\(\int x^4 (1+x^5)^5 \, dx\) [50]

Optimal result

Integrand size = 11, antiderivative size = 11 \[ \int x^4 \left (1+x^5\right )^5 \, dx=\frac {1}{30} \left (1+x^5\right )^6 \]

[Out]

1/30*(x^5+1)^6

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {267} \[ \int x^4 \left (1+x^5\right )^5 \, dx=\frac {1}{30} \left (x^5+1\right )^6 \]

[In]

Int[x^4*(1 + x^5)^5,x]

[Out]

(1 + x^5)^6/30

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}{30} \left (1+x^5\right )^6 \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(43\) vs. \(2(11)=22\).

Time = 0.00 (sec) , antiderivative size = 43, normalized size of antiderivative = 3.91 \[ \int x^4 \left (1+x^5\right )^5 \, dx=\frac {x^5}{5}+\frac {x^{10}}{2}+\frac {2 x^{15}}{3}+\frac {x^{20}}{2}+\frac {x^{25}}{5}+\frac {x^{30}}{30} \]

[In]

Integrate[x^4*(1 + x^5)^5,x]

[Out]

x^5/5 + x^10/2 + (2*x^15)/3 + x^20/2 + x^25/5 + x^30/30

Maple [A] (verified)

Time = 0.04 (sec) , antiderivative size = 10, normalized size of antiderivative = 0.91

method	result	size
default	\(\frac {\left (x^{5}+1\right )^{6}}{30}\)	\(10\)
gosper	\(\frac {x^{5} \left (x^{25}+6 x^{20}+15 x^{15}+20 x^{10}+15 x^{5}+6\right )}{30}\)	\(31\)
norman	\(\frac {1}{5} x^{25}+\frac {1}{30} x^{30}+\frac {1}{2} x^{10}+\frac {2}{3} x^{15}+\frac {1}{2} x^{20}+\frac {1}{5} x^{5}\)	\(32\)
parallelrisch	\(\frac {1}{5} x^{25}+\frac {1}{30} x^{30}+\frac {1}{2} x^{10}+\frac {2}{3} x^{15}+\frac {1}{2} x^{20}+\frac {1}{5} x^{5}\)	\(32\)
risch	\(\frac {1}{30} x^{30}+\frac {1}{5} x^{25}+\frac {1}{2} x^{20}+\frac {2}{3} x^{15}+\frac {1}{2} x^{10}+\frac {1}{5} x^{5}+\frac {1}{30}\)	\(33\)

[In]

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

[Out]

1/30*(x^5+1)^6

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (9) = 18\).

Time = 0.22 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.82 \[ \int x^4 \left (1+x^5\right )^5 \, dx=\frac {1}{30} \, x^{30} + \frac {1}{5} \, x^{25} + \frac {1}{2} \, x^{20} + \frac {2}{3} \, x^{15} + \frac {1}{2} \, x^{10} + \frac {1}{5} \, x^{5} \]

[In]

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

[Out]

1/30*x^30 + 1/5*x^25 + 1/2*x^20 + 2/3*x^15 + 1/2*x^10 + 1/5*x^5

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 31 vs. \(2 (7) = 14\).

Time = 0.02 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.82 \[ \int x^4 \left (1+x^5\right )^5 \, dx=\frac {x^{30}}{30} + \frac {x^{25}}{5} + \frac {x^{20}}{2} + \frac {2 x^{15}}{3} + \frac {x^{10}}{2} + \frac {x^{5}}{5} \]

[In]

integrate(x**4*(x**5+1)**5,x)

[Out]

x**30/30 + x**25/5 + x**20/2 + 2*x**15/3 + x**10/2 + x**5/5

Maxima [A] (verification not implemented)

none

Time = 0.18 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int x^4 \left (1+x^5\right )^5 \, dx=\frac {1}{30} \, {\left (x^{5} + 1\right )}^{6} \]

[In]

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

[Out]

1/30*(x^5 + 1)^6

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 9, normalized size of antiderivative = 0.82 \[ \int x^4 \left (1+x^5\right )^5 \, dx=\frac {1}{30} \, {\left (x^{5} + 1\right )}^{6} \]

[In]

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

[Out]

1/30*(x^5 + 1)^6

Mupad [B] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 31, normalized size of antiderivative = 2.82 \[ \int x^4 \left (1+x^5\right )^5 \, dx=\frac {x^{30}}{30}+\frac {x^{25}}{5}+\frac {x^{20}}{2}+\frac {2\,x^{15}}{3}+\frac {x^{10}}{2}+\frac {x^5}{5} \]

[In]

int(x^4*(x^5 + 1)^5,x)

[Out]

x^5/5 + x^10/2 + (2*x^15)/3 + x^20/2 + x^25/5 + x^30/30