3.1055 \(\int \frac{x^2}{\left (2+b x^2\right )^{3/4} \left (4+b x^2\right )} \, dx\)

Optimal. Leaf size=124 \[ \frac{\tanh ^{-1}\left (\frac{2\ 2^{3/4}-2 \sqrt [4]{2} \sqrt{b x^2+2}}{2 \sqrt{b} x \sqrt [4]{b x^2+2}}\right )}{\sqrt [4]{2} b^{3/2}}-\frac{\tan ^{-1}\left (\frac{2 \sqrt [4]{2} \sqrt{b x^2+2}+2\ 2^{3/4}}{2 \sqrt{b} x \sqrt [4]{b x^2+2}}\right )}{\sqrt [4]{2} b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.118746, antiderivative size = 124, 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 \[ \frac{\tanh ^{-1}\left (\frac{2^{3/4}-\sqrt [4]{2} \sqrt{b x^2+2}}{\sqrt{b} x \sqrt [4]{b x^2+2}}\right )}{\sqrt [4]{2} b^{3/2}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{2} \sqrt{b x^2+2}+2^{3/4}}{\sqrt{b} x \sqrt [4]{b x^2+2}}\right )}{\sqrt [4]{2} b^{3/2}} \]

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 = 10.3375, size = 31, normalized size = 0.25 \[ \frac{\sqrt [4]{2} x^{3} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{3}{4},1,\frac{5}{2},- \frac{b x^{2}}{2},- \frac{b x^{2}}{4} \right )}}{24} \]

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 = 0.229846, size = 150, normalized size = 1.21 \[ -\frac{20 x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right )}{3 \left (b x^2+2\right )^{3/4} \left (b x^2+4\right ) \left (b x^2 \left (2 F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right )+3 F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right )\right )-20 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};-\frac{b x^2}{2},-\frac{b x^2}{4}\right )\right )} \]

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
*AppellF1[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 = 0.07, 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 = 0.24625, size = 532, normalized size = 4.29 \[ \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} \frac{1}{b^{6}}^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x}{\sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{\frac{1}{2}} x \sqrt{\frac{\sqrt{\frac{1}{2}} b^{4} \sqrt{\frac{1}{b^{6}}} x^{2} + 2 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (b x^{2} + 2\right )}^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{b x^{2} + 2}}{x^{2}}} + 2 \,{\left (b x^{2} + 2\right )}^{\frac{1}{4}}}\right ) + \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} \frac{1}{b^{6}}^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x}{\sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x - 2 \, \sqrt{\frac{1}{2}} x \sqrt{\frac{\sqrt{\frac{1}{2}} b^{4} \sqrt{\frac{1}{b^{6}}} x^{2} - 2 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (b x^{2} + 2\right )}^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{b x^{2} + 2}}{x^{2}}} - 2 \,{\left (b x^{2} + 2\right )}^{\frac{1}{4}}}\right ) - \frac{1}{4} \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} \frac{1}{b^{6}}^{\frac{1}{4}} \log \left (\frac{\sqrt{\frac{1}{2}} b^{4} \sqrt{\frac{1}{b^{6}}} x^{2} + 2 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (b x^{2} + 2\right )}^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{b x^{2} + 2}}{2 \, x^{2}}\right ) + \frac{1}{4} \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}} \frac{1}{b^{6}}^{\frac{1}{4}} \log \left (\frac{\sqrt{\frac{1}{2}} b^{4} \sqrt{\frac{1}{b^{6}}} x^{2} - 2 \, \sqrt{2} \left (\frac{1}{8}\right )^{\frac{1}{4}}{\left (b x^{2} + 2\right )}^{\frac{1}{4}} b^{2} \frac{1}{b^{6}}^{\frac{1}{4}} x + 2 \, \sqrt{b x^{2} + 2}}{2 \, x^{2}}\right ) \]

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*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))) - 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. \[ \int \frac{x^{2}}{\left (b x^{2} + 2\right )^{\frac{3}{4}} \left (b x^{2} + 4\right )}\, dx \]

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 + 2)**(3/4)*(b*x**2 + 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)