3.315 \(\int \frac{1}{\left (-2 a+b x^2\right ) \sqrt [4]{-a+b x^2}} \, dx\)

Optimal. Leaf size=101 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}} \]

[Out]

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

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Rubi [A]  time = 0.0682901, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04 \[ -\frac{\tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{2} \sqrt [4]{a} \sqrt [4]{b x^2-a}}\right )}{2 \sqrt{2} a^{3/4} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 93.729, size = 233, normalized size = 2.31 \[ \frac{\sqrt{2} \sqrt{- \frac{1}{\sqrt{a}}} \sqrt{\frac{b x^{2}}{a}} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{- \frac{1}{\sqrt{a}}} \sqrt [4]{- a + b x^{2}}}{\sqrt{\frac{b x^{2}}{a}}} \right )}}{4 b x} + \frac{x \sqrt [4]{- a} \Pi \left (\frac{\sqrt{a}}{\sqrt{- a}}; \operatorname{asin}{\left (\frac{\sqrt [4]{- a + b x^{2}}}{\sqrt [4]{- a}} \right )}\middle | -1\right )}{a^{\frac{3}{2}} \sqrt{1 - \frac{\sqrt{- a + b x^{2}}}{\sqrt{- a}}} \sqrt{1 + \frac{\sqrt{- a + b x^{2}}}{\sqrt{- a}}}} - \frac{\sqrt{\frac{b x^{2}}{\left (\sqrt{a} + \sqrt{- a + b x^{2}}\right )^{2}}} \left (\sqrt{a} + \sqrt{- a + b x^{2}}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{- a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{4 a^{\frac{3}{4}} b x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*sqrt(-1/sqrt(a))*sqrt(b*x**2/a)*atan(sqrt(2)*sqrt(-1/sqrt(a))*(-a + b*x*
*2)**(1/4)/sqrt(b*x**2/a))/(4*b*x) + x*(-a)**(1/4)*elliptic_pi(sqrt(a)/sqrt(-a),
asin((-a + b*x**2)**(1/4)/(-a)**(1/4)), -1)/(a**(3/2)*sqrt(1 - sqrt(-a + b*x**2
)/sqrt(-a))*sqrt(1 + sqrt(-a + b*x**2)/sqrt(-a))) - sqrt(b*x**2/(sqrt(a) + sqrt(
-a + b*x**2))**2)*(sqrt(a) + sqrt(-a + b*x**2))*elliptic_f(2*atan((-a + b*x**2)*
*(1/4)/a**(1/4)), 1/2)/(4*a**(3/4)*b*x)

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Mathematica [C]  time = 0.251291, size = 163, normalized size = 1.61 \[ -\frac{6 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )}{\left (2 a-b x^2\right ) \sqrt [4]{b x^2-a} \left (b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )\right )+6 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(-6*a*x*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, (b*x^2)/(2*a)])/((2*a - b*x^2)*(-a
+ b*x^2)^(1/4)*(6*a*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, (b*x^2)/(2*a)] + b*x^
2*(2*AppellF1[3/2, 1/4, 2, 5/2, (b*x^2)/a, (b*x^2)/(2*a)] + AppellF1[3/2, 5/4, 1
, 5/2, (b*x^2)/a, (b*x^2)/(2*a)])))

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Maple [F]  time = 0.063, size = 0, normalized size = 0. \[ \int{\frac{1}{b{x}^{2}-2\,a}{\frac{1}{\sqrt [4]{b{x}^{2}-a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((b*x^2 - a)^(1/4)*(b*x^2 - 2*a)), 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 - a)^(1/4)*(b*x^2 - 2*a)),x, algorithm="fricas")

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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