3.9.6 \(\int x^3 \sqrt {5+x^4} \, dx\) [806]

Optimal. Leaf size=13 \[ \frac {1}{6} \left (5+x^4\right )^{3/2} \]

[Out]

1/6*(x^4+5)^(3/2)

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Rubi [A]
time = 0.00, antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \begin {gather*} \frac {1}{6} \left (x^4+5\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*Sqrt[5 + x^4],x]

[Out]

(5 + x^4)^(3/2)/6

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 {5+x^4} \, dx &=\frac {1}{6} \left (5+x^4\right )^{3/2}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 13, normalized size = 1.00 \begin {gather*} \frac {1}{6} \left (5+x^4\right )^{3/2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*Sqrt[5 + x^4],x]

[Out]

(5 + x^4)^(3/2)/6

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Maple [A]
time = 0.16, size = 10, normalized size = 0.77

method	result	size
gosper	\(\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{6}\)	\(10\)
derivativedivides	\(\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{6}\)	\(10\)
default	\(\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{6}\)	\(10\)
risch	\(\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{6}\)	\(10\)
trager	\(\left (\frac {x^{4}}{6}+\frac {5}{6}\right ) \sqrt {x^{4}+5}\)	\(16\)
meijerg	\(-\frac {5 \sqrt {5}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2+\frac {2 x^{4}}{5}\right ) \sqrt {1+\frac {x^{4}}{5}}}{3}\right )}{8 \sqrt {\pi }}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x^4+5)^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/6*(x^4+5)^(3/2)

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Maxima [A]
time = 0.29, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{6} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^4+5)^(1/2),x, algorithm="maxima")

[Out]

1/6*(x^4 + 5)^(3/2)

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Fricas [A]
time = 0.36, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{6} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^4+5)^(1/2),x, algorithm="fricas")

[Out]

1/6*(x^4 + 5)^(3/2)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (8) = 16\).
time = 0.06, size = 24, normalized size = 1.85 \begin {gather*} \frac {x^{4} \sqrt {x^{4} + 5}}{6} + \frac {5 \sqrt {x^{4} + 5}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(x**4+5)**(1/2),x)

[Out]

x**4*sqrt(x**4 + 5)/6 + 5*sqrt(x**4 + 5)/6

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Giac [A]
time = 0.98, size = 9, normalized size = 0.69 \begin {gather*} \frac {1}{6} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(x^4+5)^(1/2),x, algorithm="giac")

[Out]

1/6*(x^4 + 5)^(3/2)

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Mupad [B]
time = 1.12, size = 9, normalized size = 0.69 \begin {gather*} \frac {{\left (x^4+5\right )}^{3/2}}{6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(x^4 + 5)^(1/2),x)

[Out]

(x^4 + 5)^(3/2)/6

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