3.4.70 \(\int x^2 (1+x^3)^4 \, dx\) [370]

Optimal. Leaf size=11 \[ \frac {1}{15} \left (1+x^3\right )^5 \]

[Out]

1/15*(x^3+1)^5

________________________________________________________________________________________

Rubi [A]
time = 0.00, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {267} \begin {gather*} \frac {1}{15} \left (x^3+1\right )^5 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*(1 + x^3)^4,x]

[Out]

(1 + x^3)^5/15

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^2 \left (1+x^3\right )^4 \, dx &=\frac {1}{15} \left (1+x^3\right )^5\\ \end {align*}

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(36\) vs. \(2(11)=22\).
time = 0.00, size = 36, normalized size = 3.27 \begin {gather*} \frac {x^3}{3}+\frac {2 x^6}{3}+\frac {2 x^9}{3}+\frac {x^{12}}{3}+\frac {x^{15}}{15} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*(1 + x^3)^4,x]

[Out]

x^3/3 + (2*x^6)/3 + (2*x^9)/3 + x^12/3 + x^15/15

________________________________________________________________________________________

Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(27\) vs. \(2(11)=22\).
time = 1.75, size = 25, normalized size = 2.27 \begin {gather*} \frac {x^3 \left (5+10 x^3+10 x^6+5 x^9+x^{12}\right )}{15} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^2*(1 + x^3)^4,x]')

[Out]

x ^ 3 (5 + 10 x ^ 3 + 10 x ^ 6 + 5 x ^ 9 + x ^ 12) / 15

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 10, normalized size = 0.91

method	result	size
default	\(\frac {\left (x^{3}+1\right )^{5}}{15}\)	\(10\)
gosper	\(\frac {x^{3} \left (x^{12}+5 x^{9}+10 x^{6}+10 x^{3}+5\right )}{15}\)	\(26\)
norman	\(\frac {1}{3} x^{3}+\frac {2}{3} x^{6}+\frac {2}{3} x^{9}+\frac {1}{3} x^{12}+\frac {1}{15} x^{15}\)	\(27\)
risch	\(\frac {1}{15} x^{15}+\frac {1}{3} x^{12}+\frac {2}{3} x^{9}+\frac {2}{3} x^{6}+\frac {1}{3} x^{3}+\frac {1}{15}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(x^3+1)^5

________________________________________________________________________________________

Maxima [A]
time = 0.26, size = 9, normalized size = 0.82 \begin {gather*} \frac {1}{15} \, {\left (x^{3} + 1\right )}^{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(x^3 + 1)^5

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 26 vs. \(2 (9) = 18\).
time = 0.34, size = 26, normalized size = 2.36 \begin {gather*} \frac {1}{15} \, x^{15} + \frac {1}{3} \, x^{12} + \frac {2}{3} \, x^{9} + \frac {2}{3} \, x^{6} + \frac {1}{3} \, x^{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*x^15 + 1/3*x^12 + 2/3*x^9 + 2/3*x^6 + 1/3*x^3

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 27 vs. \(2 (7) = 14\)
time = 0.03, size = 27, normalized size = 2.45 \begin {gather*} \frac {x^{15}}{15} + \frac {x^{12}}{3} + \frac {2 x^{9}}{3} + \frac {2 x^{6}}{3} + \frac {x^{3}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**15/15 + x**12/3 + 2*x**9/3 + 2*x**6/3 + x**3/3

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 12, normalized size = 1.09 \begin {gather*} \frac {\left (x^{3}+1\right )^{5}}{3\cdot 5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*(x^3+1)^4,x)

[Out]

1/15*(x^3 + 1)^5

________________________________________________________________________________________

Mupad [B]
time = 0.02, size = 26, normalized size = 2.36 \begin {gather*} \frac {x^{15}}{15}+\frac {x^{12}}{3}+\frac {2\,x^9}{3}+\frac {2\,x^6}{3}+\frac {x^3}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(x^3 + 1)^4,x)

[Out]

x^3/3 + (2*x^6)/3 + (2*x^9)/3 + x^12/3 + x^15/15

________________________________________________________________________________________