3.1075 \(\int \frac{x^2}{\left (-2+b x^2\right ) \left (-1+b x^2\right )^{3/4}} \, dx\)

Optimal. Leaf size=72 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{b x^2-1}}\right )}{\sqrt{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{b x^2-1}}\right )}{\sqrt{2} b^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.091415, antiderivative size = 72, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{b x^2-1}}\right )}{\sqrt{2} b^{3/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{b x^2-1}}\right )}{\sqrt{2} b^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((-2 + b*x^2)*(-1 + b*x^2)^(3/4)),x]

[Out]

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

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Rubi in Sympy [A]  time = 28.7619, size = 41, normalized size = 0.57 \[ \frac{x^{3} \sqrt [4]{b x^{2} - 1} \operatorname{appellf_{1}}{\left (\frac{3}{2},\frac{3}{4},1,\frac{5}{2},b x^{2},\frac{b x^{2}}{2} \right )}}{6 \sqrt [4]{- b x^{2} + 1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**2-2)/(b*x**2-1)**(3/4),x)

[Out]

x**3*(b*x**2 - 1)**(1/4)*appellf1(3/2, 3/4, 1, 5/2, b*x**2, b*x**2/2)/(6*(-b*x**
2 + 1)**(1/4))

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Mathematica [C]  time = 0.297, size = 138, normalized size = 1.92 \[ \frac{10 x^3 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};b x^2,\frac{b x^2}{2}\right )}{3 \left (b x^2-2\right ) \left (b x^2-1\right )^{3/4} \left (b x^2 \left (2 F_1\left (\frac{5}{2};\frac{3}{4},2;\frac{7}{2};b x^2,\frac{b x^2}{2}\right )+3 F_1\left (\frac{5}{2};\frac{7}{4},1;\frac{7}{2};b x^2,\frac{b x^2}{2}\right )\right )+10 F_1\left (\frac{3}{2};\frac{3}{4},1;\frac{5}{2};b x^2,\frac{b x^2}{2}\right )\right )} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[x^2/((-2 + b*x^2)*(-1 + b*x^2)^(3/4)),x]

[Out]

(10*x^3*AppellF1[3/2, 3/4, 1, 5/2, b*x^2, (b*x^2)/2])/(3*(-2 + b*x^2)*(-1 + b*x^
2)^(3/4)*(10*AppellF1[3/2, 3/4, 1, 5/2, b*x^2, (b*x^2)/2] + b*x^2*(2*AppellF1[5/
2, 3/4, 2, 7/2, b*x^2, (b*x^2)/2] + 3*AppellF1[5/2, 7/4, 1, 7/2, b*x^2, (b*x^2)/
2])))

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \[ \int{\frac{{x}^{2}}{b{x}^{2}-2} \left ( b{x}^{2}-1 \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^2-2)/(b*x^2-1)^(3/4),x)

[Out]

int(x^2/(b*x^2-2)/(b*x^2-1)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b x^{2} - 1\right )}^{\frac{3}{4}}{\left (b x^{2} - 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x^2 - 1)^(3/4)*(b*x^2 - 2)),x, algorithm="maxima")

[Out]

integrate(x^2/((b*x^2 - 1)^(3/4)*(b*x^2 - 2)), x)

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Fricas [A]  time = 0.234159, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{2}{\left (2 \, \arctan \left (\frac{\sqrt{2}{\left (b x^{2} - 1\right )}^{\frac{1}{4}}}{\sqrt{b} x}\right ) - \log \left (-\frac{\sqrt{2} b^{\frac{3}{2}} x^{2} - 4 \,{\left (b x^{2} - 1\right )}^{\frac{1}{4}} b x + 2 \, \sqrt{2} \sqrt{b x^{2} - 1} \sqrt{b}}{b x^{2} - 2 \, \sqrt{b x^{2} - 1}}\right )\right )}}{4 \, b^{\frac{3}{2}}}, -\frac{\sqrt{2}{\left (2 \, \arctan \left (\frac{\sqrt{2}{\left (b x^{2} - 1\right )}^{\frac{1}{4}} \sqrt{-b}}{b x}\right ) - \log \left (-\frac{\sqrt{2} \sqrt{-b} b x^{2} + 4 \,{\left (b x^{2} - 1\right )}^{\frac{1}{4}} b x - 2 \, \sqrt{2} \sqrt{b x^{2} - 1} \sqrt{-b}}{b x^{2} + 2 \, \sqrt{b x^{2} - 1}}\right )\right )}}{4 \, \sqrt{-b} b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x^2 - 1)^(3/4)*(b*x^2 - 2)),x, algorithm="fricas")

[Out]

[-1/4*sqrt(2)*(2*arctan(sqrt(2)*(b*x^2 - 1)^(1/4)/(sqrt(b)*x)) - log(-(sqrt(2)*b
^(3/2)*x^2 - 4*(b*x^2 - 1)^(1/4)*b*x + 2*sqrt(2)*sqrt(b*x^2 - 1)*sqrt(b))/(b*x^2
- 2*sqrt(b*x^2 - 1))))/b^(3/2), -1/4*sqrt(2)*(2*arctan(sqrt(2)*(b*x^2 - 1)^(1/4
)*sqrt(-b)/(b*x)) - log(-(sqrt(2)*sqrt(-b)*b*x^2 + 4*(b*x^2 - 1)^(1/4)*b*x - 2*s
qrt(2)*sqrt(b*x^2 - 1)*sqrt(-b))/(b*x^2 + 2*sqrt(b*x^2 - 1))))/(sqrt(-b)*b)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (b x^{2} - 2\right ) \left (b x^{2} - 1\right )^{\frac{3}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**2-2)/(b*x**2-1)**(3/4),x)

[Out]

Integral(x**2/((b*x**2 - 2)*(b*x**2 - 1)**(3/4)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b x^{2} - 1\right )}^{\frac{3}{4}}{\left (b x^{2} - 2\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((b*x^2 - 1)^(3/4)*(b*x^2 - 2)),x, algorithm="giac")

[Out]

integrate(x^2/((b*x^2 - 1)^(3/4)*(b*x^2 - 2)), x)