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

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

[Out]

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

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Rubi [A]  time = 0.0178442, 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, Rules used = {195, 233, 231} \[ \frac{2 a^{3/2} \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{b} \left (a+b x^2\right )^{3/4}}+\frac{2}{3} x \sqrt [4]{a+b x^2} \]

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

Rule 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

Rule 233

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Dist[(1 + (b*x^2)/a)^(3/4)/(a + b*x^2)^(3/4), Int[1/(1 + (b*x^2
)/a)^(3/4), x], x] /; FreeQ[{a, b}, x] && PosQ[a]

Rule 231

Int[((a_) + (b_.)*(x_)^2)^(-3/4), x_Symbol] :> Simp[(2*EllipticF[(1*ArcTan[Rt[b/a, 2]*x])/2, 2])/(a^(3/4)*Rt[b
/a, 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b/a]

Rubi steps

\begin{align*} \int \sqrt [4]{a+b x^2} \, dx &=\frac{2}{3} x \sqrt [4]{a+b x^2}+\frac{1}{3} a \int \frac{1}{\left (a+b x^2\right )^{3/4}} \, dx\\ &=\frac{2}{3} x \sqrt [4]{a+b x^2}+\frac{\left (a \left (1+\frac{b x^2}{a}\right )^{3/4}\right ) \int \frac{1}{\left (1+\frac{b x^2}{a}\right )^{3/4}} \, dx}{3 \left (a+b x^2\right )^{3/4}}\\ &=\frac{2}{3} x \sqrt [4]{a+b x^2}+\frac{2 a^{3/2} \left (1+\frac{b x^2}{a}\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}}\\ \end{align*}

Mathematica [C]  time = 0.0046881, size = 46, normalized size = 0.61 \[ \frac{x \sqrt [4]{a+b x^2} \, _2F_1\left (-\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )}{\sqrt [4]{\frac{b x^2}{a}+1}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2)^(1/4),x]

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \sqrt [4]{b{x}^{2}+a}\, dx \end{align*}

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. \begin{align*} \int{\left (b x^{2} + a\right )}^{\frac{1}{4}}\,{d x} \end{align*}

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. \begin{align*}{\rm integral}\left ({\left (b x^{2} + a\right )}^{\frac{1}{4}}, x\right ) \end{align*}

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 [C]  time = 0.713273, size = 26, normalized size = 0.35 \begin{align*} \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 )} \end{align*}

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

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)