3.766 \(\int x^3 \sqrt{a+c x^4} \, dx\)

Optimal. Leaf size=18 \[ \frac{\left (a+c x^4\right )^{3/2}}{6 c} \]

[Out]

(a + c*x^4)^(3/2)/(6*c)

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Rubi [A]  time = 0.0044399, 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} \[ \frac{\left (a+c x^4\right )^{3/2}}{6 c} \]

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[a + c*x^4],x]

[Out]

(a + c*x^4)^(3/2)/(6*c)

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 x^3 \sqrt{a+c x^4} \, dx &=\frac{\left (a+c x^4\right )^{3/2}}{6 c}\\ \end{align*}

Mathematica [A]  time = 0.0031334, size = 18, normalized size = 1. \[ \frac{\left (a+c x^4\right )^{3/2}}{6 c} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[a + c*x^4],x]

[Out]

(a + c*x^4)^(3/2)/(6*c)

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Maple [A]  time = 0.003, size = 15, normalized size = 0.8 \begin{align*}{\frac{1}{6\,c} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*x^4+a)^(3/2)/c

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Maxima [A]  time = 0.948396, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*x^4 + a)^(3/2)/c

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Fricas [A]  time = 1.49359, size = 34, normalized size = 1.89 \begin{align*} \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*x^4 + a)^(3/2)/c

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Sympy [A]  time = 0.305095, size = 39, normalized size = 2.17 \begin{align*} \begin{cases} \frac{a \sqrt{a + c x^{4}}}{6 c} + \frac{x^{4} \sqrt{a + c x^{4}}}{6} & \text{for}\: c \neq 0 \\\frac{\sqrt{a} x^{4}}{4} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(c*x**4+a)**(1/2),x)

[Out]

Piecewise((a*sqrt(a + c*x**4)/(6*c) + x**4*sqrt(a + c*x**4)/6, Ne(c, 0)), (sqrt(a)*x**4/4, True))

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Giac [A]  time = 1.07666, size = 19, normalized size = 1.06 \begin{align*} \frac{{\left (c x^{4} + a\right )}^{\frac{3}{2}}}{6 \, c} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(c*x^4 + a)^(3/2)/c