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

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

[Out]

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

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Rubi [A]  time = 0.0725318, antiderivative size = 124, 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 \[ \frac{\tan ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a-b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt{b}}+\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a-b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a-b x^2}}\right )}{2 a^{3/4} \sqrt{b}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 54.2092, size = 112, normalized size = 0.9 \[ - \frac{\sqrt [4]{a} \sqrt{\frac{b x^{2}}{a}} \Pi \left (- \frac{\sqrt{a}}{\sqrt{- a}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a - b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{b x \sqrt{- a}} + \frac{\sqrt [4]{a} \sqrt{\frac{b x^{2}}{a}} \Pi \left (\frac{\sqrt{a}}{\sqrt{- a}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a - b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{b x \sqrt{- a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**(1/4)*sqrt(b*x**2/a)*elliptic_pi(-sqrt(a)/sqrt(-a), asin((a - b*x**2)**(1/4)
/a**(1/4)), -1)/(b*x*sqrt(-a)) + a**(1/4)*sqrt(b*x**2/a)*elliptic_pi(sqrt(a)/sqr
t(-a), asin((a - b*x**2)**(1/4)/a**(1/4)), -1)/(b*x*sqrt(-a))

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Mathematica [C]  time = 0.246662, size = 162, normalized size = 1.31 \[ \frac{6 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )}{\sqrt [4]{a-b x^2} \left (2 a-b x^2\right ) \left (b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )\right )+6 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};\frac{b x^2}{a},\frac{b x^2}{2 a}\right )\right )} \]

Warning: Unable to verify antiderivative.

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

[Out]

(6*a*x*AppellF1[1/2, 1/4, 1, 3/2, (b*x^2)/a, (b*x^2)/(2*a)])/((a - b*x^2)^(1/4)*
(2*a - b*x^2)*(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 = 0.064, 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+a)^(1/4)/(-b*x^2+2*a),x)

[Out]

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

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

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, algorithm="maxima")

[Out]

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

[Out]

Timed out

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

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)

[Out]

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

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

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, algorithm="giac")

[Out]

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