\(\int \frac {1}{(a+b x^3)^2 (c+d x^3)^2} \, dx\) [41]

Optimal result

Integrand size = 19, antiderivative size = 419 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx=\frac {d (b c+a d) x}{3 a c (b c-a d)^2 \left (c+d x^3\right )}+\frac {b x}{3 a (b c-a d) \left (a+b x^3\right ) \left (c+d x^3\right )}-\frac {2 b^{5/3} (b c-4 a d) \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{3 \sqrt {3} a^{5/3} (b c-a d)^3}-\frac {2 d^{5/3} (4 b c-a d) \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{3 \sqrt {3} c^{5/3} (b c-a d)^3}+\frac {2 b^{5/3} (b c-4 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{9 a^{5/3} (b c-a d)^3}+\frac {2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{9 c^{5/3} (b c-a d)^3}-\frac {b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{9 a^{5/3} (b c-a d)^3}-\frac {d^{5/3} (4 b c-a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{9 c^{5/3} (b c-a d)^3} \] Output:

1/3*d*(a*d+b*c)*x/a/c/(-a*d+b*c)^2/(d*x^3+c)+1/3*b*x/a/(-a*d+b*c)/(b*x^3+a 
)/(d*x^3+c)-2/9*b^(5/3)*(-4*a*d+b*c)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1 
/2)/a^(1/3))*3^(1/2)/a^(5/3)/(-a*d+b*c)^3-2/9*d^(5/3)*(-a*d+4*b*c)*arctan( 
1/3*(c^(1/3)-2*d^(1/3)*x)*3^(1/2)/c^(1/3))*3^(1/2)/c^(5/3)/(-a*d+b*c)^3+2/ 
9*b^(5/3)*(-4*a*d+b*c)*ln(a^(1/3)+b^(1/3)*x)/a^(5/3)/(-a*d+b*c)^3+2/9*d^(5 
/3)*(-a*d+4*b*c)*ln(c^(1/3)+d^(1/3)*x)/c^(5/3)/(-a*d+b*c)^3-1/9*b^(5/3)*(- 
4*a*d+b*c)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)/(-a*d+b*c)^3- 
1/9*d^(5/3)*(-a*d+4*b*c)*ln(c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/c^(5/3) 
/(-a*d+b*c)^3
 

Mathematica [A] (verified)

Time = 0.51 (sec) , antiderivative size = 381, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx=\frac {1}{9} \left (\frac {3 b^2 x}{a (b c-a d)^2 \left (a+b x^3\right )}+\frac {3 d^2 x}{c (b c-a d)^2 \left (c+d x^3\right )}+\frac {2 \sqrt {3} b^{5/3} (b c-4 a d) \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3} (-b c+a d)^3}+\frac {2 \sqrt {3} d^{5/3} (-4 b c+a d) \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{5/3} (b c-a d)^3}+\frac {2 b^{5/3} (-b c+4 a d) \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3} (-b c+a d)^3}+\frac {2 d^{5/3} (4 b c-a d) \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{5/3} (b c-a d)^3}+\frac {b^{5/3} (b c-4 a d) \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3} (-b c+a d)^3}+\frac {d^{5/3} (-4 b c+a d) \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{5/3} (b c-a d)^3}\right ) \] Input:

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

Output:

((3*b^2*x)/(a*(b*c - a*d)^2*(a + b*x^3)) + (3*d^2*x)/(c*(b*c - a*d)^2*(c + 
 d*x^3)) + (2*Sqrt[3]*b^(5/3)*(b*c - 4*a*d)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1 
/3))/Sqrt[3]])/(a^(5/3)*(-(b*c) + a*d)^3) + (2*Sqrt[3]*d^(5/3)*(-4*b*c + a 
*d)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/(c^(5/3)*(b*c - a*d)^3) + 
 (2*b^(5/3)*(-(b*c) + 4*a*d)*Log[a^(1/3) + b^(1/3)*x])/(a^(5/3)*(-(b*c) + 
a*d)^3) + (2*d^(5/3)*(4*b*c - a*d)*Log[c^(1/3) + d^(1/3)*x])/(c^(5/3)*(b*c 
 - a*d)^3) + (b^(5/3)*(b*c - 4*a*d)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2 
/3)*x^2])/(a^(5/3)*(-(b*c) + a*d)^3) + (d^(5/3)*(-4*b*c + a*d)*Log[c^(2/3) 
 - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(c^(5/3)*(b*c - a*d)^3))/9
 

Rubi [A] (verified)

Time = 1.09 (sec) , antiderivative size = 373, normalized size of antiderivative = 0.89, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.684, Rules used = {931, 25, 1024, 27, 1020, 750, 16, 1142, 25, 27, 1082, 217, 1103}

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.

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

\(\Big \downarrow \) 931

\(\displaystyle \frac {b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}-\frac {\int -\frac {5 b d x^3+2 b c-3 a d}{\left (b x^3+a\right ) \left (d x^3+c\right )^2}dx}{3 a (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {5 b d x^3+2 b c-3 a d}{\left (b x^3+a\right ) \left (d x^3+c\right )^2}dx}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}\)

\(\Big \downarrow \) 1024

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 1020

\(\displaystyle \frac {\frac {2 \left (\frac {b^2 c (b c-4 a d) \int \frac {1}{b x^3+a}dx}{b c-a d}+\frac {a d^2 (4 b c-a d) \int \frac {1}{d x^3+c}dx}{b c-a d}\right )}{c (b c-a d)}+\frac {d x (a d+b c)}{c \left (c+d x^3\right ) (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}\)

\(\Big \downarrow \) 750

\(\displaystyle \frac {\frac {2 \left (\frac {b^2 c (b c-4 a d) \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{b c-a d}+\frac {a d^2 (4 b c-a d) \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{d} x+\sqrt [3]{c}}dx}{3 c^{2/3}}\right )}{b c-a d}\right )}{c (b c-a d)}+\frac {d x (a d+b c)}{c \left (c+d x^3\right ) (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {\frac {2 \left (\frac {b^2 c (b c-4 a d) \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}+\frac {a d^2 (4 b c-a d) \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}\right )}{c (b c-a d)}+\frac {d x (a d+b c)}{c \left (c+d x^3\right ) (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}\)

\(\Big \downarrow \) 1142

\(\displaystyle \frac {\frac {2 \left (\frac {b^2 c (b c-4 a d) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}+\frac {a d^2 (4 b c-a d) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}\right )}{c (b c-a d)}+\frac {d x (a d+b c)}{c \left (c+d x^3\right ) (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {2 \left (\frac {b^2 c (b c-4 a d) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}+\frac {a d^2 (4 b c-a d) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {\int \frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}\right )}{c (b c-a d)}+\frac {d x (a d+b c)}{c \left (c+d x^3\right ) (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {2 \left (\frac {b^2 c (b c-4 a d) \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}+\frac {a d^2 (4 b c-a d) \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}\right )}{c (b c-a d)}+\frac {d x (a d+b c)}{c \left (c+d x^3\right ) (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}\)

\(\Big \downarrow \) 1082

\(\displaystyle \frac {\frac {2 \left (\frac {b^2 c (b c-4 a d) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}+\frac {a d^2 (4 b c-a d) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}\right )}{c (b c-a d)}+\frac {d x (a d+b c)}{c \left (c+d x^3\right ) (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}\)

\(\Big \downarrow \) 217

\(\displaystyle \frac {\frac {2 \left (\frac {b^2 c (b c-4 a d) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}+\frac {a d^2 (4 b c-a d) \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}\right )}{c (b c-a d)}+\frac {d x (a d+b c)}{c \left (c+d x^3\right ) (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}\)

\(\Big \downarrow \) 1103

\(\displaystyle \frac {\frac {2 \left (\frac {b^2 c (b c-4 a d) \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}+\frac {a d^2 (4 b c-a d) \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}\right )}{c (b c-a d)}+\frac {d x (a d+b c)}{c \left (c+d x^3\right ) (b c-a d)}}{3 a (b c-a d)}+\frac {b x}{3 a \left (a+b x^3\right ) \left (c+d x^3\right ) (b c-a d)}\)

Input:

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

Output:

(b*x)/(3*a*(b*c - a*d)*(a + b*x^3)*(c + d*x^3)) + ((d*(b*c + a*d)*x)/(c*(b 
*c - a*d)*(c + d*x^3)) + (2*((b^2*c*(b*c - 4*a*d)*(Log[a^(1/3) + b^(1/3)*x 
]/(3*a^(2/3)*b^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqr 
t[3]])/b^(1/3)) - Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2]/(2*b^(1/3 
)))/(3*a^(2/3))))/(b*c - a*d) + (a*d^2*(4*b*c - a*d)*(Log[c^(1/3) + d^(1/3 
)*x]/(3*c^(2/3)*d^(1/3)) + (-((Sqrt[3]*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/ 
Sqrt[3]])/d^(1/3)) - Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2]/(2*d^( 
1/3)))/(3*c^(2/3))))/(b*c - a*d)))/(c*(b*c - a*d)))/(3*a*(b*c - a*d))
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 217

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( 
-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & 
& (LtQ[a, 0] || LtQ[b, 0])
 

rule 750

Int[((a_) + (b_.)*(x_)^3)^(-1), x_Symbol] :> Simp[1/(3*Rt[a, 3]^2)   Int[1/ 
(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]^2)   Int[(2*Rt[a, 3] - 
 Rt[b, 3]*x)/(Rt[a, 3]^2 - Rt[a, 3]*Rt[b, 3]*x + Rt[b, 3]^2*x^2), x], x] /; 
 FreeQ[{a, b}, x]
 

rule 931

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] 
:> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - 
 a*d))), x] + Simp[1/(a*n*(p + 1)*(b*c - a*d))   Int[(a + b*x^n)^(p + 1)*(c 
 + d*x^n)^q*Simp[b*c + n*(p + 1)*(b*c - a*d) + d*b*(n*(p + q + 2) + 1)*x^n, 
 x], x], x] /; FreeQ[{a, b, c, d, n, q}, x] && NeQ[b*c - a*d, 0] && LtQ[p, 
-1] &&  !( !IntegerQ[p] && IntegerQ[q] && LtQ[q, -1]) && IntBinomialQ[a, b, 
 c, d, n, p, q, x]
 

rule 1020

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^( 
n_))), x_Symbol] :> Simp[(b*e - a*f)/(b*c - a*d)   Int[1/(a + b*x^n), x], x 
] - Simp[(d*e - c*f)/(b*c - a*d)   Int[1/(c + d*x^n), x], x] /; FreeQ[{a, b 
, c, d, e, f, n}, x]
 

rule 1024

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f 
_.)*(x_)^(n_)), x_Symbol] :> Simp[(-(b*e - a*f))*x*(a + b*x^n)^(p + 1)*((c 
+ d*x^n)^(q + 1)/(a*n*(b*c - a*d)*(p + 1))), x] + Simp[1/(a*n*(b*c - a*d)*( 
p + 1))   Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q*Simp[c*(b*e - a*f) + e*n*(b 
*c - a*d)*(p + 1) + d*(b*e - a*f)*(n*(p + q + 2) + 1)*x^n, x], x], x] /; Fr 
eeQ[{a, b, c, d, e, f, n, q}, x] && LtQ[p, -1]
 

rule 1082

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S 
implify[a*(c/b^2)]}, Simp[-2/b   Subst[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b 
)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /; Fre 
eQ[{a, b, c}, x]
 

rule 1103

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[d*(Log[RemoveContent[a + b*x + c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, 
e}, x] && EqQ[2*c*d - b*e, 0]
 

rule 1142

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S 
imp[(2*c*d - b*e)/(2*c)   Int[1/(a + b*x + c*x^2), x], x] + Simp[e/(2*c) 
Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x]
 

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 285, normalized size of antiderivative = 0.68

method	result	size
default	\(\frac {d^{2} \left (\frac {\left (a d -b c \right ) x}{3 c \left (d \,x^{3}+c \right )}+\frac {2 \left (a d -4 b c \right ) \left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right )}{3 c}\right )}{\left (a d -b c \right )^{3}}+\frac {b^{2} \left (\frac {\left (a d -b c \right ) x}{3 a \left (b \,x^{3}+a \right )}+\frac {2 \left (4 a d -b c \right ) \left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right )}{3 a}\right )}{\left (a d -b c \right )^{3}}\)	\(285\)
risch	\(\text {Expression too large to display}\)	\(1701\)

Input:

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

Output:

d^2/(a*d-b*c)^3*(1/3*(a*d-b*c)/c*x/(d*x^3+c)+2/3*(a*d-4*b*c)/c*(1/3/d/(c/d 
)^(2/3)*ln(x+(c/d)^(1/3))-1/6/d/(c/d)^(2/3)*ln(x^2-(c/d)^(1/3)*x+(c/d)^(2/ 
3))+1/3/d/(c/d)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/3)*x-1))))+b^ 
2/(a*d-b*c)^3*(1/3*(a*d-b*c)/a*x/(b*x^3+a)+2/3*(4*a*d-b*c)/a*(1/3/b/(a/b)^ 
(2/3)*ln(x+(a/b)^(1/3))-1/6/b/(a/b)^(2/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3) 
)+1/3/b/(a/b)^(2/3)*3^(1/2)*arctan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1))))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 897 vs. \(2 (341) = 682\).

Time = 47.36 (sec) , antiderivative size = 897, normalized size of antiderivative = 2.14 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

1/9*(3*(b^3*c^2*d - a^2*b*d^3)*x^4 + 2*sqrt(3)*((b^3*c^2*d - 4*a*b^2*c*d^2 
)*x^6 + a*b^2*c^3 - 4*a^2*b*c^2*d + (b^3*c^3 - 3*a*b^2*c^2*d - 4*a^2*b*c*d 
^2)*x^3)*(b^2/a^2)^(1/3)*arctan(1/3*(2*sqrt(3)*a*x*(b^2/a^2)^(2/3) - sqrt( 
3)*b)/b) + 2*sqrt(3)*((4*a*b^2*c*d^2 - a^2*b*d^3)*x^6 + 4*a^2*b*c^2*d - a^ 
3*c*d^2 + (4*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^3)*(d^2/c^2)^(1/3)*a 
rctan(1/3*(2*sqrt(3)*c*x*(d^2/c^2)^(2/3) - sqrt(3)*d)/d) - ((b^3*c^2*d - 4 
*a*b^2*c*d^2)*x^6 + a*b^2*c^3 - 4*a^2*b*c^2*d + (b^3*c^3 - 3*a*b^2*c^2*d - 
 4*a^2*b*c*d^2)*x^3)*(b^2/a^2)^(1/3)*log(b^2*x^2 - a*b*x*(b^2/a^2)^(1/3) + 
 a^2*(b^2/a^2)^(2/3)) - ((4*a*b^2*c*d^2 - a^2*b*d^3)*x^6 + 4*a^2*b*c^2*d - 
 a^3*c*d^2 + (4*a*b^2*c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^3)*(d^2/c^2)^(1/3 
)*log(d^2*x^2 - c*d*x*(d^2/c^2)^(1/3) + c^2*(d^2/c^2)^(2/3)) + 2*((b^3*c^2 
*d - 4*a*b^2*c*d^2)*x^6 + a*b^2*c^3 - 4*a^2*b*c^2*d + (b^3*c^3 - 3*a*b^2*c 
^2*d - 4*a^2*b*c*d^2)*x^3)*(b^2/a^2)^(1/3)*log(b*x + a*(b^2/a^2)^(1/3)) + 
2*((4*a*b^2*c*d^2 - a^2*b*d^3)*x^6 + 4*a^2*b*c^2*d - a^3*c*d^2 + (4*a*b^2* 
c^2*d + 3*a^2*b*c*d^2 - a^3*d^3)*x^3)*(d^2/c^2)^(1/3)*log(d*x + c*(d^2/c^2 
)^(1/3)) + 3*(b^3*c^3 - a*b^2*c^2*d + a^2*b*c*d^2 - a^3*d^3)*x)/(a^2*b^3*c 
^5 - 3*a^3*b^2*c^4*d + 3*a^4*b*c^3*d^2 - a^5*c^2*d^3 + (a*b^4*c^4*d - 3*a^ 
2*b^3*c^3*d^2 + 3*a^3*b^2*c^2*d^3 - a^4*b*c*d^4)*x^6 + (a*b^4*c^5 - 2*a^2* 
b^3*c^4*d + 2*a^4*b*c^2*d^3 - a^5*c*d^4)*x^3)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 784 vs. \(2 (341) = 682\).

Time = 0.13 (sec) , antiderivative size = 784, normalized size of antiderivative = 1.87 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx =\text {Too large to display} \] Input:

integrate(1/(b*x^3+a)^2/(d*x^3+c)^2,x, algorithm="maxima")
 

Output:

2/9*sqrt(3)*(b^2*c - 4*a*b*d)*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b) 
^(1/3))/((a*b^3*c^3*(a/b)^(1/3) - 3*a^2*b^2*c^2*d*(a/b)^(1/3) + 3*a^3*b*c* 
d^2*(a/b)^(1/3) - a^4*d^3*(a/b)^(1/3))*(a/b)^(1/3)) + 2/9*sqrt(3)*(4*b*c*d 
 - a*d^2)*arctan(1/3*sqrt(3)*(2*x - (c/d)^(1/3))/(c/d)^(1/3))/((b^3*c^4*(c 
/d)^(1/3) - 3*a*b^2*c^3*d*(c/d)^(1/3) + 3*a^2*b*c^2*d^2*(c/d)^(1/3) - a^3* 
c*d^3*(c/d)^(1/3))*(c/d)^(1/3)) - 1/9*(b^2*c - 4*a*b*d)*log(x^2 - x*(a/b)^ 
(1/3) + (a/b)^(2/3))/(a*b^3*c^3*(a/b)^(2/3) - 3*a^2*b^2*c^2*d*(a/b)^(2/3) 
+ 3*a^3*b*c*d^2*(a/b)^(2/3) - a^4*d^3*(a/b)^(2/3)) - 1/9*(4*b*c*d - a*d^2) 
*log(x^2 - x*(c/d)^(1/3) + (c/d)^(2/3))/(b^3*c^4*(c/d)^(2/3) - 3*a*b^2*c^3 
*d*(c/d)^(2/3) + 3*a^2*b*c^2*d^2*(c/d)^(2/3) - a^3*c*d^3*(c/d)^(2/3)) + 2/ 
9*(b^2*c - 4*a*b*d)*log(x + (a/b)^(1/3))/(a*b^3*c^3*(a/b)^(2/3) - 3*a^2*b^ 
2*c^2*d*(a/b)^(2/3) + 3*a^3*b*c*d^2*(a/b)^(2/3) - a^4*d^3*(a/b)^(2/3)) + 2 
/9*(4*b*c*d - a*d^2)*log(x + (c/d)^(1/3))/(b^3*c^4*(c/d)^(2/3) - 3*a*b^2*c 
^3*d*(c/d)^(2/3) + 3*a^2*b*c^2*d^2*(c/d)^(2/3) - a^3*c*d^3*(c/d)^(2/3)) + 
1/3*((b^2*c*d + a*b*d^2)*x^4 + (b^2*c^2 + a^2*d^2)*x)/(a^2*b^2*c^4 - 2*a^3 
*b*c^3*d + a^4*c^2*d^2 + (a*b^3*c^3*d - 2*a^2*b^2*c^2*d^2 + a^3*b*c*d^3)*x 
^6 + (a*b^3*c^4 - a^2*b^2*c^3*d - a^3*b*c^2*d^2 + a^4*c*d^3)*x^3)
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 664, normalized size of antiderivative = 1.58 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

-2/9*(b^3*c - 4*a*b^2*d)*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b^3* 
c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3) - 2/9*(4*b*c*d^2 - a*d^3) 
*(-c/d)^(1/3)*log(abs(x - (-c/d)^(1/3)))/(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2* 
b*c^3*d^2 - a^3*c^2*d^3) + 2/3*((-a*b^2)^(1/3)*b^2*c - 4*(-a*b^2)^(1/3)*a* 
b*d)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*a^2*b^ 
3*c^3 - 3*sqrt(3)*a^3*b^2*c^2*d + 3*sqrt(3)*a^4*b*c*d^2 - sqrt(3)*a^5*d^3) 
 + 2/3*(4*(-c*d^2)^(1/3)*b*c*d - (-c*d^2)^(1/3)*a*d^2)*arctan(1/3*sqrt(3)* 
(2*x + (-c/d)^(1/3))/(-c/d)^(1/3))/(sqrt(3)*b^3*c^5 - 3*sqrt(3)*a*b^2*c^4* 
d + 3*sqrt(3)*a^2*b*c^3*d^2 - sqrt(3)*a^3*c^2*d^3) + 1/9*((-a*b^2)^(1/3)*b 
^2*c - 4*(-a*b^2)^(1/3)*a*b*d)*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a 
^2*b^3*c^3 - 3*a^3*b^2*c^2*d + 3*a^4*b*c*d^2 - a^5*d^3) + 1/9*(4*(-c*d^2)^ 
(1/3)*b*c*d - (-c*d^2)^(1/3)*a*d^2)*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3 
))/(b^3*c^5 - 3*a*b^2*c^4*d + 3*a^2*b*c^3*d^2 - a^3*c^2*d^3) + 1/3*(b^2*c* 
d*x^4 + a*b*d^2*x^4 + b^2*c^2*x + a^2*d^2*x)/((b*d*x^6 + b*c*x^3 + a*d*x^3 
 + a*c)*(a*b^2*c^3 - 2*a^2*b*c^2*d + a^3*c*d^2))
 

Mupad [B] (verification not implemented)

Time = 25.74 (sec) , antiderivative size = 3637, normalized size of antiderivative = 8.68 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

((x*(a^2*d^2 + b^2*c^2))/(3*a*c*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)) + (b*d*x^ 
4*(a*d + b*c))/(3*a*c*(a^2*d^2 + b^2*c^2 - 2*a*b*c*d)))/(a*c + x^3*(a*d + 
b*c) + b*d*x^6) + log((2*((4*((54*b^3*d^3*x*(a*d - b*c)^2*(a^3*d^3 + b^3*c 
^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(a*c) + 54*a*b^3*c*d^3*(a*d + b*c)*(a 
*d - b*c)^4*((b^5*(4*a*d - b*c)^3)/(a^5*(a*d - b*c)^9))^(1/3))*((b^5*(4*a* 
d - b*c)^3)/(a^5*(a*d - b*c)^9))^(2/3))/81 - (8*b^4*d^4*(a^6*d^6 + b^6*c^6 
 + 37*a^2*b^4*c^4*d^2 - 27*a^3*b^3*c^3*d^3 + 37*a^4*b^2*c^2*d^4 - 11*a*b^5 
*c^5*d - 11*a^5*b*c*d^5))/(3*a^3*c^3*(a*d - b*c)^4))*((b^5*(4*a*d - b*c)^3 
)/(a^5*(a*d - b*c)^9))^(1/3))/9 - (16*b^6*d^6*x*(4*a^6*d^6 + 4*b^6*c^6 + 2 
68*a^2*b^4*c^4*d^2 - 608*a^3*b^3*c^3*d^3 + 268*a^4*b^2*c^2*d^4 - 49*a*b^5* 
c^5*d - 49*a^5*b*c*d^5))/(27*a^3*c^3*(a*d - b*c)^8))*(-(8*b^8*c^3 - 512*a^ 
3*b^5*d^3 + 384*a^2*b^6*c*d^2 - 96*a*b^7*c^2*d)/(729*a^14*d^9 - 729*a^5*b^ 
9*c^9 + 6561*a^6*b^8*c^8*d - 26244*a^7*b^7*c^7*d^2 + 61236*a^8*b^6*c^6*d^3 
 - 91854*a^9*b^5*c^5*d^4 + 91854*a^10*b^4*c^4*d^5 - 61236*a^11*b^3*c^3*d^6 
 + 26244*a^12*b^2*c^2*d^7 - 6561*a^13*b*c*d^8))^(1/3) + log((2*((4*((54*b^ 
3*d^3*x*(a*d - b*c)^2*(a^3*d^3 + b^3*c^3 - 3*a*b^2*c^2*d - 3*a^2*b*c*d^2)) 
/(a*c) + 54*a*b^3*c*d^3*(a*d + b*c)*(a*d - b*c)^4*((d^5*(a*d - 4*b*c)^3)/( 
c^5*(a*d - b*c)^9))^(1/3))*((d^5*(a*d - 4*b*c)^3)/(c^5*(a*d - b*c)^9))^(2/ 
3))/81 - (8*b^4*d^4*(a^6*d^6 + b^6*c^6 + 37*a^2*b^4*c^4*d^2 - 27*a^3*b^3*c 
^3*d^3 + 37*a^4*b^2*c^2*d^4 - 11*a*b^5*c^5*d - 11*a^5*b*c*d^5))/(3*a^3*...
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 1648, normalized size of antiderivative = 3.93 \[ \int \frac {1}{\left (a+b x^3\right )^2 \left (c+d x^3\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 8*d**(1/3)*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*s 
qrt(3)))*a**2*b**2*c**3*d - 8*d**(1/3)*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2 
*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a**2*b**2*c**2*d**2*x**3 + 2*d**(1/3)*a** 
(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**3*c* 
*4 - 6*d**(1/3)*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)* 
sqrt(3)))*a*b**3*c**3*d*x**3 - 8*d**(1/3)*a**(1/3)*sqrt(3)*atan((a**(1/3) 
- 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*a*b**3*c**2*d**2*x**6 + 2*d**(1/3)*a** 
(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt(3)))*b**4*c**4 
*x**3 + 2*d**(1/3)*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/ 
3)*sqrt(3)))*b**4*c**3*d*x**6 - 2*c**(1/3)*b**(1/3)*sqrt(3)*atan((c**(1/3) 
 - 2*d**(1/3)*x)/(c**(1/3)*sqrt(3)))*a**4*c*d**3 - 2*c**(1/3)*b**(1/3)*sqr 
t(3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c**(1/3)*sqrt(3)))*a**4*d**4*x**3 + 8 
*c**(1/3)*b**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c**(1/3)*sqrt(3 
)))*a**3*b*c**2*d**2 + 6*c**(1/3)*b**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d**( 
1/3)*x)/(c**(1/3)*sqrt(3)))*a**3*b*c*d**3*x**3 - 2*c**(1/3)*b**(1/3)*sqrt( 
3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c**(1/3)*sqrt(3)))*a**3*b*d**4*x**6 + 8 
*c**(1/3)*b**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c**(1/3)*sqrt(3 
)))*a**2*b**2*c**2*d**2*x**3 + 8*c**(1/3)*b**(1/3)*sqrt(3)*atan((c**(1/3) 
- 2*d**(1/3)*x)/(c**(1/3)*sqrt(3)))*a**2*b**2*c*d**3*x**6 - 4*d**(1/3)*a** 
(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*b**2*c**...