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

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[(Sqrt[3]*(1 - (1 + b*x^2)^(1/3)))/(Sqrt[b]*x)]/(4*Sq
rt[3]*Sqrt[b])

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Rubi [A]  time = 0.0533674, 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 \[ \frac{\tan ^{-1}\left (\frac{\left (1-\sqrt [3]{b x^2+1}\right )^2}{3 \sqrt{b} x}\right )}{12 \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{3} \left (1-\sqrt [3]{b x^2+1}\right )}{\sqrt{b} x}\right )}{4 \sqrt{3} \sqrt{b}}+\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{3}\right )}{12 \sqrt{b}} \]

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[(Sqrt[3]*(1 - (1 + b*x^2)^(1/3)))/(Sqrt[b]*x)]/(4*Sq
rt[3]*Sqrt[b])

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Rubi in Sympy [A]  time = 9.58172, size = 22, normalized size = 0.21 \[ \frac{x \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{3},1,\frac{3}{2},- b x^{2},- \frac{b x^{2}}{9} \right )}}{9} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2+1)**(1/3)/(b*x**2+9),x)

[Out]

x*appellf1(1/2, 1/3, 1, 3/2, -b*x**2, -b*x**2/9)/9

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Mathematica [C]  time = 0.16732, 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*(AppellF
1[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])))

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Maple [F]  time = 0.056, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{2}+9}{\frac{1}{\sqrt [3]{b{x}^{2}+1}}}}\, dx \]

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., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + 9\right )}{\left (b x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

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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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + 9\right )}{\left (b x^{2} + 1\right )}^{\frac{1}{3}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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