3.567 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x} \, dx\)

Optimal. Leaf size=87 \[ \frac{d x^{10}}{10}+\frac{10 d x^9}{9}+\frac{45 d x^8}{8}+\frac{120 d x^7}{7}+35 d x^6+\frac{252 d x^5}{5}+\frac{105 d x^4}{2}+40 d x^3+\frac{45 d x^2}{2}+10 d x+d \log (x)+\frac{1}{11} e (x+1)^{11} \]

[Out]

10*d*x + (45*d*x^2)/2 + 40*d*x^3 + (105*d*x^4)/2 + (252*d*x^5)/5 + 35*d*x^6 + (120*d*x^7)/7 + (45*d*x^8)/8 + (
10*d*x^9)/9 + (d*x^10)/10 + (e*(1 + x)^11)/11 + d*Log[x]

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Rubi [A]  time = 0.0217017, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {27, 80, 43} \[ \frac{d x^{10}}{10}+\frac{10 d x^9}{9}+\frac{45 d x^8}{8}+\frac{120 d x^7}{7}+35 d x^6+\frac{252 d x^5}{5}+\frac{105 d x^4}{2}+40 d x^3+\frac{45 d x^2}{2}+10 d x+d \log (x)+\frac{1}{11} e (x+1)^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

10*d*x + (45*d*x^2)/2 + 40*d*x^3 + (105*d*x^4)/2 + (252*d*x^5)/5 + 35*d*x^6 + (120*d*x^7)/7 + (45*d*x^8)/8 + (
10*d*x^9)/9 + (d*x^10)/10 + (e*(1 + x)^11)/11 + d*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 80

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(c + d*x)
^(n + 1)*(e + f*x)^(p + 1))/(d*f*(n + p + 2)), x] + Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(
d*f*(n + p + 2)), Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2,
0]

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{(d+e x) \left (1+2 x+x^2\right )^5}{x} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x} \, dx\\ &=\frac{1}{11} e (1+x)^{11}+d \int \frac{(1+x)^{10}}{x} \, dx\\ &=\frac{1}{11} e (1+x)^{11}+d \int \left (10+\frac{1}{x}+45 x+120 x^2+210 x^3+252 x^4+210 x^5+120 x^6+45 x^7+10 x^8+x^9\right ) \, dx\\ &=10 d x+\frac{45 d x^2}{2}+40 d x^3+\frac{105 d x^4}{2}+\frac{252 d x^5}{5}+35 d x^6+\frac{120 d x^7}{7}+\frac{45 d x^8}{8}+\frac{10 d x^9}{9}+\frac{d x^{10}}{10}+\frac{1}{11} e (1+x)^{11}+d \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0325468, size = 85, normalized size = 0.98 \[ d \left (\frac{x^{10}}{10}+\frac{10 x^9}{9}+\frac{45 x^8}{8}+\frac{120 x^7}{7}+35 x^6+\frac{252 x^5}{5}+\frac{105 x^4}{2}+40 x^3+\frac{45 x^2}{2}+10 x+\frac{7381}{2520}\right )+d \log (-x)+\frac{1}{11} e (x+1)^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(1 + x)^11)/11 + d*(7381/2520 + 10*x + (45*x^2)/2 + 40*x^3 + (105*x^4)/2 + (252*x^5)/5 + 35*x^6 + (120*x^7)
/7 + (45*x^8)/8 + (10*x^9)/9 + x^10/10) + d*Log[-x]

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Maple [A]  time = 0.003, size = 126, normalized size = 1.5 \begin{align*}{\frac{e{x}^{11}}{11}}+{\frac{d{x}^{10}}{10}}+e{x}^{10}+{\frac{10\,d{x}^{9}}{9}}+5\,e{x}^{9}+{\frac{45\,d{x}^{8}}{8}}+15\,e{x}^{8}+{\frac{120\,d{x}^{7}}{7}}+30\,e{x}^{7}+35\,d{x}^{6}+42\,e{x}^{6}+{\frac{252\,d{x}^{5}}{5}}+42\,e{x}^{5}+{\frac{105\,d{x}^{4}}{2}}+30\,e{x}^{4}+40\,d{x}^{3}+15\,e{x}^{3}+{\frac{45\,d{x}^{2}}{2}}+5\,e{x}^{2}+10\,dx+ex+d\ln \left ( x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*e*x^11+1/10*d*x^10+e*x^10+10/9*d*x^9+5*e*x^9+45/8*d*x^8+15*e*x^8+120/7*d*x^7+30*e*x^7+35*d*x^6+42*e*x^6+2
52/5*d*x^5+42*e*x^5+105/2*d*x^4+30*e*x^4+40*d*x^3+15*e*x^3+45/2*d*x^2+5*e*x^2+10*d*x+e*x+d*ln(x)

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Maxima [A]  time = 1.0016, size = 167, normalized size = 1.92 \begin{align*} \frac{1}{11} \, e x^{11} + \frac{1}{10} \,{\left (d + 10 \, e\right )} x^{10} + \frac{5}{9} \,{\left (2 \, d + 9 \, e\right )} x^{9} + \frac{15}{8} \,{\left (3 \, d + 8 \, e\right )} x^{8} + \frac{30}{7} \,{\left (4 \, d + 7 \, e\right )} x^{7} + 7 \,{\left (5 \, d + 6 \, e\right )} x^{6} + \frac{42}{5} \,{\left (6 \, d + 5 \, e\right )} x^{5} + \frac{15}{2} \,{\left (7 \, d + 4 \, e\right )} x^{4} + 5 \,{\left (8 \, d + 3 \, e\right )} x^{3} + \frac{5}{2} \,{\left (9 \, d + 2 \, e\right )} x^{2} +{\left (10 \, d + e\right )} x + d \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*e*x^11 + 1/10*(d + 10*e)*x^10 + 5/9*(2*d + 9*e)*x^9 + 15/8*(3*d + 8*e)*x^8 + 30/7*(4*d + 7*e)*x^7 + 7*(5*
d + 6*e)*x^6 + 42/5*(6*d + 5*e)*x^5 + 15/2*(7*d + 4*e)*x^4 + 5*(8*d + 3*e)*x^3 + 5/2*(9*d + 2*e)*x^2 + (10*d +
e)*x + d*log(x)

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Fricas [A]  time = 1.21982, size = 321, normalized size = 3.69 \begin{align*} \frac{1}{11} \, e x^{11} + \frac{1}{10} \,{\left (d + 10 \, e\right )} x^{10} + \frac{5}{9} \,{\left (2 \, d + 9 \, e\right )} x^{9} + \frac{15}{8} \,{\left (3 \, d + 8 \, e\right )} x^{8} + \frac{30}{7} \,{\left (4 \, d + 7 \, e\right )} x^{7} + 7 \,{\left (5 \, d + 6 \, e\right )} x^{6} + \frac{42}{5} \,{\left (6 \, d + 5 \, e\right )} x^{5} + \frac{15}{2} \,{\left (7 \, d + 4 \, e\right )} x^{4} + 5 \,{\left (8 \, d + 3 \, e\right )} x^{3} + \frac{5}{2} \,{\left (9 \, d + 2 \, e\right )} x^{2} +{\left (10 \, d + e\right )} x + d \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*e*x^11 + 1/10*(d + 10*e)*x^10 + 5/9*(2*d + 9*e)*x^9 + 15/8*(3*d + 8*e)*x^8 + 30/7*(4*d + 7*e)*x^7 + 7*(5*
d + 6*e)*x^6 + 42/5*(6*d + 5*e)*x^5 + 15/2*(7*d + 4*e)*x^4 + 5*(8*d + 3*e)*x^3 + 5/2*(9*d + 2*e)*x^2 + (10*d +
e)*x + d*log(x)

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Sympy [A]  time = 0.531578, size = 117, normalized size = 1.34 \begin{align*} d \log{\left (x \right )} + \frac{e x^{11}}{11} + x^{10} \left (\frac{d}{10} + e\right ) + x^{9} \left (\frac{10 d}{9} + 5 e\right ) + x^{8} \left (\frac{45 d}{8} + 15 e\right ) + x^{7} \left (\frac{120 d}{7} + 30 e\right ) + x^{6} \left (35 d + 42 e\right ) + x^{5} \left (\frac{252 d}{5} + 42 e\right ) + x^{4} \left (\frac{105 d}{2} + 30 e\right ) + x^{3} \left (40 d + 15 e\right ) + x^{2} \left (\frac{45 d}{2} + 5 e\right ) + x \left (10 d + e\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x,x)

[Out]

d*log(x) + e*x**11/11 + x**10*(d/10 + e) + x**9*(10*d/9 + 5*e) + x**8*(45*d/8 + 15*e) + x**7*(120*d/7 + 30*e)
+ x**6*(35*d + 42*e) + x**5*(252*d/5 + 42*e) + x**4*(105*d/2 + 30*e) + x**3*(40*d + 15*e) + x**2*(45*d/2 + 5*e
) + x*(10*d + e)

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Giac [A]  time = 1.15681, size = 185, normalized size = 2.13 \begin{align*} \frac{1}{11} \, x^{11} e + \frac{1}{10} \, d x^{10} + x^{10} e + \frac{10}{9} \, d x^{9} + 5 \, x^{9} e + \frac{45}{8} \, d x^{8} + 15 \, x^{8} e + \frac{120}{7} \, d x^{7} + 30 \, x^{7} e + 35 \, d x^{6} + 42 \, x^{6} e + \frac{252}{5} \, d x^{5} + 42 \, x^{5} e + \frac{105}{2} \, d x^{4} + 30 \, x^{4} e + 40 \, d x^{3} + 15 \, x^{3} e + \frac{45}{2} \, d x^{2} + 5 \, x^{2} e + 10 \, d x + x e + d \log \left ({\left | x \right |}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*x^11*e + 1/10*d*x^10 + x^10*e + 10/9*d*x^9 + 5*x^9*e + 45/8*d*x^8 + 15*x^8*e + 120/7*d*x^7 + 30*x^7*e + 3
5*d*x^6 + 42*x^6*e + 252/5*d*x^5 + 42*x^5*e + 105/2*d*x^4 + 30*x^4*e + 40*d*x^3 + 15*x^3*e + 45/2*d*x^2 + 5*x^
2*e + 10*d*x + x*e + d*log(abs(x))