\(\int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx\) [3036]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.02 (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 = {124} \[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\frac {\sqrt {3} \arctan \left (\frac {1}{\sqrt {3}}-\frac {(d-3 e x)^{2/3}}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d+3 e x}}\right )}{4 d^{2/3} e}+\frac {\log (d+e x)}{4 d^{2/3} e}-\frac {3 \log \left (-\frac {(d-3 e x)^{2/3}}{2 \sqrt [3]{d}}-\sqrt [3]{d+3 e x}\right )}{8 d^{2/3} e} \]

[In]

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

[Out]

(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)

Rule 124

Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(1/3)*((e_.) + (f_.)*(x_))^(1/3)), x_Symbol] :> 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]

Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {3} \tan ^{-1}\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} \\ \end{align*}

Mathematica [B] (verified)

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

Time = 1.25 (sec) , antiderivative size = 439, normalized size of antiderivative = 3.66 \[ \int \frac {1}{\sqrt [3]{d-3 e x} (d+e x) \sqrt [3]{d+3 e x}} \, dx=\frac {-2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d-3 e x}}{2^{2/3} \sqrt [3]{d}+\sqrt [3]{d-3 e x}-\sqrt [3]{2} \sqrt [3]{d+3 e x}}\right )-4 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d-3 e x}}{-2 2^{2/3} \sqrt [3]{d}+\sqrt [3]{d-3 e x}+2 \sqrt [3]{2} \sqrt [3]{d+3 e x}}\right )-4 \log \left (2^{2/3} \sqrt [3]{d}+\sqrt [3]{d-3 e x}-\sqrt [3]{2} \sqrt [3]{d+3 e x}\right )-2 \log \left (-2^{2/3} \sqrt [3]{d}+2 \sqrt [3]{d-3 e x}+\sqrt [3]{2} \sqrt [3]{d+3 e x}\right )+\log \left (2 \sqrt [3]{2} d^{2/3}+4 (d-3 e x)^{2/3}-2 \sqrt [3]{2} \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}+2^{2/3} (d+3 e x)^{2/3}+2 \sqrt [3]{d} \left (2^{2/3} \sqrt [3]{d-3 e x}-2 \sqrt [3]{d+3 e x}\right )\right )+2 \log \left (2 \sqrt [3]{2} d^{2/3}+(d-3 e x)^{2/3}+\sqrt [3]{2} \sqrt [3]{d-3 e x} \sqrt [3]{d+3 e x}+2^{2/3} (d+3 e x)^{2/3}-\sqrt [3]{d} \left (2^{2/3} \sqrt [3]{d-3 e x}+4 \sqrt [3]{d+3 e x}\right )\right )}{8 d^{2/3} e} \]

[In]

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

[Out]

(-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)

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-1)]

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

[In]

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

[Out]

Timed out

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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