\(\int \frac {1}{(d+e x) \sqrt [3]{d^2+3 e^2 x^2}} \, dx\) [699]

Optimal result

Integrand size = 24, antiderivative size = 151 \[ \int \frac {1}{(d+e x) \sqrt [3]{d^2+3 e^2 x^2}} \, dx=-\frac {\arctan \left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (d-e x)}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}\right )}{2^{2/3} \sqrt {3} d^{2/3} e}-\frac {\log (d+e x)}{2\ 2^{2/3} d^{2/3} e}+\frac {\log \left (3 d e^2-3 e^3 x-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}\right )}{2\ 2^{2/3} d^{2/3} e} \]

[Out]

-1/4*ln(e*x+d)*2^(1/3)/d^(2/3)/e+1/4*ln(3*d*e^2-3*e^3*x-3*2^(1/3)*d^(1/3)*e^2*(3*e^2*x^2+d^2)^(1/3))*2^(1/3)/d
^(2/3)/e-1/6*arctan(1/3*3^(1/2)+1/3*2^(2/3)*(-e*x+d)/d^(1/3)/(3*e^2*x^2+d^2)^(1/3)*3^(1/2))*2^(1/3)/d^(2/3)/e*
3^(1/2)

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {765} \[ \int \frac {1}{(d+e x) \sqrt [3]{d^2+3 e^2 x^2}} \, dx=-\frac {\arctan \left (\frac {2^{2/3} (d-e x)}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}+\frac {1}{\sqrt {3}}\right )}{2^{2/3} \sqrt {3} d^{2/3} e}-\frac {\log (d+e x)}{2\ 2^{2/3} d^{2/3} e}+\frac {\log \left (-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}+3 d e^2-3 e^3 x\right )}{2\ 2^{2/3} d^{2/3} e} \]

[In]

Int[1/((d + e*x)*(d^2 + 3*e^2*x^2)^(1/3)),x]

[Out]

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

Rule 765

Int[1/(((d_) + (e_.)*(x_))*((a_) + (c_.)*(x_)^2)^(1/3)), x_Symbol] :> With[{q = Rt[6*c^2*(e^2/d^2), 3]}, Simp[
(-Sqrt[3])*c*e*(ArcTan[1/Sqrt[3] + 2*c*((d - e*x)/(Sqrt[3]*d*q*(a + c*x^2)^(1/3)))]/(d^2*q^2)), x] + (-Simp[3*
c*e*(Log[d + e*x]/(2*d^2*q^2)), x] + Simp[3*c*e*(Log[c*d - c*e*x - d*q*(a + c*x^2)^(1/3)]/(2*d^2*q^2)), x])] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - 3*a*e^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (d-e x)}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}\right )}{2^{2/3} \sqrt {3} d^{2/3} e}-\frac {\log (d+e x)}{2\ 2^{2/3} d^{2/3} e}+\frac {\log \left (3 d e^2-3 e^3 x-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}\right )}{2\ 2^{2/3} d^{2/3} e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.53 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(d+e x) \sqrt [3]{d^2+3 e^2 x^2}} \, dx=\frac {2 \sqrt {3} \arctan \left (\frac {\sqrt {3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}{2^{2/3} d-2^{2/3} e x+\sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}\right )+2 \log \left (\sqrt {e} \left (-2^{2/3} d+2^{2/3} e x+2 \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}\right )\right )-\log \left (e \left (\sqrt [3]{2} d^2-2 \sqrt [3]{2} d e x+\sqrt [3]{2} e^2 x^2+2^{2/3} d^{4/3} \sqrt [3]{d^2+3 e^2 x^2}-2^{2/3} \sqrt [3]{d} e x \sqrt [3]{d^2+3 e^2 x^2}+2 d^{2/3} \left (d^2+3 e^2 x^2\right )^{2/3}\right )\right )}{6\ 2^{2/3} d^{2/3} e} \]

[In]

Integrate[1/((d + e*x)*(d^2 + 3*e^2*x^2)^(1/3)),x]

[Out]

(2*Sqrt[3]*ArcTan[(Sqrt[3]*d^(1/3)*(d^2 + 3*e^2*x^2)^(1/3))/(2^(2/3)*d - 2^(2/3)*e*x + d^(1/3)*(d^2 + 3*e^2*x^
2)^(1/3))] + 2*Log[Sqrt[e]*(-(2^(2/3)*d) + 2^(2/3)*e*x + 2*d^(1/3)*(d^2 + 3*e^2*x^2)^(1/3))] - Log[e*(2^(1/3)*
d^2 - 2*2^(1/3)*d*e*x + 2^(1/3)*e^2*x^2 + 2^(2/3)*d^(4/3)*(d^2 + 3*e^2*x^2)^(1/3) - 2^(2/3)*d^(1/3)*e*x*(d^2 +
3*e^2*x^2)^(1/3) + 2*d^(2/3)*(d^2 + 3*e^2*x^2)^(2/3))])/(6*2^(2/3)*d^(2/3)*e)

Maple [F]

\[\int \frac {1}{\left (e x +d \right ) \left (3 x^{2} e^{2}+d^{2}\right )^{\frac {1}{3}}}d x\]

[In]

int(1/(e*x+d)/(3*e^2*x^2+d^2)^(1/3),x)

[Out]

int(1/(e*x+d)/(3*e^2*x^2+d^2)^(1/3),x)

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 337 vs. \(2 (121) = 242\).

Time = 25.57 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.23 \[ \int \frac {1}{(d+e x) \sqrt [3]{d^2+3 e^2 x^2}} \, dx=-\frac {4 \, \sqrt {3} d \sqrt {4^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}}} \arctan \left (\frac {\sqrt {3} {\left (2 \cdot 4^{\frac {2}{3}} {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} {\left (e x - d\right )} + 4^{\frac {1}{3}} {\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 4 \, {\left (d e^{2} x^{2} - 2 \, d^{2} e x + d^{3}\right )} {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}}\right )} \sqrt {4^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}}}}{6 \, {\left (d e^{3} x^{3} - 9 \, d^{2} e^{2} x^{2} + 3 \, d^{3} e x - 3 \, d^{4}\right )}}\right ) + 4^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {4^{\frac {2}{3}} {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (e^{2} x^{2} - 2 \, d e x + d^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} - 2 \, {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} {\left (d e x - d^{2}\right )}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \cdot 4^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} \log \left (\frac {4^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}} {\left (e x - d\right )} + 2 \, {\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} d}{e x + d}\right )}{24 \, d^{2} e} \]

[In]

integrate(1/(e*x+d)/(3*e^2*x^2+d^2)^(1/3),x, algorithm="fricas")

[Out]

-1/24*(4*sqrt(3)*d*sqrt(4^(1/3)*(d^2)^(1/3))*arctan(1/6*sqrt(3)*(2*4^(2/3)*(3*e^2*x^2 + d^2)^(2/3)*(d^2)^(2/3)
*(e*x - d) + 4^(1/3)*(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*(d^2)^(1/3) + 4*(d*e^2*x^2 - 2*d^2*e*x + d^3)*(
3*e^2*x^2 + d^2)^(1/3))*sqrt(4^(1/3)*(d^2)^(1/3))/(d*e^3*x^3 - 9*d^2*e^2*x^2 + 3*d^3*e*x - 3*d^4)) + 4^(2/3)*(
d^2)^(2/3)*log((4^(2/3)*(3*e^2*x^2 + d^2)^(2/3)*(d^2)^(2/3) + 4^(1/3)*(e^2*x^2 - 2*d*e*x + d^2)*(d^2)^(1/3) -
2*(3*e^2*x^2 + d^2)^(1/3)*(d*e*x - d^2))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*4^(2/3)*(d^2)^(2/3)*log((4^(1/3)*(d^2)
^(1/3)*(e*x - d) + 2*(3*e^2*x^2 + d^2)^(1/3)*d)/(e*x + d)))/(d^2*e)

Sympy [F]

\[ \int \frac {1}{(d+e x) \sqrt [3]{d^2+3 e^2 x^2}} \, dx=\int \frac {1}{\left (d + e x\right ) \sqrt [3]{d^{2} + 3 e^{2} x^{2}}}\, dx \]

[In]

integrate(1/(e*x+d)/(3*e**2*x**2+d**2)**(1/3),x)

[Out]

Integral(1/((d + e*x)*(d**2 + 3*e**2*x**2)**(1/3)), x)

Maxima [F]

\[ \int \frac {1}{(d+e x) \sqrt [3]{d^2+3 e^2 x^2}} \, dx=\int { \frac {1}{{\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}} \,d x } \]

[In]

integrate(1/(e*x+d)/(3*e^2*x^2+d^2)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((3*e^2*x^2 + d^2)^(1/3)*(e*x + d)), x)

Giac [F]

\[ \int \frac {1}{(d+e x) \sqrt [3]{d^2+3 e^2 x^2}} \, dx=\int { \frac {1}{{\left (3 \, e^{2} x^{2} + d^{2}\right )}^{\frac {1}{3}} {\left (e x + d\right )}} \,d x } \]

[In]

integrate(1/(e*x+d)/(3*e^2*x^2+d^2)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((3*e^2*x^2 + d^2)^(1/3)*(e*x + d)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{(d+e x) \sqrt [3]{d^2+3 e^2 x^2}} \, dx=\int \frac {1}{{\left (d^2+3\,e^2\,x^2\right )}^{1/3}\,\left (d+e\,x\right )} \,d x \]

[In]

int(1/((d^2 + 3*e^2*x^2)^(1/3)*(d + e*x)),x)

[Out]

int(1/((d^2 + 3*e^2*x^2)^(1/3)*(d + e*x)), x)