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

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

________________________________________________________________________________________

Rubi [A]  time = 0.01, antiderivative size = 97, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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}} \end {gather*}

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 = 0.08, size = 121, normalized size = 1.25 \begin {gather*} -\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}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-1/2*((((-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)])/(3^(2/3)*(4 + 27*x^2)^(1/3))

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 0.24, size = 187, normalized size = 1.93 \begin {gather*} \frac {\log \left (2 \sqrt [3]{27 x^2+4}+3 \sqrt [3]{2} x-2 \sqrt [3]{2}\right )}{18 \sqrt [3]{2}}-\frac {\log \left (9\ 2^{2/3} x^2+4 \left (27 x^2+4\right )^{2/3}+\left (4 \sqrt [3]{2}-6 \sqrt [3]{2} x\right ) \sqrt [3]{27 x^2+4}-12\ 2^{2/3} x+4\ 2^{2/3}\right )}{36 \sqrt [3]{2}}-\frac {\tan ^{-1}\left (\frac {\frac {\sqrt [3]{27 x^2+4}}{\sqrt {3}}-\sqrt [3]{2} \sqrt {3} x+\frac {2 \sqrt [3]{2}}{\sqrt {3}}}{\sqrt [3]{27 x^2+4}}\right )}{6 \sqrt [3]{2} \sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*ArcTan[((2*2^(1/3))/Sqrt[3] - 2^(1/3)*Sqrt[3]*x + (4 + 27*x^2)^(1/3)/Sqrt[3])/(4 + 27*x^2)^(1/3)]/(2^(1/3
)*Sqrt[3]) + Log[-2*2^(1/3) + 3*2^(1/3)*x + 2*(4 + 27*x^2)^(1/3)]/(18*2^(1/3)) - Log[4*2^(2/3) - 12*2^(2/3)*x
+ 9*2^(2/3)*x^2 + (4*2^(1/3) - 6*2^(1/3)*x)*(4 + 27*x^2)^(1/3) + 4*(4 + 27*x^2)^(2/3)]/(36*2^(1/3))

________________________________________________________________________________________

fricas [B]  time = 2.68, size = 200, normalized size = 2.06 \begin {gather*} -\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 {gather*}

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))

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}}\,{d x} \end {gather*}

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)

________________________________________________________________________________________

maple [F]  time = 2.80, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 x +2\right ) \left (27 x^{2}+4\right )^{\frac {1}{3}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (27 \, x^{2} + 4\right )}^{\frac {1}{3}} {\left (3 \, x + 2\right )}}\,{d x} \end {gather*}

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)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (3\,x+2\right )\,{\left (27\,x^2+4\right )}^{1/3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (3 x + 2\right ) \sqrt [3]{27 x^{2} + 4}}\, dx \end {gather*}

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)

________________________________________________________________________________________