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]

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

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Rubi [A]  time = 0.00, antiderivative size = 18, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 18, normalized size = 1.00 \[ \frac {\left (a+c x^4\right )^{3/2}}{6 c} \]

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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fricas [A]  time = 0.83, size = 14, normalized size = 0.78 \[ \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, c} \]

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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giac [A]  time = 0.18, size = 14, normalized size = 0.78 \[ \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, c} \]

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

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maple [A]  time = 0.00, size = 15, normalized size = 0.83 \[ \frac {\left (c \,x^{4}+a \right )^{\frac {3}{2}}}{6 c} \]

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 = 1.22, size = 14, normalized size = 0.78 \[ \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, c} \]

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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mupad [B]  time = 1.12, size = 14, normalized size = 0.78 \[ \frac {{\left (c\,x^4+a\right )}^{3/2}}{6\,c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + c*x^4)^(1/2),x)

[Out]

(a + c*x^4)^(3/2)/(6*c)

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sympy [A]  time = 0.40, size = 39, normalized size = 2.17 \[ \begin {cases} \frac {a \sqrt {a + c x^{4}}}{6 c} + \frac {x^{4} \sqrt {a + c x^{4}}}{6} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{4}}{4} & \text {otherwise} \end {cases} \]

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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