Integrand size = 20, antiderivative size = 264 \[ \int \frac {x^2}{(c+d x) \left (a+b x^3\right )} \, dx=-\frac {\sqrt [3]{a} d \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{2/3} \left (b^{2/3} c^2+\sqrt [3]{a} \sqrt [3]{b} c d+a^{2/3} d^2\right )}+\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} \left (b c^3-a d^3\right )}-\frac {c^2 \log (c+d x)}{b c^3-a d^3}-\frac {\sqrt [3]{a} d \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} \left (b c^3-a d^3\right )}+\frac {c^2 \log \left (a+b x^3\right )}{3 \left (b c^3-a d^3\right )} \] Output:
-1/3*a^(1/3)*d*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/b ^(2/3)/(b^(2/3)*c^2+a^(1/3)*b^(1/3)*c*d+a^(2/3)*d^2)+1/3*a^(1/3)*d*(b^(1/3 )*c+a^(1/3)*d)*ln(a^(1/3)+b^(1/3)*x)/b^(2/3)/(-a*d^3+b*c^3)-c^2*ln(d*x+c)/ (-a*d^3+b*c^3)-1/6*a^(1/3)*d*(b^(1/3)*c+a^(1/3)*d)*ln(a^(2/3)-a^(1/3)*b^(1 /3)*x+b^(2/3)*x^2)/b^(2/3)/(-a*d^3+b*c^3)+c^2*ln(b*x^3+a)/(-3*a*d^3+3*b*c^ 3)
Time = 0.11 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{(c+d x) \left (a+b x^3\right )} \, dx=\frac {2 \sqrt {3} \sqrt [3]{a} d \left (-\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )+2 \sqrt [3]{a} d \left (\sqrt [3]{b} c+\sqrt [3]{a} d\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )-6 b^{2/3} c^2 \log (c+d x)-\sqrt [3]{a} \sqrt [3]{b} c d \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )-a^{2/3} d^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )+2 b^{2/3} c^2 \log \left (a+b x^3\right )}{6 b^{2/3} \left (b c^3-a d^3\right )} \] Input:
Integrate[x^2/((c + d*x)*(a + b*x^3)),x]
Output:
(2*Sqrt[3]*a^(1/3)*d*(-(b^(1/3)*c) + a^(1/3)*d)*ArcTan[(1 - (2*b^(1/3)*x)/ a^(1/3))/Sqrt[3]] + 2*a^(1/3)*d*(b^(1/3)*c + a^(1/3)*d)*Log[a^(1/3) + b^(1 /3)*x] - 6*b^(2/3)*c^2*Log[c + d*x] - a^(1/3)*b^(1/3)*c*d*Log[a^(2/3) - a^ (1/3)*b^(1/3)*x + b^(2/3)*x^2] - a^(2/3)*d^2*Log[a^(2/3) - a^(1/3)*b^(1/3) *x + b^(2/3)*x^2] + 2*b^(2/3)*c^2*Log[a + b*x^3])/(6*b^(2/3)*(b*c^3 - a*d^ 3))
Time = 1.08 (sec) , antiderivative size = 264, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {7276, 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.
| \(\displaystyle \int \frac {x^2}{\left (a+b x^3\right ) (c+d x)} \, dx\) |
| \(\Big \downarrow \) 7276 |
| \(\displaystyle \int \left (\frac {a c d-a d^2 x+b c^2 x^2}{\left (a+b x^3\right ) \left (b c^3-a d^3\right )}-\frac {c^2 d}{(c+d x) \left (b c^3-a d^3\right )}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [3]{a} d \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{2/3} \left (a^{2/3} d^2+\sqrt [3]{a} \sqrt [3]{b} c d+b^{2/3} c^2\right )}-\frac {\sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} \left (b c^3-a d^3\right )}+\frac {\sqrt [3]{a} d \left (\sqrt [3]{a} d+\sqrt [3]{b} c\right ) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} \left (b c^3-a d^3\right )}+\frac {c^2 \log \left (a+b x^3\right )}{3 \left (b c^3-a d^3\right )}-\frac {c^2 \log (c+d x)}{b c^3-a d^3}\) |
Input:
Int[x^2/((c + d*x)*(a + b*x^3)),x]
Output:
-((a^(1/3)*d*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3))])/(Sqrt[3]*b ^(2/3)*(b^(2/3)*c^2 + a^(1/3)*b^(1/3)*c*d + a^(2/3)*d^2))) + (a^(1/3)*d*(b ^(1/3)*c + a^(1/3)*d)*Log[a^(1/3) + b^(1/3)*x])/(3*b^(2/3)*(b*c^3 - a*d^3) ) - (c^2*Log[c + d*x])/(b*c^3 - a*d^3) - (a^(1/3)*d*(b^(1/3)*c + a^(1/3)*d )*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/(6*b^(2/3)*(b*c^3 - a*d^ 3)) + (c^2*Log[a + b*x^3])/(3*(b*c^3 - a*d^3))
Int[(u_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = RationalFunctionE xpand[u/(a + b*x^n), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && IGtQ [n, 0]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.29 (sec) , antiderivative size = 177, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(\frac {c^{2} \ln \left (-d x -c \right )}{d^{3} a -b \,c^{3}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (1+\left (a \,b^{2} d^{3}-b^{3} c^{3}\right ) \textit {\_Z}^{3}+3 b^{2} c^{2} \textit {\_Z}^{2}-3 b c \textit {\_Z} \right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a \,b^{2} d^{4}-2 b^{3} c^{3} d \right ) \textit {\_R}^{3}-3 \textit {\_R}^{2} b^{2} c^{2} d +8 c b d \textit {\_R} -3 d \right ) x +\left (-5 a \,b^{2} c \,d^{3}-b^{3} c^{4}\right ) \textit {\_R}^{3}+\left (d^{3} a b -b^{2} c^{3}\right ) \textit {\_R}^{2}+5 b \,c^{2} \textit {\_R} -3 c \right )\right )}{3}\) | \(177\) |
| default | \(\frac {c^{2} \ln \left (d x +c \right )}{d^{3} a -b \,c^{3}}+\frac {-a c d \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )+a \,d^{2} \left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )-\frac {c^{2} \ln \left (b \,x^{3}+a \right )}{3}}{d^{3} a -b \,c^{3}}\) | \(245\) |
Input:
int(x^2/(d*x+c)/(b*x^3+a),x,method=_RETURNVERBOSE)
Output:
c^2/(a*d^3-b*c^3)*ln(-d*x-c)+1/3*sum(_R*ln(((-4*a*b^2*d^4-2*b^3*c^3*d)*_R^ 3-3*_R^2*b^2*c^2*d+8*c*b*d*_R-3*d)*x+(-5*a*b^2*c*d^3-b^3*c^4)*_R^3+(a*b*d^ 3-b^2*c^3)*_R^2+5*b*c^2*_R-3*c),_R=RootOf(1+(a*b^2*d^3-b^3*c^3)*_Z^3+3*b^2 *c^2*_Z^2-3*b*c*_Z))
Result contains complex when optimal does not.
Time = 0.86 (sec) , antiderivative size = 5975, normalized size of antiderivative = 22.63 \[ \int \frac {x^2}{(c+d x) \left (a+b x^3\right )} \, dx=\text {Too large to display} \] Input:
integrate(x^2/(d*x+c)/(b*x^3+a),x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {x^2}{(c+d x) \left (a+b x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(x**2/(d*x+c)/(b*x**3+a),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.06 \[ \int \frac {x^2}{(c+d x) \left (a+b x^3\right )} \, dx=-\frac {c^{2} \log \left (d x + c\right )}{b c^{3} - a d^{3}} - \frac {\sqrt {3} {\left (a d^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a c d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b^{2} c^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {{\left (2 \, b c^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} - a c d\right )} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {{\left (b c^{2} \left (\frac {a}{b}\right )^{\frac {2}{3}} + a d^{2} \left (\frac {a}{b}\right )^{\frac {1}{3}} + a c d\right )} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b^{2} c^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b d^{3} \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} \] Input:
integrate(x^2/(d*x+c)/(b*x^3+a),x, algorithm="maxima")
Output:
-c^2*log(d*x + c)/(b*c^3 - a*d^3) - 1/3*sqrt(3)*(a*d^2*(a/b)^(2/3) - a*c*d *(a/b)^(1/3))*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/((b^2*c^ 3*(a/b)^(2/3) - a*b*d^3*(a/b)^(2/3))*(a/b)^(1/3)) + 1/6*(2*b*c^2*(a/b)^(2/ 3) - a*d^2*(a/b)^(1/3) - a*c*d)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^ 2*c^3*(a/b)^(2/3) - a*b*d^3*(a/b)^(2/3)) + 1/3*(b*c^2*(a/b)^(2/3) + a*d^2* (a/b)^(1/3) + a*c*d)*log(x + (a/b)^(1/3))/(b^2*c^3*(a/b)^(2/3) - a*b*d^3*( a/b)^(2/3))
Time = 0.15 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.21 \[ \int \frac {x^2}{(c+d x) \left (a+b x^3\right )} \, dx=-\frac {c^{2} d \log \left ({\left | d x + c \right |}\right )}{b c^{3} d - a d^{4}} + \frac {c^{2} \log \left ({\left | b x^{3} + a \right |}\right )}{3 \, {\left (b c^{3} - a d^{3}\right )}} + \frac {\left (-a b^{2}\right )^{\frac {1}{3}} d \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{2} c^{2} - \sqrt {3} \left (-a b^{2}\right )^{\frac {1}{3}} b c d + \sqrt {3} \left (-a b^{2}\right )^{\frac {2}{3}} d^{2}} + \frac {{\left (a b^{2} c^{3} d^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} b d^{5} \left (-\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} c^{4} d + a^{2} b c d^{4}\right )} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a b^{3} c^{6} - 2 \, a^{2} b^{2} c^{3} d^{3} + a^{3} b d^{6}\right )}} + \frac {{\left (\left (-a b^{2}\right )^{\frac {1}{3}} b c d - \left (-a b^{2}\right )^{\frac {2}{3}} d^{2}\right )} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{3} c^{3} - a b^{2} d^{3}\right )}} \] Input:
integrate(x^2/(d*x+c)/(b*x^3+a),x, algorithm="giac")
Output:
-c^2*d*log(abs(d*x + c))/(b*c^3*d - a*d^4) + 1/3*c^2*log(abs(b*x^3 + a))/( b*c^3 - a*d^3) + (-a*b^2)^(1/3)*d*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/ (-a/b)^(1/3))/(sqrt(3)*b^2*c^2 - sqrt(3)*(-a*b^2)^(1/3)*b*c*d + sqrt(3)*(- a*b^2)^(2/3)*d^2) + 1/3*(a*b^2*c^3*d^2*(-a/b)^(1/3) - a^2*b*d^5*(-a/b)^(1/ 3) - a*b^2*c^4*d + a^2*b*c*d^4)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a *b^3*c^6 - 2*a^2*b^2*c^3*d^3 + a^3*b*d^6) + 1/6*((-a*b^2)^(1/3)*b*c*d - (- a*b^2)^(2/3)*d^2)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(b^3*c^3 - a*b^ 2*d^3)
Time = 0.35 (sec) , antiderivative size = 570, normalized size of antiderivative = 2.16 \[ \int \frac {x^2}{(c+d x) \left (a+b x^3\right )} \, dx=\left (\sum _{k=1}^3\ln \left (-a\,b\,d\,\left (c+d\,x+{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^2\,b^2\,c^3\,3+{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^3\,b^3\,c^4\,9-\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )\,b\,c^2\,5-{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^2\,a\,b\,d^3\,3-\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )\,b\,c\,d\,x\,8+{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^3\,a\,b^2\,c\,d^3\,45+{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^3\,a\,b^2\,d^4\,x\,36+{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^2\,b^2\,c^2\,d\,x\,9+{\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )}^3\,b^3\,c^3\,d\,x\,18\right )\right )\,\mathrm {root}\left (27\,a\,b^2\,d^3\,z^3-27\,b^3\,c^3\,z^3+27\,b^2\,c^2\,z^2-9\,b\,c\,z+1,z,k\right )\right )+\frac {c^2\,\ln \left (c+d\,x\right )}{a\,d^3-b\,c^3} \] Input:
int(x^2/((a + b*x^3)*(c + d*x)),x)
Output:
symsum(log(-a*b*d*(c + d*x + 3*root(27*a*b^2*d^3*z^3 - 27*b^3*c^3*z^3 + 27 *b^2*c^2*z^2 - 9*b*c*z + 1, z, k)^2*b^2*c^3 + 9*root(27*a*b^2*d^3*z^3 - 27 *b^3*c^3*z^3 + 27*b^2*c^2*z^2 - 9*b*c*z + 1, z, k)^3*b^3*c^4 - 5*root(27*a *b^2*d^3*z^3 - 27*b^3*c^3*z^3 + 27*b^2*c^2*z^2 - 9*b*c*z + 1, z, k)*b*c^2 - 3*root(27*a*b^2*d^3*z^3 - 27*b^3*c^3*z^3 + 27*b^2*c^2*z^2 - 9*b*c*z + 1, z, k)^2*a*b*d^3 - 8*root(27*a*b^2*d^3*z^3 - 27*b^3*c^3*z^3 + 27*b^2*c^2*z ^2 - 9*b*c*z + 1, z, k)*b*c*d*x + 45*root(27*a*b^2*d^3*z^3 - 27*b^3*c^3*z^ 3 + 27*b^2*c^2*z^2 - 9*b*c*z + 1, z, k)^3*a*b^2*c*d^3 + 36*root(27*a*b^2*d ^3*z^3 - 27*b^3*c^3*z^3 + 27*b^2*c^2*z^2 - 9*b*c*z + 1, z, k)^3*a*b^2*d^4* x + 9*root(27*a*b^2*d^3*z^3 - 27*b^3*c^3*z^3 + 27*b^2*c^2*z^2 - 9*b*c*z + 1, z, k)^2*b^2*c^2*d*x + 18*root(27*a*b^2*d^3*z^3 - 27*b^3*c^3*z^3 + 27*b^ 2*c^2*z^2 - 9*b*c*z + 1, z, k)^3*b^3*c^3*d*x))*root(27*a*b^2*d^3*z^3 - 27* b^3*c^3*z^3 + 27*b^2*c^2*z^2 - 9*b*c*z + 1, z, k), k, 1, 3) + (c^2*log(c + d*x))/(a*d^3 - b*c^3)
Time = 0.15 (sec) , antiderivative size = 1118, normalized size of antiderivative = 4.23 \[ \int \frac {x^2}{(c+d x) \left (a+b x^3\right )} \, dx =\text {Too large to display} \] Input:
int(x^2/(d*x+c)/(b*x^3+a),x)
Output:
(16*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a* *2*b*c*d**8 - 140*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3 )*sqrt(3)))*a*b**2*c**4*d**5 + 16*a**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**( 1/3)*x)/(a**(1/3)*sqrt(3)))*b**3*c**7*d**2 - 56*b**(1/3)*a**(1/3)*sqrt(3)* atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b*c**2*d**7 + 112* b**(1/3)*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3) ))*a*b**2*c**5*d**4 - 2*b**(1/3)*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1 /3)*x)/(a**(1/3)*sqrt(3)))*b**3*c**8*d - 2*b**(2/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**3*d**9 + 112*b**(2/3)*sqrt(3)*atan ((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b*c**3*d**6 - 56*b**(2 /3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**2*c**6 *d**3 - 6*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a** 2*b*c*d**8 - 42*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x** 2)*a*b**2*c**4*d**5 + 48*a**(2/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b** (2/3)*x**2)*b**3*c**7*d**2 + 12*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*a**2*b *c*d**8 - 42*a**(2/3)*log(a**(1/3) + b**(1/3)*x)*a*b**2*c**4*d**5 + 30*a** (2/3)*log(a**(1/3) + b**(1/3)*x)*b**3*c**7*d**2 + 126*a**(2/3)*log(c + d*x )*a*b**2*c**4*d**5 - 126*a**(2/3)*log(c + d*x)*b**3*c**7*d**2 + 12*b**(1/3 )*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b*c**2 *d**7 + 84*b**(1/3)*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2...