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

Optimal. Leaf size=120 \[ -\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt{b}} \]

[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])

_______________________________________________________________________________________

Rubi [A]  time = 0.0723392, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043 \[ -\frac{\tan ^{-1}\left (\frac{a^{3/4} \left (\frac{\sqrt{a+b x^2}}{\sqrt{a}}+1\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt{b}}-\frac{\tanh ^{-1}\left (\frac{a^{3/4} \left (1-\frac{\sqrt{a+b x^2}}{\sqrt{a}}\right )}{\sqrt{b} x \sqrt [4]{a+b x^2}}\right )}{2 a^{3/4} \sqrt{b}} \]

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])

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 52.7652, size = 116, normalized size = 0.97 \[ \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \Pi \left (- \frac{\sqrt{a}}{\sqrt{- a}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{b x \sqrt{- a}} - \frac{\sqrt [4]{a} \sqrt{- \frac{b x^{2}}{a}} \Pi \left (\frac{\sqrt{a}}{\sqrt{- a}}; \operatorname{asin}{\left (\frac{\sqrt [4]{a + b x^{2}}}{\sqrt [4]{a}} \right )}\middle | -1\right )}{b x \sqrt{- a}} \]

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)/sq
rt(-a), asin((a + b*x**2)**(1/4)/a**(1/4)), -1)/(b*x*sqrt(-a))

_______________________________________________________________________________________

Mathematica [C]  time = 0.236927, size = 165, normalized size = 1.38 \[ \frac{6 a x F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )}{\sqrt [4]{a+b x^2} \left (2 a+b x^2\right ) \left (6 a F_1\left (\frac{1}{2};\frac{1}{4},1;\frac{3}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )-b x^2 \left (2 F_1\left (\frac{3}{2};\frac{1}{4},2;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )+F_1\left (\frac{3}{2};\frac{5}{4},1;\frac{5}{2};-\frac{b x^2}{a},-\frac{b x^2}{2 a}\right )\right )\right )} \]

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)])))

_______________________________________________________________________________________

Maple [F]  time = 0.074, 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)

_______________________________________________________________________________________

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)

_______________________________________________________________________________________

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

_______________________________________________________________________________________

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

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

_______________________________________________________________________________________

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)