Integrand size = 22, antiderivative size = 301 \[ \int \frac {x^7}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {x^2}{2 b d}-\frac {a^{5/3} \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{5/3} (b c-a d)}+\frac {c^{5/3} \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} d^{5/3} (b c-a d)}-\frac {a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{5/3} (b c-a d)}+\frac {c^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{5/3} (b c-a d)}+\frac {a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{5/3} (b c-a d)}-\frac {c^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{5/3} (b c-a d)} \] Output:
1/2*x^2/b/d-1/3*a^(5/3)*arctan(1/3*(a^(1/3)-2*b^(1/3)*x)*3^(1/2)/a^(1/3))* 3^(1/2)/b^(5/3)/(-a*d+b*c)+1/3*c^(5/3)*arctan(1/3*(c^(1/3)-2*d^(1/3)*x)*3^ (1/2)/c^(1/3))*3^(1/2)/d^(5/3)/(-a*d+b*c)-1/3*a^(5/3)*ln(a^(1/3)+b^(1/3)*x )/b^(5/3)/(-a*d+b*c)+1/3*c^(5/3)*ln(c^(1/3)+d^(1/3)*x)/d^(5/3)/(-a*d+b*c)+ 1/6*a^(5/3)*ln(a^(2/3)-a^(1/3)*b^(1/3)*x+b^(2/3)*x^2)/b^(5/3)/(-a*d+b*c)-1 /6*c^(5/3)*ln(c^(2/3)-c^(1/3)*d^(1/3)*x+d^(2/3)*x^2)/d^(5/3)/(-a*d+b*c)
Time = 0.13 (sec) , antiderivative size = 242, normalized size of antiderivative = 0.80 \[ \int \frac {x^7}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {-\frac {3 a x^2}{b}+\frac {3 c x^2}{d}-\frac {2 \sqrt {3} a^{5/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{b} x}{\sqrt [3]{a}}}{\sqrt {3}}\right )}{b^{5/3}}+\frac {2 \sqrt {3} c^{5/3} \arctan \left (\frac {1-\frac {2 \sqrt [3]{d} x}{\sqrt [3]{c}}}{\sqrt {3}}\right )}{d^{5/3}}-\frac {2 a^{5/3} \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{b^{5/3}}+\frac {2 c^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{d^{5/3}}+\frac {a^{5/3} \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{b^{5/3}}-\frac {c^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{d^{5/3}}}{6 b c-6 a d} \] Input:
Integrate[x^7/((a + b*x^3)*(c + d*x^3)),x]
Output:
((-3*a*x^2)/b + (3*c*x^2)/d - (2*Sqrt[3]*a^(5/3)*ArcTan[(1 - (2*b^(1/3)*x) /a^(1/3))/Sqrt[3]])/b^(5/3) + (2*Sqrt[3]*c^(5/3)*ArcTan[(1 - (2*d^(1/3)*x) /c^(1/3))/Sqrt[3]])/d^(5/3) - (2*a^(5/3)*Log[a^(1/3) + b^(1/3)*x])/b^(5/3) + (2*c^(5/3)*Log[c^(1/3) + d^(1/3)*x])/d^(5/3) + (a^(5/3)*Log[a^(2/3) - a ^(1/3)*b^(1/3)*x + b^(2/3)*x^2])/b^(5/3) - (c^(5/3)*Log[c^(2/3) - c^(1/3)* d^(1/3)*x + d^(2/3)*x^2])/d^(5/3))/(6*b*c - 6*a*d)
Time = 0.84 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {979, 27, 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.
| \(\displaystyle \int \frac {x^7}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx\) |
| \(\Big \downarrow \) 979 |
| \(\displaystyle \frac {x^2}{2 b d}-\frac {\int \frac {2 x \left ((b c+a d) x^3+a c\right )}{\left (b x^3+a\right ) \left (d x^3+c\right )}dx}{2 b d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {x^2}{2 b d}-\frac {\int \frac {x \left ((b c+a d) x^3+a c\right )}{\left (b x^3+a\right ) \left (d x^3+c\right )}dx}{b d}\) |
| \(\Big \downarrow \) 1054 |
| \(\displaystyle \frac {x^2}{2 b d}-\frac {\int \left (\frac {d x a^2}{(a d-b c) \left (b x^3+a\right )}+\frac {b c^2 x}{(b c-a d) \left (d x^3+c\right )}\right )dx}{b d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {x^2}{2 b d}-\frac {\frac {a^{5/3} d \arctan \left (\frac {\sqrt [3]{a}-2 \sqrt [3]{b} x}{\sqrt {3} \sqrt [3]{a}}\right )}{\sqrt {3} b^{2/3} (b c-a d)}-\frac {a^{5/3} d \log \left (a^{2/3}-\sqrt [3]{a} \sqrt [3]{b} x+b^{2/3} x^2\right )}{6 b^{2/3} (b c-a d)}+\frac {a^{5/3} d \log \left (\sqrt [3]{a}+\sqrt [3]{b} x\right )}{3 b^{2/3} (b c-a d)}-\frac {b c^{5/3} \arctan \left (\frac {\sqrt [3]{c}-2 \sqrt [3]{d} x}{\sqrt {3} \sqrt [3]{c}}\right )}{\sqrt {3} d^{2/3} (b c-a d)}+\frac {b c^{5/3} \log \left (c^{2/3}-\sqrt [3]{c} \sqrt [3]{d} x+d^{2/3} x^2\right )}{6 d^{2/3} (b c-a d)}-\frac {b c^{5/3} \log \left (\sqrt [3]{c}+\sqrt [3]{d} x\right )}{3 d^{2/3} (b c-a d)}}{b d}\) |
Input:
Int[x^7/((a + b*x^3)*(c + d*x^3)),x]
Output:
x^2/(2*b*d) - ((a^(5/3)*d*ArcTan[(a^(1/3) - 2*b^(1/3)*x)/(Sqrt[3]*a^(1/3)) ])/(Sqrt[3]*b^(2/3)*(b*c - a*d)) - (b*c^(5/3)*ArcTan[(c^(1/3) - 2*d^(1/3)* x)/(Sqrt[3]*c^(1/3))])/(Sqrt[3]*d^(2/3)*(b*c - a*d)) + (a^(5/3)*d*Log[a^(1 /3) + b^(1/3)*x])/(3*b^(2/3)*(b*c - a*d)) - (b*c^(5/3)*Log[c^(1/3) + d^(1/ 3)*x])/(3*d^(2/3)*(b*c - a*d)) - (a^(5/3)*d*Log[a^(2/3) - a^(1/3)*b^(1/3)* x + b^(2/3)*x^2])/(6*b^(2/3)*(b*c - a*d)) + (b*c^(5/3)*Log[c^(2/3) - c^(1/ 3)*d^(1/3)*x + d^(2/3)*x^2])/(6*d^(2/3)*(b*c - a*d)))/(b*d)
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^(2*n - 1)*(e*x)^(m - 2*n + 1)*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(b*d*(m + n*(p + q) + 1))), x] - Simp[e^(2*n)/(b*d *(m + n*(p + q) + 1)) Int[(e*x)^(m - 2*n)*(a + b*x^n)^p*(c + d*x^n)^q*Sim p[a*c*(m - 2*n + 1) + (a*d*(m + n*(q - 1) + 1) + b*c*(m + n*(p - 1) + 1))*x ^n, x], x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && I GtQ[n, 0] && GtQ[m - n + 1, n] && IntBinomialQ[a, b, c, d, e, m, n, p, q, x ]
Int[(((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((e_) + (f_.)*(x_)^(n _)))/((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[(g*x)^m*(a + b*x^n)^p*((e + f*x^n)/(c + d*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && IGtQ[n, 0]
Time = 1.14 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.76
| method | result | size |
| default | \(\frac {x^{2}}{2 b d}+\frac {\left (-\frac {\ln \left (x +\left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {c}{d}\right )^{\frac {1}{3}} x +\left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {c}{d}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 d \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right ) c^{2}}{\left (a d -b c \right ) d}-\frac {\left (-\frac {\ln \left (x +\left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\ln \left (x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{3}} x +\left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}+\frac {\sqrt {3}\, \arctan \left (\frac {\sqrt {3}\, \left (\frac {2 x}{\left (\frac {a}{b}\right )^{\frac {1}{3}}}-1\right )}{3}\right )}{3 b \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right ) a^{2}}{\left (a d -b c \right ) b}\) | \(228\) |
| risch | \(\frac {x^{2}}{2 b d}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} b^{2} d^{3}-3 a^{2} b^{3} c \,d^{2}+3 a \,b^{4} c^{2} d -b^{5} c^{3}\right ) \textit {\_Z}^{3}-a^{5} d^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{5} b^{2} c \,d^{6}+2 a^{4} b^{3} c^{2} d^{5}-2 a^{3} b^{4} c^{3} d^{4}+2 a^{2} b^{5} c^{4} d^{3}-a \,b^{6} c^{5} d^{2}\right ) \textit {\_R}^{3}+a^{5} b^{2} c^{3} d^{4}+a^{4} b^{3} c^{4} d^{3}+a^{3} b^{4} c^{5} d^{2}\right ) x +\left (-a^{5} b^{2} d^{7}+3 a^{4} b^{3} c \,d^{6}-2 a^{3} b^{4} c^{2} d^{5}-2 a^{2} b^{5} c^{3} d^{4}+3 a \,b^{6} c^{4} d^{3}-b^{7} c^{5} d^{2}\right ) \textit {\_R}^{5}+\left (a^{7} d^{7}-a^{6} b c \,d^{6}-a \,b^{6} c^{6} d +b^{7} c^{7}\right ) \textit {\_R}^{2}\right )}{3 d b}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a^{3} d^{5}-3 a^{2} b c \,d^{4}+3 a \,b^{2} c^{2} d^{3}-b^{3} c^{3} d^{2}\right ) \textit {\_Z}^{3}+c^{5} b^{3}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (-a^{5} b^{2} c \,d^{6}+2 a^{4} b^{3} c^{2} d^{5}-2 a^{3} b^{4} c^{3} d^{4}+2 a^{2} b^{5} c^{4} d^{3}-a \,b^{6} c^{5} d^{2}\right ) \textit {\_R}^{3}+a^{5} b^{2} c^{3} d^{4}+a^{4} b^{3} c^{4} d^{3}+a^{3} b^{4} c^{5} d^{2}\right ) x +\left (-a^{5} b^{2} d^{7}+3 a^{4} b^{3} c \,d^{6}-2 a^{3} b^{4} c^{2} d^{5}-2 a^{2} b^{5} c^{3} d^{4}+3 a \,b^{6} c^{4} d^{3}-b^{7} c^{5} d^{2}\right ) \textit {\_R}^{5}+\left (a^{7} d^{7}-a^{6} b c \,d^{6}-a \,b^{6} c^{6} d +b^{7} c^{7}\right ) \textit {\_R}^{2}\right )}{3 d b}\) | \(614\) |
Input:
int(x^7/(b*x^3+a)/(d*x^3+c),x,method=_RETURNVERBOSE)
Output:
1/2*x^2/b/d+(-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^2/(a*d-b*c)/d-(-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^2/(a*d-b*c)/b
Time = 0.27 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.91 \[ \int \frac {x^7}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {2 \, \sqrt {3} a d \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} b x \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} - \sqrt {3} a}{3 \, a}\right ) - 2 \, \sqrt {3} b c \left (-\frac {c^{2}}{d^{2}}\right )^{\frac {1}{3}} \arctan \left (\frac {2 \, \sqrt {3} d x \left (-\frac {c^{2}}{d^{2}}\right )^{\frac {1}{3}} + \sqrt {3} c}{3 \, c}\right ) + a d \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x^{2} - b x \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}} + a \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}}\right ) + b c \left (-\frac {c^{2}}{d^{2}}\right )^{\frac {1}{3}} \log \left (c x^{2} - d x \left (-\frac {c^{2}}{d^{2}}\right )^{\frac {2}{3}} - c \left (-\frac {c^{2}}{d^{2}}\right )^{\frac {1}{3}}\right ) - 2 \, a d \left (\frac {a^{2}}{b^{2}}\right )^{\frac {1}{3}} \log \left (a x + b \left (\frac {a^{2}}{b^{2}}\right )^{\frac {2}{3}}\right ) - 2 \, b c \left (-\frac {c^{2}}{d^{2}}\right )^{\frac {1}{3}} \log \left (c x + d \left (-\frac {c^{2}}{d^{2}}\right )^{\frac {2}{3}}\right ) + 3 \, {\left (b c - a d\right )} x^{2}}{6 \, {\left (b^{2} c d - a b d^{2}\right )}} \] Input:
integrate(x^7/(b*x^3+a)/(d*x^3+c),x, algorithm="fricas")
Output:
1/6*(2*sqrt(3)*a*d*(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)*b*c*(-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*d*(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)) + b*c*(-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*d*(a^2/b^2)^(1/3)*log(a*x + b*(a^2/b^2)^(2/3)) - 2*b*c*(-c^2/d^2)^(1/3)*log(c*x + d*(-c^2/d^2)^(2/3) ) + 3*(b*c - a*d)*x^2)/(b^2*c*d - a*b*d^2)
Timed out. \[ \int \frac {x^7}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\text {Timed out} \] Input:
integrate(x**7/(b*x**3+a)/(d*x**3+c),x)
Output:
Timed out
Time = 0.11 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.08 \[ \int \frac {x^7}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {\sqrt {3} a^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b^{3} c - a b^{2} d\right )} \left (\frac {a}{b}\right )^{\frac {1}{3}}} - \frac {\sqrt {3} c^{2} \arctan \left (\frac {\sqrt {3} {\left (2 \, x - \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{3 \, {\left (b c d^{2} - a d^{3}\right )} \left (\frac {c}{d}\right )^{\frac {1}{3}}} + \frac {a^{2} \log \left (x^{2} - x \left (\frac {a}{b}\right )^{\frac {1}{3}} + \left (\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{3} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} - \frac {c^{2} \log \left (x^{2} - x \left (\frac {c}{d}\right )^{\frac {1}{3}} + \left (\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d^{3} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} - \frac {a^{2} \log \left (x + \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b^{3} c \left (\frac {a}{b}\right )^{\frac {1}{3}} - a b^{2} d \left (\frac {a}{b}\right )^{\frac {1}{3}}\right )}} + \frac {c^{2} \log \left (x + \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}{3 \, {\left (b c d^{2} \left (\frac {c}{d}\right )^{\frac {1}{3}} - a d^{3} \left (\frac {c}{d}\right )^{\frac {1}{3}}\right )}} + \frac {x^{2}}{2 \, b d} \] Input:
integrate(x^7/(b*x^3+a)/(d*x^3+c),x, algorithm="maxima")
Output:
1/3*sqrt(3)*a^2*arctan(1/3*sqrt(3)*(2*x - (a/b)^(1/3))/(a/b)^(1/3))/((b^3* c - a*b^2*d)*(a/b)^(1/3)) - 1/3*sqrt(3)*c^2*arctan(1/3*sqrt(3)*(2*x - (c/d )^(1/3))/(c/d)^(1/3))/((b*c*d^2 - a*d^3)*(c/d)^(1/3)) + 1/6*a^2*log(x^2 - x*(a/b)^(1/3) + (a/b)^(2/3))/(b^3*c*(a/b)^(1/3) - a*b^2*d*(a/b)^(1/3)) - 1 /6*c^2*log(x^2 - x*(c/d)^(1/3) + (c/d)^(2/3))/(b*c*d^2*(c/d)^(1/3) - a*d^3 *(c/d)^(1/3)) - 1/3*a^2*log(x + (a/b)^(1/3))/(b^3*c*(a/b)^(1/3) - a*b^2*d* (a/b)^(1/3)) + 1/3*c^2*log(x + (c/d)^(1/3))/(b*c*d^2*(c/d)^(1/3) - a*d^3*( c/d)^(1/3)) + 1/2*x^2/(b*d)
Time = 0.13 (sec) , antiderivative size = 311, normalized size of antiderivative = 1.03 \[ \int \frac {x^7}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=-\frac {a^{2} \left (-\frac {a}{b}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {a}{b}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (a b^{2} c - a^{2} b d\right )}} + \frac {c^{2} \left (-\frac {c}{d}\right )^{\frac {2}{3}} \log \left ({\left | x - \left (-\frac {c}{d}\right )^{\frac {1}{3}} \right |}\right )}{3 \, {\left (b c^{2} d - a c d^{2}\right )}} - \frac {\left (-a b^{2}\right )^{\frac {2}{3}} a \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {a}{b}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {a}{b}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b^{4} c - \sqrt {3} a b^{3} d} + \frac {\left (-c d^{2}\right )^{\frac {2}{3}} c \arctan \left (\frac {\sqrt {3} {\left (2 \, x + \left (-\frac {c}{d}\right )^{\frac {1}{3}}\right )}}{3 \, \left (-\frac {c}{d}\right )^{\frac {1}{3}}}\right )}{\sqrt {3} b c d^{3} - \sqrt {3} a d^{4}} + \frac {\left (-a b^{2}\right )^{\frac {2}{3}} a \log \left (x^{2} + x \left (-\frac {a}{b}\right )^{\frac {1}{3}} + \left (-\frac {a}{b}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b^{4} c - a b^{3} d\right )}} - \frac {\left (-c d^{2}\right )^{\frac {2}{3}} c \log \left (x^{2} + x \left (-\frac {c}{d}\right )^{\frac {1}{3}} + \left (-\frac {c}{d}\right )^{\frac {2}{3}}\right )}{6 \, {\left (b c d^{3} - a d^{4}\right )}} + \frac {x^{2}}{2 \, b d} \] Input:
integrate(x^7/(b*x^3+a)/(d*x^3+c),x, algorithm="giac")
Output:
-1/3*a^2*(-a/b)^(2/3)*log(abs(x - (-a/b)^(1/3)))/(a*b^2*c - a^2*b*d) + 1/3 *c^2*(-c/d)^(2/3)*log(abs(x - (-c/d)^(1/3)))/(b*c^2*d - a*c*d^2) - (-a*b^2 )^(2/3)*a*arctan(1/3*sqrt(3)*(2*x + (-a/b)^(1/3))/(-a/b)^(1/3))/(sqrt(3)*b ^4*c - sqrt(3)*a*b^3*d) + (-c*d^2)^(2/3)*c*arctan(1/3*sqrt(3)*(2*x + (-c/d )^(1/3))/(-c/d)^(1/3))/(sqrt(3)*b*c*d^3 - sqrt(3)*a*d^4) + 1/6*(-a*b^2)^(2 /3)*a*log(x^2 + x*(-a/b)^(1/3) + (-a/b)^(2/3))/(b^4*c - a*b^3*d) - 1/6*(-c *d^2)^(2/3)*c*log(x^2 + x*(-c/d)^(1/3) + (-c/d)^(2/3))/(b*c*d^3 - a*d^4) + 1/2*x^2/(b*d)
Time = 10.70 (sec) , antiderivative size = 1751, normalized size of antiderivative = 5.82 \[ \int \frac {x^7}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\text {Too large to display} \] Input:
int(x^7/((a + b*x^3)*(c + d*x^3)),x)
Output:
log(((((27*a^2*b*c^2*d*x*(a^2*d^2 + b^2*c^2)*(a*d - b*c)^2 + 27*a*b^3*c*d^ 3*(a*d + b*c)*(a*d - b*c)^4*(a^5/(b^5*(a*d - b*c)^3))^(2/3))*(a^5/(b^5*(a* d - b*c)^3))^(1/3))/3 - (9*(a*b^7*c^8 + a^8*c*d^7 - a^2*b^6*c^7*d - a^7*b* c^2*d^6))/(b^2*d^2))*(a^5/(b^5*(a*d - b*c)^3))^(2/3))/9 - (a^4*c^4*x*(a^2* d^2 + b^2*c^2 + a*b*c*d))/(b^2*d^2))*(-a^5/(27*b^8*c^3 - 27*a^3*b^5*d^3 + 81*a^2*b^6*c*d^2 - 81*a*b^7*c^2*d))^(1/3) + log(((((27*a^2*b*c^2*d*x*(a^2* d^2 + b^2*c^2)*(a*d - b*c)^2 + 27*a*b^3*c*d^3*(a*d + b*c)*(a*d - b*c)^4*(- c^5/(d^5*(a*d - b*c)^3))^(2/3))*(-c^5/(d^5*(a*d - b*c)^3))^(1/3))/3 - (9*( a*b^7*c^8 + a^8*c*d^7 - a^2*b^6*c^7*d - a^7*b*c^2*d^6))/(b^2*d^2))*(-c^5/( d^5*(a*d - b*c)^3))^(2/3))/9 - (a^4*c^4*x*(a^2*d^2 + b^2*c^2 + a*b*c*d))/( b^2*d^2))*(-c^5/(27*a^3*d^8 - 27*b^3*c^3*d^5 + 81*a*b^2*c^2*d^6 - 81*a^2*b *c*d^7))^(1/3) - (log(((3^(1/2)*1i + 1)^2*(((3^(1/2)*1i + 1)*(27*a^2*b*c^2 *d*x*(a^2*d^2 + b^2*c^2)*(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^5/(b^5*(a*d - b*c)^3))^(2/3))/4)*(a^5/(b^5* (a*d - b*c)^3))^(1/3))/6 + (9*(a*b^7*c^8 + a^8*c*d^7 - a^2*b^6*c^7*d - a^7 *b*c^2*d^6))/(b^2*d^2))*(a^5/(b^5*(a*d - b*c)^3))^(2/3))/36 + (a^4*c^4*x*( a^2*d^2 + b^2*c^2 + a*b*c*d))/(b^2*d^2))*(-a^5/(27*b^8*c^3 - 27*a^3*b^5*d^ 3 + 81*a^2*b^6*c*d^2 - 81*a*b^7*c^2*d))^(1/3)*(3^(1/2)*1i + 1))/2 + (log(( (3^(1/2)*1i - 1)^2*(((3^(1/2)*1i - 1)*(27*a^2*b*c^2*d*x*(a^2*d^2 + b^2*c^2 )*(a*d - b*c)^2 + (27*a*b^3*c*d^3*(3^(1/2)*1i - 1)^2*(a*d + b*c)*(a*d -...
Time = 0.26 (sec) , antiderivative size = 228, normalized size of antiderivative = 0.76 \[ \int \frac {x^7}{\left (a+b x^3\right ) \left (c+d x^3\right )} \, dx=\frac {2 d^{\frac {5}{3}} c^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {a^{\frac {1}{3}}-2 b^{\frac {1}{3}} x}{a^{\frac {1}{3}} \sqrt {3}}\right ) a^{2}-2 b^{\frac {5}{3}} a^{\frac {1}{3}} \sqrt {3}\, \mathit {atan} \left (\frac {c^{\frac {1}{3}}-2 d^{\frac {1}{3}} x}{c^{\frac {1}{3}} \sqrt {3}}\right ) c^{2}+3 d^{\frac {5}{3}} c^{\frac {1}{3}} b^{\frac {2}{3}} a^{\frac {4}{3}} x^{2}-3 d^{\frac {2}{3}} c^{\frac {4}{3}} b^{\frac {5}{3}} a^{\frac {1}{3}} x^{2}+b^{\frac {5}{3}} a^{\frac {1}{3}} \mathrm {log}\left (c^{\frac {2}{3}}-d^{\frac {1}{3}} c^{\frac {1}{3}} x +d^{\frac {2}{3}} x^{2}\right ) c^{2}-2 b^{\frac {5}{3}} a^{\frac {1}{3}} \mathrm {log}\left (c^{\frac {1}{3}}+d^{\frac {1}{3}} x \right ) c^{2}-d^{\frac {5}{3}} c^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {2}{3}}-b^{\frac {1}{3}} a^{\frac {1}{3}} x +b^{\frac {2}{3}} x^{2}\right ) a^{2}+2 d^{\frac {5}{3}} c^{\frac {1}{3}} \mathrm {log}\left (a^{\frac {1}{3}}+b^{\frac {1}{3}} x \right ) a^{2}}{6 d^{\frac {5}{3}} c^{\frac {1}{3}} b^{\frac {5}{3}} a^{\frac {1}{3}} \left (a d -b c \right )} \] Input:
int(x^7/(b*x^3+a)/(d*x^3+c),x)
Output:
(2*d**(2/3)*c**(1/3)*sqrt(3)*atan((a**(1/3) - 2*b**(1/3)*x)/(a**(1/3)*sqrt (3)))*a**2*d - 2*b**(2/3)*a**(1/3)*sqrt(3)*atan((c**(1/3) - 2*d**(1/3)*x)/ (c**(1/3)*sqrt(3)))*b*c**2 + 3*d**(2/3)*c**(1/3)*b**(2/3)*a**(1/3)*a*d*x** 2 - 3*d**(2/3)*c**(1/3)*b**(2/3)*a**(1/3)*b*c*x**2 + b**(2/3)*a**(1/3)*log (c**(2/3) - d**(1/3)*c**(1/3)*x + d**(2/3)*x**2)*b*c**2 - 2*b**(2/3)*a**(1 /3)*log(c**(1/3) + d**(1/3)*x)*b*c**2 - d**(2/3)*c**(1/3)*log(a**(2/3) - b **(1/3)*a**(1/3)*x + b**(2/3)*x**2)*a**2*d + 2*d**(2/3)*c**(1/3)*log(a**(1 /3) + b**(1/3)*x)*a**2*d)/(6*d**(2/3)*c**(1/3)*b**(2/3)*a**(1/3)*b*d*(a*d - b*c))