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

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

[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])

________________________________________________________________________________________

Rubi [A]  time = 0.0163418, antiderivative size = 104, normalized size of antiderivative = 1., 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} \[ \frac{\tan ^{-1}\left (\frac{\left (1-\sqrt [3]{b x^2+1}\right )^2}{3 \sqrt{b} x}\right )}{12 \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{b x^2+1}\right )}{\sqrt{b} x}\right )}{4 \sqrt{3} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{12 \sqrt{b}} \]

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*}

Mathematica [C]  time = 0.108447, size = 137, normalized size = 1.32 \[ -\frac{27 x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-b x^2,-\frac{b x^2}{9}\right )}{\sqrt [3]{b x^2+1} \left (b x^2+9\right ) \left (2 b x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-b x^2,-\frac{b x^2}{9}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-b x^2,-\frac{b x^2}{9}\right )\right )-27 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-b x^2,-\frac{b x^2}{9}\right )\right )} \]

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

________________________________________________________________________________________

Maple [F]  time = 0.037, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{b{x}^{2}+9}{\frac{1}{\sqrt [3]{b{x}^{2}+1}}}}\, dx \end{align*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

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

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)