3.605 \(\int \frac{(1+x) (1+2 x+x^2)^5}{x^6} \, dx\)

Optimal. Leaf size=72 \[ \frac{x^6}{6}+\frac{11 x^5}{5}+\frac{55 x^4}{4}+55 x^3+165 x^2-\frac{165}{2 x^2}-\frac{55}{3 x^3}-\frac{11}{4 x^4}-\frac{1}{5 x^5}+462 x-\frac{330}{x}+462 \log (x) \]

[Out]

-1/(5*x^5) - 11/(4*x^4) - 55/(3*x^3) - 165/(2*x^2) - 330/x + 462*x + 165*x^2 + 55*x^3 + (55*x^4)/4 + (11*x^5)/
5 + x^6/6 + 462*Log[x]

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Rubi [A]  time = 0.0215513, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {27, 43} \[ \frac{x^6}{6}+\frac{11 x^5}{5}+\frac{55 x^4}{4}+55 x^3+165 x^2-\frac{165}{2 x^2}-\frac{55}{3 x^3}-\frac{11}{4 x^4}-\frac{1}{5 x^5}+462 x-\frac{330}{x}+462 \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((1 + x)*(1 + 2*x + x^2)^5)/x^6,x]

[Out]

-1/(5*x^5) - 11/(4*x^4) - 55/(3*x^3) - 165/(2*x^2) - 330/x + 462*x + 165*x^2 + 55*x^3 + (55*x^4)/4 + (11*x^5)/
5 + x^6/6 + 462*Log[x]

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

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 \frac{(1+x) \left (1+2 x+x^2\right )^5}{x^6} \, dx &=\int \frac{(1+x)^{11}}{x^6} \, dx\\ &=\int \left (462+\frac{1}{x^6}+\frac{11}{x^5}+\frac{55}{x^4}+\frac{165}{x^3}+\frac{330}{x^2}+\frac{462}{x}+330 x+165 x^2+55 x^3+11 x^4+x^5\right ) \, dx\\ &=-\frac{1}{5 x^5}-\frac{11}{4 x^4}-\frac{55}{3 x^3}-\frac{165}{2 x^2}-\frac{330}{x}+462 x+165 x^2+55 x^3+\frac{55 x^4}{4}+\frac{11 x^5}{5}+\frac{x^6}{6}+462 \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0026145, size = 72, normalized size = 1. \[ \frac{x^6}{6}+\frac{11 x^5}{5}+\frac{55 x^4}{4}+55 x^3+165 x^2-\frac{165}{2 x^2}-\frac{55}{3 x^3}-\frac{11}{4 x^4}-\frac{1}{5 x^5}+462 x-\frac{330}{x}+462 \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[((1 + x)*(1 + 2*x + x^2)^5)/x^6,x]

[Out]

-1/(5*x^5) - 11/(4*x^4) - 55/(3*x^3) - 165/(2*x^2) - 330/x + 462*x + 165*x^2 + 55*x^3 + (55*x^4)/4 + (11*x^5)/
5 + x^6/6 + 462*Log[x]

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Maple [A]  time = 0.007, size = 59, normalized size = 0.8 \begin{align*} -{\frac{1}{5\,{x}^{5}}}-{\frac{11}{4\,{x}^{4}}}-{\frac{55}{3\,{x}^{3}}}-{\frac{165}{2\,{x}^{2}}}-330\,{x}^{-1}+462\,x+165\,{x}^{2}+55\,{x}^{3}+{\frac{55\,{x}^{4}}{4}}+{\frac{11\,{x}^{5}}{5}}+{\frac{{x}^{6}}{6}}+462\,\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1+x)*(x^2+2*x+1)^5/x^6,x)

[Out]

-1/5/x^5-11/4/x^4-55/3/x^3-165/2/x^2-330/x+462*x+165*x^2+55*x^3+55/4*x^4+11/5*x^5+1/6*x^6+462*ln(x)

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Maxima [A]  time = 1.0053, size = 78, normalized size = 1.08 \begin{align*} \frac{1}{6} \, x^{6} + \frac{11}{5} \, x^{5} + \frac{55}{4} \, x^{4} + 55 \, x^{3} + 165 \, x^{2} + 462 \, x - \frac{19800 \, x^{4} + 4950 \, x^{3} + 1100 \, x^{2} + 165 \, x + 12}{60 \, x^{5}} + 462 \, \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^6,x, algorithm="maxima")

[Out]

1/6*x^6 + 11/5*x^5 + 55/4*x^4 + 55*x^3 + 165*x^2 + 462*x - 1/60*(19800*x^4 + 4950*x^3 + 1100*x^2 + 165*x + 12)
/x^5 + 462*log(x)

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Fricas [A]  time = 1.26942, size = 190, normalized size = 2.64 \begin{align*} \frac{10 \, x^{11} + 132 \, x^{10} + 825 \, x^{9} + 3300 \, x^{8} + 9900 \, x^{7} + 27720 \, x^{6} + 27720 \, x^{5} \log \left (x\right ) - 19800 \, x^{4} - 4950 \, x^{3} - 1100 \, x^{2} - 165 \, x - 12}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^6,x, algorithm="fricas")

[Out]

1/60*(10*x^11 + 132*x^10 + 825*x^9 + 3300*x^8 + 9900*x^7 + 27720*x^6 + 27720*x^5*log(x) - 19800*x^4 - 4950*x^3
- 1100*x^2 - 165*x - 12)/x^5

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Sympy [A]  time = 0.126323, size = 61, normalized size = 0.85 \begin{align*} \frac{x^{6}}{6} + \frac{11 x^{5}}{5} + \frac{55 x^{4}}{4} + 55 x^{3} + 165 x^{2} + 462 x + 462 \log{\left (x \right )} - \frac{19800 x^{4} + 4950 x^{3} + 1100 x^{2} + 165 x + 12}{60 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x**2+2*x+1)**5/x**6,x)

[Out]

x**6/6 + 11*x**5/5 + 55*x**4/4 + 55*x**3 + 165*x**2 + 462*x + 462*log(x) - (19800*x**4 + 4950*x**3 + 1100*x**2
+ 165*x + 12)/(60*x**5)

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Giac [A]  time = 1.11834, size = 80, normalized size = 1.11 \begin{align*} \frac{1}{6} \, x^{6} + \frac{11}{5} \, x^{5} + \frac{55}{4} \, x^{4} + 55 \, x^{3} + 165 \, x^{2} + 462 \, x - \frac{19800 \, x^{4} + 4950 \, x^{3} + 1100 \, x^{2} + 165 \, x + 12}{60 \, x^{5}} + 462 \, \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1+x)*(x^2+2*x+1)^5/x^6,x, algorithm="giac")

[Out]

1/6*x^6 + 11/5*x^5 + 55/4*x^4 + 55*x^3 + 165*x^2 + 462*x - 1/60*(19800*x^4 + 4950*x^3 + 1100*x^2 + 165*x + 12)
/x^5 + 462*log(abs(x))