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

Optimal result

Integrand size = 13, antiderivative size = 13 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {1}{6} \left (5+x^4\right )^{3/2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {267} \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {1}{6} \left (x^4+5\right )^{3/2} \]

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 13, normalized size of antiderivative = 1.00 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {1}{6} \left (5+x^4\right )^{3/2} \]

[In]

Integrate[x^3*Sqrt[5 + x^4],x]

[Out]

(5 + x^4)^(3/2)/6

Maple [A] (verified)

Time = 4.00 (sec) , antiderivative size = 10, normalized size of antiderivative = 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	\(\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{6}\)	\(10\)
pseudoelliptic	\(\frac {\left (x^{4}+5\right )^{\frac {3}{2}}}{6}\)	\(10\)
trager	\(\left (\frac {x^{4}}{6}+\frac {5}{6}\right ) \sqrt {x^{4}+5}\)	\(16\)
meijerg	\(-\frac {5 \sqrt {5}\, \left (\frac {4 \sqrt {\pi }}{3}-\frac {2 \sqrt {\pi }\, \left (2+\frac {2 x^{4}}{5}\right ) \sqrt {1+\frac {x^{4}}{5}}}{3}\right )}{8 \sqrt {\pi }}\)	\(36\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {1}{6} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

1/6*(x^4 + 5)^(3/2)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 24 vs. \(2 (8) = 16\).

Time = 0.08 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.85 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {x^{4} \sqrt {x^{4} + 5}}{6} + \frac {5 \sqrt {x^{4} + 5}}{6} \]

[In]

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

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {1}{6} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

1/6*(x^4 + 5)^(3/2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {1}{6} \, {\left (x^{4} + 5\right )}^{\frac {3}{2}} \]

[In]

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

[Out]

1/6*(x^4 + 5)^(3/2)

Mupad [B] (verification not implemented)

Time = 5.57 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt {5+x^4} \, dx=\frac {{\left (x^4+5\right )}^{3/2}}{6} \]

[In]

int(x^3*(x^4 + 5)^(1/2),x)

[Out]

(x^4 + 5)^(3/2)/6