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

Optimal. Leaf size=151 \[ -\frac {\tan ^{-1}\left (\frac {1}{\sqrt {3}}+\frac {2^{2/3} (d-e x)}{\sqrt {3} \sqrt [3]{d} \sqrt [3]{d^2+3 e^2 x^2}}\right )}{2^{2/3} \sqrt {3} d^{2/3} e}-\frac {\log (d+e x)}{2\ 2^{2/3} d^{2/3} e}+\frac {\log \left (3 d e^2-3 e^3 x-3 \sqrt [3]{2} \sqrt [3]{d} e^2 \sqrt [3]{d^2+3 e^2 x^2}\right )}{2\ 2^{2/3} d^{2/3} e} \]

[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)

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Rubi [A]
time = 0.03, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.042, Rules used = {765} \begin {gather*} -\frac {\text {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} \end {gather*}

Antiderivative was successfully verified.

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

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Mathematica [A]
time = 0.33, size = 260, normalized size = 1.72 \begin {gather*} \frac {2 \sqrt {3} \tan ^{-1}\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} \end {gather*}

Antiderivative was successfully verified.

[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)

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e x +d \right ) \left (3 e^{2} x^{2}+d^{2}\right )^{\frac {1}{3}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 333 vs. \(2 (115) = 230\).
time = 62.13, size = 333, normalized size = 2.21 \begin {gather*} -\frac {{\left (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 \, x^{2} e^{2} + d^{2}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} {\left (x e - d\right )} + 4^{\frac {1}{3}} {\left (x^{3} e^{3} + 3 \, d x^{2} e^{2} + 3 \, d^{2} x e + d^{3}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} + 4 \, {\left (d x^{2} e^{2} - 2 \, d^{2} x e + d^{3}\right )} {\left (3 \, x^{2} e^{2} + d^{2}\right )}^{\frac {1}{3}}\right )} \sqrt {4^{\frac {1}{3}} {\left (d^{2}\right )}^{\frac {1}{3}}}}{6 \, {\left (d x^{3} e^{3} - 9 \, d^{2} x^{2} e^{2} + 3 \, d^{3} x e - 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 \, x^{2} e^{2} + d^{2}\right )}^{\frac {2}{3}} {\left (d^{2}\right )}^{\frac {2}{3}} + 4^{\frac {1}{3}} {\left (x^{2} e^{2} - 2 \, d x e + d^{2}\right )} {\left (d^{2}\right )}^{\frac {1}{3}} - 2 \, {\left (3 \, x^{2} e^{2} + d^{2}\right )}^{\frac {1}{3}} {\left (d x e - d^{2}\right )}}{x^{2} e^{2} + 2 \, d x e + 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 (x e - d\right )} + 2 \, {\left (3 \, x^{2} e^{2} + d^{2}\right )}^{\frac {1}{3}} d}{x e + d}\right )\right )} e^{\left (-1\right )}}{24 \, d^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x\right ) \sqrt [3]{d^{2} + 3 e^{2} x^{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (d^2+3\,e^2\,x^2\right )}^{1/3}\,\left (d+e\,x\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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