### 3.703 $$\int \frac{1}{(2+3 x) \sqrt [3]{4+27 x^2}} \, dx$$

Optimal. Leaf size=97 $\frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{12 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{6 \sqrt [3]{2} \sqrt{3}}-\frac{\log (3 x+2)}{12 \sqrt [3]{2}}$

[Out]

-ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - 3*x))/(Sqrt[3]*(4 + 27*x^2)^(1/3))]/(6*2^(1/3)*Sqrt[3]) - Log[2 + 3*x]/(12*2
^(1/3)) + Log[54 - 81*x - 27*2^(2/3)*(4 + 27*x^2)^(1/3)]/(12*2^(1/3))

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Rubi [A]  time = 0.0143885, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 19, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.053, Rules used = {751} $\frac{\log \left (-27\ 2^{2/3} \sqrt [3]{27 x^2+4}-81 x+54\right )}{12 \sqrt [3]{2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{27 x^2+4}}+\frac{1}{\sqrt{3}}\right )}{6 \sqrt [3]{2} \sqrt{3}}-\frac{\log (3 x+2)}{12 \sqrt [3]{2}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((2 + 3*x)*(4 + 27*x^2)^(1/3)),x]

[Out]

-ArcTan[1/Sqrt[3] + (2^(1/3)*(2 - 3*x))/(Sqrt[3]*(4 + 27*x^2)^(1/3))]/(6*2^(1/3)*Sqrt[3]) - Log[2 + 3*x]/(12*2
^(1/3)) + Log[54 - 81*x - 27*2^(2/3)*(4 + 27*x^2)^(1/3)]/(12*2^(1/3))

Rule 751

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}{(2+3 x) \sqrt [3]{4+27 x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{\sqrt [3]{2} (2-3 x)}{\sqrt{3} \sqrt [3]{4+27 x^2}}\right )}{6 \sqrt [3]{2} \sqrt{3}}-\frac{\log (2+3 x)}{12 \sqrt [3]{2}}+\frac{\log \left (54-81 x-27\ 2^{2/3} \sqrt [3]{4+27 x^2}\right )}{12 \sqrt [3]{2}}\\ \end{align*}

Mathematica [C]  time = 0.0776071, size = 121, normalized size = 1.25 $-\frac{\sqrt [3]{\frac{9 x-2 i \sqrt{3}}{3 x+2}} \sqrt [3]{\frac{9 x+2 i \sqrt{3}}{3 x+2}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{6-2 i \sqrt{3}}{9 x+6},\frac{6+2 i \sqrt{3}}{9 x+6}\right )}{2\ 3^{2/3} \sqrt [3]{27 x^2+4}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((2 + 3*x)*(4 + 27*x^2)^(1/3)),x]

[Out]

-((((-2*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*(((2*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3
, (6 - (2*I)*Sqrt[3])/(6 + 9*x), (6 + (2*I)*Sqrt[3])/(6 + 9*x)])/(2*3^(2/3)*(4 + 27*x^2)^(1/3))

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Maple [F]  time = 0.362, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{2+3\,x}{\frac{1}{\sqrt [3]{27\,{x}^{2}+4}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2+3*x)/(27*x^2+4)^(1/3),x)

[Out]

int(1/(2+3*x)/(27*x^2+4)^(1/3),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(27*x^2+4)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((27*x^2 + 4)^(1/3)*(3*x + 2)), x)

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Fricas [B]  time = 28.3654, size = 571, normalized size = 5.89 \begin{align*} -\frac{1}{36} \, \sqrt{6} 2^{\frac{1}{6}} \arctan \left (\frac{2^{\frac{1}{6}}{\left (4 \, \sqrt{6} 2^{\frac{2}{3}}{\left (27 \, x^{2} + 4\right )}^{\frac{2}{3}}{\left (3 \, x - 2\right )} + \sqrt{6} 2^{\frac{1}{3}}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} + 4 \, \sqrt{6}{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (9 \, x^{2} - 12 \, x + 4\right )}\right )}}{18 \,{\left (9 \, x^{3} - 54 \, x^{2} + 12 \, x - 8\right )}}\right ) - \frac{1}{72} \cdot 2^{\frac{2}{3}} \log \left (\frac{2 \cdot 2^{\frac{2}{3}}{\left (27 \, x^{2} + 4\right )}^{\frac{2}{3}} + 2^{\frac{1}{3}}{\left (9 \, x^{2} - 12 \, x + 4\right )} - 2 \,{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 \, x - 2\right )}}{9 \, x^{2} + 12 \, x + 4}\right ) + \frac{1}{36} \cdot 2^{\frac{2}{3}} \log \left (\frac{2^{\frac{1}{3}}{\left (3 \, x - 2\right )} + 2 \,{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}}{3 \, x + 2}\right ) \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(27*x^2+4)^(1/3),x, algorithm="fricas")

[Out]

-1/36*sqrt(6)*2^(1/6)*arctan(1/18*2^(1/6)*(4*sqrt(6)*2^(2/3)*(27*x^2 + 4)^(2/3)*(3*x - 2) + sqrt(6)*2^(1/3)*(2
7*x^3 + 54*x^2 + 36*x + 8) + 4*sqrt(6)*(27*x^2 + 4)^(1/3)*(9*x^2 - 12*x + 4))/(9*x^3 - 54*x^2 + 12*x - 8)) - 1
/72*2^(2/3)*log((2*2^(2/3)*(27*x^2 + 4)^(2/3) + 2^(1/3)*(9*x^2 - 12*x + 4) - 2*(27*x^2 + 4)^(1/3)*(3*x - 2))/(
9*x^2 + 12*x + 4)) + 1/36*2^(2/3)*log((2^(1/3)*(3*x - 2) + 2*(27*x^2 + 4)^(1/3))/(3*x + 2))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (3 x + 2\right ) \sqrt [3]{27 x^{2} + 4}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(27*x**2+4)**(1/3),x)

[Out]

Integral(1/((3*x + 2)*(27*x**2 + 4)**(1/3)), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (27 \, x^{2} + 4\right )}^{\frac{1}{3}}{\left (3 \, x + 2\right )}}\,{d x} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(27*x^2+4)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((27*x^2 + 4)^(1/3)*(3*x + 2)), x)