Integrand size = 22, antiderivative size = 352 \[ \int \frac {1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {1}{7 a c x^7}+\frac {b c+a d}{4 a^2 c^2 x^4}-\frac {b^2 c^2+a b c d+a^2 d^2}{a^3 c^3 x}+\frac {b^{10/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} a^{10/3} (b c-a d)}-\frac {d^{10/3} \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} c^{10/3} (b c-a d)}+\frac {b^{10/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 a^{10/3} (b c-a d)}-\frac {d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 c^{10/3} (b c-a d)}-\frac {b^{10/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 a^{10/3} (b c-a d)}+\frac {d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 c^{10/3} (b c-a d)} \] Output:
-1/7/a/c/x^7+1/4*(a*d+b*c)/a^2/c^2/x^4-(a^2*d^2+a*b*c*d+b^2*c^2)/a^3/c^3/x +1/3*b^(10/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))*3^(1/2)/a^ (10/3)/(-a*d+b*c)-1/3*d^(10/3)*arctan(1/3*(c^(1/3)-2*d^(1/3)*x)*3^(1/2)/c^ (1/3))*3^(1/2)/c^(10/3)/(-a*d+b*c)+1/3*b^(10/3)*ln(a^(1/3)+b^(1/3)*x)/a^(1 0/3)/(-a*d+b*c)-1/3*d^(10/3)*ln(c^(1/3)+d^(1/3)*x)/c^(10/3)/(-a*d+b*c)-1/6 *b^(10/3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/a^(10/3)/(-a*d+b*c)+1/ 6*d^(10/3)*ln(c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/c^(10/3)/(-a*d+b*c)
Time = 0.31 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {\frac {12 b}{a}-\frac {12 d}{c}-\frac {21 b^2 x^3}{a^2}+\frac {21 d^2 x^3}{c^2}+\frac {84 b^3 x^6}{a^3}-\frac {84 d^3 x^6}{c^3}-\frac {28 \sqrt {3} b^{10/3} x^7 \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{a^{10/3}}+\frac {28 \sqrt {3} d^{10/3} x^7 \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{c^{10/3}}-\frac {28 b^{10/3} x^7 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{a^{10/3}}+\frac {28 d^{10/3} x^7 \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{c^{10/3}}+\frac {14 b^{10/3} x^7 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{a^{10/3}}-\frac {14 d^{10/3} x^7 \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{c^{10/3}}}{84 (-b c+a d) x^7} \] Input:
Integrate[1/(x^8*(a + b*x^3)*(c + d*x^3)),x]
Output:
((12*b)/a - (12*d)/c - (21*b^2*x^3)/a^2 + (21*d^2*x^3)/c^2 + (84*b^3*x^6)/ a^3 - (84*d^3*x^6)/c^3 - (28*Sqrt[3]*b^(10/3)*x^7*ArcTan[(1 - (2*b^(1/3)*x )/a^(1/3))/Sqrt[3]])/a^(10/3) + (28*Sqrt[3]*d^(10/3)*x^7*ArcTan[(1 - (2*d^ (1/3)*x)/c^(1/3))/Sqrt[3]])/c^(10/3) - (28*b^(10/3)*x^7*Log[a^(1/3) + b^(1 /3)*x])/a^(10/3) + (28*d^(10/3)*x^7*Log[c^(1/3) + d^(1/3)*x])/c^(10/3) + ( 14*b^(10/3)*x^7*Log[a^(2/3) - a^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/a^(10/3) - (14*d^(10/3)*x^7*Log[c^(2/3) - c^(1/3)*d^(1/3)*x + d^(2/3)*x^2])/c^(10/3) )/(84*(-(b*c) + a*d)*x^7)
Time = 1.12 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {980, 27, 1053, 27, 1053, 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.
| \(\displaystyle \int \frac {1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 980 |
| \(\displaystyle \frac {\int -\frac {7 \left (b d x^3+b c+a d\right )}{x^5 \left (b x^3+a\right ) \left (d x^3+c\right )}dx}{7 a c}-\frac {1}{7 a c x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {b d x^3+b c+a d}{x^5 \left (b x^3+a\right ) \left (d x^3+c\right )}dx}{a c}-\frac {1}{7 a c x^7}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle -\frac {-\frac {\int \frac {4 \left (b d (b c+a d) x^3+b^2 c^2+a^2 d^2+a b c d\right )}{x^2 \left (b x^3+a\right ) \left (d x^3+c\right )}dx}{4 a c}-\frac {a d+b c}{4 a c x^4}}{a c}-\frac {1}{7 a c x^7}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {-\frac {\int \frac {b d (b c+a d) x^3+b^2 c^2+a^2 d^2+a b c d}{x^2 \left (b x^3+a\right ) \left (d x^3+c\right )}dx}{a c}-\frac {a d+b c}{4 a c x^4}}{a c}-\frac {1}{7 a c x^7}\) |
| \(\Big \downarrow \) 1053 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \frac {x \left (b d \left (b^2 c^2+a b d c+a^2 d^2\right ) x^3+(b c+a d) \left (b^2 c^2+a^2 d^2\right )\right )}{\left (b x^3+a\right ) \left (d x^3+c\right )}dx}{a c}-\frac {\frac {b^2 c}{a}+\frac {a d^2}{c}+b d}{x}}{a c}-\frac {a d+b c}{4 a c x^4}}{a c}-\frac {1}{7 a c x^7}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle -\frac {-\frac {-\frac {\int \left (\frac {c^3 x b^4}{(b c-a d) \left (b x^3+a\right )}+\frac {a^3 d^4 x}{(a d-b c) \left (d x^3+c\right )}\right )dx}{a c}-\frac {\frac {b^2 c}{a}+\frac {a d^2}{c}+b d}{x}}{a c}-\frac {a d+b c}{4 a c x^4}}{a c}-\frac {1}{7 a c x^7}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {-\frac {-\frac {\frac {b^{10/3} c^3 \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 \sqrt [3]{a} (b c-a d)}+\frac {a^3 d^{10/3} \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} \sqrt [3]{c} (b c-a d)}-\frac {a^3 d^{10/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 \sqrt [3]{c} (b c-a d)}+\frac {a^3 d^{10/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 \sqrt [3]{c} (b c-a d)}-\frac {b^{10/3} c^3 \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} \sqrt [3]{a} (b c-a d)}-\frac {b^{10/3} c^3 \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 \sqrt [3]{a} (b c-a d)}}{a c}-\frac {\frac {b^2 c}{a}+\frac {a d^2}{c}+b d}{x}}{a c}-\frac {a d+b c}{4 a c x^4}}{a c}-\frac {1}{7 a c x^7}\) |
Input:
Int[1/(x^8*(a + b*x^3)*(c + d*x^3)),x]
Output:
-1/7*1/(a*c*x^7) - (-1/4*(b*c + a*d)/(a*c*x^4) - (-(((b^2*c)/a + b*d + (a* d^2)/c)/x) - (-((b^(10/3)*c^3*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1 /3))])/(Sqrt[3]*a^(1/3)*(b*c - a*d))) + (a^3*d^(10/3)*ArcTan[(c^(1/3) - 2* d^(1/3)*x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*c^(1/3)*(b*c - a*d)) - (b^(10/3)*c ^3*Log[a^(1/3) + b^(1/3)*x])/(3*a^(1/3)*(b*c - a*d)) + (a^3*d^(10/3)*Log[c ^(1/3) + d^(1/3)*x])/(3*c^(1/3)*(b*c - a*d)) + (b^(10/3)*c^3*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^3*d^(10/3) *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))/(a*c))/(a*c)
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[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[((g_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_ ))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[e*(g*x)^(m + 1)*(a + b *x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*c*g*(m + 1))), x] + Simp[1/(a*c*g^n*( m + 1)) Int[(g*x)^(m + n)*(a + b*x^n)^p*(c + d*x^n)^q*Simp[a*f*c*(m + 1) - e*(b*c + a*d)*(m + n + 1) - e*n*(b*c*p + a*d*q) - b*e*d*(m + n*(p + q + 2 ) + 1)*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p, q}, x] && IGtQ[n, 0] && LtQ[m, -1]
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 = 1.19 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.79
| method | result | size |
| default | \(-\frac {\left (-\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right ) d^{4}}{c^{3} \left (a d -b c \right )}-\frac {1}{7 a c \,x^{7}}-\frac {-a d -b c}{4 a^{2} c^{2} x^{4}}-\frac {a^{2} d^{2}+a b c d +b^{2} c^{2}}{a^{3} c^{3} x}+\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) b^{4}}{a^{3} \left (a d -b c \right )}\) | \(279\) |
| risch | \(\frac {-\frac {\left (a^{2} d^{2}+a b c d +b^{2} c^{2}\right ) x^{6}}{c^{3} a^{3}}+\frac {\left (a d +b c \right ) x^{3}}{4 a^{2} c^{2}}-\frac {1}{7 a c}}{x^{7}}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (d^{3} a^{13}-3 c \,d^{2} a^{12} b +3 c^{2} d \,a^{11} b^{2}-a^{10} b^{3} c^{3}\right ) \textit {\_Z}^{3}+b^{10}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a^{16} c^{10} d^{6}+22 a^{15} b \,c^{11} d^{5}-52 a^{14} b^{2} c^{12} d^{4}+68 a^{13} b^{3} c^{13} d^{3}-52 a^{12} b^{4} c^{14} d^{2}+22 a^{11} b^{5} c^{15} d -4 a^{10} b^{6} c^{16}\right ) \textit {\_R}^{6}+\left (3 d^{13} a^{13}-9 d^{12} c b \,a^{12}+10 d^{11} c^{2} b^{2} a^{11}-4 d^{10} c^{3} b^{3} a^{10}-4 d^{3} c^{10} b^{10} a^{3}+10 d^{2} c^{11} b^{11} a^{2}-9 d \,c^{12} b^{12} a +3 c^{13} b^{13}\right ) \textit {\_R}^{3}+3 b^{10} d^{10}\right ) x +\left (-a^{15} c^{7} d^{8}+3 a^{14} b \,c^{8} d^{7}-3 a^{13} b^{2} c^{9} d^{6}+a^{12} b^{3} c^{10} d^{5}+a^{10} b^{5} c^{12} d^{3}-3 a^{9} b^{6} c^{13} d^{2}+3 a^{8} b^{7} c^{14} d -a^{7} b^{8} c^{15}\right ) \textit {\_R}^{5}+\left (-a^{7} b^{5} c^{2} d^{10}-a^{2} b^{10} c^{7} d^{5}\right ) \textit {\_R}^{2}\right )\right )}{3}+\frac {\left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (d^{3} c^{10} a^{3}-3 a^{2} b \,c^{11} d^{2}+3 a \,b^{2} c^{12} d -b^{3} c^{13}\right ) \textit {\_Z}^{3}-d^{10}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-4 a^{16} c^{10} d^{6}+22 a^{15} b \,c^{11} d^{5}-52 a^{14} b^{2} c^{12} d^{4}+68 a^{13} b^{3} c^{13} d^{3}-52 a^{12} b^{4} c^{14} d^{2}+22 a^{11} b^{5} c^{15} d -4 a^{10} b^{6} c^{16}\right ) \textit {\_R}^{6}+\left (3 d^{13} a^{13}-9 d^{12} c b \,a^{12}+10 d^{11} c^{2} b^{2} a^{11}-4 d^{10} c^{3} b^{3} a^{10}-4 d^{3} c^{10} b^{10} a^{3}+10 d^{2} c^{11} b^{11} a^{2}-9 d \,c^{12} b^{12} a +3 c^{13} b^{13}\right ) \textit {\_R}^{3}+3 b^{10} d^{10}\right ) x +\left (-a^{15} c^{7} d^{8}+3 a^{14} b \,c^{8} d^{7}-3 a^{13} b^{2} c^{9} d^{6}+a^{12} b^{3} c^{10} d^{5}+a^{10} b^{5} c^{12} d^{3}-3 a^{9} b^{6} c^{13} d^{2}+3 a^{8} b^{7} c^{14} d -a^{7} b^{8} c^{15}\right ) \textit {\_R}^{5}+\left (-a^{7} b^{5} c^{2} d^{10}-a^{2} b^{10} c^{7} d^{5}\right ) \textit {\_R}^{2}\right )\right )}{3}\) | \(862\) |
Input:
int(1/x^8/(b*x^3+a)/(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
-(-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^4/c^3/(a*d-b*c)-1/7/a/c/x^7-1/4*(-a*d-b*c)/a^2/c^2/x^4-(a^2*d^ 2+a*b*c*d+b^2*c^2)/a^3/c^3/x+(-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)*arc tan(1/3*3^(1/2)*(2/(a/b)^(1/3)*x-1)))*b^4/a^3/(a*d-b*c)
Time = 0.24 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.94 \[ \int \frac {1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {28 \, \sqrt {3} b^{3} c^{3} x^{7} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (-\frac {b}{a}\right )^{\frac {1}{3}} + \frac {1}{3} \, \sqrt {3}\right ) - 28 \, \sqrt {3} a^{3} d^{3} x^{7} \left (\frac {d}{c}\right )^{\frac {1}{3}} \arctan \left (\frac {2}{3} \, \sqrt {3} x \left (\frac {d}{c}\right )^{\frac {1}{3}} - \frac {1}{3} \, \sqrt {3}\right ) - 14 \, b^{3} c^{3} x^{7} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x^{2} - a x \left (-\frac {b}{a}\right )^{\frac {2}{3}} - a \left (-\frac {b}{a}\right )^{\frac {1}{3}}\right ) - 14 \, a^{3} d^{3} x^{7} \left (\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x^{2} - c x \left (\frac {d}{c}\right )^{\frac {2}{3}} + c \left (\frac {d}{c}\right )^{\frac {1}{3}}\right ) + 28 \, b^{3} c^{3} x^{7} \left (-\frac {b}{a}\right )^{\frac {1}{3}} \log \left (b x + a \left (-\frac {b}{a}\right )^{\frac {2}{3}}\right ) + 28 \, a^{3} d^{3} x^{7} \left (\frac {d}{c}\right )^{\frac {1}{3}} \log \left (d x + c \left (\frac {d}{c}\right )^{\frac {2}{3}}\right ) + 84 \, {\left (b^{3} c^{3} - a^{3} d^{3}\right )} x^{6} + 12 \, a^{2} b c^{3} - 12 \, a^{3} c^{2} d - 21 \, {\left (a b^{2} c^{3} - a^{3} c d^{2}\right )} x^{3}}{84 \, {\left (a^{3} b c^{4} - a^{4} c^{3} d\right )} x^{7}} \] Input:
integrate(1/x^8/(b*x^3+a)/(d*x^3+c),x, algorithm="fricas")
Output:
-1/84*(28*sqrt(3)*b^3*c^3*x^7*(-b/a)^(1/3)*arctan(2/3*sqrt(3)*x*(-b/a)^(1/ 3) + 1/3*sqrt(3)) - 28*sqrt(3)*a^3*d^3*x^7*(d/c)^(1/3)*arctan(2/3*sqrt(3)* x*(d/c)^(1/3) - 1/3*sqrt(3)) - 14*b^3*c^3*x^7*(-b/a)^(1/3)*log(b*x^2 - a*x *(-b/a)^(2/3) - a*(-b/a)^(1/3)) - 14*a^3*d^3*x^7*(d/c)^(1/3)*log(d*x^2 - c *x*(d/c)^(2/3) + c*(d/c)^(1/3)) + 28*b^3*c^3*x^7*(-b/a)^(1/3)*log(b*x + a* (-b/a)^(2/3)) + 28*a^3*d^3*x^7*(d/c)^(1/3)*log(d*x + c*(d/c)^(2/3)) + 84*( b^3*c^3 - a^3*d^3)*x^6 + 12*a^2*b*c^3 - 12*a^3*c^2*d - 21*(a*b^2*c^3 - a^3 *c*d^2)*x^3)/((a^3*b*c^4 - a^4*c^3*d)*x^7)
Timed out. \[ \int \frac {1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(1/x**8/(b*x**3+a)/(d*x**3+c),x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {\sqrt {3} b^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (a^{3} b c - a^{4} d\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} + \frac {\sqrt {3} d^{3} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c^{4} - a c^{3} d\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} - \frac {b^{3} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{3} b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{4} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} + \frac {d^{3} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{4} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c^{3} d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} + \frac {b^{3} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (a^{3} b c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a^{4} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} - \frac {d^{3} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c^{4} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a c^{3} d \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} - \frac {28 \, {\left (b^{2} c^{2} + a b c d + a^{2} d^{2}\right )} x^{6} + 4 \, a^{2} c^{2} - 7 \, {\left (a b c^{2} + a^{2} c d\right )} x^{3}}{28 \, a^{3} c^{3} x^{7}} \] Input:
integrate(1/x^8/(b*x^3+a)/(d*x^3+c),x, algorithm="maxima")
Output:
-1/3*sqrt(3)*b^3*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/((a^3 *b*c - a^4*d)*(a/b)^(1/3)) + 1/3*sqrt(3)*d^3*arctan(1/3*sqrt(3)*(2*x - (c/ d)^(1/3))/(c/d)^(1/3))/((b*c^4 - a*c^3*d)*(c/d)^(1/3)) - 1/6*b^3*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(a^3*b*c*(a/b)^(1/3) - a^4*d*(a/b)^(1/3)) + 1/6*d^3*log(x^2 - x*(c/d)^(1/3) + (c/d)^(2/3))/(b*c^4*(c/d)^(1/3) - a*c^3* d*(c/d)^(1/3)) + 1/3*b^3*log(x + (a/b)^(1/3))/(a^3*b*c*(a/b)^(1/3) - a^4*d *(a/b)^(1/3)) - 1/3*d^3*log(x + (c/d)^(1/3))/(b*c^4*(c/d)^(1/3) - a*c^3*d* (c/d)^(1/3)) - 1/28*(28*(b^2*c^2 + a*b*c*d + a^2*d^2)*x^6 + 4*a^2*c^2 - 7* (a*b*c^2 + a^2*c*d)*x^3)/(a^3*c^3*x^7)
Time = 0.15 (sec) , antiderivative size = 377, normalized size of antiderivative = 1.07 \[ \int \frac {1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {b^{4} \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a^{4} b c - a^{5} d\right )}} - \frac {d^{4} \left (-\frac {c}{d}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{5} - a c^{4} d\right )}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} b^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} a^{4} b c - \sqrt {3} a^{5} d} - \frac {\left (-c d^{2}\right )^{\frac {2}{3}} d^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c^{5} - \sqrt {3} a c^{4} d} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} b^{2} \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (a^{4} b c - a^{5} d\right )}} + \frac {\left (-c d^{2}\right )^{\frac {2}{3}} d^{2} \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c^{5} - a c^{4} d\right )}} - \frac {28 \, b^{2} c^{2} x^{6} + 28 \, a b c d x^{6} + 28 \, a^{2} d^{2} x^{6} - 7 \, a b c^{2} x^{3} - 7 \, a^{2} c d x^{3} + 4 \, a^{2} c^{2}}{28 \, a^{3} c^{3} x^{7}} \] Input:
integrate(1/x^8/(b*x^3+a)/(d*x^3+c),x, algorithm="giac")
Output:
1/3*b^4*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a^4*b*c - a^5*d) - 1/3*d^ 4*(-c/d)^(2/3)*log(abs(x - (-c/d)^(1/3)))/(b*c^5 - a*c^4*d) + (-a*b^2)^(2/ 3)*b^2*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*a^4* b*c - sqrt(3)*a^5*d) - (-c*d^2)^(2/3)*d^2*arctan(1/3*sqrt(3)*(2*x + (-c/d) ^(1/3))/(-c/d)^(1/3))/(sqrt(3)*b*c^5 - sqrt(3)*a*c^4*d) - 1/6*(-a*b^2)^(2/ 3)*b^2*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(a^4*b*c - a^5*d) + 1/6*(- c*d^2)^(2/3)*d^2*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(b*c^5 - a*c^4*d ) - 1/28*(28*b^2*c^2*x^6 + 28*a*b*c*d*x^6 + 28*a^2*d^2*x^6 - 7*a*b*c^2*x^3 - 7*a^2*c*d*x^3 + 4*a^2*c^2)/(a^3*c^3*x^7)
Time = 10.78 (sec) , antiderivative size = 1814, normalized size of antiderivative = 5.15 \[ \int \frac {1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:
int(1/(x^8*(a + b*x^3)*(c + d*x^3)),x)
Output:
log(((-b^10/(a^10*(a*d - b*c)^3))^(2/3)*(((27*a^21*b^3*c^21*d^3*x*(a^8*d^8 + b^8*c^8)*(a*d - b*c)^2 + 27*a^28*b^3*c^28*d^3*(a*d + b*c)*(a*d - b*c)^4 *(-b^10/(a^10*(a*d - b*c)^3))^(2/3))*(-b^10/(a^10*(a*d - b*c)^3))^(1/3))/3 - 9*a^19*b^14*c^29*d^4 + 9*a^20*b^13*c^28*d^5 + 9*a^28*b^5*c^20*d^13 - 9* a^29*b^4*c^19*d^14))/9 - a^19*b^11*c^19*d^11*x*(a*d + b*c))*(-b^10/(27*a^1 3*d^3 - 27*a^10*b^3*c^3 + 81*a^11*b^2*c^2*d - 81*a^12*b*c*d^2))^(1/3) + lo g(((d^10/(c^10*(a*d - b*c)^3))^(2/3)*(((27*a^21*b^3*c^21*d^3*x*(a^8*d^8 + b^8*c^8)*(a*d - b*c)^2 + 27*a^28*b^3*c^28*d^3*(a*d + b*c)*(a*d - b*c)^4*(d ^10/(c^10*(a*d - b*c)^3))^(2/3))*(d^10/(c^10*(a*d - b*c)^3))^(1/3))/3 - 9* a^19*b^14*c^29*d^4 + 9*a^20*b^13*c^28*d^5 + 9*a^28*b^5*c^20*d^13 - 9*a^29* b^4*c^19*d^14))/9 - a^19*b^11*c^19*d^11*x*(a*d + b*c))*(-d^10/(27*b^3*c^13 - 27*a^3*c^10*d^3 + 81*a^2*b*c^11*d^2 - 81*a*b^2*c^12*d))^(1/3) - (1/(7*a *c) - (x^3*(a*d + b*c))/(4*a^2*c^2) + (x^6*(a^2*d^2 + b^2*c^2 + a*b*c*d))/ (a^3*c^3))/x^7 - (log(((3^(1/2)*1i + 1)^2*(-b^10/(a^10*(a*d - b*c)^3))^(2/ 3)*(((3^(1/2)*1i + 1)*(27*a^21*b^3*c^21*d^3*x*(a^8*d^8 + b^8*c^8)*(a*d - b *c)^2 + (27*a^28*b^3*c^28*d^3*(3^(1/2)*1i + 1)^2*(a*d + b*c)*(a*d - b*c)^4 *(-b^10/(a^10*(a*d - b*c)^3))^(2/3))/4)*(-b^10/(a^10*(a*d - b*c)^3))^(1/3) )/6 + 9*a^19*b^14*c^29*d^4 - 9*a^20*b^13*c^28*d^5 - 9*a^28*b^5*c^20*d^13 + 9*a^29*b^4*c^19*d^14))/36 + a^19*b^11*c^19*d^11*x*(a*d + b*c))*(-b^10/(27 *a^13*d^3 - 27*a^10*b^3*c^3 + 81*a^11*b^2*c^2*d - 81*a^12*b*c*d^2))^(1/...
Time = 0.22 (sec) , antiderivative size = 312, normalized size of antiderivative = 0.89 \[ \int \frac {1}{x^8 \left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {-28 d^{\frac {2}{3}} c^{\frac {10}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) b^{4} x^{7}+28 b^{\frac {2}{3}} a^{\frac {10}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x}{c^{\frac {1}{3}} \sqrt {3}}\right ) d^{4} x^{7}-12 d^{\frac {5}{3}} c^{\frac {7}{3}} b^{\frac {2}{3}} a^{\frac {10}{3}}+21 d^{\frac {8}{3}} c^{\frac {4}{3}} b^{\frac {2}{3}} a^{\frac {10}{3}} x^{3}-84 d^{\frac {11}{3}} c^{\frac {1}{3}} b^{\frac {2}{3}} a^{\frac {10}{3}} x^{6}+12 d^{\frac {2}{3}} c^{\frac {10}{3}} b^{\frac {5}{3}} a^{\frac {7}{3}}-21 d^{\frac {2}{3}} c^{\frac {10}{3}} b^{\frac {8}{3}} a^{\frac {4}{3}} x^{3}+84 d^{\frac {2}{3}} c^{\frac {10}{3}} b^{\frac {11}{3}} a^{\frac {1}{3}} x^{6}-14 b^{\frac {2}{3}} a^{\frac {10}{3}} \mathrm {log}\left (c^{\frac {2}{3}}-d^{\frac {1}{3}} c^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right ) d^{4} x^{7}+28 b^{\frac {2}{3}} a^{\frac {10}{3}} \mathrm {log}\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) d^{4} x^{7}+14 d^{\frac {2}{3}} c^{\frac {10}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) b^{4} x^{7}-28 d^{\frac {2}{3}} c^{\frac {10}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) b^{4} x^{7}}{84 d^{\frac {2}{3}} c^{\frac {10}{3}} b^{\frac {2}{3}} a^{\frac {10}{3}} x^{7} \left (a d -b c \right )} \] Input:
int(1/x^8/(b*x^3+a)/(d*x^3+c),x)
Output:
( - 28*d**(2/3)*c**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)* sqrt(3)))*b**4*c**3*x**7 + 28*b**(2/3)*a**(1/3)*sqrt(3)*atan((c**(1/3) - 2 *d**(1/3)*x)/(c**(1/3)*sqrt(3)))*a**3*d**4*x**7 - 12*d**(2/3)*c**(1/3)*b** (2/3)*a**(1/3)*a**3*c**2*d + 21*d**(2/3)*c**(1/3)*b**(2/3)*a**(1/3)*a**3*c *d**2*x**3 - 84*d**(2/3)*c**(1/3)*b**(2/3)*a**(1/3)*a**3*d**3*x**6 + 12*d* *(2/3)*c**(1/3)*b**(2/3)*a**(1/3)*a**2*b*c**3 - 21*d**(2/3)*c**(1/3)*b**(2 /3)*a**(1/3)*a*b**2*c**3*x**3 + 84*d**(2/3)*c**(1/3)*b**(2/3)*a**(1/3)*b** 3*c**3*x**6 - 14*b**(2/3)*a**(1/3)*log(c**(2/3) - d**(1/3)*c**(1/3)*x + d* *(2/3)*x**2)*a**3*d**4*x**7 + 28*b**(2/3)*a**(1/3)*log(c**(1/3) + d**(1/3) *x)*a**3*d**4*x**7 + 14*d**(2/3)*c**(1/3)*log(a**(2/3) - b**(1/3)*a**(1/3) *x + b**(2/3)*x**2)*b**4*c**3*x**7 - 28*d**(2/3)*c**(1/3)*log(a**(1/3) + b **(1/3)*x)*b**4*c**3*x**7)/(84*d**(2/3)*c**(1/3)*b**(2/3)*a**(1/3)*a**3*c* *3*x**7*(a*d - b*c))