\(\int x^3 \sqrt [4]{1+x^4} \, dx\) [33]

Optimal result

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

[Out]

1/5*(x^4+1)^(5/4)

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} \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {1}{5} \left (x^4+1\right )^{5/4} \]

[In]

Int[x^3*(1 + x^4)^(1/4),x]

[Out]

(1 + x^4)^(5/4)/5

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}{5} \left (1+x^4\right )^{5/4} \\ \end{align*}

Mathematica [A] (verified)

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

[In]

Integrate[x^3*(1 + x^4)^(1/4),x]

[Out]

(1 + x^4)^(5/4)/5

Maple [A] (verified)

Time = 0.76 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.77

method	result	size
gosper	\(\frac {\left (x^{4}+1\right )^{\frac {5}{4}}}{5}\)	\(10\)
derivativedivides	\(\frac {\left (x^{4}+1\right )^{\frac {5}{4}}}{5}\)	\(10\)
default	\(\frac {\left (x^{4}+1\right )^{\frac {5}{4}}}{5}\)	\(10\)
risch	\(\frac {\left (x^{4}+1\right )^{\frac {5}{4}}}{5}\)	\(10\)
pseudoelliptic	\(\frac {\left (x^{4}+1\right )^{\frac {5}{4}}}{5}\)	\(10\)
trager	\(\left (\frac {x^{4}}{5}+\frac {1}{5}\right ) \left (x^{4}+1\right )^{\frac {1}{4}}\)	\(16\)
meijerg	\(\frac {x^{4} \operatorname {hypergeom}\left (\left [-\frac {1}{4}, 1\right ], \left [2\right ], -x^{4}\right )}{4}\)	\(17\)

[In]

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

[Out]

1/5*(x^4+1)^(5/4)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {1}{5} \, {\left (x^{4} + 1\right )}^{\frac {5}{4}} \]

[In]

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

[Out]

1/5*(x^4 + 1)^(5/4)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 22 vs. \(2 (8) = 16\).

Time = 0.09 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.69 \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {x^{4} \sqrt [4]{x^{4} + 1}}{5} + \frac {\sqrt [4]{x^{4} + 1}}{5} \]

[In]

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

[Out]

x**4*(x**4 + 1)**(1/4)/5 + (x**4 + 1)**(1/4)/5

Maxima [A] (verification not implemented)

none

Time = 0.17 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {1}{5} \, {\left (x^{4} + 1\right )}^{\frac {5}{4}} \]

[In]

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

[Out]

1/5*(x^4 + 1)^(5/4)

Giac [A] (verification not implemented)

none

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

[In]

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

[Out]

1/5*(x^4 + 1)^(5/4)

Mupad [B] (verification not implemented)

Time = 5.14 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.69 \[ \int x^3 \sqrt [4]{1+x^4} \, dx=\frac {{\left (x^4+1\right )}^{5/4}}{5} \]

[In]

int(x^3*(x^4 + 1)^(1/4),x)

[Out]

(x^4 + 1)^(5/4)/5