3.66 \(\int x^5 (1+x^2) (1+2 x^2+x^4)^5 \, dx\)

Optimal. Leaf size=34 \[ \frac{1}{28} \left (x^2+1\right )^{14}-\frac{1}{13} \left (x^2+1\right )^{13}+\frac{1}{24} \left (x^2+1\right )^{12} \]

[Out]

(1 + x^2)^12/24 - (1 + x^2)^13/13 + (1 + x^2)^14/28

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Rubi [A]  time = 0.0467126, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143, Rules used = {28, 266, 43} \[ \frac{1}{28} \left (x^2+1\right )^{14}-\frac{1}{13} \left (x^2+1\right )^{13}+\frac{1}{24} \left (x^2+1\right )^{12} \]

Antiderivative was successfully verified.

[In]

Int[x^5*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]

[Out]

(1 + x^2)^12/24 - (1 + x^2)^13/13 + (1 + x^2)^14/28

Rule 28

Int[(u_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/c^p, Int[u*(b/2 + c*x^n)^(2*
p), x], x] /; FreeQ[{a, b, c, n}, x] && EqQ[n2, 2*n] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x^5 \left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5 \, dx &=\int x^5 \left (1+x^2\right )^{11} \, dx\\ &=\frac{1}{2} \operatorname{Subst}\left (\int x^2 (1+x)^{11} \, dx,x,x^2\right )\\ &=\frac{1}{2} \operatorname{Subst}\left (\int \left ((1+x)^{11}-2 (1+x)^{12}+(1+x)^{13}\right ) \, dx,x,x^2\right )\\ &=\frac{1}{24} \left (1+x^2\right )^{12}-\frac{1}{13} \left (1+x^2\right )^{13}+\frac{1}{28} \left (1+x^2\right )^{14}\\ \end{align*}

Mathematica [B]  time = 0.0019485, size = 85, normalized size = 2.5 \[ \frac{x^{28}}{28}+\frac{11 x^{26}}{26}+\frac{55 x^{24}}{24}+\frac{15 x^{22}}{2}+\frac{33 x^{20}}{2}+\frac{77 x^{18}}{3}+\frac{231 x^{16}}{8}+\frac{165 x^{14}}{7}+\frac{55 x^{12}}{4}+\frac{11 x^{10}}{2}+\frac{11 x^8}{8}+\frac{x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[x^5*(1 + x^2)*(1 + 2*x^2 + x^4)^5,x]

[Out]

x^6/6 + (11*x^8)/8 + (11*x^10)/2 + (55*x^12)/4 + (165*x^14)/7 + (231*x^16)/8 + (77*x^18)/3 + (33*x^20)/2 + (15
*x^22)/2 + (55*x^24)/24 + (11*x^26)/26 + x^28/28

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Maple [B]  time = 0.001, size = 62, normalized size = 1.8 \begin{align*}{\frac{{x}^{28}}{28}}+{\frac{11\,{x}^{26}}{26}}+{\frac{55\,{x}^{24}}{24}}+{\frac{15\,{x}^{22}}{2}}+{\frac{33\,{x}^{20}}{2}}+{\frac{77\,{x}^{18}}{3}}+{\frac{231\,{x}^{16}}{8}}+{\frac{165\,{x}^{14}}{7}}+{\frac{55\,{x}^{12}}{4}}+{\frac{11\,{x}^{10}}{2}}+{\frac{11\,{x}^{8}}{8}}+{\frac{{x}^{6}}{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^5*(x^2+1)*(x^4+2*x^2+1)^5,x)

[Out]

1/28*x^28+11/26*x^26+55/24*x^24+15/2*x^22+33/2*x^20+77/3*x^18+231/8*x^16+165/7*x^14+55/4*x^12+11/2*x^10+11/8*x
^8+1/6*x^6

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Maxima [B]  time = 0.953037, size = 82, normalized size = 2.41 \begin{align*} \frac{1}{28} \, x^{28} + \frac{11}{26} \, x^{26} + \frac{55}{24} \, x^{24} + \frac{15}{2} \, x^{22} + \frac{33}{2} \, x^{20} + \frac{77}{3} \, x^{18} + \frac{231}{8} \, x^{16} + \frac{165}{7} \, x^{14} + \frac{55}{4} \, x^{12} + \frac{11}{2} \, x^{10} + \frac{11}{8} \, x^{8} + \frac{1}{6} \, x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*x^28 + 11/26*x^26 + 55/24*x^24 + 15/2*x^22 + 33/2*x^20 + 77/3*x^18 + 231/8*x^16 + 165/7*x^14 + 55/4*x^12
+ 11/2*x^10 + 11/8*x^8 + 1/6*x^6

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Fricas [B]  time = 1.25211, size = 194, normalized size = 5.71 \begin{align*} \frac{1}{28} x^{28} + \frac{11}{26} x^{26} + \frac{55}{24} x^{24} + \frac{15}{2} x^{22} + \frac{33}{2} x^{20} + \frac{77}{3} x^{18} + \frac{231}{8} x^{16} + \frac{165}{7} x^{14} + \frac{55}{4} x^{12} + \frac{11}{2} x^{10} + \frac{11}{8} x^{8} + \frac{1}{6} x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*x^28 + 11/26*x^26 + 55/24*x^24 + 15/2*x^22 + 33/2*x^20 + 77/3*x^18 + 231/8*x^16 + 165/7*x^14 + 55/4*x^12
+ 11/2*x^10 + 11/8*x^8 + 1/6*x^6

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Sympy [B]  time = 0.068427, size = 76, normalized size = 2.24 \begin{align*} \frac{x^{28}}{28} + \frac{11 x^{26}}{26} + \frac{55 x^{24}}{24} + \frac{15 x^{22}}{2} + \frac{33 x^{20}}{2} + \frac{77 x^{18}}{3} + \frac{231 x^{16}}{8} + \frac{165 x^{14}}{7} + \frac{55 x^{12}}{4} + \frac{11 x^{10}}{2} + \frac{11 x^{8}}{8} + \frac{x^{6}}{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**28/28 + 11*x**26/26 + 55*x**24/24 + 15*x**22/2 + 33*x**20/2 + 77*x**18/3 + 231*x**16/8 + 165*x**14/7 + 55*x
**12/4 + 11*x**10/2 + 11*x**8/8 + x**6/6

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Giac [B]  time = 1.11737, size = 82, normalized size = 2.41 \begin{align*} \frac{1}{28} \, x^{28} + \frac{11}{26} \, x^{26} + \frac{55}{24} \, x^{24} + \frac{15}{2} \, x^{22} + \frac{33}{2} \, x^{20} + \frac{77}{3} \, x^{18} + \frac{231}{8} \, x^{16} + \frac{165}{7} \, x^{14} + \frac{55}{4} \, x^{12} + \frac{11}{2} \, x^{10} + \frac{11}{8} \, x^{8} + \frac{1}{6} \, x^{6} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/28*x^28 + 11/26*x^26 + 55/24*x^24 + 15/2*x^22 + 33/2*x^20 + 77/3*x^18 + 231/8*x^16 + 165/7*x^14 + 55/4*x^12
+ 11/2*x^10 + 11/8*x^8 + 1/6*x^6