Integrand size = 22, antiderivative size = 288 \[ \int \frac {x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {\sqrt [3]{a} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{b} (b c-a d)}-\frac {\sqrt [3]{c} \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [3]{d} (b c-a d)}-\frac {\sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{b} (b c-a d)}+\frac {\sqrt [3]{c} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{d} (b c-a d)}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{b} (b c-a d)}-\frac {\sqrt [3]{c} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{d} (b c-a d)} \] Output:
1/3*a^(1/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/b^(1 /3)/(-a*d+b*c)-1/3*c^(1/3)*arctan(1/3*(c^(1/3)-2*d^(1/3)*x)*3^(1/2)/c^(1/3 ))*3^(1/2)/d^(1/3)/(-a*d+b*c)-1/3*a^(1/3)*ln(a^(1/3)+b^(1/3)*x)/b^(1/3)/(- a*d+b*c)+1/3*c^(1/3)*ln(c^(1/3)+d^(1/3)*x)/d^(1/3)/(-a*d+b*c)+1/6*a^(1/3)* ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(1/3)/(-a*d+b*c)-1/6*c^(1/3)*l n(c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/d^(1/3)/(-a*d+b*c)
Time = 0.09 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.78 \[ \int \frac {x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {\frac {2 \sqrt {3} \sqrt [3]{a} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}-\frac {2 \sqrt {3} \sqrt [3]{c} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {2 \sqrt [3]{a} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{\sqrt [3]{b}}+\frac {2 \sqrt [3]{c} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{\sqrt [3]{d}}+\frac {\sqrt [3]{a} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{\sqrt [3]{b}}-\frac {\sqrt [3]{c} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{\sqrt [3]{d}}}{6 b c-6 a d} \] Input:
Integrate[x^3/((a + b*x^3)*(c + d*x^3)),x]
Output:
((2*Sqrt[3]*a^(1/3)*ArcTan[(1 - (2*b^(1/3)*x)/a^(1/3))/Sqrt[3]])/b^(1/3) - (2*Sqrt[3]*c^(1/3)*ArcTan[(1 - (2*d^(1/3)*x)/c^(1/3))/Sqrt[3]])/d^(1/3) - (2*a^(1/3)*Log[a^(1/3) + b^(1/3)*x])/b^(1/3) + (2*c^(1/3)*Log[c^(1/3) + d ^(1/3)*x])/d^(1/3) + (a^(1/3)*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^ 2])/b^(1/3) - (c^(1/3)*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/d^( 1/3))/(6*b*c - 6*a*d)
Time = 0.73 (sec) , antiderivative size = 248, normalized size of antiderivative = 0.86, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.409, Rules used = {981, 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.
| \(\displaystyle \int \frac {x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 981 |
| \(\displaystyle \frac {c \int \frac {1}{d x^3+c}dx}{b c-a d}-\frac {a \int \frac {1}{b x^3+a}dx}{b c-a d}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{d} x+\sqrt [3]{c}}dx}{3 c^{2/3}}\right )}{b c-a d}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {c \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle \frac {c \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {c \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {\int \frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {c \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {c \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {c \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {c \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}-\frac {a \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}\) |
Input:
Int[x^3/((a + b*x^3)*(c + d*x^3)),x]
Output:
-((a*(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))/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^(2/3))))/(b*c - a*d)) + (c*(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)
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[((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]
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 = 1.12 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.72
| method | result | size |
| default | \(-\frac {\left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right ) c}{a d -b c}+\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) a}{a d -b c}\) | \(207\) |
| risch | \(\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (d^{4} a^{3}-3 a^{2} c \,d^{3} b +3 a \,c^{2} d^{2} b^{2}-d \,c^{3} b^{3}\right ) \textit {\_Z}^{3}+c \right )}{\sum }\textit {\_R} \ln \left (\left (\left (a^{4} b \,d^{5}-4 a^{3} b^{2} c \,d^{4}+6 a^{2} b^{3} c^{2} d^{3}-4 a \,b^{4} c^{3} d^{2}+b^{5} c^{4} d \right ) \textit {\_R}^{3}-a^{2} d^{2}-b^{2} c^{2}\right ) x +\left (-a^{5} b \,d^{6}+3 a^{4} b^{2} c \,d^{5}-2 a^{3} b^{3} c^{2} d^{4}-2 a^{2} b^{4} c^{3} d^{3}+3 a \,b^{5} c^{4} d^{2}-b^{6} c^{5} d \right ) \textit {\_R}^{4}+\left (a^{3} d^{3}-a^{2} b c \,d^{2}-a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \textit {\_R} \right )\right )}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (b \,d^{3} a^{3}-3 a^{2} b^{2} c \,d^{2}+3 a \,b^{3} c^{2} d -b^{4} c^{3}\right ) \textit {\_Z}^{3}-a \right )}{\sum }\textit {\_R} \ln \left (\left (\left (a^{4} b \,d^{5}-4 a^{3} b^{2} c \,d^{4}+6 a^{2} b^{3} c^{2} d^{3}-4 a \,b^{4} c^{3} d^{2}+b^{5} c^{4} d \right ) \textit {\_R}^{3}-a^{2} d^{2}-b^{2} c^{2}\right ) x +\left (-a^{5} b \,d^{6}+3 a^{4} b^{2} c \,d^{5}-2 a^{3} b^{3} c^{2} d^{4}-2 a^{2} b^{4} c^{3} d^{3}+3 a \,b^{5} c^{4} d^{2}-b^{6} c^{5} d \right ) \textit {\_R}^{4}+\left (a^{3} d^{3}-a^{2} b c \,d^{2}-a \,b^{2} c^{2} d +b^{3} c^{3}\right ) \textit {\_R} \right )\right )}{3}\) | \(494\) |
Input:
int(x^3/(b*x^3+a)/(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
-(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)))*c/(a*d-b*c)+(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)))*a/(a*d-b*c)
Time = 0.13 (sec) , antiderivative size = 199, normalized size of antiderivative = 0.69 \[ \int \frac {x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {2 \, \sqrt {3} \left (\frac {a}{b}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a}{b}\right )^{\frac {2}{3}} - \sqrt {3} a}{3 \, a}\right ) + 2 \, \sqrt {3} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} d x \left (-\frac {c}{d}\right )^{\frac {2}{3}} - \sqrt {3} c}{3 \, c}\right ) - \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right ) - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right ) + 2 \, \left (\frac {a}{b}\right )^{\frac {1}{3}} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right ) + 2 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left (x - \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}{6 \, {\left (b c - a d\right )}} \] Input:
integrate(x^3/(b*x^3+a)/(d*x^3+c),x, algorithm="fricas")
Output:
-1/6*(2*sqrt(3)*(a/b)^(1/3)*arctan(1/3*(2*sqrt(3)*b*x*(a/b)^(2/3) - sqrt(3 )*a)/a) + 2*sqrt(3)*(-c/d)^(1/3)*arctan(1/3*(2*sqrt(3)*d*x*(-c/d)^(2/3) - sqrt(3)*c)/c) - (a/b)^(1/3)*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3)) - (-c/d )^(1/3)*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3)) + 2*(a/b)^(1/3)*log(x + ( a/b)^(1/3)) + 2*(-c/d)^(1/3)*log(x - (-c/d)^(1/3)))/(b*c - a*d)
Time = 9.18 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.19 \[ \int \frac {x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\operatorname {RootSum} {\left (t^{3} \cdot \left (27 a^{3} d^{4} - 81 a^{2} b c d^{3} + 81 a b^{2} c^{2} d^{2} - 27 b^{3} c^{3} d\right ) + c, \left ( t \mapsto t \log {\left (x + \frac {162 t^{4} a^{4} b d^{5} - 648 t^{4} a^{3} b^{2} c d^{4} + 972 t^{4} a^{2} b^{3} c^{2} d^{3} - 648 t^{4} a b^{4} c^{3} d^{2} + 162 t^{4} b^{5} c^{4} d - 3 t a^{2} d^{2} + 6 t a b c d - 3 t b^{2} c^{2}}{a d + b c} \right )} \right )\right )} + \operatorname {RootSum} {\left (t^{3} \cdot \left (27 a^{3} b d^{3} - 81 a^{2} b^{2} c d^{2} + 81 a b^{3} c^{2} d - 27 b^{4} c^{3}\right ) - a, \left ( t \mapsto t \log {\left (x + \frac {162 t^{4} a^{4} b d^{5} - 648 t^{4} a^{3} b^{2} c d^{4} + 972 t^{4} a^{2} b^{3} c^{2} d^{3} - 648 t^{4} a b^{4} c^{3} d^{2} + 162 t^{4} b^{5} c^{4} d - 3 t a^{2} d^{2} + 6 t a b c d - 3 t b^{2} c^{2}}{a d + b c} \right )} \right )\right )} \] Input:
integrate(x**3/(b*x**3+a)/(d*x**3+c),x)
Output:
RootSum(_t**3*(27*a**3*d**4 - 81*a**2*b*c*d**3 + 81*a*b**2*c**2*d**2 - 27* b**3*c**3*d) + c, Lambda(_t, _t*log(x + (162*_t**4*a**4*b*d**5 - 648*_t**4 *a**3*b**2*c*d**4 + 972*_t**4*a**2*b**3*c**2*d**3 - 648*_t**4*a*b**4*c**3* d**2 + 162*_t**4*b**5*c**4*d - 3*_t*a**2*d**2 + 6*_t*a*b*c*d - 3*_t*b**2*c **2)/(a*d + b*c)))) + RootSum(_t**3*(27*a**3*b*d**3 - 81*a**2*b**2*c*d**2 + 81*a*b**3*c**2*d - 27*b**4*c**3) - a, Lambda(_t, _t*log(x + (162*_t**4*a **4*b*d**5 - 648*_t**4*a**3*b**2*c*d**4 + 972*_t**4*a**2*b**3*c**2*d**3 - 648*_t**4*a*b**4*c**3*d**2 + 162*_t**4*b**5*c**4*d - 3*_t*a**2*d**2 + 6*_t *a*b*c*d - 3*_t*b**2*c**2)/(a*d + b*c))))
Time = 0.12 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.10 \[ \int \frac {x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {\sqrt {3} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} c \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c d \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} + \frac {a \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} - \frac {c \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c d \left (\frac {c}{d}\right )^{\frac {2}{3}} - a d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} - \frac {a \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b^{2} c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a b d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {c \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c d \left (\frac {c}{d}\right )^{\frac {2}{3}} - a d^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} \] Input:
integrate(x^3/(b*x^3+a)/(d*x^3+c),x, algorithm="maxima")
Output:
-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)^(1/3) - a*b*d*(a/b)^(1/3))*(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*(c/d)^(1/3) - a*d^2*(c/d) ^(1/3))*(c/d)^(1/3)) + 1/6*a*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^2*c *(a/b)^(2/3) - a*b*d*(a/b)^(2/3)) - 1/6*c*log(x^2 - x*(c/d)^(1/3) + (c/d)^ (2/3))/(b*c*d*(c/d)^(2/3) - a*d^2*(c/d)^(2/3)) - 1/3*a*log(x + (a/b)^(1/3) )/(b^2*c*(a/b)^(2/3) - a*b*d*(a/b)^(2/3)) + 1/3*c*log(x + (c/d)^(1/3))/(b* c*d*(c/d)^(2/3) - a*d^2*(c/d)^(2/3))
Time = 0.14 (sec) , antiderivative size = 278, normalized size of antiderivative = 0.97 \[ \int \frac {x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {a \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a b c - a^{2} d\right )}} - \frac {c \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{2} - a c d\right )}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{2} c - \sqrt {3} a b d} + \frac {\left (-c d^{2}\right )^{\frac {1}{3}} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c d - \sqrt {3} a d^{2}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{2} c - a b d\right )}} + \frac {\left (-c d^{2}\right )^{\frac {1}{3}} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c d - a d^{2}\right )}} \] Input:
integrate(x^3/(b*x^3+a)/(d*x^3+c),x, algorithm="giac")
Output:
1/3*a*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a*b*c - a^2*d) - 1/3*c*(-c/ d)^(1/3)*log(abs(x - (-c/d)^(1/3)))/(b*c^2 - a*c*d) - (-a*b^2)^(1/3)*arcta n(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*b^2*c - sqrt(3)* a*b*d) + (-c*d^2)^(1/3)*arctan(1/3*sqrt(3)*(2*x + (-c/d)^(1/3))/(-c/d)^(1/ 3))/(sqrt(3)*b*c*d - sqrt(3)*a*d^2) - 1/6*(-a*b^2)^(1/3)*log(x^2 + x*(-a/b )^(1/3) + (-a/b)^(2/3))/(b^2*c - a*b*d) + 1/6*(-c*d^2)^(1/3)*log(x^2 + x*( -c/d)^(1/3) + (-c/d)^(2/3))/(b*c*d - a*d^2)
Time = 6.75 (sec) , antiderivative size = 1265, normalized size of antiderivative = 4.39 \[ \int \frac {x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:
int(x^3/((a + b*x^3)*(c + d*x^3)),x)
Output:
log(x + a*d*(a/(b*(a*d - b*c)^3))^(1/3) - b*c*(a/(b*(a*d - b*c)^3))^(1/3)) *(-a/(27*b^4*c^3 - 27*a^3*b*d^3 + 81*a^2*b^2*c*d^2 - 81*a*b^3*c^2*d))^(1/3 ) + log(x - a*d*(-c/(d*(a*d - b*c)^3))^(1/3) + b*c*(-c/(d*(a*d - b*c)^3))^ (1/3))*(-c/(27*a^3*d^4 - 27*b^3*c^3*d + 81*a*b^2*c^2*d^2 - 81*a^2*b*c*d^3) )^(1/3) + (log(((3^(1/2)*1i - 1)*(a/(b*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i - 1)^2*(81*a*b^3*c*d^3*x*(a*d - b*c)^4 - (81*a*b^3*c*d^3*(3^(1/2)*1i - 1) *(a*d + b*c)*(a*d - b*c)^4*(a/(b*(a*d - b*c)^3))^(1/3))/2)*(a/(b*(a*d - b* c)^3))^(2/3))/36 + 9*a*b^5*c^4*d^2 + 9*a^4*b^2*c*d^5 - 9*a^2*b^4*c^3*d^3 - 9*a^3*b^3*c^2*d^4))/6 - 3*a*b^2*c*d^2*x*(a^2*d^2 + b^2*c^2))*(3^(1/2)*1i - 1)*(-a/(27*b^4*c^3 - 27*a^3*b*d^3 + 81*a^2*b^2*c*d^2 - 81*a*b^3*c^2*d))^ (1/3))/2 - (log(((3^(1/2)*1i + 1)*(a/(b*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1 i + 1)^2*(81*a*b^3*c*d^3*x*(a*d - b*c)^4 + (81*a*b^3*c*d^3*(3^(1/2)*1i + 1 )*(a*d + b*c)*(a*d - b*c)^4*(a/(b*(a*d - b*c)^3))^(1/3))/2)*(a/(b*(a*d - b *c)^3))^(2/3))/36 + 9*a*b^5*c^4*d^2 + 9*a^4*b^2*c*d^5 - 9*a^2*b^4*c^3*d^3 - 9*a^3*b^3*c^2*d^4))/6 + 3*a*b^2*c*d^2*x*(a^2*d^2 + b^2*c^2))*(3^(1/2)*1i + 1)*(-a/(27*b^4*c^3 - 27*a^3*b*d^3 + 81*a^2*b^2*c*d^2 - 81*a*b^3*c^2*d)) ^(1/3))/2 + (log(((3^(1/2)*1i - 1)*(-c/(d*(a*d - b*c)^3))^(1/3)*(((3^(1/2) *1i - 1)^2*(81*a*b^3*c*d^3*x*(a*d - b*c)^4 - (81*a*b^3*c*d^3*(3^(1/2)*1i - 1)*(a*d + b*c)*(a*d - b*c)^4*(-c/(d*(a*d - b*c)^3))^(1/3))/2)*(-c/(d*(a*d - b*c)^3))^(2/3))/36 + 9*a*b^5*c^4*d^2 + 9*a^4*b^2*c*d^5 - 9*a^2*b^4*c...
Time = 0.21 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.59 \[ \int \frac {x^3}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {-2 d^{\frac {1}{3}} a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right )+2 c^{\frac {1}{3}} b^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x}{c^{\frac {1}{3}} \sqrt {3}}\right )-d^{\frac {1}{3}} a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right )+2 d^{\frac {1}{3}} a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right )+c^{\frac {1}{3}} b^{\frac {1}{3}} \mathrm {log}\left (c^{\frac {2}{3}}-d^{\frac {1}{3}} c^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right )-2 c^{\frac {1}{3}} b^{\frac {1}{3}} \mathrm {log}\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right )}{6 d^{\frac {1}{3}} b^{\frac {1}{3}} \left (a d -b c \right )} \] Input:
int(x^3/(b*x^3+a)/(d*x^3+c),x)
Output:
( - 2*d**(1/3)*a**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*s qrt(3))) + 2*c**(1/3)*b**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)*x)/(c** (1/3)*sqrt(3))) - d**(1/3)*a**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3)*x + b **(2/3)*x**2) + 2*d**(1/3)*a**(1/3)*log(a**(1/3) + b**(1/3)*x) + c**(1/3)* b**(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2) - 2*c**(1/3)* b**(1/3)*log(c**(1/3) + d**(1/3)*x))/(6*d**(1/3)*b**(1/3)*(a*d - b*c))