3.142 \(\int \frac{1}{(3 a-b x^2) \sqrt [3]{a+b x^2}} \, dx\)

Optimal. Leaf size=202 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a+b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0287481, antiderivative size = 202, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {392} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt [6]{a} \left (\sqrt [3]{2} \sqrt [3]{a+b x^2}+\sqrt [3]{a}\right )}\right )}{2\ 2^{2/3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt [6]{a} \left (\sqrt [3]{a}-\sqrt [3]{2} \sqrt [3]{a+b x^2}\right )}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}}-\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{6\ 2^{2/3} a^{5/6} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \sqrt{a}}{\sqrt{b} x}\right )}{2\ 2^{2/3} \sqrt{3} a^{5/6} \sqrt{b}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 392

Int[1/(((a_) + (b_.)*(x_)^2)^(1/3)*((c_) + (d_.)*(x_)^2)), x_Symbol] :> With[{q = Rt[b/a, 2]}, Simp[(q*ArcTanh
[Sqrt[3]/(q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x] + (-Simp[(q*ArcTan[(a^(1/3)*q*x)/(a^(1/3) + 2^(1/3)*(a + b*
x^2)^(1/3))])/(2*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTan[q*x])/(6*2^(2/3)*a^(1/3)*d), x] + Simp[(q*ArcTanh[(Sq
rt[3]*(a^(1/3) - 2^(1/3)*(a + b*x^2)^(1/3)))/(a^(1/3)*q*x)])/(2*2^(2/3)*Sqrt[3]*a^(1/3)*d), x])] /; FreeQ[{a,
b, c, d}, x] && NeQ[b*c - a*d, 0] && EqQ[b*c + 3*a*d, 0] && PosQ[b/a]

Rubi steps

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

Mathematica [C]  time = 0.151177, size = 166, normalized size = 0.82 \[ \frac{9 a x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{a},\frac{b x^2}{3 a}\right )}{\left (3 a-b x^2\right ) \sqrt [3]{a+b x^2} \left (2 b x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};-\frac{b x^2}{a},\frac{b x^2}{3 a}\right )-F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};-\frac{b x^2}{a},\frac{b x^2}{3 a}\right )\right )+9 a F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};-\frac{b x^2}{a},\frac{b x^2}{3 a}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

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

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Maple [F]  time = 0.046, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{-b{x}^{2}+3\,a}{\frac{1}{\sqrt [3]{b{x}^{2}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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