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

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

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Rubi [A]  time = 0.02, antiderivative size = 123, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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} \end {gather*}

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*}

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Mathematica [C]  time = 0.12, size = 136, normalized size = 1.11 \begin {gather*} -\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 )} \end {gather*}

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

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IntegrateAlgebraic [F]  time = 4.56, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{2+3 x^2} \left (6 d+d x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (d x^{2} + 6 \, d\right )} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

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)

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maple [C]  time = 58.10, size = 549, normalized size = 4.46 \begin {gather*} -\frac {24 \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \ln \left (-\frac {192 x \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )^{6}-4 x \RootOf \left (\textit {\_Z}^{6}+54\right )^{7}+9 x^{2} \RootOf \left (\textit {\_Z}^{6}+54\right )^{4}-288 \left (3 x^{2}+2\right )^{\frac {1}{3}} x \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )^{4}+6 \left (3 x^{2}+2\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{6}+54\right )^{5}-18 \RootOf \left (\textit {\_Z}^{6}+54\right )^{4}-108 \left (3 x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+54\right )^{2}+324 \left (3 x^{2}+2\right )^{\frac {2}{3}}}{x^{2}+6}\right )-\RootOf \left (\textit {\_Z}^{6}+54\right ) \ln \left (-\frac {768 x \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{6}+54\right )^{5}-16 x \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )^{6}-36 x^{2} \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )^{3}+1152 \left (3 x^{2}+2\right )^{\frac {1}{3}} x \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right )^{2} \RootOf \left (\textit {\_Z}^{6}+54\right )^{3}-72 \left (3 x^{2}+2\right )^{\frac {1}{3}} x \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )^{4}+\left (3 x^{2}+2\right )^{\frac {1}{3}} x \RootOf \left (\textit {\_Z}^{6}+54\right )^{5}+72 \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )^{3}-432 \left (3 x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (576 \textit {\_Z}^{2}-24 \textit {\_Z} \RootOf \left (\textit {\_Z}^{6}+54\right )+\RootOf \left (\textit {\_Z}^{6}+54\right )^{2}\right ) \RootOf \left (\textit {\_Z}^{6}+54\right )+18 \left (3 x^{2}+2\right )^{\frac {1}{3}} \RootOf \left (\textit {\_Z}^{6}+54\right )^{2}+54 \left (3 x^{2}+2\right )^{\frac {2}{3}}}{x^{2}+6}\right )}{24 d} \end {gather*}

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]

-1/24*(24*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*ln(-(192*RootOf(RootOf(_Z^6+54)^2-24*_Z*Roo
tOf(_Z^6+54)+576*_Z^2)*RootOf(_Z^6+54)^6*x-4*RootOf(_Z^6+54)^7*x-288*(3*x^2+2)^(1/3)*RootOf(_Z^6+54)^4*RootOf(
RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*x+6*(3*x^2+2)^(1/3)*RootOf(_Z^6+54)^5*x+9*x^2*RootOf(_Z^6+54
)^4-18*RootOf(_Z^6+54)^4-108*RootOf(_Z^6+54)^2*(3*x^2+2)^(1/3)+324*(3*x^2+2)^(2/3))/(x^2+6))-RootOf(_Z^6+54)*l
n(-(768*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)^2*RootOf(_Z^6+54)^5*x-16*RootOf(RootOf(_Z^6+5
4)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*RootOf(_Z^6+54)^6*x+1152*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+5
76*_Z^2)^2*RootOf(_Z^6+54)^3*(3*x^2+2)^(1/3)*x-72*(3*x^2+2)^(1/3)*RootOf(_Z^6+54)^4*RootOf(RootOf(_Z^6+54)^2-2
4*_Z*RootOf(_Z^6+54)+576*_Z^2)*x+(3*x^2+2)^(1/3)*RootOf(_Z^6+54)^5*x-36*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(
_Z^6+54)+576*_Z^2)*RootOf(_Z^6+54)^3*x^2+72*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*RootOf(_Z
^6+54)^3-432*RootOf(RootOf(_Z^6+54)^2-24*_Z*RootOf(_Z^6+54)+576*_Z^2)*RootOf(_Z^6+54)*(3*x^2+2)^(1/3)+18*RootO
f(_Z^6+54)^2*(3*x^2+2)^(1/3)+54*(3*x^2+2)^(2/3))/(x^2+6)))/d

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (d x^{2} + 6 \, d\right )} {\left (3 \, x^{2} + 2\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (3\,x^2+2\right )}^{1/3}\,\left (d\,x^2+6\,d\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{x^{2} \sqrt [3]{3 x^{2} + 2} + 6 \sqrt [3]{3 x^{2} + 2}}\, dx}{d} \end {gather*}

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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