3.12.85 \(\int x \sqrt [4]{a-b x^4} \, dx\) [1185]

Optimal. Leaf size=82 \[ \frac {1}{3} x^2 \sqrt [4]{a-b x^4}+\frac {a^{3/2} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \sin ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{3 \sqrt {b} \left (a-b x^4\right )^{3/4}} \]

[Out]

1/3*x^2*(-b*x^4+a)^(1/4)+1/3*a^(3/2)*(1-b*x^4/a)^(3/4)*(cos(1/2*arcsin(x^2*b^(1/2)/a^(1/2)))^2)^(1/2)/cos(1/2*
arcsin(x^2*b^(1/2)/a^(1/2)))*EllipticF(sin(1/2*arcsin(x^2*b^(1/2)/a^(1/2))),2^(1/2))/(-b*x^4+a)^(3/4)/b^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 82, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {281, 201, 239, 238} \begin {gather*} \frac {a^{3/2} \left (1-\frac {b x^4}{a}\right )^{3/4} F\left (\left .\frac {1}{2} \text {ArcSin}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )\right |2\right )}{3 \sqrt {b} \left (a-b x^4\right )^{3/4}}+\frac {1}{3} x^2 \sqrt [4]{a-b x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x*(a - b*x^4)^(1/4),x]

[Out]

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

Rule 201

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 238

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

Rule 239

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 281

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rubi steps

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

________________________________________________________________________________________

Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
time = 6.50, size = 52, normalized size = 0.63 \begin {gather*} \frac {x^2 \sqrt [4]{a-b x^4} \, _2F_1\left (-\frac {1}{4},\frac {1}{2};\frac {3}{2};\frac {b x^4}{a}\right )}{2 \sqrt [4]{1-\frac {b x^4}{a}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x*(a - b*x^4)^(1/4),x]

[Out]

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

________________________________________________________________________________________

Maple [F]
time = 0.00, size = 0, normalized size = 0.00 \[\int x \left (-b \,x^{4}+a \right )^{\frac {1}{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-b*x^4 + a)^(1/4)*x, x)

________________________________________________________________________________________

Fricas [F]
time = 0.08, size = 14, normalized size = 0.17 \begin {gather*} {\rm integral}\left ({\left (-b x^{4} + a\right )}^{\frac {1}{4}} x, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((-b*x^4 + a)^(1/4)*x, x)

________________________________________________________________________________________

Sympy [C] Result contains complex when optimal does not.
time = 0.40, size = 31, normalized size = 0.38 \begin {gather*} \frac {\sqrt [4]{a} x^{2} {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {1}{2} \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{4} e^{2 i \pi }}{a}} \right )}}{2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((-b*x^4 + a)^(1/4)*x, x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x\,{\left (a-b\,x^4\right )}^{1/4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x*(a - b*x^4)^(1/4),x)

[Out]

int(x*(a - b*x^4)^(1/4), x)

________________________________________________________________________________________