3.789 \(\int \sqrt [4]{a+b x^2} \, dx\)

Optimal. Leaf size=75 \[ \frac{2 a^{3/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{b} \left (a+b x^2\right )^{3/4}}+\frac{2}{3} x \sqrt [4]{a+b x^2} \]

[Out]

(2*x*(a + b*x^2)^(1/4))/3 + (2*a^(3/2)*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(S
qrt[b]*x)/Sqrt[a]]/2, 2])/(3*Sqrt[b]*(a + b*x^2)^(3/4))

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Rubi [A]  time = 0.0518148, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{2 a^{3/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{b} \left (a+b x^2\right )^{3/4}}+\frac{2}{3} x \sqrt [4]{a+b x^2} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(1/4),x]

[Out]

(2*x*(a + b*x^2)^(1/4))/3 + (2*a^(3/2)*(1 + (b*x^2)/a)^(3/4)*EllipticF[ArcTan[(S
qrt[b]*x)/Sqrt[a]]/2, 2])/(3*Sqrt[b]*(a + b*x^2)^(3/4))

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Rubi in Sympy [A]  time = 5.66467, size = 66, normalized size = 0.88 \[ \frac{2 a^{\frac{3}{2}} \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{3 \sqrt{b} \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 x \sqrt [4]{a + b x^{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*a**(3/2)*(1 + b*x**2/a)**(3/4)*elliptic_f(atan(sqrt(b)*x/sqrt(a))/2, 2)/(3*sqr
t(b)*(a + b*x**2)**(3/4)) + 2*x*(a + b*x**2)**(1/4)/3

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Mathematica [C]  time = 0.033219, size = 62, normalized size = 0.83 \[ \frac{a x \left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )+2 x \left (a+b x^2\right )}{3 \left (a+b x^2\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(1/4),x]

[Out]

(2*x*(a + b*x^2) + a*x*(1 + (b*x^2)/a)^(3/4)*Hypergeometric2F1[1/2, 3/4, 3/2, -(
(b*x^2)/a)])/(3*(a + b*x^2)^(3/4))

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Maple [F]  time = 0.047, size = 0, normalized size = 0. \[ \int \sqrt [4]{b{x}^{2}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(1/4),x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(1/4), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{1}{4}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integral((b*x^2 + a)^(1/4), x)

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Sympy [A]  time = 2.32004, size = 26, normalized size = 0.35 \[ \sqrt [4]{a} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**2+a)**(1/4),x)

[Out]

a**(1/4)*x*hyper((-1/4, 1/2), (3/2,), b*x**2*exp_polar(I*pi)/a)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x^2 + a)^(1/4), x)