3.155 \(\int \frac{1}{\sqrt [3]{-2+b x^2} \left (-\frac{18 d}{b}+d x^2\right )} \, dx\)

Optimal. Leaf size=147 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt [6]{2} \sqrt{3} \left (\sqrt [3]{b x^2-2}+\sqrt [3]{2}\right )}{\sqrt{b} x}\right )}{4\ 2^{5/6} \sqrt{3} d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{b x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt{b} x}\right )}{12\ 2^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{2}}\right )}{12\ 2^{5/6} d} \]

[Out]

(Sqrt[b]*ArcTan[(2^(1/6)*Sqrt[3]*(2^(1/3) + (-2 + b*x^2)^(1/3)))/(Sqrt[b]*x)])/(
4*2^(5/6)*Sqrt[3]*d) + (Sqrt[b]*ArcTanh[(Sqrt[b]*x)/(3*Sqrt[2])])/(12*2^(5/6)*d)
- (Sqrt[b]*ArcTanh[(2^(1/3) + (-2 + b*x^2)^(1/3))^2/(3*2^(1/6)*Sqrt[b]*x)])/(12
*2^(5/6)*d)

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Rubi [A]  time = 0.0797945, antiderivative size = 147, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.038 \[ \frac{\sqrt{b} \tan ^{-1}\left (\frac{\sqrt [6]{2} \sqrt{3} \left (\sqrt [3]{b x^2-2}+\sqrt [3]{2}\right )}{\sqrt{b} x}\right )}{4\ 2^{5/6} \sqrt{3} d}-\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\left (\sqrt [3]{b x^2-2}+\sqrt [3]{2}\right )^2}{3 \sqrt [6]{2} \sqrt{b} x}\right )}{12\ 2^{5/6} d}+\frac{\sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} x}{3 \sqrt{2}}\right )}{12\ 2^{5/6} d} \]

Antiderivative was successfully verified.

[In]  Int[1/((-2 + b*x^2)^(1/3)*((-18*d)/b + d*x^2)),x]

[Out]

(Sqrt[b]*ArcTan[(2^(1/6)*Sqrt[3]*(2^(1/3) + (-2 + b*x^2)^(1/3)))/(Sqrt[b]*x)])/(
4*2^(5/6)*Sqrt[3]*d) + (Sqrt[b]*ArcTanh[(Sqrt[b]*x)/(3*Sqrt[2])])/(12*2^(5/6)*d)
- (Sqrt[b]*ArcTanh[(2^(1/3) + (-2 + b*x^2)^(1/3))^2/(3*2^(1/6)*Sqrt[b]*x)])/(12
*2^(5/6)*d)

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Rubi in Sympy [A]  time = 25.286, size = 46, normalized size = 0.31 \[ \frac{b x \left (b x^{2} - 2\right )^{\frac{2}{3}} \operatorname{appellf_{1}}{\left (\frac{1}{2},\frac{1}{3},1,\frac{3}{2},\frac{b x^{2}}{2},\frac{b x^{2}}{18} \right )}}{36 d \left (- \frac{b x^{2}}{2} + 1\right )^{\frac{2}{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(b*x**2-2)**(1/3)/(-18*d/b+d*x**2),x)

[Out]

b*x*(b*x**2 - 2)**(2/3)*appellf1(1/2, 1/3, 1, 3/2, b*x**2/2, b*x**2/18)/(36*d*(-
b*x**2/2 + 1)**(2/3))

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Mathematica [C]  time = 0.273102, size = 148, normalized size = 1.01 \[ \frac{27 b x F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right )}{d \left (b x^2-18\right ) \sqrt [3]{b x^2-2} \left (b x^2 \left (F_1\left (\frac{3}{2};\frac{1}{3},2;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right )+3 F_1\left (\frac{3}{2};\frac{4}{3},1;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right )\right )+27 F_1\left (\frac{1}{2};\frac{1}{3},1;\frac{3}{2};\frac{b x^2}{2},\frac{b x^2}{18}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[1/((-2 + b*x^2)^(1/3)*((-18*d)/b + d*x^2)),x]

[Out]

(27*b*x*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/2, (b*x^2)/18])/(d*(-18 + b*x^2)*(-2
+ b*x^2)^(1/3)*(27*AppellF1[1/2, 1/3, 1, 3/2, (b*x^2)/2, (b*x^2)/18] + b*x^2*(Ap
pellF1[3/2, 1/3, 2, 5/2, (b*x^2)/2, (b*x^2)/18] + 3*AppellF1[3/2, 4/3, 1, 5/2, (
b*x^2)/2, (b*x^2)/18])))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{1{\frac{1}{\sqrt [3]{b{x}^{2}-2}}} \left ( -18\,{\frac{d}{b}}+d{x}^{2} \right ) ^{-1}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x^2-2)^(1/3)/(-18/b*d+d*x^2),x)

[Out]

int(1/(b*x^2-2)^(1/3)/(-18/b*d+d*x^2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} - 2\right )}^{\frac{1}{3}}{\left (d x^{2} - \frac{18 \, d}{b}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - 2)^(1/3)*(d*x^2 - 18*d/b)),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 - 2)^(1/3)*(d*x^2 - 18*d/b)), x)

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Fricas [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - 2)^(1/3)*(d*x^2 - 18*d/b)),x, algorithm="fricas")

[Out]

Exception raised: TypeError

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{b \int \frac{1}{b x^{2} \sqrt [3]{b x^{2} - 2} - 18 \sqrt [3]{b x^{2} - 2}}\, dx}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x**2-2)**(1/3)/(-18*d/b+d*x**2),x)

[Out]

b*Integral(1/(b*x**2*(b*x**2 - 2)**(1/3) - 18*(b*x**2 - 2)**(1/3)), x)/d

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} - 2\right )}^{\frac{1}{3}}{\left (d x^{2} - \frac{18 \, d}{b}\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 - 2)^(1/3)*(d*x^2 - 18*d/b)),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 - 2)^(1/3)*(d*x^2 - 18*d/b)), x)