3.1056 \(\int \frac{x^2}{\left (2-b x^2\right )^{3/4} \left (4-b x^2\right )} \, dx\)

Optimal. Leaf size=119 \[ \frac{\tan ^{-1}\left (\frac{2-\sqrt{2} \sqrt{2-b x^2}}{\sqrt [4]{2} \sqrt{b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{2} \sqrt{2-b x^2}+2}{\sqrt [4]{2} \sqrt{b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.128891, antiderivative size = 119, 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 \[ \frac{\tan ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{2-b x^2}}{\sqrt{b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{2-b x^2}+2^{3/4}}{\sqrt{b} x \sqrt [4]{2-b x^2}}\right )}{\sqrt [4]{2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((2 - b*x^2)^(3/4)*(4 - b*x^2)),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.6322, size = 27, normalized size = 0.23 \[ \frac{\sqrt [4]{2} x^{3} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{3}{4},1,\frac{5}{2},\frac{b x^{2}}{2},\frac{b x^{2}}{4} \right )}}{24} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(-b*x**2+2)**(3/4)/(-b*x**2+4),x)

[Out]

2**(1/4)*x**3*appellf1(3/2, 3/4, 1, 5/2, b*x**2/2, b*x**2/4)/24

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Mathematica [C]  time = 0.222151, size = 151, normalized size = 1.27 \[ -\frac{20 x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{4}\right )}{3 \left (2-b x^2\right )^{3/4} \left (b x^2-4\right ) \left (b x^2 \left (2 F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};\frac{b x^2}{2},\frac{b x^2}{4}\right )+3 F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};\frac{b x^2}{2},\frac{b x^2}{4}\right )\right )+20 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};\frac{b x^2}{2},\frac{b x^2}{4}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^2/((2 - b*x^2)^(3/4)*(4 - b*x^2)),x]

[Out]

(-20*x^3*AppellF1[3/2, 3/4, 1, 5/2, (b*x^2)/2, (b*x^2)/4])/(3*(2 - b*x^2)^(3/4)*
(-4 + b*x^2)*(20*AppellF1[3/2, 3/4, 1, 5/2, (b*x^2)/2, (b*x^2)/4] + b*x^2*(2*App
ellF1[5/2, 3/4, 2, 7/2, (b*x^2)/2, (b*x^2)/4] + 3*AppellF1[5/2, 7/4, 1, 7/2, (b*
x^2)/2, (b*x^2)/4])))

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Maple [F]  time = 0.074, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{-b{x}^{2}+4} \left ( -b{x}^{2}+2 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(-b*x^2+2)^(3/4)/(-b*x^2+4),x)

[Out]

int(x^2/(-b*x^2+2)^(3/4)/(-b*x^2+4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ -\int \frac{x^{2}}{{\left (b x^{2} - 4\right )}{\left (-b x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/((b*x^2 - 4)*(-b*x^2 + 2)^(3/4)),x, algorithm="maxima")

[Out]

-integrate(x^2/((b*x^2 - 4)*(-b*x^2 + 2)^(3/4)), x)

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Fricas [A]  time = 0.246712, size = 545, normalized size = 4.58 \[ \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} \frac{1}{b^{6}}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x}{\sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{\frac{1}{2}} x \sqrt{\frac{\sqrt{\frac{1}{2}} b^{4} \sqrt{\frac{1}{b^{6}}} x^{2} + 2 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (-b x^{2} + 2\right )}^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{-b x^{2} + 2}}{x^{2}}} + 2 \,{\left (-b x^{2} + 2\right )}^{\frac{1}{4}}}\right ) + \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} \frac{1}{b^{6}}^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x}{\sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x - 2 \, \sqrt{\frac{1}{2}} x \sqrt{\frac{\sqrt{\frac{1}{2}} b^{4} \sqrt{\frac{1}{b^{6}}} x^{2} - 2 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (-b x^{2} + 2\right )}^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{-b x^{2} + 2}}{x^{2}}} - 2 \,{\left (-b x^{2} + 2\right )}^{\frac{1}{4}}}\right ) - \frac{1}{4} \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} \frac{1}{b^{6}}^{\frac{1}{4}} \log \left (\frac{\sqrt{\frac{1}{2}} b^{4} \sqrt{\frac{1}{b^{6}}} x^{2} + 2 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (-b x^{2} + 2\right )}^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{-b x^{2} + 2}}{2 \, x^{2}}\right ) + \frac{1}{4} \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} \frac{1}{b^{6}}^{\frac{1}{4}} \log \left (\frac{\sqrt{\frac{1}{2}} b^{4} \sqrt{\frac{1}{b^{6}}} x^{2} - 2 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (-b x^{2} + 2\right )}^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{-b x^{2} + 2}}{2 \, x^{2}}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/((b*x^2 - 4)*(-b*x^2 + 2)^(3/4)),x, algorithm="fricas")

[Out]

sqrt(2)*(1/8)^(1/4)*(b^(-6))^(1/4)*arctan(sqrt(2)*(1/8)^(1/4)*b^2*(b^(-6))^(1/4)
*x/(sqrt(2)*(1/8)^(1/4)*b^2*(b^(-6))^(1/4)*x + 2*sqrt(1/2)*x*sqrt((sqrt(1/2)*b^4
*sqrt(b^(-6))*x^2 + 2*sqrt(2)*(1/8)^(1/4)*(-b*x^2 + 2)^(1/4)*b^2*(b^(-6))^(1/4)*
x + 2*sqrt(-b*x^2 + 2))/x^2) + 2*(-b*x^2 + 2)^(1/4))) + sqrt(2)*(1/8)^(1/4)*(b^(
-6))^(1/4)*arctan(-sqrt(2)*(1/8)^(1/4)*b^2*(b^(-6))^(1/4)*x/(sqrt(2)*(1/8)^(1/4)
*b^2*(b^(-6))^(1/4)*x - 2*sqrt(1/2)*x*sqrt((sqrt(1/2)*b^4*sqrt(b^(-6))*x^2 - 2*s
qrt(2)*(1/8)^(1/4)*(-b*x^2 + 2)^(1/4)*b^2*(b^(-6))^(1/4)*x + 2*sqrt(-b*x^2 + 2))
/x^2) - 2*(-b*x^2 + 2)^(1/4))) - 1/4*sqrt(2)*(1/8)^(1/4)*(b^(-6))^(1/4)*log(1/2*
(sqrt(1/2)*b^4*sqrt(b^(-6))*x^2 + 2*sqrt(2)*(1/8)^(1/4)*(-b*x^2 + 2)^(1/4)*b^2*(
b^(-6))^(1/4)*x + 2*sqrt(-b*x^2 + 2))/x^2) + 1/4*sqrt(2)*(1/8)^(1/4)*(b^(-6))^(1
/4)*log(1/2*(sqrt(1/2)*b^4*sqrt(b^(-6))*x^2 - 2*sqrt(2)*(1/8)^(1/4)*(-b*x^2 + 2)
^(1/4)*b^2*(b^(-6))^(1/4)*x + 2*sqrt(-b*x^2 + 2))/x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \int \frac{x^{2}}{b x^{2} \left (- b x^{2} + 2\right )^{\frac{3}{4}} - 4 \left (- b x^{2} + 2\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(-b*x**2+2)**(3/4)/(-b*x**2+4),x)

[Out]

-Integral(x**2/(b*x**2*(-b*x**2 + 2)**(3/4) - 4*(-b*x**2 + 2)**(3/4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int -\frac{x^{2}}{{\left (b x^{2} - 4\right )}{\left (-b x^{2} + 2\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-x^2/((b*x^2 - 4)*(-b*x^2 + 2)^(3/4)),x, algorithm="giac")

[Out]

integrate(-x^2/((b*x^2 - 4)*(-b*x^2 + 2)^(3/4)), x)