Integrand size = 21, antiderivative size = 267 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^3} \, dx=-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c (b c-a d) \left (c+d x^3\right )^2}+\frac {(6 b c-5 a d) x \left (a+b x^3\right )^{2/3}}{18 c^2 (b c-a d) \left (c+d x^3\right )}+\frac {a (6 b c-5 a d) \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{9 \sqrt {3} c^{8/3} (b c-a d)^{4/3}}+\frac {a (6 b c-5 a d) \log \left (c+d x^3\right )}{54 c^{8/3} (b c-a d)^{4/3}}-\frac {a (6 b c-5 a d) \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{18 c^{8/3} (b c-a d)^{4/3}} \]
-1/6*d*x*(b*x^3+a)^(5/3)/c/(-a*d+b*c)/(d*x^3+c)^2+1/18*(-5*a*d+6*b*c)*x*(b *x^3+a)^(2/3)/c^2/(-a*d+b*c)/(d*x^3+c)+1/54*a*(-5*a*d+6*b*c)*ln(d*x^3+c)/c ^(8/3)/(-a*d+b*c)^(4/3)-1/18*a*(-5*a*d+6*b*c)*ln((-a*d+b*c)^(1/3)*x/c^(1/3 )-(b*x^3+a)^(1/3))/c^(8/3)/(-a*d+b*c)^(4/3)+1/27*a*(-5*a*d+6*b*c)*arctan(1 /3*(1+2*(-a*d+b*c)^(1/3)*x/c^(1/3)/(b*x^3+a)^(1/3))*3^(1/2))/c^(8/3)/(-a*d +b*c)^(4/3)*3^(1/2)
Result contains complex when optimal does not.
Time = 4.30 (sec) , antiderivative size = 366, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^3} \, dx=\frac {\frac {6 c^{2/3} x \left (a+b x^3\right )^{2/3} \left (3 b c \left (2 c+d x^3\right )-a d \left (8 c+5 d x^3\right )\right )}{(b c-a d) \left (c+d x^3\right )^2}+\frac {2 i \left (3 i+\sqrt {3}\right ) a (-6 b c+5 a d) \text {arctanh}\left (\frac {i+\frac {\left (-i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}{\sqrt [3]{b c-a d} x}}{\sqrt {3}}\right )}{(b c-a d)^{4/3}}+\frac {2 \left (1+i \sqrt {3}\right ) a (6 b c-5 a d) \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )}{(b c-a d)^{4/3}}+\frac {\left (1+i \sqrt {3}\right ) a (-6 b c+5 a d) \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{(b c-a d)^{4/3}}}{108 c^{8/3}} \]
((6*c^(2/3)*x*(a + b*x^3)^(2/3)*(3*b*c*(2*c + d*x^3) - a*d*(8*c + 5*d*x^3) ))/((b*c - a*d)*(c + d*x^3)^2) + ((2*I)*(3*I + Sqrt[3])*a*(-6*b*c + 5*a*d) *ArcTanh[(I + ((-I + Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3))/((b*c - a*d)^(1/3 )*x))/Sqrt[3]])/(b*c - a*d)^(4/3) + (2*(1 + I*Sqrt[3])*a*(6*b*c - 5*a*d)*L og[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3)])/(b* c - a*d)^(4/3) + ((1 + I*Sqrt[3])*a*(-6*b*c + 5*a*d)*Log[2*(b*c - a*d)^(2/ 3)*x^2 + (-1 - I*Sqrt[3])*c^(1/3)*(b*c - a*d)^(1/3)*x*(a + b*x^3)^(1/3) + I*(I + Sqrt[3])*c^(2/3)*(a + b*x^3)^(2/3)])/(b*c - a*d)^(4/3))/(108*c^(8/3 ))
Time = 0.31 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.94, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {907, 903, 901}
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.
| \(\displaystyle \int \frac {\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^3} \, dx\) |
| \(\Big \downarrow \) 907 |
| \(\displaystyle \frac {(6 b c-5 a d) \int \frac {\left (b x^3+a\right )^{2/3}}{\left (d x^3+c\right )^2}dx}{6 c (b c-a d)}-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 903 |
| \(\displaystyle \frac {(6 b c-5 a d) \left (\frac {2 a \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{3 c}+\frac {x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}\right )}{6 c (b c-a d)}-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2 (b c-a d)}\) |
| \(\Big \downarrow \) 901 |
| \(\displaystyle \frac {(6 b c-5 a d) \left (\frac {2 a \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{3 c}+\frac {x \left (a+b x^3\right )^{2/3}}{3 c \left (c+d x^3\right )}\right )}{6 c (b c-a d)}-\frac {d x \left (a+b x^3\right )^{5/3}}{6 c \left (c+d x^3\right )^2 (b c-a d)}\) |
-1/6*(d*x*(a + b*x^3)^(5/3))/(c*(b*c - a*d)*(c + d*x^3)^2) + ((6*b*c - 5*a *d)*((x*(a + b*x^3)^(2/3))/(3*c*(c + d*x^3)) + (2*a*(ArcTan[(1 + (2*(b*c - a*d)^(1/3)*x)/(c^(1/3)*(a + b*x^3)^(1/3)))/Sqrt[3]]/(Sqrt[3]*c^(2/3)*(b*c - a*d)^(1/3)) + Log[c + d*x^3]/(6*c^(2/3)*(b*c - a*d)^(1/3)) - Log[((b*c - a*d)^(1/3)*x)/c^(1/3) - (a + b*x^3)^(1/3)]/(2*c^(2/3)*(b*c - a*d)^(1/3)) ))/(3*c)))/(6*c*(b*c - a*d))
3.2.13.3.1 Defintions of rubi rules used
Int[1/(((a_) + (b_.)*(x_)^3)^(1/3)*((c_) + (d_.)*(x_)^3)), x_Symbol] :> Wit h[{q = Rt[(b*c - a*d)/c, 3]}, Simp[ArcTan[(1 + (2*q*x)/(a + b*x^3)^(1/3))/S qrt[3]]/(Sqrt[3]*c*q), x] + (-Simp[Log[q*x - (a + b*x^3)^(1/3)]/(2*c*q), x] + Simp[Log[c + d*x^3]/(6*c*q), x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[(-x)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*n*(p + 1))), x] - Simp[ c*(q/(a*(p + 1))) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(-b)*x*(a + b*x^n)^(p + 1)*((c + d*x^n)^(q + 1)/(a*n*(p + 1)*(b*c - a*d))), x] + Simp[(b*c + n*(p + 1)*(b*c - a*d))/(a*n*(p + 1)*(b*c - a*d)) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, n, q} , x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 2) + 1, 0] && (LtQ[p, -1] || ! LtQ[q, -1]) && NeQ[p, -1]
Time = 4.32 (sec) , antiderivative size = 297, normalized size of antiderivative = 1.11
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {5 \left (a d -\frac {6 b c}{5}\right ) a \left (d \,x^{3}+c \right )^{2} \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right )}{54}+\frac {5 \left (a d -\frac {6 b c}{5}\right ) a \left (d \,x^{3}+c \right )^{2} \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right )}{27}+\frac {4 \left (\frac {\left (5 a \,d^{2}-3 b c d \right ) x^{3}}{8}+c \left (a d -\frac {3 b c}{4}\right )\right ) x c \left (b \,x^{3}+a \right )^{\frac {2}{3}} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}{9}+\frac {5 \left (a d -\frac {6 b c}{5}\right ) \sqrt {3}\, a \arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \left (d \,x^{3}+c \right )^{2}}{27}}{\left (a d -b c \right ) c^{3} \left (d \,x^{3}+c \right )^{2} \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}\) | \(297\) |
5/27/((a*d-b*c)/c)^(1/3)*(-1/2*(a*d-6/5*b*c)*a*(d*x^3+c)^2*ln((((a*d-b*c)/ c)^(2/3)*x^2-((a*d-b*c)/c)^(1/3)*(b*x^3+a)^(1/3)*x+(b*x^3+a)^(2/3))/x^2)+( a*d-6/5*b*c)*a*(d*x^3+c)^2*ln((((a*d-b*c)/c)^(1/3)*x+(b*x^3+a)^(1/3))/x)+1 2/5*(1/8*(5*a*d^2-3*b*c*d)*x^3+c*(a*d-3/4*b*c))*x*c*(b*x^3+a)^(2/3)*((a*d- b*c)/c)^(1/3)+(a*d-6/5*b*c)*3^(1/2)*a*arctan(1/3*3^(1/2)*(((a*d-b*c)/c)^(1 /3)*x-2*(b*x^3+a)^(1/3))/((a*d-b*c)/c)^(1/3)/x)*(d*x^3+c)^2)/(a*d-b*c)/c^3 /(d*x^3+c)^2
Timed out. \[ \int \frac {\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^3} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^3} \, dx=\int \frac {\left (a + b x^{3}\right )^{\frac {2}{3}}}{\left (c + d x^{3}\right )^{3}}\, dx \]
\[ \int \frac {\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \]
\[ \int \frac {\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^3} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {2}{3}}}{{\left (d x^{3} + c\right )}^{3}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{2/3}}{\left (c+d x^3\right )^3} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{2/3}}{{\left (d\,x^3+c\right )}^3} \,d x \]