3.73 \(\int \frac{(1+x^2) (1+2 x^2+x^4)^5}{x^2} \, dx\)

Optimal. Leaf size=73 \[ \frac{x^{21}}{21}+\frac{11 x^{19}}{19}+\frac{55 x^{17}}{17}+11 x^{15}+\frac{330 x^{13}}{13}+42 x^{11}+\frac{154 x^9}{3}+\frac{330 x^7}{7}+33 x^5+\frac{55 x^3}{3}+11 x-\frac{1}{x} \]

[Out]

-x^(-1) + 11*x + (55*x^3)/3 + 33*x^5 + (330*x^7)/7 + (154*x^9)/3 + 42*x^11 + (330*x^13)/13 + 11*x^15 + (55*x^1
7)/17 + (11*x^19)/19 + x^21/21

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Rubi [A]  time = 0.0247976, antiderivative size = 73, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {28, 270} \[ \frac{x^{21}}{21}+\frac{11 x^{19}}{19}+\frac{55 x^{17}}{17}+11 x^{15}+\frac{330 x^{13}}{13}+42 x^{11}+\frac{154 x^9}{3}+\frac{330 x^7}{7}+33 x^5+\frac{55 x^3}{3}+11 x-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) + 11*x + (55*x^3)/3 + 33*x^5 + (330*x^7)/7 + (154*x^9)/3 + 42*x^11 + (330*x^13)/13 + 11*x^15 + (55*x^1
7)/17 + (11*x^19)/19 + x^21/21

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 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin{align*} \int \frac{\left (1+x^2\right ) \left (1+2 x^2+x^4\right )^5}{x^2} \, dx &=\int \frac{\left (1+x^2\right )^{11}}{x^2} \, dx\\ &=\int \left (11+\frac{1}{x^2}+55 x^2+165 x^4+330 x^6+462 x^8+462 x^{10}+330 x^{12}+165 x^{14}+55 x^{16}+11 x^{18}+x^{20}\right ) \, dx\\ &=-\frac{1}{x}+11 x+\frac{55 x^3}{3}+33 x^5+\frac{330 x^7}{7}+\frac{154 x^9}{3}+42 x^{11}+\frac{330 x^{13}}{13}+11 x^{15}+\frac{55 x^{17}}{17}+\frac{11 x^{19}}{19}+\frac{x^{21}}{21}\\ \end{align*}

Mathematica [A]  time = 0.0030978, size = 73, normalized size = 1. \[ \frac{x^{21}}{21}+\frac{11 x^{19}}{19}+\frac{55 x^{17}}{17}+11 x^{15}+\frac{330 x^{13}}{13}+42 x^{11}+\frac{154 x^9}{3}+\frac{330 x^7}{7}+33 x^5+\frac{55 x^3}{3}+11 x-\frac{1}{x} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-x^(-1) + 11*x + (55*x^3)/3 + 33*x^5 + (330*x^7)/7 + (154*x^9)/3 + 42*x^11 + (330*x^13)/13 + 11*x^15 + (55*x^1
7)/17 + (11*x^19)/19 + x^21/21

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Maple [A]  time = 0.005, size = 60, normalized size = 0.8 \begin{align*} -{x}^{-1}+11\,x+{\frac{55\,{x}^{3}}{3}}+33\,{x}^{5}+{\frac{330\,{x}^{7}}{7}}+{\frac{154\,{x}^{9}}{3}}+42\,{x}^{11}+{\frac{330\,{x}^{13}}{13}}+11\,{x}^{15}+{\frac{55\,{x}^{17}}{17}}+{\frac{11\,{x}^{19}}{19}}+{\frac{{x}^{21}}{21}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/x+11*x+55/3*x^3+33*x^5+330/7*x^7+154/3*x^9+42*x^11+330/13*x^13+11*x^15+55/17*x^17+11/19*x^19+1/21*x^21

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Maxima [A]  time = 0.927436, size = 80, normalized size = 1.1 \begin{align*} \frac{1}{21} \, x^{21} + \frac{11}{19} \, x^{19} + \frac{55}{17} \, x^{17} + 11 \, x^{15} + \frac{330}{13} \, x^{13} + 42 \, x^{11} + \frac{154}{3} \, x^{9} + \frac{330}{7} \, x^{7} + 33 \, x^{5} + \frac{55}{3} \, x^{3} + 11 \, x - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*x^21 + 11/19*x^19 + 55/17*x^17 + 11*x^15 + 330/13*x^13 + 42*x^11 + 154/3*x^9 + 330/7*x^7 + 33*x^5 + 55/3*
x^3 + 11*x - 1/x

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Fricas [A]  time = 1.39501, size = 232, normalized size = 3.18 \begin{align*} \frac{4199 \, x^{22} + 51051 \, x^{20} + 285285 \, x^{18} + 969969 \, x^{16} + 2238390 \, x^{14} + 3703518 \, x^{12} + 4526522 \, x^{10} + 4157010 \, x^{8} + 2909907 \, x^{6} + 1616615 \, x^{4} + 969969 \, x^{2} - 88179}{88179 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/88179*(4199*x^22 + 51051*x^20 + 285285*x^18 + 969969*x^16 + 2238390*x^14 + 3703518*x^12 + 4526522*x^10 + 415
7010*x^8 + 2909907*x^6 + 1616615*x^4 + 969969*x^2 - 88179)/x

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Sympy [A]  time = 0.100858, size = 66, normalized size = 0.9 \begin{align*} \frac{x^{21}}{21} + \frac{11 x^{19}}{19} + \frac{55 x^{17}}{17} + 11 x^{15} + \frac{330 x^{13}}{13} + 42 x^{11} + \frac{154 x^{9}}{3} + \frac{330 x^{7}}{7} + 33 x^{5} + \frac{55 x^{3}}{3} + 11 x - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**21/21 + 11*x**19/19 + 55*x**17/17 + 11*x**15 + 330*x**13/13 + 42*x**11 + 154*x**9/3 + 330*x**7/7 + 33*x**5
+ 55*x**3/3 + 11*x - 1/x

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Giac [A]  time = 1.11949, size = 80, normalized size = 1.1 \begin{align*} \frac{1}{21} \, x^{21} + \frac{11}{19} \, x^{19} + \frac{55}{17} \, x^{17} + 11 \, x^{15} + \frac{330}{13} \, x^{13} + 42 \, x^{11} + \frac{154}{3} \, x^{9} + \frac{330}{7} \, x^{7} + 33 \, x^{5} + \frac{55}{3} \, x^{3} + 11 \, x - \frac{1}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/21*x^21 + 11/19*x^19 + 55/17*x^17 + 11*x^15 + 330/13*x^13 + 42*x^11 + 154/3*x^9 + 330/7*x^7 + 33*x^5 + 55/3*
x^3 + 11*x - 1/x