Integrand size = 24, antiderivative size = 206 \[ \int \frac {1}{(d+e x) \sqrt [3]{d^2-9 e^2 x^2}} \, dx=\frac {\sqrt {3} \sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \arctan \left (\frac {1}{\sqrt {3}}-\frac {\left (1-\frac {3 e x}{d}\right )^{2/3}}{\sqrt {3} \sqrt [3]{1+\frac {3 e x}{d}}}\right )}{4 e \sqrt [3]{d^2-9 e^2 x^2}}+\frac {\sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \log (d+e x)}{4 e \sqrt [3]{d^2-9 e^2 x^2}}-\frac {3 \sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \log \left (-\frac {1}{2} \left (1-\frac {3 e x}{d}\right )^{2/3}-\sqrt [3]{1+\frac {3 e x}{d}}\right )}{8 e \sqrt [3]{d^2-9 e^2 x^2}} \]
1/4*(1-9*e^2*x^2/d^2)^(1/3)*ln(e*x+d)/e/(-9*e^2*x^2+d^2)^(1/3)-3/8*(1-9*e^ 2*x^2/d^2)^(1/3)*ln(-1/2*(1-3*e*x/d)^(2/3)-(1+3*e*x/d)^(1/3))/e/(-9*e^2*x^ 2+d^2)^(1/3)-1/4*(1-9*e^2*x^2/d^2)^(1/3)*arctan(-1/3*3^(1/2)+1/3*(1-3*e*x/ d)^(2/3)/(1+3*e*x/d)^(1/3)*3^(1/2))*3^(1/2)/e/(-9*e^2*x^2+d^2)^(1/3)
Result contains complex when optimal does not.
Time = 1.13 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.32 \[ \int \frac {1}{(d+e x) \sqrt [3]{d^2-9 e^2 x^2}} \, dx=\frac {2 \sqrt {-6+6 i \sqrt {3}} \arctan \left (\frac {2 \sqrt {3} \sqrt [3]{d} \sqrt [3]{d^2-9 e^2 x^2}}{d-i \sqrt {3} d+3 i \left (i+\sqrt {3}\right ) e x+2 \sqrt [3]{d} \sqrt [3]{d^2-9 e^2 x^2}}\right )+\left (1+i \sqrt {3}\right ) \left (2 \log \left (\sqrt {e} \left (-d+i \sqrt {3} d+3 e x-3 i \sqrt {3} e x+4 \sqrt [3]{d} \sqrt [3]{d^2-9 e^2 x^2}\right )\right )-\log \left (e \left (d-3 e x+2 \sqrt [3]{d} \sqrt [3]{d^2-9 e^2 x^2}\right ) \left (-d-i \sqrt {3} d+3 e x+3 i \sqrt {3} e x+4 \sqrt [3]{d} \sqrt [3]{d^2-9 e^2 x^2}\right )\right )\right )}{16 d^{2/3} e} \]
(2*Sqrt[-6 + (6*I)*Sqrt[3]]*ArcTan[(2*Sqrt[3]*d^(1/3)*(d^2 - 9*e^2*x^2)^(1 /3))/(d - I*Sqrt[3]*d + (3*I)*(I + Sqrt[3])*e*x + 2*d^(1/3)*(d^2 - 9*e^2*x ^2)^(1/3))] + (1 + I*Sqrt[3])*(2*Log[Sqrt[e]*(-d + I*Sqrt[3]*d + 3*e*x - ( 3*I)*Sqrt[3]*e*x + 4*d^(1/3)*(d^2 - 9*e^2*x^2)^(1/3))] - Log[e*(d - 3*e*x + 2*d^(1/3)*(d^2 - 9*e^2*x^2)^(1/3))*(-d - I*Sqrt[3]*d + 3*e*x + (3*I)*Sqr t[3]*e*x + 4*d^(1/3)*(d^2 - 9*e^2*x^2)^(1/3))]))/(16*d^(2/3)*e)
Time = 0.24 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.68, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {503, 502, 133}
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.
\(\displaystyle \int \frac {1}{(d+e x) \sqrt [3]{d^2-9 e^2 x^2}} \, dx\) |
\(\Big \downarrow \) 503 |
\(\displaystyle \frac {\sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \int \frac {1}{(d+e x) \sqrt [3]{1-\frac {9 e^2 x^2}{d^2}}}dx}{\sqrt [3]{d^2-9 e^2 x^2}}\) |
\(\Big \downarrow \) 502 |
\(\displaystyle \frac {\sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \int \frac {1}{(d+e x) \sqrt [3]{1-\frac {3 e x}{d}} \sqrt [3]{\frac {3 e x}{d}+1}}dx}{\sqrt [3]{d^2-9 e^2 x^2}}\) |
\(\Big \downarrow \) 133 |
\(\displaystyle \frac {\sqrt [3]{1-\frac {9 e^2 x^2}{d^2}} \left (\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {\left (1-\frac {3 e x}{d}\right )^{2/3}}{\sqrt {3} \sqrt [3]{\frac {3 e x}{d}+1}}\right )}{4 e}+\frac {\log (d+e x)}{4 e}-\frac {3 \log \left (-\frac {1}{2} \left (1-\frac {3 e x}{d}\right )^{2/3}-\sqrt [3]{\frac {3 e x}{d}+1}\right )}{8 e}\right )}{\sqrt [3]{d^2-9 e^2 x^2}}\) |
((1 - (9*e^2*x^2)/d^2)^(1/3)*((Sqrt[3]*ArcTan[1/Sqrt[3] - (1 - (3*e*x)/d)^ (2/3)/(Sqrt[3]*(1 + (3*e*x)/d)^(1/3))])/(4*e) + Log[d + e*x]/(4*e) - (3*Lo g[-1/2*(1 - (3*e*x)/d)^(2/3) - (1 + (3*e*x)/d)^(1/3)])/(8*e)))/(d^2 - 9*e^ 2*x^2)^(1/3)
3.8.16.3.1 Defintions of rubi rules used
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_)) ^(1/3)), x_] :> With[{q = Rt[b*((b*e - a*f)/(b*c - a*d)^2), 3]}, Simp[-Log[ a + b*x]/(2*q*(b*c - a*d)), x] + (-Simp[Sqrt[3]*(ArcTan[1/Sqrt[3] + 2*q*((c + d*x)^(2/3)/(Sqrt[3]*(e + f*x)^(1/3)))]/(2*q*(b*c - a*d))), x] + Simp[3*( Log[q*(c + d*x)^(2/3) - (e + f*x)^(1/3)]/(4*q*(b*c - a*d))), x])] /; FreeQ[ {a, b, c, d, e, f}, x] && EqQ[2*b*d*e - b*c*f - a*d*f, 0]
Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(1/3)), x_Symbol] :> Simp[ a^(1/3) Int[1/((c + d*x)*(1 - 3*d*(x/c))^(1/3)*(1 + 3*d*(x/c))^(1/3)), x] , x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + 9*a*d^2, 0] && GtQ[a, 0]
Int[1/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)^(1/3)), x_Symbol] :> Simp[ (1 + b*(x^2/a))^(1/3)/(a + b*x^2)^(1/3) Int[1/((c + d*x)*(1 + b*(x^2/a))^ (1/3)), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c^2 + 9*a*d^2, 0] && !Gt Q[a, 0]
\[\int \frac {1}{\left (e x +d \right ) \left (-9 x^{2} e^{2}+d^{2}\right )^{\frac {1}{3}}}d x\]
Timed out. \[ \int \frac {1}{(d+e x) \sqrt [3]{d^2-9 e^2 x^2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(d+e x) \sqrt [3]{d^2-9 e^2 x^2}} \, dx=\int \frac {1}{\sqrt [3]{- \left (- d + 3 e x\right ) \left (d + 3 e x\right )} \left (d + e x\right )}\, dx \]
\[ \int \frac {1}{(d+e x) \sqrt [3]{d^2-9 e^2 x^2}} \, dx=\int { \frac {1}{{\left (-9 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}} \,d x } \]
\[ \int \frac {1}{(d+e x) \sqrt [3]{d^2-9 e^2 x^2}} \, dx=\int { \frac {1}{{\left (-9 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}} \,d x } \]
Timed out. \[ \int \frac {1}{(d+e x) \sqrt [3]{d^2-9 e^2 x^2}} \, dx=\int \frac {1}{{\left (d^2-9\,e^2\,x^2\right )}^{1/3}\,\left (d+e\,x\right )} \,d x \]