+1/3/b^3*sum((a^2*d^2-2*a*b*c*d+b^2*c^2 )/_R^2*ln(x-_R),_R=RootOf(_Z^3*b+a))
Time = 0.09 (sec) , antiderivative size = 507, normalized size of antiderivative = 2.93 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=\left [\frac {3 \, a^{2} b^{2} d^{2} x^{4} + 6 \, \sqrt {\frac {1}{3}} {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \log \left (\frac {2 \, a b x^{3} - 3 \, \left (a^{2} b\right )^{\frac {1}{3}} a x - a^{2} + 3 \, \sqrt {\frac {1}{3}} {\left (2 \, a b x^{2} + \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {-\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{b x^{3} + a}\right ) - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 12 \, {\left (2 \, a^{2} b^{2} c d - a^{3} b d^{2}\right )} x}{12 \, a^{2} b^{3}}, \frac {3 \, a^{2} b^{2} d^{2} x^{4} + 12 \, \sqrt {\frac {1}{3}} {\left (a b^{3} c^{2} - 2 \, a^{2} b^{2} c d + a^{3} b d^{2}\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}} \arctan \left (\frac {\sqrt {\frac {1}{3}} {\left (2 \, \left (a^{2} b\right )^{\frac {2}{3}} x - \left (a^{2} b\right )^{\frac {1}{3}} a\right )} \sqrt {\frac {\left (a^{2} b\right )^{\frac {1}{3}}}{b}}}{a^{2}}\right ) - 2 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x^{2} - \left (a^{2} b\right )^{\frac {2}{3}} x + \left (a^{2} b\right )^{\frac {1}{3}} a\right ) + 4 \, {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \left (a^{2} b\right )^{\frac {2}{3}} \log \left (a b x + \left (a^{2} b\right )^{\frac {2}{3}}\right ) + 12 \, {\left (2 \, a^{2} b^{2} c d - a^{3} b d^{2}\right )} x}{12 \, a^{2} b^{3}}\right ] \] Input:
integrate((d*x^3+c)^2/(b*x^3+a),x, algorithm="fricas")
Output:
[1/12*(3*a^2*b^2*d^2*x^4 + 6*sqrt(1/3)*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b* d^2)*sqrt(-(a^2*b)^(1/3)/b)*log((2*a*b*x^3 - 3*(a^2*b)^(1/3)*a*x - a^2 + 3 *sqrt(1/3)*(2*a*b*x^2 + (a^2*b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt(-(a^2*b)^( 1/3)/b))/(b*x^3 + a)) - 2*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(a^2*b)^(2/3)*lo g(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^(1/3)*a) + 4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(a^2*b)^(2/3)*log(a*b*x + (a^2*b)^(2/3)) + 12*(2*a^2*b^2*c*d - a^ 3*b*d^2)*x)/(a^2*b^3), 1/12*(3*a^2*b^2*d^2*x^4 + 12*sqrt(1/3)*(a*b^3*c^2 - 2*a^2*b^2*c*d + a^3*b*d^2)*sqrt((a^2*b)^(1/3)/b)*arctan(sqrt(1/3)*(2*(a^2 *b)^(2/3)*x - (a^2*b)^(1/3)*a)*sqrt((a^2*b)^(1/3)/b)/a^2) - 2*(b^2*c^2 - 2 *a*b*c*d + a^2*d^2)*(a^2*b)^(2/3)*log(a*b*x^2 - (a^2*b)^(2/3)*x + (a^2*b)^ (1/3)*a) + 4*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*(a^2*b)^(2/3)*log(a*b*x + (a^ 2*b)^(2/3)) + 12*(2*a^2*b^2*c*d - a^3*b*d^2)*x)/(a^2*b^3)]
Time = 0.37 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.90 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=x \left (- \frac {a d^{2}}{b^{2}} + \frac {2 c d}{b}\right ) + \operatorname {RootSum} {\left (27 t^{3} a^{2} b^{7} - a^{6} d^{6} + 6 a^{5} b c d^{5} - 15 a^{4} b^{2} c^{2} d^{4} + 20 a^{3} b^{3} c^{3} d^{3} - 15 a^{2} b^{4} c^{4} d^{2} + 6 a b^{5} c^{5} d - b^{6} c^{6}, \left ( t \mapsto t \log {\left (\frac {3 t a b^{2}}{a^{2} d^{2} - 2 a b c d + b^{2} c^{2}} + x \right )} \right )\right )} + \frac {d^{2} x^{4}}{4 b} \] Input:
integrate((d*x**3+c)**2/(b*x**3+a),x)
Output:
x*(-a*d**2/b**2 + 2*c*d/b) + RootSum(27*_t**3*a**2*b**7 - a**6*d**6 + 6*a* *5*b*c*d**5 - 15*a**4*b**2*c**2*d**4 + 20*a**3*b**3*c**3*d**3 - 15*a**2*b* *4*c**4*d**2 + 6*a*b**5*c**5*d - b**6*c**6, Lambda(_t, _t*log(3*_t*a*b**2/ (a**2*d**2 - 2*a*b*c*d + b**2*c**2) + x))) + d**2*x**4/(4*b)
Time = 0.11 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.10 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=\frac {b d^{2} x^{4} + 4 \, {\left (2 \, b c d - a d^{2}\right )} x}{4 \, b^{2}} + \frac {\sqrt {3} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} + \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, b^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}} \] Input:
integrate((d*x^3+c)^2/(b*x^3+a),x, algorithm="maxima")
Output:
1/4*(b*d^2*x^4 + 4*(2*b*c*d - a*d^2)*x)/b^2 + 1/3*sqrt(3)*(b^2*c^2 - 2*a*b *c*d + a^2*d^2)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/(b^3*( a/b)^(2/3)) - 1/6*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*(a/b)^(2/3)) + 1/3*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*log (x + (a/b)^(1/3))/(b^3*(a/b)^(2/3))
Time = 0.12 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.22 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=-\frac {\sqrt {3} {\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} - \frac {{\left (b^{2} c^{2} - 2 \, a b c d + a^{2} d^{2}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, \left (-a b^{2}\right )^{\frac {2}{3}} b} - \frac {{\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, a b^{4}} + \frac {b^{3} d^{2} x^{4} + 8 \, b^{3} c d x - 4 \, a b^{2} d^{2} x}{4 \, b^{4}} \] Input:
integrate((d*x^3+c)^2/(b*x^3+a),x, algorithm="giac")
Output:
-1/3*sqrt(3)*(b^2*c^2 - 2*a*b*c*d + a^2*d^2)*arctan(1/3*sqrt(3)*(2*x + (-a /b)^(1/3))/(-a/b)^(1/3))/((-a*b^2)^(2/3)*b) - 1/6*(b^2*c^2 - 2*a*b*c*d + a ^2*d^2)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/((-a*b^2)^(2/3)*b) - 1/3* (b^4*c^2 - 2*a*b^3*c*d + a^2*b^2*d^2)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3 )))/(a*b^4) + 1/4*(b^3*d^2*x^4 + 8*b^3*c*d*x - 4*a*b^2*d^2*x)/b^4
Time = 0.28 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.88 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=\frac {d^2\,x^4}{4\,b}-x\,\left (\frac {a\,d^2}{b^2}-\frac {2\,c\,d}{b}\right )+\frac {\ln \left (b^{1/3}\,x+a^{1/3}\right )\,{\left (a\,d-b\,c\right )}^2}{3\,a^{2/3}\,b^{7/3}}+\frac {\ln \left (2\,b^{1/3}\,x-a^{1/3}+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (-\frac {1}{6}+\frac {\sqrt {3}\,1{}\mathrm {i}}{6}\right )\,{\left (a\,d-b\,c\right )}^2}{a^{2/3}\,b^{7/3}}-\frac {\ln \left (a^{1/3}-2\,b^{1/3}\,x+\sqrt {3}\,a^{1/3}\,1{}\mathrm {i}\right )\,\left (\frac {1}{2}+\frac {\sqrt {3}\,1{}\mathrm {i}}{2}\right )\,{\left (a\,d-b\,c\right )}^2}{3\,a^{2/3}\,b^{7/3}} \] Input:
int((c + d*x^3)^2/(a + b*x^3),x)
Output:
(d^2*x^4)/(4*b) - x*((a*d^2)/b^2 - (2*c*d)/b) + (log(b^(1/3)*x + a^(1/3))* (a*d - b*c)^2)/(3*a^(2/3)*b^(7/3)) + (log(3^(1/2)*a^(1/3)*1i + 2*b^(1/3)*x - a^(1/3))*((3^(1/2)*1i)/6 - 1/6)*(a*d - b*c)^2)/(a^(2/3)*b^(7/3)) - (log (3^(1/2)*a^(1/3)*1i - 2*b^(1/3)*x + a^(1/3))*((3^(1/2)*1i)/2 + 1/2)*(a*d - b*c)^2)/(3*a^(2/3)*b^(7/3))
Time = 0.23 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.61 \[ \int \frac {\left (c+d x^3\right )^2}{a+b x^3} \, dx=\frac {-4 a^{\frac {7}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) d^{2}+8 a^{\frac {4}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b c d -4 a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} c^{2}-2 a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) d^{2}+4 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b c d -2 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{2} c^{2}+4 a^{\frac {7}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) d^{2}-8 a^{\frac {4}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b c d +4 a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{2} c^{2}-12 b^{\frac {1}{3}} a^{2} d^{2} x +24 b^{\frac {4}{3}} a c d x +3 b^{\frac {4}{3}} a \,d^{2} x^{4}}{12 b^{\frac {7}{3}} a} \] Input:
int((d*x^3+c)^2/(b*x^3+a),x)
Output:
( - 4*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))* a**2*d**2 + 8*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sq rt(3)))*a*b*c*d - 4*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1 /3)*sqrt(3)))*b**2*c**2 - 2*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*d**2 + 4*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a*b*c*d - 2*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b**2*c**2 + 4*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a**2*d**2 - 8*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*a*b*c*d + 4*a**(1/3)*log(a**(1/3) + b**(1/3)*x)*b**2*c**2 - 12*b**(1/3)*a**2*d**2*x + 24*b**(1/3)*a*b*c*d*x + 3*b**(1/3)*a*b*d**2*x**4)/(12*b**(1/3)*a*b**2)