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

Optimal. Leaf size=19 \[ -\frac{\sqrt{a-b x^4}}{2 b} \]

[Out]

-Sqrt[a - b*x^4]/(2*b)

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Rubi [A]  time = 0.0043524, 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} \[ -\frac{\sqrt{a-b x^4}}{2 b} \]

Antiderivative was successfully verified.

[In]

Int[x^3/Sqrt[a - b*x^4],x]

[Out]

-Sqrt[a - b*x^4]/(2*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{a-b x^4}} \, dx &=-\frac{\sqrt{a-b x^4}}{2 b}\\ \end{align*}

Mathematica [A]  time = 0.0042814, size = 19, normalized size = 1. \[ -\frac{\sqrt{a-b x^4}}{2 b} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/Sqrt[a - b*x^4],x]

[Out]

-Sqrt[a - b*x^4]/(2*b)

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Maple [A]  time = 0.005, size = 16, normalized size = 0.8 \begin{align*} -{\frac{1}{2\,b}\sqrt{-b{x}^{4}+a}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(-b*x^4+a)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.972779, size = 20, normalized size = 1.05 \begin{align*} -\frac{\sqrt{-b x^{4} + a}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-b*x^4+a)^(1/2),x, algorithm="maxima")

[Out]

-1/2*sqrt(-b*x^4 + a)/b

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Fricas [A]  time = 1.44779, size = 34, normalized size = 1.79 \begin{align*} -\frac{\sqrt{-b x^{4} + a}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-b*x^4+a)^(1/2),x, algorithm="fricas")

[Out]

-1/2*sqrt(-b*x^4 + a)/b

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Sympy [A]  time = 0.609188, size = 24, normalized size = 1.26 \begin{align*} \begin{cases} - \frac{\sqrt{a - b x^{4}}}{2 b} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 \sqrt{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)**(1/2),x)

[Out]

Piecewise((-sqrt(a - b*x**4)/(2*b), Ne(b, 0)), (x**4/(4*sqrt(a)), True))

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Giac [A]  time = 1.11519, size = 20, normalized size = 1.05 \begin{align*} -\frac{\sqrt{-b x^{4} + a}}{2 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(-b*x^4+a)^(1/2),x, algorithm="giac")

[Out]

-1/2*sqrt(-b*x^4 + a)/b