Integrand size = 22, antiderivative size = 288 \[ \int \frac {x^4}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {a^{2/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{2/3} (b c-a d)}-\frac {c^{2/3} \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} d^{2/3} (b c-a d)}+\frac {a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} (b c-a d)}-\frac {c^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{2/3} (b c-a d)}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} (b c-a d)}+\frac {c^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{2/3} (b c-a d)} \]
1/3*a^(2/3)*ln(a^(1/3)+b^(1/3)*x)/b^(2/3)/(-a*d+b*c)-1/3*c^(2/3)*ln(c^(1/3 )+d^(1/3)*x)/d^(2/3)/(-a*d+b*c)-1/6*a^(2/3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b ^(2/3)*x^2)/b^(2/3)/(-a*d+b*c)+1/6*c^(2/3)*ln(c^(2/3)-c^(1/3)*d^(1/3)*x+d^ (2/3)*x^2)/d^(2/3)/(-a*d+b*c)+1/3*a^(2/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x) /a^(1/3)*3^(1/2))/b^(2/3)/(-a*d+b*c)*3^(1/2)-1/3*c^(2/3)*arctan(1/3*(c^(1/ 3)-2*d^(1/3)*x)/c^(1/3)*3^(1/2))/d^(2/3)/(-a*d+b*c)*3^(1/2)
Time = 0.08 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.78 \[ \int \frac {x^4}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {\frac {2 \sqrt {3} a^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{2/3}}-\frac {2 \sqrt {3} c^{2/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{d^{2/3}}+\frac {2 a^{2/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{2/3}}-\frac {2 c^{2/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{d^{2/3}}-\frac {a^{2/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{2/3}}+\frac {c^{2/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{d^{2/3}}}{6 b c-6 a d} \]
((2*Sqrt[3]*a^(2/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(2/3) - (2*Sqrt[3]*c^(2/3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/d^(2/3) + (2*a^(2/3)*Log[a^(1/3) + b^(1/3)*x])/b^(2/3) - (2*c^(2/3)*Log[c^(1/3) + d ^(1/3)*x])/d^(2/3) - (a^(2/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^ 2])/b^(2/3) + (c^(2/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/d^( 2/3))/(6*b*c - 6*a*d)
Time = 0.44 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.90, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {981, 821, 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.
| \(\displaystyle \int \frac {x^4}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 981 |
| \(\displaystyle \frac {c \int \frac {x}{d x^3+c}dx}{b c-a d}-\frac {a \int \frac {x}{b x^3+a}dx}{b c-a d}\) |
| \(\Big \downarrow \) 821 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {\sqrt [3]{d} x+\sqrt [3]{c}}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 \sqrt [3]{c} \sqrt [3]{d}}-\frac {\int \frac {1}{\sqrt [3]{d} x+\sqrt [3]{c}}dx}{3 \sqrt [3]{c} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {\sqrt [3]{d} x+\sqrt [3]{c}}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 \sqrt [3]{c} \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{b c-a d}-\frac {a \left (\frac {\int \frac {\sqrt [3]{b} x+\sqrt [3]{a}}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {c \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 \sqrt [3]{c} \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{b c-a d}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\int \frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 \sqrt [3]{c} \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{b c-a d}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 \sqrt [3]{c} \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{b c-a d}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {c \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{d}}-\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 \sqrt [3]{c} \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{b c-a d}-\frac {a \left (\frac {\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {c \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 \sqrt [3]{c} \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{b c-a d}-\frac {a \left (\frac {-\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {c \left (\frac {\frac {\log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 \sqrt [3]{c} \sqrt [3]{d}}-\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} d^{2/3}}\right )}{b c-a d}-\frac {a \left (\frac {\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 \sqrt [3]{a} \sqrt [3]{b}}-\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} b^{2/3}}\right )}{b c-a d}\) |
-((a*(-1/3*Log[a^(1/3) + b^(1/3)*x]/(a^(1/3)*b^(2/3)) + (-((Sqrt[3]*ArcTan [(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[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^(1/3)*b^(1/3))))/(b*c - a*d)) + ( c*(-1/3*Log[c^(1/3) + d^(1/3)*x]/(c^(1/3)*d^(2/3)) + (-((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^(1/3)*d^(1/3))))/(b*c - a*d)
3.2.11.3.1 Defintions of rubi rules used
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[(x_)/((a_) + (b_.)*(x_)^3), x_Symbol] :> Simp[-(3*Rt[a, 3]*Rt[b, 3])^(- 1) Int[1/(Rt[a, 3] + Rt[b, 3]*x), x], x] + Simp[1/(3*Rt[a, 3]*Rt[b, 3]) Int[(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]
Int[((e_.)*(x_))^(m_.)/(((a_) + (b_.)*(x_)^(n_))*((c_) + (d_.)*(x_)^(n_))), x_Symbol] :> Simp[(-a)*(e^n/(b*c - a*d)) Int[(e*x)^(m - n)/(a + b*x^n), x], x] + Simp[c*(e^n/(b*c - a*d)) Int[(e*x)^(m - n)/(c + d*x^n), x], x] / ; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && LeQ[n, m, 2*n - 1]
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]
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]
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]
Time = 4.29 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(-\frac {\left (-\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right ) c}{a d -b c}+\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a}{a d -b c}\) | \(207\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} b^{2} d^{3}-3 a^{2} b^{3} c \,d^{2}+3 a \,b^{4} c^{2} d -b^{5} c^{3}\right ) \textit {\_Z}^{3}+a^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-2 a^{3} b^{2} c \,d^{4}+4 a^{2} b^{3} c^{2} d^{3}-2 a \,b^{4} c^{3} d^{2}\right ) \textit {\_R}^{3}-a^{2} c d -b \,c^{2} a \right ) x +\left (-a^{5} b^{2} d^{7}+3 a^{4} b^{3} c \,d^{6}-2 a^{3} b^{4} c^{2} d^{5}-2 a^{2} b^{5} c^{3} d^{4}+3 a \,b^{6} c^{4} d^{3}-b^{7} c^{5} d^{2}\right ) \textit {\_R}^{5}+\left (-a^{4} d^{4}+a^{3} b c \,d^{3}+a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \textit {\_R}^{2}\right )\right )}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} d^{5}-3 a^{2} b c \,d^{4}+3 a \,b^{2} c^{2} d^{3}-b^{3} c^{3} d^{2}\right ) \textit {\_Z}^{3}-c^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-2 a^{3} b^{2} c \,d^{4}+4 a^{2} b^{3} c^{2} d^{3}-2 a \,b^{4} c^{3} d^{2}\right ) \textit {\_R}^{3}-a^{2} c d -b \,c^{2} a \right ) x +\left (-a^{5} b^{2} d^{7}+3 a^{4} b^{3} c \,d^{6}-2 a^{3} b^{4} c^{2} d^{5}-2 a^{2} b^{5} c^{3} d^{4}+3 a \,b^{6} c^{4} d^{3}-b^{7} c^{5} d^{2}\right ) \textit {\_R}^{5}+\left (-a^{4} d^{4}+a^{3} b c \,d^{3}+a \,b^{3} c^{3} d -b^{4} c^{4}\right ) \textit {\_R}^{2}\right )\right )}{3}\) | \(478\) |
-(-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)))*c/(a*d-b*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)))*a/(a*d-b*c)
Time = 0.27 (sec) , antiderivative size = 244, normalized size of antiderivative = 0.85 \[ \int \frac {x^4}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {2 \, \sqrt {3} \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} + \sqrt {3} a}{3 \, a}\right ) - 2 \, \sqrt {3} \left (\frac {c^{2}}{d^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} d x \left (\frac {c^{2}}{d^{2}}\right )^{\frac {1}{3}} - \sqrt {3} c}{3 \, c}\right ) - \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} - a \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) - \left (\frac {c^{2}}{d^{2}}\right )^{\frac {1}{3}} \log \left (c x^{2} - d x \left (\frac {c^{2}}{d^{2}}\right )^{\frac {2}{3}} + c \left (\frac {c^{2}}{d^{2}}\right )^{\frac {1}{3}}\right ) + 2 \, \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x + b \left (-\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, \left (\frac {c^{2}}{d^{2}}\right )^{\frac {1}{3}} \log \left (c x + d \left (\frac {c^{2}}{d^{2}}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c - a d\right )}} \]
-1/6*(2*sqrt(3)*(-a^2/b^2)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(-a^2/b^2)^(1/3 ) + sqrt(3)*a)/a) - 2*sqrt(3)*(c^2/d^2)^(1/3)*arctan(1/3*(2*sqrt(3)*d*x*(c ^2/d^2)^(1/3) - sqrt(3)*c)/c) - (-a^2/b^2)^(1/3)*log(a*x^2 - b*x*(-a^2/b^2 )^(2/3) - a*(-a^2/b^2)^(1/3)) - (c^2/d^2)^(1/3)*log(c*x^2 - d*x*(c^2/d^2)^ (2/3) + c*(c^2/d^2)^(1/3)) + 2*(-a^2/b^2)^(1/3)*log(a*x + b*(-a^2/b^2)^(2/ 3)) + 2*(c^2/d^2)^(1/3)*log(c*x + d*(c^2/d^2)^(2/3)))/(b*c - a*d)
Timed out. \[ \int \frac {x^4}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.00 \[ \int \frac {x^4}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b^{2} c - a b d\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} c \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c d - a d^{2}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} - \frac {a \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} + \frac {c \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c d \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} + \frac {a \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} - \frac {c \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c d \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} \]
-1/3*sqrt(3)*a*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/((b^2*c - a*b*d)*(a/b)^(1/3)) + 1/3*sqrt(3)*c*arctan(1/3*sqrt(3)*(2*x - (c/d)^(1/ 3))/(c/d)^(1/3))/((b*c*d - a*d^2)*(c/d)^(1/3)) - 1/6*a*log(x^2 - x*(a/b)^( 1/3) + (a/b)^(2/3))/(b^2*c*(a/b)^(1/3) - a*b*d*(a/b)^(1/3)) + 1/6*c*log(x^ 2 - x*(c/d)^(1/3) + (c/d)^(2/3))/(b*c*d*(c/d)^(1/3) - a*d^2*(c/d)^(1/3)) + 1/3*a*log(x + (a/b)^(1/3))/(b^2*c*(a/b)^(1/3) - a*b*d*(a/b)^(1/3)) - 1/3* c*log(x + (c/d)^(1/3))/(b*c*d*(c/d)^(1/3) - a*d^2*(c/d)^(1/3))
Time = 0.30 (sec) , antiderivative size = 286, normalized size of antiderivative = 0.99 \[ \int \frac {x^4}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {a \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a b c - a^{2} d\right )}} - \frac {c \left (-\frac {c}{d}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{2} - a c d\right )}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{3} c - \sqrt {3} a b^{2} d} - \frac {\left (-c d^{2}\right )^{\frac {2}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c d^{2} - \sqrt {3} a d^{3}} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{3} c - a b^{2} d\right )}} + \frac {\left (-c d^{2}\right )^{\frac {2}{3}} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c d^{2} - a d^{3}\right )}} \]
1/3*a*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a*b*c - a^2*d) - 1/3*c*(-c/ d)^(2/3)*log(abs(x - (-c/d)^(1/3)))/(b*c^2 - a*c*d) + (-a*b^2)^(2/3)*arcta n(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*b^3*c - sqrt(3)* a*b^2*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*d^2 - sqrt(3)*a*d^3) - 1/6*(-a*b^2)^(2/3)*log(x^2 + x*( -a/b)^(1/3) + (-a/b)^(2/3))/(b^3*c - a*b^2*d) + 1/6*(-c*d^2)^(2/3)*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(b*c*d^2 - a*d^3)
Time = 14.05 (sec) , antiderivative size = 1364, normalized size of antiderivative = 4.74 \[ \int \frac {x^4}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\text {Too large to display} \]
log(a*x + b^3*c^2*(-a^2/(b^2*(a*d - b*c)^3))^(2/3) + a^2*b*d^2*(-a^2/(b^2* (a*d - b*c)^3))^(2/3) - 2*a*b^2*c*d*(-a^2/(b^2*(a*d - b*c)^3))^(2/3))*(a^2 /(27*b^5*c^3 - 27*a^3*b^2*d^3 + 81*a^2*b^3*c*d^2 - 81*a*b^4*c^2*d))^(1/3) + log(c*x + a^2*d^3*(c^2/(d^2*(a*d - b*c)^3))^(2/3) + b^2*c^2*d*(c^2/(d^2* (a*d - b*c)^3))^(2/3) - 2*a*b*c*d^2*(c^2/(d^2*(a*d - b*c)^3))^(2/3))*(c^2/ (27*a^3*d^5 - 27*b^3*c^3*d^2 + 81*a*b^2*c^2*d^3 - 81*a^2*b*c*d^4))^(1/3) + (log(((3^(1/2)*1i - 1)^2*(-a^2/(b^2*(a*d - b*c)^3))^(2/3)*(((3^(1/2)*1i - 1)*(54*a^2*b^3*c^2*d^3*x*(a*d - b*c)^2 + (27*a*b^3*c*d^3*(3^(1/2)*1i - 1) ^2*(a*d + b*c)*(a*d - b*c)^4*(-a^2/(b^2*(a*d - b*c)^3))^(2/3))/4)*(-a^2/(b ^2*(a*d - b*c)^3))^(1/3))/6 - 9*a^2*b^4*c^4*d^2 - 9*a^4*b^2*c^2*d^4 + 9*a* b^5*c^5*d + 9*a^5*b*c*d^5))/36 + a^2*b*c^2*d*x*(a*d + b*c))*(a^2/(27*b^5*c ^3 - 27*a^3*b^2*d^3 + 81*a^2*b^3*c*d^2 - 81*a*b^4*c^2*d))^(1/3)*(3^(1/2)*1 i - 1))/2 - (log(((3^(1/2)*1i + 1)^2*(-a^2/(b^2*(a*d - b*c)^3))^(2/3)*(((3 ^(1/2)*1i + 1)*(54*a^2*b^3*c^2*d^3*x*(a*d - b*c)^2 + (27*a*b^3*c*d^3*(3^(1 /2)*1i + 1)^2*(a*d + b*c)*(a*d - b*c)^4*(-a^2/(b^2*(a*d - b*c)^3))^(2/3))/ 4)*(-a^2/(b^2*(a*d - b*c)^3))^(1/3))/6 + 9*a^2*b^4*c^4*d^2 + 9*a^4*b^2*c^2 *d^4 - 9*a*b^5*c^5*d - 9*a^5*b*c*d^5))/36 - a^2*b*c^2*d*x*(a*d + b*c))*(a^ 2/(27*b^5*c^3 - 27*a^3*b^2*d^3 + 81*a^2*b^3*c*d^2 - 81*a*b^4*c^2*d))^(1/3) *(3^(1/2)*1i + 1))/2 + (log(((3^(1/2)*1i - 1)^2*(c^2/(d^2*(a*d - b*c)^3))^ (2/3)*(((3^(1/2)*1i - 1)*(54*a^2*b^3*c^2*d^3*x*(a*d - b*c)^2 + (27*a*b^...