3.807 \(\int (a-b x^2)^{3/4} \, dx\)

Optimal. Leaf size=78 \[ \frac{6 a^{3/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{b} \sqrt [4]{a-b x^2}}+\frac{2}{5} x \left (a-b x^2\right )^{3/4} \]

[Out]

(2*x*(a - b*x^2)^(3/4))/5 + (6*a^(3/2)*(1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*S
qrt[b]*(a - b*x^2)^(1/4))

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Rubi [A]  time = 0.0187742, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {195, 229, 228} \[ \frac{6 a^{3/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{b} \sqrt [4]{a-b x^2}}+\frac{2}{5} x \left (a-b x^2\right )^{3/4} \]

Antiderivative was successfully verified.

[In]

Int[(a - b*x^2)^(3/4),x]

[Out]

(2*x*(a - b*x^2)^(3/4))/5 + (6*a^(3/2)*(1 - (b*x^2)/a)^(1/4)*EllipticE[ArcSin[(Sqrt[b]*x)/Sqrt[a]]/2, 2])/(5*S
qrt[b]*(a - b*x^2)^(1/4))

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 229

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

Rule 228

Int[((a_) + (b_.)*(x_)^2)^(-1/4), x_Symbol] :> Simp[(2*EllipticE[(1*ArcSin[Rt[-(b/a), 2]*x])/2, 2])/(a^(1/4)*R
t[-(b/a), 2]), x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && NegQ[b/a]

Rubi steps

\begin{align*} \int \left (a-b x^2\right )^{3/4} \, dx &=\frac{2}{5} x \left (a-b x^2\right )^{3/4}+\frac{1}{5} (3 a) \int \frac{1}{\sqrt [4]{a-b x^2}} \, dx\\ &=\frac{2}{5} x \left (a-b x^2\right )^{3/4}+\frac{\left (3 a \sqrt [4]{1-\frac{b x^2}{a}}\right ) \int \frac{1}{\sqrt [4]{1-\frac{b x^2}{a}}} \, dx}{5 \sqrt [4]{a-b x^2}}\\ &=\frac{2}{5} x \left (a-b x^2\right )^{3/4}+\frac{6 a^{3/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{5 \sqrt{b} \sqrt [4]{a-b x^2}}\\ \end{align*}

Mathematica [C]  time = 0.0062014, size = 47, normalized size = 0.6 \[ \frac{x \left (a-b x^2\right )^{3/4} \, _2F_1\left (-\frac{3}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )}{\left (1-\frac{b x^2}{a}\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a - b*x^2)^(3/4),x]

[Out]

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

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Maple [F]  time = 0.033, size = 0, normalized size = 0. \begin{align*} \int \left ( -b{x}^{2}+a \right ) ^{{\frac{3}{4}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-b*x^2+a)^(3/4),x)

[Out]

int((-b*x^2+a)^(3/4),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (-b x^{2} + a\right )}^{\frac{3}{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(3/4),x, algorithm="maxima")

[Out]

integrate((-b*x^2 + a)^(3/4), 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{3}{4}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(3/4),x, algorithm="fricas")

[Out]

integral((-b*x^2 + a)^(3/4), x)

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Sympy [C]  time = 0.886928, size = 27, normalized size = 0.35 \begin{align*} a^{\frac{3}{4}} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{3}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x**2+a)**(3/4),x)

[Out]

a**(3/4)*x*hyper((-3/4, 1/2), (3/2,), b*x**2*exp_polar(2*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{3}{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-b*x^2+a)^(3/4),x, algorithm="giac")

[Out]

integrate((-b*x^2 + a)^(3/4), x)