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

Optimal. Leaf size=18 \[ \frac{\sqrt{a+b x^4}}{2 b} \]

[Out]

Sqrt[a + b*x^4]/(2*b)

________________________________________________________________________________________

Rubi [A]  time = 0.0044366, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, 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 = 0.0028254, size = 18, 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)

________________________________________________________________________________________

Maple [A]  time = 0.004, size = 15, 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

________________________________________________________________________________________

Maxima [A]  time = 0.966064, size = 19, normalized size = 1.06 \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

________________________________________________________________________________________

Fricas [A]  time = 1.46437, size = 31, normalized size = 1.72 \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

________________________________________________________________________________________

Sympy [A]  time = 0.650496, size = 22, normalized size = 1.22 \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))

________________________________________________________________________________________

Giac [A]  time = 1.1088, size = 19, normalized size = 1.06 \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