3.1208 \(\int \frac{x^3}{\sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=19 \[ -\frac{\left (a-b x^4\right )^{3/4}}{3 b} \]

[Out]

-(a - b*x^4)^(3/4)/(3*b)

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Rubi [A]  time = 0.0041099, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {261} \[ -\frac{\left (a-b x^4\right )^{3/4}}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a - b*x^4)^(1/4),x]

[Out]

-(a - b*x^4)^(3/4)/(3*b)

Rule 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

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

Mathematica [A]  time = 0.0038246, size = 19, normalized size = 1. \[ -\frac{\left (a-b x^4\right )^{3/4}}{3 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a - b*x^4)^(1/4),x]

[Out]

-(a - b*x^4)^(3/4)/(3*b)

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Maple [A]  time = 0.004, size = 16, normalized size = 0.8 \begin{align*} -{\frac{1}{3\,b} \left ( -b{x}^{4}+a \right ) ^{{\frac{3}{4}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(-b*x^4+a)^(1/4),x)

[Out]

-1/3*(-b*x^4+a)^(3/4)/b

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Maxima [A]  time = 0.98267, size = 20, normalized size = 1.05 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-b*x^4+a)^(1/4),x, algorithm="maxima")

[Out]

-1/3*(-b*x^4 + a)^(3/4)/b

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Fricas [A]  time = 1.4713, size = 36, normalized size = 1.89 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-b*x^4+a)^(1/4),x, algorithm="fricas")

[Out]

-1/3*(-b*x^4 + a)^(3/4)/b

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Sympy [A]  time = 0.601348, size = 24, normalized size = 1.26 \begin{align*} \begin{cases} - \frac{\left (a - b x^{4}\right )^{\frac{3}{4}}}{3 b} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 \sqrt [4]{a}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(-b*x**4+a)**(1/4),x)

[Out]

Piecewise((-(a - b*x**4)**(3/4)/(3*b), Ne(b, 0)), (x**4/(4*a**(1/4)), True))

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Giac [A]  time = 1.12089, size = 20, normalized size = 1.05 \begin{align*} -\frac{{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{3 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-b*x^4+a)^(1/4),x, algorithm="giac")

[Out]

-1/3*(-b*x^4 + a)^(3/4)/b