3.2500 \(\int \frac{1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx\)

Optimal. Leaf size=108 \[ \frac{\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{6\ 10^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} 10^{2/3}}-\frac{\log (3 x+2)}{6\ 10^{2/3}} \]

[Out]

-ArcTan[1/Sqrt[3] + (2^(2/3)*(8 - 3*x))/(Sqrt[3]*5^(1/3)*(52 - 54*x + 27*x^2)^(1/3))]/(3*Sqrt[3]*10^(2/3)) - L
og[2 + 3*x]/(6*10^(2/3)) + Log[216 - 81*x - 27*10^(1/3)*(52 - 54*x + 27*x^2)^(1/3)]/(6*10^(2/3))

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Rubi [A]  time = 0.0188848, antiderivative size = 108, 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 = {750} \[ \frac{\log \left (-27 \sqrt [3]{10} \sqrt [3]{27 x^2-54 x+52}-81 x+216\right )}{6\ 10^{2/3}}-\frac{\tan ^{-1}\left (\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{27 x^2-54 x+52}}+\frac{1}{\sqrt{3}}\right )}{3 \sqrt{3} 10^{2/3}}-\frac{\log (3 x+2)}{6\ 10^{2/3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((2 + 3*x)*(52 - 54*x + 27*x^2)^(1/3)),x]

[Out]

-ArcTan[1/Sqrt[3] + (2^(2/3)*(8 - 3*x))/(Sqrt[3]*5^(1/3)*(52 - 54*x + 27*x^2)^(1/3))]/(3*Sqrt[3]*10^(2/3)) - L
og[2 + 3*x]/(6*10^(2/3)) + Log[216 - 81*x - 27*10^(1/3)*(52 - 54*x + 27*x^2)^(1/3)]/(6*10^(2/3))

Rule 750

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] && PosQ[c*e^2*(2*c*d - b*e)]

Rubi steps

\begin{align*} \int \frac{1}{(2+3 x) \sqrt [3]{52-54 x+27 x^2}} \, dx &=-\frac{\tan ^{-1}\left (\frac{1}{\sqrt{3}}+\frac{2^{2/3} (8-3 x)}{\sqrt{3} \sqrt [3]{5} \sqrt [3]{52-54 x+27 x^2}}\right )}{3 \sqrt{3} 10^{2/3}}-\frac{\log (2+3 x)}{6\ 10^{2/3}}+\frac{\log \left (216-81 x-27 \sqrt [3]{10} \sqrt [3]{52-54 x+27 x^2}\right )}{6\ 10^{2/3}}\\ \end{align*}

Mathematica [C]  time = 0.0770016, size = 126, normalized size = 1.17 \[ -\frac{\sqrt [3]{\frac{9 x-5 i \sqrt{3}-9}{3 x+2}} \sqrt [3]{\frac{9 x+5 i \sqrt{3}-9}{3 x+2}} F_1\left (\frac{2}{3};\frac{1}{3},\frac{1}{3};\frac{5}{3};\frac{15-5 i \sqrt{3}}{9 x+6},\frac{15+5 i \sqrt{3}}{9 x+6}\right )}{2\ 3^{2/3} \sqrt [3]{27 x^2-54 x+52}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((2 + 3*x)*(52 - 54*x + 27*x^2)^(1/3)),x]

[Out]

-(((-9 - (5*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*((-9 + (5*I)*Sqrt[3] + 9*x)/(2 + 3*x))^(1/3)*AppellF1[2/3, 1/3,
1/3, 5/3, (15 - (5*I)*Sqrt[3])/(6 + 9*x), (15 + (5*I)*Sqrt[3])/(6 + 9*x)])/(2*3^(2/3)*(52 - 54*x + 27*x^2)^(1
/3))

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Maple [F]  time = 1.619, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{2+3\,x}{\frac{1}{\sqrt [3]{27\,{x}^{2}-54\,x+52}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(2+3*x)/(27*x^2-54*x+52)^(1/3),x)

[Out]

int(1/(2+3*x)/(27*x^2-54*x+52)^(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 + 52\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+52)^(1/3),x, algorithm="maxima")

[Out]

integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)), x)

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Fricas [B]  time = 22.4995, size = 663, normalized size = 6.14 \begin{align*} -\frac{1}{90} \cdot 100^{\frac{1}{6}} \sqrt{3} \arctan \left (\frac{100^{\frac{1}{6}}{\left (2 \cdot 100^{\frac{2}{3}} \sqrt{3}{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{2}{3}}{\left (3 \, x - 8\right )} + 100^{\frac{1}{3}} \sqrt{3}{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} + 20 \, \sqrt{3}{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}{\left (9 \, x^{2} - 48 \, x + 64\right )}\right )}}{90 \,{\left (9 \, x^{3} - 162 \, x^{2} + 372 \, x - 344\right )}}\right ) - \frac{1}{1800} \cdot 100^{\frac{2}{3}} \log \left (\frac{100^{\frac{2}{3}}{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{2}{3}} + 100^{\frac{1}{3}}{\left (9 \, x^{2} - 48 \, x + 64\right )} - 10 \,{\left (27 \, x^{2} - 54 \, x + 52\right )}^{\frac{1}{3}}{\left (3 \, x - 8\right )}}{9 \, x^{2} + 12 \, x + 4}\right ) + \frac{1}{900} \cdot 100^{\frac{2}{3}} \log \left (\frac{100^{\frac{1}{3}}{\left (3 \, x - 8\right )} + 10 \,{\left (27 \, x^{2} - 54 \, x + 52\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+52)^(1/3),x, algorithm="fricas")

[Out]

-1/90*100^(1/6)*sqrt(3)*arctan(1/90*100^(1/6)*(2*100^(2/3)*sqrt(3)*(27*x^2 - 54*x + 52)^(2/3)*(3*x - 8) + 100^
(1/3)*sqrt(3)*(27*x^3 + 54*x^2 + 36*x + 8) + 20*sqrt(3)*(27*x^2 - 54*x + 52)^(1/3)*(9*x^2 - 48*x + 64))/(9*x^3
- 162*x^2 + 372*x - 344)) - 1/1800*100^(2/3)*log((100^(2/3)*(27*x^2 - 54*x + 52)^(2/3) + 100^(1/3)*(9*x^2 - 4
8*x + 64) - 10*(27*x^2 - 54*x + 52)^(1/3)*(3*x - 8))/(9*x^2 + 12*x + 4)) + 1/900*100^(2/3)*log((100^(1/3)*(3*x
- 8) + 10*(27*x^2 - 54*x + 52)^(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 + 52}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(2+3*x)/(27*x**2-54*x+52)**(1/3),x)

[Out]

Integral(1/((3*x + 2)*(27*x**2 - 54*x + 52)**(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 + 52\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+52)^(1/3),x, algorithm="giac")

[Out]

integrate(1/((27*x^2 - 54*x + 52)^(1/3)*(3*x + 2)), x)