3.156 \(\int \frac{1}{\sqrt [3]{2+3 x^2} (6 d+d x^2)} \, dx\)

Optimal. Leaf size=123 \[ \frac{\tan ^{-1}\left (\frac{\left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )^2}{3 \sqrt [6]{2} \sqrt{3} x}\right )}{4\ 2^{5/6} \sqrt{3} d}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )}{x}\right )}{4\ 2^{5/6} d}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{6}}\right )}{4\ 2^{5/6} \sqrt{3} d} \]

[Out]

ArcTan[x/Sqrt[6]]/(4*2^(5/6)*Sqrt[3]*d) + ArcTan[(2^(1/3) - (2 + 3*x^2)^(1/3))^2/(3*2^(1/6)*Sqrt[3]*x)]/(4*2^(
5/6)*Sqrt[3]*d) - ArcTanh[(2^(1/6)*(2^(1/3) - (2 + 3*x^2)^(1/3)))/x]/(4*2^(5/6)*d)

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Rubi [A]  time = 0.0204762, antiderivative size = 123, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {394} \[ \frac{\tan ^{-1}\left (\frac{\left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )^2}{3 \sqrt [6]{2} \sqrt{3} x}\right )}{4\ 2^{5/6} \sqrt{3} d}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{3 x^2+2}\right )}{x}\right )}{4\ 2^{5/6} d}+\frac{\tan ^{-1}\left (\frac{x}{\sqrt{6}}\right )}{4\ 2^{5/6} \sqrt{3} d} \]

Antiderivative was successfully verified.

[In]

Int[1/((2 + 3*x^2)^(1/3)*(6*d + d*x^2)),x]

[Out]

ArcTan[x/Sqrt[6]]/(4*2^(5/6)*Sqrt[3]*d) + ArcTan[(2^(1/3) - (2 + 3*x^2)^(1/3))^2/(3*2^(1/6)*Sqrt[3]*x)]/(4*2^(
5/6)*Sqrt[3]*d) - ArcTanh[(2^(1/6)*(2^(1/3) - (2 + 3*x^2)^(1/3)))/x]/(4*2^(5/6)*d)

Rule 394

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b/a, 2]}, Simp[(q*ArcTan[
(q*x)/3])/(12*Rt[a, 3]*d), x] + (Simp[(q*ArcTan[(Rt[a, 3] - (a + b*x^2)^(1/3))^2/(3*Rt[a, 3]^2*q*x)])/(12*Rt[a
, 3]*d), x] - Simp[(q*ArcTanh[(Sqrt[3]*(Rt[a, 3] - (a + b*x^2)^(1/3)))/(Rt[a, 3]*q*x)])/(4*Sqrt[3]*Rt[a, 3]*d)
, x])] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c - 9*a*d, 0] && PosQ[b/a]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [3]{2+3 x^2} \left (6 d+d x^2\right )} \, dx &=\frac{\tan ^{-1}\left (\frac{x}{\sqrt{6}}\right )}{4\ 2^{5/6} \sqrt{3} d}+\frac{\tan ^{-1}\left (\frac{\left (\sqrt [3]{2}-\sqrt [3]{2+3 x^2}\right )^2}{3 \sqrt [6]{2} \sqrt{3} x}\right )}{4\ 2^{5/6} \sqrt{3} d}-\frac{\tanh ^{-1}\left (\frac{\sqrt [6]{2} \left (\sqrt [3]{2}-\sqrt [3]{2+3 x^2}\right )}{x}\right )}{4\ 2^{5/6} d}\\ \end{align*}

Mathematica [C]  time = 0.112846, size = 136, normalized size = 1.11 \[ -\frac{9 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right )}{d \left (x^2+6\right ) \sqrt [3]{3 x^2+2} \left (x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right )\right )-9 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{3 x^2}{2},-\frac{x^2}{6}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((2 + 3*x^2)^(1/3)*(6*d + d*x^2)),x]

[Out]

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

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Maple [F]  time = 0.028, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{d{x}^{2}+6\,d}{\frac{1}{\sqrt [3]{3\,{x}^{2}+2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x^2+2)^(1/3)/(d*x^2+6*d),x)

[Out]

int(1/(3*x^2+2)^(1/3)/(d*x^2+6*d),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x^{2} + 6 \, d\right )}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+2)^(1/3)/(d*x^2+6*d),x, algorithm="maxima")

[Out]

integrate(1/((d*x^2 + 6*d)*(3*x^2 + 2)^(1/3)), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+2)^(1/3)/(d*x^2+6*d),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{1}{x^{2} \sqrt [3]{3 x^{2} + 2} + 6 \sqrt [3]{3 x^{2} + 2}}\, dx}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x**2+2)**(1/3)/(d*x**2+6*d),x)

[Out]

Integral(1/(x**2*(3*x**2 + 2)**(1/3) + 6*(3*x**2 + 2)**(1/3)), x)/d

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (d x^{2} + 6 \, d\right )}{\left (3 \, x^{2} + 2\right )}^{\frac{1}{3}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(3*x^2+2)^(1/3)/(d*x^2+6*d),x, algorithm="giac")

[Out]

integrate(1/((d*x^2 + 6*d)*(3*x^2 + 2)^(1/3)), x)