### 3.2506 $$\int \frac{1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx$$

Optimal. Leaf size=103 $\frac{\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{6\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt [3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt{3}}-\frac{\log (3 x+2)}{6\ 2^{2/3}}$

[Out]

-ArcTan[1/Sqrt[3] + (2^(2/3)*(4 + 3*x))/(Sqrt[3]*(28 + 54*x + 27*x^2)^(1/3))]/(3*2^(2/3)*Sqrt[3]) - Log[2 + 3*
x]/(6*2^(2/3)) + Log[-108 - 81*x + 27*2^(1/3)*(28 + 54*x + 27*x^2)^(1/3)]/(6*2^(2/3))

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Rubi [A]  time = 0.0187359, antiderivative size = 103, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 22, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.045, Rules used = {752} $\frac{\log \left (27 \sqrt [3]{2} \sqrt [3]{27 x^2+54 x+28}-81 x-108\right )}{6\ 2^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (3 x+4)}{\sqrt{3} \sqrt [3]{27 x^2+54 x+28}}+\frac{1}{\sqrt{3}}\right )}{3\ 2^{2/3} \sqrt{3}}-\frac{\log (3 x+2)}{6\ 2^{2/3}}$

Antiderivative was successfully veriﬁed.

[In]

Int[1/((2 + 3*x)*(28 + 54*x + 27*x^2)^(1/3)),x]

[Out]

-ArcTan[1/Sqrt[3] + (2^(2/3)*(4 + 3*x))/(Sqrt[3]*(28 + 54*x + 27*x^2)^(1/3))]/(3*2^(2/3)*Sqrt[3]) - Log[2 + 3*
x]/(6*2^(2/3)) + Log[-108 - 81*x + 27*2^(1/3)*(28 + 54*x + 27*x^2)^(1/3)]/(6*2^(2/3))

Rule 752

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

Rubi steps

\begin{align*} \int \frac{1}{(2+3 x) \sqrt [3]{28+54 x+27 x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2^{2/3} (4+3 x)}{\sqrt{3} \sqrt [3]{28+54 x+27 x^2}}\right )}{3\ 2^{2/3} \sqrt{3}}-\frac{\log (2+3 x)}{6\ 2^{2/3}}+\frac{\log \left (-108-81 x+27 \sqrt [3]{2} \sqrt [3]{28+54 x+27 x^2}\right )}{6\ 2^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0767898, size = 127, normalized size = 1.23 $-\frac{\sqrt [3]{\frac{9 x-i \sqrt{3}+9}{3 x+2}} \sqrt [3]{\frac{9 x+i \sqrt{3}+9}{3 x+2}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};-\frac{3+i \sqrt{3}}{9 x+6},\frac{-3+i \sqrt{3}}{9 x+6}\right )}{2\ 3^{2/3} \sqrt [3]{27 x^2+54 x+28}}$

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((2 + 3*x)*(28 + 54*x + 27*x^2)^(1/3)),x]

[Out]

-(((9 - I*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*((9 + I*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3, 1/3, 5/3,
-((3 + I*Sqrt[3])/(6 + 9*x)), (-3 + I*Sqrt[3])/(6 + 9*x)])/(2*3^(2/3)*(28 + 54*x + 27*x^2)^(1/3))

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Maple [F]  time = 1.678, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{2+3\,x}{\frac{1}{\sqrt [3]{27\,{x}^{2}+54\,x+28}}}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2+3*x)/(27*x^2+54*x+28)^(1/3),x)

[Out]

int(1/(2+3*x)/(27*x^2+54*x+28)^(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} + 54 \, x + 28\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+54*x+28)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((27*x^2 + 54*x + 28)^(1/3)*(3*x + 2)), x)

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Fricas [B]  time = 25.4418, size = 626, normalized size = 6.08 \begin{align*} -\frac{1}{18} \cdot 4^{\frac{1}{6}} \sqrt{3} \arctan \left (\frac{4^{\frac{1}{6}}{\left (2 \cdot 4^{\frac{2}{3}} \sqrt{3}{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{2}{3}}{\left (3 \, x + 4\right )} + 4^{\frac{1}{3}} \sqrt{3}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} - 4 \, \sqrt{3}{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}{\left (9 \, x^{2} + 24 \, x + 16\right )}\right )}}{18 \,{\left (9 \, x^{3} + 54 \, x^{2} + 84 \, x + 40\right )}}\right ) - \frac{1}{72} \cdot 4^{\frac{2}{3}} \log \left (\frac{4^{\frac{2}{3}}{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{2}{3}} + 4^{\frac{1}{3}}{\left (9 \, x^{2} + 24 \, x + 16\right )} + 2 \,{\left (27 \, x^{2} + 54 \, x + 28\right )}^{\frac{1}{3}}{\left (3 \, x + 4\right )}}{9 \, x^{2} + 12 \, x + 4}\right ) + \frac{1}{36} \cdot 4^{\frac{2}{3}} \log \left (\frac{4^{\frac{1}{3}}{\left (3 \, x + 4\right )} - 2 \,{\left (27 \, x^{2} + 54 \, x + 28\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+54*x+28)^(1/3),x, algorithm="fricas")

[Out]

-1/18*4^(1/6)*sqrt(3)*arctan(1/18*4^(1/6)*(2*4^(2/3)*sqrt(3)*(27*x^2 + 54*x + 28)^(2/3)*(3*x + 4) + 4^(1/3)*sq
rt(3)*(27*x^3 + 54*x^2 + 36*x + 8) - 4*sqrt(3)*(27*x^2 + 54*x + 28)^(1/3)*(9*x^2 + 24*x + 16))/(9*x^3 + 54*x^2
+ 84*x + 40)) - 1/72*4^(2/3)*log((4^(2/3)*(27*x^2 + 54*x + 28)^(2/3) + 4^(1/3)*(9*x^2 + 24*x + 16) + 2*(27*x^
2 + 54*x + 28)^(1/3)*(3*x + 4))/(9*x^2 + 12*x + 4)) + 1/36*4^(2/3)*log((4^(1/3)*(3*x + 4) - 2*(27*x^2 + 54*x +
28)^(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} + 54 x + 28}}\, dx \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(27*x**2+54*x+28)**(1/3),x)

[Out]

Integral(1/((3*x + 2)*(27*x**2 + 54*x + 28)**(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} + 54 \, x + 28\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+54*x+28)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((27*x^2 + 54*x + 28)^(1/3)*(3*x + 2)), x)