3.8.66 \(\int x^3 \sqrt {a+c x^4} \, dx\) [766]

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

________________________________________________________________________________________

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 = {267} \begin {gather*} \frac {\left (a+c x^4\right )^{3/2}}{6 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[a + c*x^4],x]

[Out]

(a + c*x^4)^(3/2)/(6*c)

Rule 267

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 = 0.02, size = 18, normalized size = 1.00 \begin {gather*} \frac {\left (a+c x^4\right )^{3/2}}{6 c} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[a + c*x^4],x]

[Out]

(a + c*x^4)^(3/2)/(6*c)

________________________________________________________________________________________

Maple [A]
time = 0.14, size = 15, normalized size = 0.83

method	result	size
gosper	\(\frac {\left (x^{4} c +a \right )^{\frac {3}{2}}}{6 c}\)	\(15\)
derivativedivides	\(\frac {\left (x^{4} c +a \right )^{\frac {3}{2}}}{6 c}\)	\(15\)
default	\(\frac {\left (x^{4} c +a \right )^{\frac {3}{2}}}{6 c}\)	\(15\)
trager	\(\frac {\left (x^{4} c +a \right )^{\frac {3}{2}}}{6 c}\)	\(15\)
risch	\(\frac {\left (x^{4} c +a \right )^{\frac {3}{2}}}{6 c}\)	\(15\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [A]
time = 0.30, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, c} \end {gather*}

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

________________________________________________________________________________________

Fricas [A]
time = 0.35, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, c} \end {gather*}

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

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (12) = 24\).
time = 0.08, size = 39, normalized size = 2.17 \begin {gather*} \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 {gather*}

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))

________________________________________________________________________________________

Giac [A]
time = 0.48, size = 14, normalized size = 0.78 \begin {gather*} \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}}}{6 \, c} \end {gather*}

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

________________________________________________________________________________________

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

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)

________________________________________________________________________________________