Integrand size = 22, antiderivative size = 299 \[ \int \frac {1}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {1}{a c x}+\frac {b^{4/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{4/3} (b c-a d)}-\frac {d^{4/3} \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{4/3} (b c-a d)}+\frac {b^{4/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{4/3} (b c-a d)}-\frac {d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{4/3} (b c-a d)}-\frac {b^{4/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{4/3} (b c-a d)}+\frac {d^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{4/3} (b c-a d)} \]
-1/a/c/x+1/3*b^(4/3)*ln(a^(1/3)+b^(1/3)*x)/a^(4/3)/(-a*d+b*c)-1/3*d^(4/3)* ln(c^(1/3)+d^(1/3)*x)/c^(4/3)/(-a*d+b*c)-1/6*b^(4/3)*ln(a^(2/3)-a^(1/3)*b^ (1/3)*x+b^(2/3)*x^2)/a^(4/3)/(-a*d+b*c)+1/6*d^(4/3)*ln(c^(2/3)-c^(1/3)*d^( 1/3)*x+d^(2/3)*x^2)/c^(4/3)/(-a*d+b*c)+1/3*b^(4/3)*arctan(1/3*(a^(1/3)-2*b ^(1/3)*x)/a^(1/3)*3^(1/2))/a^(4/3)/(-a*d+b*c)*3^(1/2)-1/3*d^(4/3)*arctan(1 /3*(c^(1/3)-2*d^(1/3)*x)/c^(1/3)*3^(1/2))/c^(4/3)/(-a*d+b*c)*3^(1/2)
Time = 0.14 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {\frac {6 b}{a}-\frac {6 d}{c}-\frac {2 \sqrt {3} b^{4/3} x \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{4/3}}+\frac {2 \sqrt {3} d^{4/3} x \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{4/3}}-\frac {2 b^{4/3} x \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{4/3}}+\frac {2 d^{4/3} x \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{4/3}}+\frac {b^{4/3} x \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{4/3}}-\frac {d^{4/3} x \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{4/3}}}{-6 b c x+6 a d x} \]
((6*b)/a - (6*d)/c - (2*Sqrt[3]*b^(4/3)*x*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3 ))/Sqrt[3]])/a^(4/3) + (2*Sqrt[3]*d^(4/3)*x*ArcTan[(1 - (2*d^(1/3)*x)/c^(1 /3))/Sqrt[3]])/c^(4/3) - (2*b^(4/3)*x*Log[a^(1/3) + b^(1/3)*x])/a^(4/3) + (2*d^(4/3)*x*Log[c^(1/3) + d^(1/3)*x])/c^(4/3) + (b^(4/3)*x*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(4/3) - (d^(4/3)*x*Log[c^(2/3) - c^(1/ 3)*d^(1/3)*x + d^(2/3)*x^2])/c^(4/3))/(-6*b*c*x + 6*a*d*x)
Time = 0.46 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {980, 25, 1054, 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.
\(\displaystyle \int \frac {1}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\) |
\(\Big \downarrow \) 980 |
\(\displaystyle \frac {\int -\frac {x \left (b d x^3+b c+a d\right )}{\left (b x^3+a\right ) \left (d x^3+c\right )}dx}{a c}-\frac {1}{a c x}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle -\frac {\int \frac {x \left (b d x^3+b c+a d\right )}{\left (b x^3+a\right ) \left (d x^3+c\right )}dx}{a c}-\frac {1}{a c x}\) |
\(\Big \downarrow \) 1054 |
\(\displaystyle -\frac {\int \left (\frac {c x b^2}{(b c-a d) \left (b x^3+a\right )}+\frac {a d^2 x}{(a d-b c) \left (d x^3+c\right )}\right )dx}{a c}-\frac {1}{a c x}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {\frac {b^{4/3} c \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}-\frac {b^{4/3} c \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} (b c-a d)}+\frac {a d^{4/3} \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [3]{c} (b c-a d)}-\frac {b^{4/3} c \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}-\frac {a d^{4/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}+\frac {a d^{4/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}}{a c}-\frac {1}{a c x}\) |
-(1/(a*c*x)) - (-((b^(4/3)*c*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/ 3))])/(Sqrt[3]*a^(1/3)*(b*c - a*d))) + (a*d^(4/3)*ArcTan[(c^(1/3) - 2*d^(1 /3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(1/3)*(b*c - a*d)) - (b^(4/3)*c*Log[ a^(1/3) + b^(1/3)*x])/(3*a^(1/3)*(b*c - a*d)) + (a*d^(4/3)*Log[c^(1/3) + d ^(1/3)*x])/(3*c^(1/3)*(b*c - a*d)) + (b^(4/3)*c*Log[a^(2/3) - a^(1/3)*b^(1 /3)*x + b^(2/3)*x^2])/(6*a^(1/3)*(b*c - a*d)) - (a*d^(4/3)*Log[c^(2/3) - c ^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/(6*c^(1/3)*(b*c - a*d)))/(a*c)
3.2.17.3.1 Defintions of rubi rules used
Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_) )^(q_), x_Symbol] :> Simp[(e*x)^(m + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*e*(m + 1))), x] - Simp[1/(a*c*e^n*(m + 1)) Int[(e*x)^(m + n)*( a + b*x^n)^p*(c + d*x^n)^q*Simp[(b*c + a*d)*(m + n + 1) + n*(b*c*p + a*d*q) + b*d*(m + n*(p + q + 2) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LtQ[m, -1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Time = 4.21 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.76
method | result | size |
default | \(-\frac {\left (-\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right ) d^{2}}{\left (a d -b c \right ) c}+\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b^{2}}{\left (a d -b c \right ) a}-\frac {1}{a c x}\) | \(228\) |
risch | \(-\frac {1}{a c x}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (d^{3} a^{7}-3 a^{6} b c \,d^{2}+3 a^{5} b^{2} c^{2} d -a^{4} b^{3} c^{3}\right ) \textit {\_Z}^{3}+b^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a^{10} c^{4} d^{6}+22 a^{9} b \,c^{5} d^{5}-52 a^{8} b^{2} c^{6} d^{4}+68 a^{7} b^{3} c^{7} d^{3}-52 a^{6} b^{4} c^{8} d^{2}+22 a^{5} b^{5} c^{9} d -4 a^{4} b^{6} c^{10}\right ) \textit {\_R}^{6}+\left (3 a^{7} d^{7}-9 d^{6} c b \,a^{6}+10 b^{2} c^{2} d^{5} a^{5}-4 d^{4} c^{3} b^{3} a^{4}-4 b^{4} c^{4} d^{3} a^{3}+10 d^{2} c^{5} b^{5} a^{2}-9 d \,c^{6} b^{6} a +3 c^{7} b^{7}\right ) \textit {\_R}^{3}+3 b^{4} d^{4}\right ) x +\left (-a^{9} c^{3} d^{6}+3 a^{8} b \,c^{4} d^{5}-3 a^{7} b^{2} c^{5} d^{4}+2 a^{6} b^{3} c^{6} d^{3}-3 a^{5} b^{4} c^{7} d^{2}+3 a^{4} b^{5} c^{8} d -a^{3} b^{6} c^{9}\right ) \textit {\_R}^{5}+\left (-d^{4} c^{2} b^{3} a^{3}-d^{3} c^{3} b^{4} a^{2}\right ) \textit {\_R}^{2}\right )\right )}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (d^{3} c^{4} a^{3}-3 d^{2} c^{5} a^{2} b +3 d \,c^{6} a \,b^{2}-b^{3} c^{7}\right ) \textit {\_Z}^{3}-d^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a^{10} c^{4} d^{6}+22 a^{9} b \,c^{5} d^{5}-52 a^{8} b^{2} c^{6} d^{4}+68 a^{7} b^{3} c^{7} d^{3}-52 a^{6} b^{4} c^{8} d^{2}+22 a^{5} b^{5} c^{9} d -4 a^{4} b^{6} c^{10}\right ) \textit {\_R}^{6}+\left (3 a^{7} d^{7}-9 d^{6} c b \,a^{6}+10 b^{2} c^{2} d^{5} a^{5}-4 d^{4} c^{3} b^{3} a^{4}-4 b^{4} c^{4} d^{3} a^{3}+10 d^{2} c^{5} b^{5} a^{2}-9 d \,c^{6} b^{6} a +3 c^{7} b^{7}\right ) \textit {\_R}^{3}+3 b^{4} d^{4}\right ) x +\left (-a^{9} c^{3} d^{6}+3 a^{8} b \,c^{4} d^{5}-3 a^{7} b^{2} c^{5} d^{4}+2 a^{6} b^{3} c^{6} d^{3}-3 a^{5} b^{4} c^{7} d^{2}+3 a^{4} b^{5} c^{8} d -a^{3} b^{6} c^{9}\right ) \textit {\_R}^{5}+\left (-d^{4} c^{2} b^{3} a^{3}-d^{3} c^{3} b^{4} a^{2}\right ) \textit {\_R}^{2}\right )\right )}{3}\) | \(787\) |
-(-1/3/d/(c/d)^(1/3)*ln(x+(c/d)^(1/3))+1/6/d/(c/d)^(1/3)*ln(x^2-(c/d)^(1/3 )*x+(c/d)^(2/3))+1/3*3^(1/2)/d/(c/d)^(1/3)*arctan(1/3*3^(1/2)*(2/(c/d)^(1/ 3)*x-1)))*d^2/(a*d-b*c)/c+(-1/3/b/(a/b)^(1/3)*ln(x+(a/b)^(1/3))+1/6/b/(a/b )^(1/3)*ln(x^2-(a/b)^(1/3)*x+(a/b)^(2/3))+1/3*3^(1/2)/b/(a/b)^(1/3)*arctan (1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))*b^2/(a*d-b*c)/a-1/a/c/x
Time = 0.30 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.80 \[ \int \frac {1}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {2 \, \sqrt {3} b c x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 2 \, \sqrt {3} a d x \left (\frac {d}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {d}{c}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - b c x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) - a d x \left (\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x^{2} - c x \left (\frac {d}{c}\right )^{\frac {2}{3}} + c \left (\frac {d}{c}\right )^{\frac {1}{3}}\right ) + 2 \, b c x \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 2 \, a d x \left (\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x + c \left (\frac {d}{c}\right )^{\frac {2}{3}}\right ) + 6 \, b c - 6 \, a d}{6 \, {\left (a b c^{2} - a^{2} c d\right )} x} \]
-1/6*(2*sqrt(3)*b*c*x*(-b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/3) + 1/3 *sqrt(3)) - 2*sqrt(3)*a*d*x*(d/c)^(1/3)*arctan(2/3*sqrt(3)*x*(d/c)^(1/3) - 1/3*sqrt(3)) - b*c*x*(-b/a)^(1/3)*log(b*x^2 - a*x*(-b/a)^(2/3) - a*(-b/a) ^(1/3)) - a*d*x*(d/c)^(1/3)*log(d*x^2 - c*x*(d/c)^(2/3) + c*(d/c)^(1/3)) + 2*b*c*x*(-b/a)^(1/3)*log(b*x + a*(-b/a)^(2/3)) + 2*a*d*x*(d/c)^(1/3)*log( d*x + c*(d/c)^(2/3)) + 6*b*c - 6*a*d)/((a*b*c^2 - a^2*c*d)*x)
Timed out. \[ \int \frac {1}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\text {Timed out} \]
Time = 0.28 (sec) , antiderivative size = 300, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (a b c - a^{2} d\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} d \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c^{2} - a c d\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} - \frac {b \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} + \frac {d \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} + \frac {b \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (a b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} - \frac {d \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} - \frac {1}{a c x} \]
-1/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/((a*b*c - a^2*d)*(a/b)^(1/3)) + 1/3*sqrt(3)*d*arctan(1/3*sqrt(3)*(2*x - (c/d)^(1/ 3))/(c/d)^(1/3))/((b*c^2 - a*c*d)*(c/d)^(1/3)) - 1/6*b*log(x^2 - x*(a/b)^( 1/3) + (a/b)^(2/3))/(a*b*c*(a/b)^(1/3) - a^2*d*(a/b)^(1/3)) + 1/6*d*log(x^ 2 - x*(c/d)^(1/3) + (c/d)^(2/3))/(b*c^2*(c/d)^(1/3) - a*c*d*(c/d)^(1/3)) + 1/3*b*log(x + (a/b)^(1/3))/(a*b*c*(a/b)^(1/3) - a^2*d*(a/b)^(1/3)) - 1/3* d*log(x + (c/d)^(1/3))/(b*c^2*(c/d)^(1/3) - a*c*d*(c/d)^(1/3)) - 1/(a*c*x)
Time = 0.29 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.02 \[ \int \frac {1}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {b^{2} \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a^{2} b c - a^{3} d\right )}} - \frac {d^{2} \left (-\frac {c}{d}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{3} - a c^{2} d\right )}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a^{2} b c - \sqrt {3} a^{3} d} - \frac {\left (-c d^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c^{3} - \sqrt {3} a c^{2} d} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} b c - a^{3} d\right )}} + \frac {\left (-c d^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {1}{a c x} \]
1/3*b^2*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b*c - a^3*d) - 1/3*d^ 2*(-c/d)^(2/3)*log(abs(x - (-c/d)^(1/3)))/(b*c^3 - a*c^2*d) + (-a*b^2)^(2/ 3)*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*a^2*b*c - sqrt(3)*a^3*d) - (-c*d^2)^(2/3)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/ (-c/d)^(1/3))/(sqrt(3)*b*c^3 - sqrt(3)*a*c^2*d) - 1/6*(-a*b^2)^(2/3)*log(x ^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b*c - a^3*d) + 1/6*(-c*d^2)^(2/3) *log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(b*c^3 - a*c^2*d) - 1/(a*c*x)
Time = 8.43 (sec) , antiderivative size = 716, normalized size of antiderivative = 2.39 \[ \int \frac {1}{x^2 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\ln \left (b-a^2\,d\,x\,{\left (-\frac {b^4}{a^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}+a\,b\,c\,x\,{\left (-\frac {b^4}{a^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\right )\,{\left (-\frac {b^4}{27\,a^7\,d^3-81\,a^6\,b\,c\,d^2+81\,a^5\,b^2\,c^2\,d-27\,a^4\,b^3\,c^3}\right )}^{1/3}+\ln \left (d-b\,c^2\,x\,{\left (\frac {d^4}{c^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}+a\,c\,d\,x\,{\left (\frac {d^4}{c^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}\right )\,{\left (-\frac {d^4}{-27\,a^3\,c^4\,d^3+81\,a^2\,b\,c^5\,d^2-81\,a\,b^2\,c^6\,d+27\,b^3\,c^7}\right )}^{1/3}-\frac {1}{a\,c\,x}-\frac {\ln \left (b+2\,a^2\,d\,x\,{\left (-\frac {b^4}{a^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-2\,a\,b\,c\,x\,{\left (-\frac {b^4}{a^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-\sqrt {3}\,b\,1{}\mathrm {i}\right )\,{\left (-\frac {b^4}{27\,a^7\,d^3-81\,a^6\,b\,c\,d^2+81\,a^5\,b^2\,c^2\,d-27\,a^4\,b^3\,c^3}\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+\frac {\ln \left (b+2\,a^2\,d\,x\,{\left (-\frac {b^4}{a^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-2\,a\,b\,c\,x\,{\left (-\frac {b^4}{a^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}+\sqrt {3}\,b\,1{}\mathrm {i}\right )\,{\left (-\frac {b^4}{27\,a^7\,d^3-81\,a^6\,b\,c\,d^2+81\,a^5\,b^2\,c^2\,d-27\,a^4\,b^3\,c^3}\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}-\frac {\ln \left (d+2\,b\,c^2\,x\,{\left (\frac {d^4}{c^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-2\,a\,c\,d\,x\,{\left (\frac {d^4}{c^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-\sqrt {3}\,d\,1{}\mathrm {i}\right )\,{\left (-\frac {d^4}{-27\,a^3\,c^4\,d^3+81\,a^2\,b\,c^5\,d^2-81\,a\,b^2\,c^6\,d+27\,b^3\,c^7}\right )}^{1/3}\,\left (1+\sqrt {3}\,1{}\mathrm {i}\right )}{2}+\frac {\ln \left (d+2\,b\,c^2\,x\,{\left (\frac {d^4}{c^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}-2\,a\,c\,d\,x\,{\left (\frac {d^4}{c^4\,{\left (a\,d-b\,c\right )}^3}\right )}^{1/3}+\sqrt {3}\,d\,1{}\mathrm {i}\right )\,{\left (-\frac {d^4}{-27\,a^3\,c^4\,d^3+81\,a^2\,b\,c^5\,d^2-81\,a\,b^2\,c^6\,d+27\,b^3\,c^7}\right )}^{1/3}\,\left (-1+\sqrt {3}\,1{}\mathrm {i}\right )}{2} \]
log(b - a^2*d*x*(-b^4/(a^4*(a*d - b*c)^3))^(1/3) + a*b*c*x*(-b^4/(a^4*(a*d - b*c)^3))^(1/3))*(-b^4/(27*a^7*d^3 - 27*a^4*b^3*c^3 + 81*a^5*b^2*c^2*d - 81*a^6*b*c*d^2))^(1/3) + log(d - b*c^2*x*(d^4/(c^4*(a*d - b*c)^3))^(1/3) + a*c*d*x*(d^4/(c^4*(a*d - b*c)^3))^(1/3))*(-d^4/(27*b^3*c^7 - 27*a^3*c^4* d^3 + 81*a^2*b*c^5*d^2 - 81*a*b^2*c^6*d))^(1/3) - 1/(a*c*x) - (log(b - 3^( 1/2)*b*1i + 2*a^2*d*x*(-b^4/(a^4*(a*d - b*c)^3))^(1/3) - 2*a*b*c*x*(-b^4/( a^4*(a*d - b*c)^3))^(1/3))*(-b^4/(27*a^7*d^3 - 27*a^4*b^3*c^3 + 81*a^5*b^2 *c^2*d - 81*a^6*b*c*d^2))^(1/3)*(3^(1/2)*1i + 1))/2 + (log(b + 3^(1/2)*b*1 i + 2*a^2*d*x*(-b^4/(a^4*(a*d - b*c)^3))^(1/3) - 2*a*b*c*x*(-b^4/(a^4*(a*d - b*c)^3))^(1/3))*(-b^4/(27*a^7*d^3 - 27*a^4*b^3*c^3 + 81*a^5*b^2*c^2*d - 81*a^6*b*c*d^2))^(1/3)*(3^(1/2)*1i - 1))/2 - (log(d - 3^(1/2)*d*1i + 2*b* c^2*x*(d^4/(c^4*(a*d - b*c)^3))^(1/3) - 2*a*c*d*x*(d^4/(c^4*(a*d - b*c)^3) )^(1/3))*(-d^4/(27*b^3*c^7 - 27*a^3*c^4*d^3 + 81*a^2*b*c^5*d^2 - 81*a*b^2* c^6*d))^(1/3)*(3^(1/2)*1i + 1))/2 + (log(d + 3^(1/2)*d*1i + 2*b*c^2*x*(d^4 /(c^4*(a*d - b*c)^3))^(1/3) - 2*a*c*d*x*(d^4/(c^4*(a*d - b*c)^3))^(1/3))*( -d^4/(27*b^3*c^7 - 27*a^3*c^4*d^3 + 81*a^2*b*c^5*d^2 - 81*a*b^2*c^6*d))^(1 /3)*(3^(1/2)*1i - 1))/2