\(\int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx\) [1661]

Optimal result

Integrand size = 28, antiderivative size = 120 \[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e}+\frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e} \] Output:

-1/4*3^(1/2)*arctan(-1/3*3^(1/2)+1/3*(-3*e*x+d)^(2/3)*3^(1/2)/d^(1/3)/(3*e 
*x+d)^(1/3))/d^(2/3)/e+1/4*ln(e*x+d)/d^(2/3)/e-3/8*ln(-1/2*(-3*e*x+d)^(2/3 
)/d^(1/3)-(3*e*x+d)^(1/3))/d^(2/3)/e
                                                                                    
                                                                                    
 

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(439\) vs. \(2(120)=240\).

Time = 1.14 (sec) , antiderivative size = 439, normalized size of antiderivative = 3.66 \[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d-3 e x}}{2^{2/3} \sqrt [3]{d}+\sqrt [3]{d-3 e x}-\sqrt [3]{2} \sqrt [3]{d+3 e x}}\right )-4 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d-3 e x}}{-2 2^{2/3} \sqrt [3]{d}+\sqrt [3]{d-3 e x}+2 \sqrt [3]{2} \sqrt [3]{d+3 e x}}\right )-4 \log \left (2^{2/3} \sqrt [3]{d}+\sqrt [3]{d-3 e x}-\sqrt [3]{2} \sqrt [3]{d+3 e x}\right )-2 \log \left (-2^{2/3} \sqrt [3]{d}+2 \sqrt [3]{d-3 e x}+\sqrt [3]{2} \sqrt [3]{d+3 e x}\right )+\log \left (2 \sqrt [3]{2} d^{2/3}+4 (d-3 e x)^{2/3}-2 \sqrt [3]{2} \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}+2^{2/3} (d+3 e x)^{2/3}+2 \sqrt [3]{d} \left (2^{2/3} \sqrt [3]{d-3 e x}-2 \sqrt [3]{d+3 e x}\right )\right )+2 \log \left (2 \sqrt [3]{2} d^{2/3}+(d-3 e x)^{2/3}+\sqrt [3]{2} \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}+2^{2/3} (d+3 e x)^{2/3}-\sqrt [3]{d} \left (2^{2/3} \sqrt [3]{d-3 e x}+4 \sqrt [3]{d+3 e x}\right )\right )}{8 d^{2/3} e} \] Input:

Integrate[1/((d - 3*e*x)^(1/3)*(d + e*x)*(d + 3*e*x)^(1/3)),x]
 

Output:

(-2*Sqrt[3]*ArcTan[(Sqrt[3]*(d - 3*e*x)^(1/3))/(2^(2/3)*d^(1/3) + (d - 3*e 
*x)^(1/3) - 2^(1/3)*(d + 3*e*x)^(1/3))] - 4*Sqrt[3]*ArcTan[(Sqrt[3]*(d - 3 
*e*x)^(1/3))/(-2*2^(2/3)*d^(1/3) + (d - 3*e*x)^(1/3) + 2*2^(1/3)*(d + 3*e* 
x)^(1/3))] - 4*Log[2^(2/3)*d^(1/3) + (d - 3*e*x)^(1/3) - 2^(1/3)*(d + 3*e* 
x)^(1/3)] - 2*Log[-(2^(2/3)*d^(1/3)) + 2*(d - 3*e*x)^(1/3) + 2^(1/3)*(d + 
3*e*x)^(1/3)] + Log[2*2^(1/3)*d^(2/3) + 4*(d - 3*e*x)^(2/3) - 2*2^(1/3)*(d 
 - 3*e*x)^(1/3)*(d + 3*e*x)^(1/3) + 2^(2/3)*(d + 3*e*x)^(2/3) + 2*d^(1/3)* 
(2^(2/3)*(d - 3*e*x)^(1/3) - 2*(d + 3*e*x)^(1/3))] + 2*Log[2*2^(1/3)*d^(2/ 
3) + (d - 3*e*x)^(2/3) + 2^(1/3)*(d - 3*e*x)^(1/3)*(d + 3*e*x)^(1/3) + 2^( 
2/3)*(d + 3*e*x)^(2/3) - d^(1/3)*(2^(2/3)*(d - 3*e*x)^(1/3) + 4*(d + 3*e*x 
)^(1/3))])/(8*d^(2/3)*e)
 

Rubi [A] (verified)

Time = 0.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {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.

\(\displaystyle \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx\)

\(\Big \downarrow \) 133

\(\displaystyle \frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e}+\frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e}\)

Input:

Int[1/((d - 3*e*x)^(1/3)*(d + e*x)*(d + 3*e*x)^(1/3)),x]
 

Output:

(Sqrt[3]*ArcTan[1/Sqrt[3] - (d - 3*e*x)^(2/3)/(Sqrt[3]*d^(1/3)*(d + 3*e*x) 
^(1/3))])/(4*d^(2/3)*e) + Log[d + e*x]/(4*d^(2/3)*e) - (3*Log[-1/2*(d - 3* 
e*x)^(2/3)/d^(1/3) - (d + 3*e*x)^(1/3)])/(8*d^(2/3)*e)
 

Defintions of rubi rules used

rule 133

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]
 

Maple [F]

\[\int \frac {1}{\left (-3 e x +d \right )^{\frac {1}{3}} \left (e x +d \right ) \left (3 e x +d \right )^{\frac {1}{3}}}d x\]

Input:

int(1/(-3*e*x+d)^(1/3)/(e*x+d)/(3*e*x+d)^(1/3),x)
 

Output:

int(1/(-3*e*x+d)^(1/3)/(e*x+d)/(3*e*x+d)^(1/3),x)
 

Fricas [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\text {Timed out} \] Input:

integrate(1/(-3*e*x+d)^(1/3)/(e*x+d)/(3*e*x+d)^(1/3),x, algorithm="fricas" 
)
 

Output:

Timed out
 

Sympy [F]

\[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\int \frac {1}{\sqrt [3]{d - 3 e x} \left (d + e x\right ) \sqrt [3]{d + 3 e x}}\, dx \] Input:

integrate(1/(-3*e*x+d)**(1/3)/(e*x+d)/(3*e*x+d)**(1/3),x)
 

Output:

Integral(1/((d - 3*e*x)**(1/3)*(d + e*x)*(d + 3*e*x)**(1/3)), x)
 

Maxima [F]

\[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\int { \frac {1}{{\left (3 \, e x + d\right )}^{\frac {1}{3}} {\left (e x + d\right )} {\left (-3 \, e x + d\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(-3*e*x+d)^(1/3)/(e*x+d)/(3*e*x+d)^(1/3),x, algorithm="maxima" 
)
 

Output:

integrate(1/((3*e*x + d)^(1/3)*(e*x + d)*(-3*e*x + d)^(1/3)), x)
 

Giac [F]

\[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\int { \frac {1}{{\left (3 \, e x + d\right )}^{\frac {1}{3}} {\left (e x + d\right )} {\left (-3 \, e x + d\right )}^{\frac {1}{3}}} \,d x } \] Input:

integrate(1/(-3*e*x+d)^(1/3)/(e*x+d)/(3*e*x+d)^(1/3),x, algorithm="giac")
 

Output:

integrate(1/((3*e*x + d)^(1/3)*(e*x + d)*(-3*e*x + d)^(1/3)), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\int \frac {1}{\left (d+e\,x\right )\,{\left (d-3\,e\,x\right )}^{1/3}\,{\left (d+3\,e\,x\right )}^{1/3}} \,d x \] Input:

int(1/((d + e*x)*(d - 3*e*x)^(1/3)*(d + 3*e*x)^(1/3)),x)
 

Output:

int(1/((d + e*x)*(d - 3*e*x)^(1/3)*(d + 3*e*x)^(1/3)), x)
 

Reduce [F]

\[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\int \frac {1}{\left (3 e x +d \right )^{\frac {1}{3}} \left (-3 e x +d \right )^{\frac {1}{3}} d +\left (3 e x +d \right )^{\frac {1}{3}} \left (-3 e x +d \right )^{\frac {1}{3}} e x}d x \] Input:

int(1/(-3*e*x+d)^(1/3)/(e*x+d)/(3*e*x+d)^(1/3),x)
 

Output:

int(1/((d + 3*e*x)**(1/3)*(d - 3*e*x)**(1/3)*d + (d + 3*e*x)**(1/3)*(d - 3 
*e*x)**(1/3)*e*x),x)