Integrand size = 24, antiderivative size = 176 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=-\frac {\left (a+b x^3\right )^{2/3}}{2 a c x^2}-\frac {d \arctan \left (\frac {1+\frac {2 \sqrt [3]{b c-a d} x}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}}{\sqrt {3}}\right )}{\sqrt {3} c^{5/3} \sqrt [3]{b c-a d}}-\frac {d \log \left (c+d x^3\right )}{6 c^{5/3} \sqrt [3]{b c-a d}}+\frac {d \log \left (\frac {\sqrt [3]{b c-a d} x}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{5/3} \sqrt [3]{b c-a d}} \]
-1/2*(b*x^3+a)^(2/3)/a/c/x^2-1/6*d*ln(d*x^3+c)/c^(5/3)/(-a*d+b*c)^(1/3)+1/ 2*d*ln((-a*d+b*c)^(1/3)*x/c^(1/3)-(b*x^3+a)^(1/3))/c^(5/3)/(-a*d+b*c)^(1/3 )-1/3*d*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^(5/3)/(-a*d+b*c)^(1/3)*3^(1/2)
Result contains complex when optimal does not.
Time = 2.14 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.78 \[ \int \frac {1}{x^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\frac {-6 c^{2/3} \sqrt [3]{b c-a d} \left (a+b x^3\right )^{2/3}+2 \sqrt {-6+6 i \sqrt {3}} a d x^2 \arctan \left (\frac {3 \sqrt [3]{b c-a d} x}{\sqrt {3} \sqrt [3]{b c-a d} x-\left (3 i+\sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}}\right )-2 i \left (-i+\sqrt {3}\right ) a d x^2 \log \left (2 \sqrt [3]{b c-a d} x+\left (1+i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{a+b x^3}\right )+a \left (d+i \sqrt {3} d\right ) x^2 \log \left (2 (b c-a d)^{2/3} x^2+\left (-1-i \sqrt {3}\right ) \sqrt [3]{c} \sqrt [3]{b c-a d} x \sqrt [3]{a+b x^3}+i \left (i+\sqrt {3}\right ) c^{2/3} \left (a+b x^3\right )^{2/3}\right )}{12 a c^{5/3} \sqrt [3]{b c-a d} x^2} \]
(-6*c^(2/3)*(b*c - a*d)^(1/3)*(a + b*x^3)^(2/3) + 2*Sqrt[-6 + (6*I)*Sqrt[3 ]]*a*d*x^2*ArcTan[(3*(b*c - a*d)^(1/3)*x)/(Sqrt[3]*(b*c - a*d)^(1/3)*x - ( 3*I + Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3))] - (2*I)*(-I + Sqrt[3])*a*d*x^2* Log[2*(b*c - a*d)^(1/3)*x + (1 + I*Sqrt[3])*c^(1/3)*(a + b*x^3)^(1/3)] + a *(d + I*Sqrt[3]*d)*x^2*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)])/(12*a*c^(5/3)*(b*c - a*d)^(1/3)*x^2)
Time = 0.26 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.02, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {980, 27, 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.
\(\displaystyle \int \frac {1}{x^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx\) |
\(\Big \downarrow \) 980 |
\(\displaystyle \frac {\int -\frac {2 a d}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{2 a c}-\frac {\left (a+b x^3\right )^{2/3}}{2 a c x^2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {d \int \frac {1}{\sqrt [3]{b x^3+a} \left (d x^3+c\right )}dx}{c}-\frac {\left (a+b x^3\right )^{2/3}}{2 a c x^2}\) |
\(\Big \downarrow \) 901 |
\(\displaystyle -\frac {d \left (\frac {\arctan \left (\frac {\frac {2 x \sqrt [3]{b c-a d}}{\sqrt [3]{c} \sqrt [3]{a+b x^3}}+1}{\sqrt {3}}\right )}{\sqrt {3} c^{2/3} \sqrt [3]{b c-a d}}+\frac {\log \left (c+d x^3\right )}{6 c^{2/3} \sqrt [3]{b c-a d}}-\frac {\log \left (\frac {x \sqrt [3]{b c-a d}}{\sqrt [3]{c}}-\sqrt [3]{a+b x^3}\right )}{2 c^{2/3} \sqrt [3]{b c-a d}}\right )}{c}-\frac {\left (a+b x^3\right )^{2/3}}{2 a c x^2}\) |
-1/2*(a + b*x^3)^(2/3)/(a*c*x^2) - (d*(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))))/c
3.8.24.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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[((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]
Time = 4.86 (sec) , antiderivative size = 216, normalized size of antiderivative = 1.23
method | result | size |
pseudoelliptic | \(-\frac {\arctan \left (\frac {\sqrt {3}\, \left (\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x -2 \left (b \,x^{3}+a \right )^{\frac {1}{3}}\right )}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x}\right ) \sqrt {3}\, a d \,x^{2}+a \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {1}{3}}}{x}\right ) d \,x^{2}-\frac {a \ln \left (\frac {\left (\frac {a d -b c}{c}\right )^{\frac {2}{3}} x^{2}-\left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} \left (b \,x^{3}+a \right )^{\frac {1}{3}} x +\left (b \,x^{3}+a \right )^{\frac {2}{3}}}{x^{2}}\right ) d \,x^{2}}{2}+\frac {3 \left (b \,x^{3}+a \right )^{\frac {2}{3}} c \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}}}{2}}{3 \left (\frac {a d -b c}{c}\right )^{\frac {1}{3}} a \,c^{2} x^{2}}\) | \(216\) |
-1/3/((a*d-b*c)/c)^(1/3)*(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)*3^(1/2)*a*d*x^2+a*ln((((a*d-b*c)/c)^(1 /3)*x+(b*x^3+a)^(1/3))/x)*d*x^2-1/2*a*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)*d*x^2+3/2*(b*x^3+a)^(2 /3)*c*((a*d-b*c)/c)^(1/3))/a/c^2/x^2
Timed out. \[ \int \frac {1}{x^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\text {Timed out} \]
\[ \int \frac {1}{x^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int \frac {1}{x^{3} \sqrt [3]{a + b x^{3}} \left (c + d x^{3}\right )}\, dx \]
\[ \int \frac {1}{x^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )} x^{3}} \,d x } \]
\[ \int \frac {1}{x^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int { \frac {1}{{\left (b x^{3} + a\right )}^{\frac {1}{3}} {\left (d x^{3} + c\right )} x^{3}} \,d x } \]
Timed out. \[ \int \frac {1}{x^3 \sqrt [3]{a+b x^3} \left (c+d x^3\right )} \, dx=\int \frac {1}{x^3\,{\left (b\,x^3+a\right )}^{1/3}\,\left (d\,x^3+c\right )} \,d x \]