3.2.31 \(\int \frac {1}{\sqrt [3]{1+b x^2} (9+b x^2)} \, dx\)

Optimal. Leaf size=104 \[ \frac {\tan ^{-1}\left (\frac {\left (1-\sqrt [3]{b x^2+1}\right )^2}{3 \sqrt {b} x}\right )}{12 \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{b x^2+1}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{3}\right )}{12 \sqrt {b}} \]

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Rubi [A]  time = 0.02, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.048, Rules used = {394} \begin {gather*} \frac {\tan ^{-1}\left (\frac {\left (1-\sqrt [3]{b x^2+1}\right )^2}{3 \sqrt {b} x}\right )}{12 \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{b x^2+1}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{3}\right )}{12 \sqrt {b}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 + b*x^2)^(1/3)*(9 + b*x^2)),x]

[Out]

ArcTan[(Sqrt[b]*x)/3]/(12*Sqrt[b]) + ArcTan[(1 - (1 + b*x^2)^(1/3))^2/(3*Sqrt[b]*x)]/(12*Sqrt[b]) - ArcTanh[(S
qrt[3]*(1 - (1 + b*x^2)^(1/3)))/(Sqrt[b]*x)]/(4*Sqrt[3]*Sqrt[b])

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]{1+b x^2} \left (9+b x^2\right )} \, dx &=\frac {\tan ^{-1}\left (\frac {\sqrt {b} x}{3}\right )}{12 \sqrt {b}}+\frac {\tan ^{-1}\left (\frac {\left (1-\sqrt [3]{1+b x^2}\right )^2}{3 \sqrt {b} x}\right )}{12 \sqrt {b}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {3} \left (1-\sqrt [3]{1+b x^2}\right )}{\sqrt {b} x}\right )}{4 \sqrt {3} \sqrt {b}}\\ \end {align*}

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Mathematica [C]  time = 0.11, size = 137, normalized size = 1.32 \begin {gather*} -\frac {27 x F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-b x^2,-\frac {b x^2}{9}\right )}{\sqrt [3]{b x^2+1} \left (b x^2+9\right ) \left (2 b x^2 \left (F_1\left (\frac {3}{2};\frac {1}{3},2;\frac {5}{2};-b x^2,-\frac {b x^2}{9}\right )+3 F_1\left (\frac {3}{2};\frac {4}{3},1;\frac {5}{2};-b x^2,-\frac {b x^2}{9}\right )\right )-27 F_1\left (\frac {1}{2};\frac {1}{3},1;\frac {3}{2};-b x^2,-\frac {b x^2}{9}\right )\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((1 + b*x^2)^(1/3)*(9 + b*x^2)),x]

[Out]

(-27*x*AppellF1[1/2, 1/3, 1, 3/2, -(b*x^2), -1/9*(b*x^2)])/((1 + b*x^2)^(1/3)*(9 + b*x^2)*(-27*AppellF1[1/2, 1
/3, 1, 3/2, -(b*x^2), -1/9*(b*x^2)] + 2*b*x^2*(AppellF1[3/2, 1/3, 2, 5/2, -(b*x^2), -1/9*(b*x^2)] + 3*AppellF1
[3/2, 4/3, 1, 5/2, -(b*x^2), -1/9*(b*x^2)])))

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IntegrateAlgebraic [F]  time = 4.40, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{1+b x^2} \left (9+b x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[1/((1 + b*x^2)^(1/3)*(9 + b*x^2)),x]

[Out]

Defer[IntegrateAlgebraic][1/((1 + b*x^2)^(1/3)*(9 + b*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/(b*x^2+1)^(1/3)/(b*x^2+9),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 (b x^{2} + 9\right )} {\left (b x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+1)^(1/3)/(b*x^2+9),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + 9)*(b*x^2 + 1)^(1/3)), x)

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maple [F]  time = 0.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (b \,x^{2}+1\right )^{\frac {1}{3}} \left (b \,x^{2}+9\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^2+1)^(1/3)/(b*x^2+9),x)

[Out]

int(1/(b*x^2+1)^(1/3)/(b*x^2+9),x)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{{\left (b x^{2} + 9\right )} {\left (b x^{2} + 1\right )}^{\frac {1}{3}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^2+1)^(1/3)/(b*x^2+9),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + 9)*(b*x^2 + 1)^(1/3)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\left (b\,x^2+1\right )}^{1/3}\,\left (b\,x^2+9\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((b*x^2 + 1)^(1/3)*(b*x^2 + 9)),x)

[Out]

int(1/((b*x^2 + 1)^(1/3)*(b*x^2 + 9)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt [3]{b x^{2} + 1} \left (b x^{2} + 9\right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**2+1)**(1/3)/(b*x**2+9),x)

[Out]

Integral(1/((b*x**2 + 1)**(1/3)*(b*x**2 + 9)), x)

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