Integrand size = 22, antiderivative size = 301 \[ \int \frac {1}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {1}{2 a c x^2}+\frac {b^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{5/3} (b c-a d)}-\frac {d^{5/3} \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{5/3} (b c-a d)}-\frac {b^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{5/3} (b c-a d)}+\frac {d^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{5/3} (b c-a d)}+\frac {b^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{5/3} (b c-a d)}-\frac {d^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{5/3} (b c-a d)} \] Output:
-1/2/a/c/x^2+1/3*b^(5/3)*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)-1/3*d^(5/3)*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)-1/3*b^(5/3)*ln(a^(1/3)+b^(1/3)* x)/a^(5/3)/(-a*d+b*c)+1/3*d^(5/3)*ln(c^(1/3)+d^(1/3)*x)/c^(5/3)/(-a*d+b*c) +1/6*b^(5/3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(5/3)/(-a*d+b*c)- 1/6*d^(5/3)*ln(c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/c^(5/3)/(-a*d+b*c)
Time = 0.20 (sec) , antiderivative size = 259, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {\frac {3 b}{a}-\frac {3 d}{c}-\frac {2 \sqrt {3} b^{5/3} x^2 \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{5/3}}+\frac {2 \sqrt {3} d^{5/3} x^2 \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{5/3}}+\frac {2 b^{5/3} x^2 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{5/3}}-\frac {2 d^{5/3} x^2 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{5/3}}-\frac {b^{5/3} x^2 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{5/3}}+\frac {d^{5/3} x^2 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{5/3}}}{6 (-b c+a d) x^2} \] Input:
Integrate[1/(x^3*(a + b*x^3)*(c + d*x^3)),x]
Output:
((3*b)/a - (3*d)/c - (2*Sqrt[3]*b^(5/3)*x^2*ArcTan[(1 - (2*b^(1/3)*x)/a^(1 /3))/Sqrt[3]])/a^(5/3) + (2*Sqrt[3]*d^(5/3)*x^2*ArcTan[(1 - (2*d^(1/3)*x)/ c^(1/3))/Sqrt[3]])/c^(5/3) + (2*b^(5/3)*x^2*Log[a^(1/3) + b^(1/3)*x])/a^(5 /3) - (2*d^(5/3)*x^2*Log[c^(1/3) + d^(1/3)*x])/c^(5/3) - (b^(5/3)*x^2*Log[ a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(5/3) + (d^(5/3)*x^2*Log[c^( 2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(5/3))/(6*(-(b*c) + a*d)*x^2)
Time = 0.83 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.92, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {980, 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.
| \(\displaystyle \int \frac {1}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 980 |
| \(\displaystyle \frac {\int -\frac {2 \left (b d x^3+b c+a d\right )}{\left (b x^3+a\right ) \left (d x^3+c\right )}dx}{2 a c}-\frac {1}{2 a c x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {b d x^3+b c+a d}{\left (b x^3+a\right ) \left (d x^3+c\right )}dx}{a c}-\frac {1}{2 a c x^2}\) |
| \(\Big \downarrow \) 1020 |
| \(\displaystyle -\frac {\frac {b^2 c \int \frac {1}{b x^3+a}dx}{b c-a d}-\frac {a d^2 \int \frac {1}{d x^3+c}dx}{b c-a d}}{a c}-\frac {1}{2 a c x^2}\) |
| \(\Big \downarrow \) 750 |
| \(\displaystyle -\frac {\frac {b^2 c \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{b} x+\sqrt [3]{a}}dx}{3 a^{2/3}}\right )}{b c-a d}-\frac {a d^2 \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\int \frac {1}{\sqrt [3]{d} x+\sqrt [3]{c}}dx}{3 c^{2/3}}\right )}{b c-a d}}{a c}-\frac {1}{2 a c x^2}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle -\frac {\frac {b^2 c \left (\frac {\int \frac {2 \sqrt [3]{a}-\sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}-\frac {a d^2 \left (\frac {\int \frac {2 \sqrt [3]{c}-\sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}}{a c}-\frac {1}{2 a c x^2}\) |
| \(\Big \downarrow \) 1142 |
| \(\displaystyle -\frac {\frac {b^2 c \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}-\frac {a d^2 \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\int -\frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}}{a c}-\frac {1}{2 a c x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\frac {b^2 c \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {\int \frac {\sqrt [3]{b} \left (\sqrt [3]{a}-2 \sqrt [3]{b} x\right )}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{2 \sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}-\frac {a d^2 \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {\int \frac {\sqrt [3]{d} \left (\sqrt [3]{c}-2 \sqrt [3]{d} x\right )}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}}{a c}-\frac {1}{2 a c x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\frac {b^2 c \left (\frac {\frac {3}{2} \sqrt [3]{a} \int \frac {1}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}-\frac {a d^2 \left (\frac {\frac {3}{2} \sqrt [3]{c} \int \frac {1}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}}{a c}-\frac {1}{2 a c x^2}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {\frac {b^2 c \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}-\frac {a d^2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx+\frac {3 \int \frac {1}{-\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )^2-3}d\left (1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}}{a c}-\frac {1}{2 a c x^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\frac {b^2 c \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{b^{2/3} x^2-\sqrt [3]{a} \sqrt [3]{b} x+a^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}-\frac {a d^2 \left (\frac {\frac {1}{2} \int \frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{d^{2/3} x^2-\sqrt [3]{c} \sqrt [3]{d} x+c^{2/3}}dx-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}}{a c}-\frac {1}{2 a c x^2}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {\frac {b^2 c \left (\frac {-\frac {\log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{2 \sqrt [3]{b}}-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{\sqrt [3]{b}}}{3 a^{2/3}}+\frac {\log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{2/3} \sqrt [3]{b}}\right )}{b c-a d}-\frac {a d^2 \left (\frac {-\frac {\sqrt {3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{\sqrt [3]{d}}-\frac {\log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{2 \sqrt [3]{d}}}{3 c^{2/3}}+\frac {\log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{2/3} \sqrt [3]{d}}\right )}{b c-a d}}{a c}-\frac {1}{2 a c x^2}\) |
Input:
Int[1/(x^3*(a + b*x^3)*(c + d*x^3)),x]
Output:
-1/2*1/(a*c*x^2) - ((b^2*c*(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) - (a*d^2*(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))/(a* c)
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_))^(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[((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]
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 = 1.32 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(-\frac {1}{2 a c \,x^{2}}-\frac {\left (\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {2}{3}}}\right ) d^{2}}{c \left (a d -b c \right )}+\frac {\left (\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}-\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {2}{3}}}\right ) b^{2}}{a \left (a d -b c \right )}\) | \(228\) |
| risch | \(-\frac {1}{2 a c \,x^{2}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (c^{5} d^{3} a^{3}-3 a^{2} b \,c^{6} d^{2}+3 a \,b^{2} c^{7} d -b^{3} c^{8}\right ) \textit {\_Z}^{3}+d^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a^{11} c^{5} d^{6}+22 a^{10} b \,c^{6} d^{5}-52 a^{9} b^{2} c^{7} d^{4}+68 a^{8} b^{3} c^{8} d^{3}-52 a^{7} b^{4} c^{9} d^{2}+22 a^{6} b^{5} c^{10} d -4 a^{5} b^{6} c^{11}\right ) \textit {\_R}^{6}+\left (-3 a^{8} d^{8}+9 a^{7} b c \,d^{7}-10 a^{6} c^{2} d^{6} b^{2}+4 a^{5} c^{3} d^{5} b^{3}+4 a^{3} c^{5} d^{3} b^{5}-10 a^{2} c^{6} d^{2} b^{6}+9 a \,c^{7} d \,b^{7}-3 c^{8} b^{8}\right ) \textit {\_R}^{3}+3 b^{5} d^{5}\right ) x +\left (-a^{9} c^{2} d^{7}+2 a^{8} b \,c^{3} d^{6}-a^{7} b^{2} c^{4} d^{5}-a^{4} b^{5} c^{7} d^{2}+2 a^{3} b^{6} c^{8} d -a^{2} b^{7} c^{9}\right ) \textit {\_R}^{4}\right )\right )}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (d^{3} a^{8}-3 c \,d^{2} a^{7} b +3 c^{2} d \,a^{6} b^{2}-a^{5} b^{3} c^{3}\right ) \textit {\_Z}^{3}-b^{5}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a^{11} c^{5} d^{6}+22 a^{10} b \,c^{6} d^{5}-52 a^{9} b^{2} c^{7} d^{4}+68 a^{8} b^{3} c^{8} d^{3}-52 a^{7} b^{4} c^{9} d^{2}+22 a^{6} b^{5} c^{10} d -4 a^{5} b^{6} c^{11}\right ) \textit {\_R}^{6}+\left (-3 a^{8} d^{8}+9 a^{7} b c \,d^{7}-10 a^{6} c^{2} d^{6} b^{2}+4 a^{5} c^{3} d^{5} b^{3}+4 a^{3} c^{5} d^{3} b^{5}-10 a^{2} c^{6} d^{2} b^{6}+9 a \,c^{7} d \,b^{7}-3 c^{8} b^{8}\right ) \textit {\_R}^{3}+3 b^{5} d^{5}\right ) x +\left (-a^{9} c^{2} d^{7}+2 a^{8} b \,c^{3} d^{6}-a^{7} b^{2} c^{4} d^{5}-a^{4} b^{5} c^{7} d^{2}+2 a^{3} b^{6} c^{8} d -a^{2} b^{7} c^{9}\right ) \textit {\_R}^{4}\right )\right )}{3}\) | \(693\) |
Input:
int(1/x^3/(b*x^3+a)/(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
-1/2/a/c/x^2-(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*d^2/(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*b^2/(a*d-b*c)
Time = 0.92 (sec) , antiderivative size = 301, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {2 \, \sqrt {3} b c x^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} a x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}} - \sqrt {3} b}{3 \, b}\right ) + 2 \, \sqrt {3} a d x^{2} \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} c x \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}} - \sqrt {3} d}{3 \, d}\right ) - b c x^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b^{2} x^{2} - a b x \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} + a^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {2}{3}}\right ) - a d x^{2} \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d^{2} x^{2} + c d x \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} + c^{2} \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {2}{3}}\right ) + 2 \, b c x^{2} \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}} \log \left (b x + a \left (\frac {b^{2}}{a^{2}}\right )^{\frac {1}{3}}\right ) + 2 \, a d x^{2} \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}} \log \left (d x - c \left (-\frac {d^{2}}{c^{2}}\right )^{\frac {1}{3}}\right ) + 3 \, b c - 3 \, a d}{6 \, {\left (a b c^{2} - a^{2} c d\right )} x^{2}} \] Input:
integrate(1/x^3/(b*x^3+a)/(d*x^3+c),x, algorithm="fricas")
Output:
-1/6*(2*sqrt(3)*b*c*x^2*(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)*a*d*x^2*(-d^2/c^2)^(1/3)*arctan(1/3*(2 *sqrt(3)*c*x*(-d^2/c^2)^(2/3) - sqrt(3)*d)/d) - b*c*x^2*(b^2/a^2)^(1/3)*lo g(b^2*x^2 - a*b*x*(b^2/a^2)^(1/3) + a^2*(b^2/a^2)^(2/3)) - a*d*x^2*(-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*c*x^2*(b^2/a^2)^(1/3)*log(b*x + a*(b^2/a^2)^(1/3)) + 2*a*d*x^2*(-d^2/c^ 2)^(1/3)*log(d*x - c*(-d^2/c^2)^(1/3)) + 3*b*c - 3*a*d)/((a*b*c^2 - a^2*c* d)*x^2)
Timed out. \[ \int \frac {1}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x**3/(b*x**3+a)/(d*x**3+c),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.09 \[ \int \frac {1}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {\sqrt {3} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (a b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{2} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} d \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} + \frac {b \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} - \frac {d \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - a c d \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} - \frac {b \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (a b c \left (\frac {a}{b}\right )^{\frac {2}{3}} - a^{2} d \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}} + \frac {d \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c^{2} \left (\frac {c}{d}\right )^{\frac {2}{3}} - a c d \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}} - \frac {1}{2 \, a c x^{2}} \] Input:
integrate(1/x^3/(b*x^3+a)/(d*x^3+c),x, algorithm="maxima")
Output:
-1/3*sqrt(3)*b*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/((a*b*c *(a/b)^(1/3) - a^2*d*(a/b)^(1/3))*(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*(c/d)^(1/3) - a*c*d*(c/d) ^(1/3))*(c/d)^(1/3)) + 1/6*b*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a*b*c *(a/b)^(2/3) - a^2*d*(a/b)^(2/3)) - 1/6*d*log(x^2 - x*(c/d)^(1/3) + (c/d)^ (2/3))/(b*c^2*(c/d)^(2/3) - a*c*d*(c/d)^(2/3)) - 1/3*b*log(x + (a/b)^(1/3) )/(a*b*c*(a/b)^(2/3) - a^2*d*(a/b)^(2/3)) + 1/3*d*log(x + (c/d)^(1/3))/(b* c^2*(c/d)^(2/3) - a*c*d*(c/d)^(2/3)) - 1/2/(a*c*x^2)
Time = 0.14 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.03 \[ \int \frac {1}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {b^{2} \left (-\frac {a}{b}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a^{2} b c - a^{3} d\right )}} - \frac {d^{2} \left (-\frac {c}{d}\right )^{\frac {1}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} b \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a^{2} b c - \sqrt {3} a^{3} d} + \frac {\left (-c d^{2}\right )^{\frac {1}{3}} d \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c^{3} - \sqrt {3} a c^{2} d} - \frac {\left (-a b^{2}\right )^{\frac {1}{3}} b \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{2} b c - a^{3} d\right )}} + \frac {\left (-c d^{2}\right )^{\frac {1}{3}} d \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{3} - a c^{2} d\right )}} - \frac {1}{2 \, a c x^{2}} \] Input:
integrate(1/x^3/(b*x^3+a)/(d*x^3+c),x, algorithm="giac")
Output:
1/3*b^2*(-a/b)^(1/3)*log(abs(x - (-a/b)^(1/3)))/(a^2*b*c - a^3*d) - 1/3*d^ 2*(-c/d)^(1/3)*log(abs(x - (-c/d)^(1/3)))/(b*c^3 - a*c^2*d) - (-a*b^2)^(1/ 3)*b*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)^(1/3)*d*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)^(1/3)*b *log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^2*b*c - a^3*d) + 1/6*(-c*d^2) ^(1/3)*d*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(b*c^3 - a*c^2*d) - 1/2/ (a*c*x^2)
Time = 10.63 (sec) , antiderivative size = 1829, normalized size of antiderivative = 6.08 \[ \int \frac {1}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x^3*(a + b*x^3)*(c + d*x^3)),x)
Output:
log(((b^5/(a^5*(a*d - b*c)^3))^(1/3)*(((81*a^10*b^3*c^10*d^3*(a*d + b*c)*( a*d - b*c)^4*(b^5/(a^5*(a*d - b*c)^3))^(1/3) - 81*a^8*b^3*c^8*d^3*x*(a*d - b*c)^4*(a^2*d^2 + b^2*c^2 + a*b*c*d))*(b^5/(a^5*(a*d - b*c)^3))^(2/3))/9 + 9*a^6*b^9*c^11*d^4 - 9*a^7*b^8*c^10*d^5 - 9*a^10*b^5*c^7*d^8 + 9*a^11*b^ 4*c^6*d^9))/3 + 3*a^6*b^6*c^6*d^6*x*(a^2*d^2 + b^2*c^2))*(b^5/(27*a^8*d^3 - 27*a^5*b^3*c^3 + 81*a^6*b^2*c^2*d - 81*a^7*b*c*d^2))^(1/3) + log(((-d^5/ (c^5*(a*d - b*c)^3))^(1/3)*(((81*a^10*b^3*c^10*d^3*(a*d + b*c)*(a*d - b*c) ^4*(-d^5/(c^5*(a*d - b*c)^3))^(1/3) - 81*a^8*b^3*c^8*d^3*x*(a*d - b*c)^4*( a^2*d^2 + b^2*c^2 + a*b*c*d))*(-d^5/(c^5*(a*d - b*c)^3))^(2/3))/9 + 9*a^6* b^9*c^11*d^4 - 9*a^7*b^8*c^10*d^5 - 9*a^10*b^5*c^7*d^8 + 9*a^11*b^4*c^6*d^ 9))/3 + 3*a^6*b^6*c^6*d^6*x*(a^2*d^2 + b^2*c^2))*(d^5/(27*b^3*c^8 - 27*a^3 *c^5*d^3 + 81*a^2*b*c^6*d^2 - 81*a*b^2*c^7*d))^(1/3) + (log(((3^(1/2)*1i - 1)*(b^5/(a^5*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i - 1)^2*(81*a^8*b^3*c^8*d ^3*x*(a*d - b*c)^4*(a^2*d^2 + b^2*c^2 + a*b*c*d) - (81*a^10*b^3*c^10*d^3*( 3^(1/2)*1i - 1)*(a*d + b*c)*(a*d - b*c)^4*(b^5/(a^5*(a*d - b*c)^3))^(1/3)) /2)*(b^5/(a^5*(a*d - b*c)^3))^(2/3))/36 - 9*a^6*b^9*c^11*d^4 + 9*a^7*b^8*c ^10*d^5 + 9*a^10*b^5*c^7*d^8 - 9*a^11*b^4*c^6*d^9))/6 - 3*a^6*b^6*c^6*d^6* x*(a^2*d^2 + b^2*c^2))*(b^5/(27*a^8*d^3 - 27*a^5*b^3*c^3 + 81*a^6*b^2*c^2* d - 81*a^7*b*c*d^2))^(1/3)*(3^(1/2)*1i - 1))/2 - (log(((3^(1/2)*1i + 1)*(b ^5/(a^5*(a*d - b*c)^3))^(1/3)*(((3^(1/2)*1i + 1)^2*(81*a^8*b^3*c^8*d^3*...
Time = 0.24 (sec) , antiderivative size = 257, normalized size of antiderivative = 0.85 \[ \int \frac {1}{x^3 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {-2 d^{\frac {1}{3}} a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{2} c^{2} x^{2}+2 c^{\frac {1}{3}} b^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x}{c^{\frac {1}{3}} \sqrt {3}}\right ) a^{2} d^{2} x^{2}-d^{\frac {1}{3}} a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{2} c^{2} x^{2}+2 d^{\frac {1}{3}} a^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{2} c^{2} x^{2}+c^{\frac {1}{3}} b^{\frac {1}{3}} \mathrm {log}\left (c^{\frac {2}{3}}-d^{\frac {1}{3}} c^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right ) a^{2} d^{2} x^{2}-2 c^{\frac {1}{3}} b^{\frac {1}{3}} \mathrm {log}\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) a^{2} d^{2} x^{2}-3 d^{\frac {4}{3}} b^{\frac {1}{3}} a^{2} c +3 d^{\frac {1}{3}} b^{\frac {4}{3}} a \,c^{2}}{6 d^{\frac {1}{3}} b^{\frac {1}{3}} a^{2} c^{2} x^{2} \left (a d -b c \right )} \] Input:
int(1/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)))*b**2*c**2*x**2 + 2*c**(1/3)*b**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d **(1/3)*x)/(c**(1/3)*sqrt(3)))*a**2*d**2*x**2 - d**(1/3)*a**(1/3)*log(a**( 2/3) - b**(1/3)*a**(1/3)*x + b**(2/3)*x**2)*b**2*c**2*x**2 + 2*d**(1/3)*a* *(1/3)*log(a**(1/3) + b**(1/3)*x)*b**2*c**2*x**2 + c**(1/3)*b**(1/3)*log(c **(2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2)*a**2*d**2*x**2 - 2*c**(1/3) *b**(1/3)*log(c**(1/3) + d**(1/3)*x)*a**2*d**2*x**2 - 3*d**(1/3)*b**(1/3)* a**2*c*d + 3*d**(1/3)*b**(1/3)*a*b*c**2)/(6*d**(1/3)*b**(1/3)*a**2*c**2*x* *2*(a*d - b*c))