3.578 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^{12}} \, dx\)

Optimal. Leaf size=92 \[ -\frac{d (x+1)^{11}}{11 x^{11}}-\frac{45 e}{2 x^2}-\frac{40 e}{x^3}-\frac{105 e}{2 x^4}-\frac{252 e}{5 x^5}-\frac{35 e}{x^6}-\frac{120 e}{7 x^7}-\frac{45 e}{8 x^8}-\frac{10 e}{9 x^9}-\frac{e}{10 x^{10}}-\frac{10 e}{x}+e \log (x) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0316115, antiderivative size = 92, 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, 78, 43} \[ -\frac{d (x+1)^{11}}{11 x^{11}}-\frac{45 e}{2 x^2}-\frac{40 e}{x^3}-\frac{105 e}{2 x^4}-\frac{252 e}{5 x^5}-\frac{35 e}{x^6}-\frac{120 e}{7 x^7}-\frac{45 e}{8 x^8}-\frac{10 e}{9 x^9}-\frac{e}{10 x^{10}}-\frac{10 e}{x}+e \log (x) \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^12,x]

[Out]

-e/(10*x^10) - (10*e)/(9*x^9) - (45*e)/(8*x^8) - (120*e)/(7*x^7) - (35*e)/x^6 - (252*e)/(5*x^5) - (105*e)/(2*x
^4) - (40*e)/x^3 - (45*e)/(2*x^2) - (10*e)/x - (d*(1 + x)^11)/(11*x^11) + e*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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, 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 \frac{(d+e x) \left (1+2 x+x^2\right )^5}{x^{12}} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^{12}} \, dx\\ &=-\frac{d (1+x)^{11}}{11 x^{11}}+e \int \frac{(1+x)^{10}}{x^{11}} \, dx\\ &=-\frac{d (1+x)^{11}}{11 x^{11}}+e \int \left (\frac{1}{x^{11}}+\frac{10}{x^{10}}+\frac{45}{x^9}+\frac{120}{x^8}+\frac{210}{x^7}+\frac{252}{x^6}+\frac{210}{x^5}+\frac{120}{x^4}+\frac{45}{x^3}+\frac{10}{x^2}+\frac{1}{x}\right ) \, dx\\ &=-\frac{e}{10 x^{10}}-\frac{10 e}{9 x^9}-\frac{45 e}{8 x^8}-\frac{120 e}{7 x^7}-\frac{35 e}{x^6}-\frac{252 e}{5 x^5}-\frac{105 e}{2 x^4}-\frac{40 e}{x^3}-\frac{45 e}{2 x^2}-\frac{10 e}{x}-\frac{d (1+x)^{11}}{11 x^{11}}+e \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0488074, size = 143, normalized size = 1.55 \[ -\frac{5 (2 d+9 e)}{2 x^2}-\frac{5 (3 d+8 e)}{x^3}-\frac{15 (4 d+7 e)}{2 x^4}-\frac{42 (5 d+6 e)}{5 x^5}-\frac{7 (6 d+5 e)}{x^6}-\frac{30 (7 d+4 e)}{7 x^7}-\frac{15 (8 d+3 e)}{8 x^8}-\frac{5 (9 d+2 e)}{9 x^9}-\frac{10 d+e}{10 x^{10}}-\frac{d+10 e}{x}-\frac{d}{11 x^{11}}+e \log (x) \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^12,x]

[Out]

-d/(11*x^11) - (10*d + e)/(10*x^10) - (5*(9*d + 2*e))/(9*x^9) - (15*(8*d + 3*e))/(8*x^8) - (30*(7*d + 4*e))/(7
*x^7) - (7*(6*d + 5*e))/x^6 - (42*(5*d + 6*e))/(5*x^5) - (15*(4*d + 7*e))/(2*x^4) - (5*(3*d + 8*e))/x^3 - (5*(
2*d + 9*e))/(2*x^2) - (d + 10*e)/x + e*Log[x]

________________________________________________________________________________________

Maple [A]  time = 0.009, size = 132, normalized size = 1.4 \begin{align*} -5\,{\frac{d}{{x}^{9}}}-{\frac{10\,e}{9\,{x}^{9}}}+e\ln \left ( x \right ) -30\,{\frac{d}{{x}^{4}}}-{\frac{105\,e}{2\,{x}^{4}}}-15\,{\frac{d}{{x}^{3}}}-40\,{\frac{e}{{x}^{3}}}-42\,{\frac{d}{{x}^{5}}}-{\frac{252\,e}{5\,{x}^{5}}}-5\,{\frac{d}{{x}^{2}}}-{\frac{45\,e}{2\,{x}^{2}}}-{\frac{d}{x}}-10\,{\frac{e}{x}}-30\,{\frac{d}{{x}^{7}}}-{\frac{120\,e}{7\,{x}^{7}}}-{\frac{d}{11\,{x}^{11}}}-15\,{\frac{d}{{x}^{8}}}-{\frac{45\,e}{8\,{x}^{8}}}-{\frac{d}{{x}^{10}}}-{\frac{e}{10\,{x}^{10}}}-42\,{\frac{d}{{x}^{6}}}-35\,{\frac{e}{{x}^{6}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^12,x)

[Out]

-5*d/x^9-10/9*e/x^9+e*ln(x)-30*d/x^4-105/2*e/x^4-15*d/x^3-40*e/x^3-42*d/x^5-252/5*e/x^5-5*d/x^2-45/2*e/x^2-d/x
-10*e/x-30*d/x^7-120/7*e/x^7-1/11*d/x^11-15*d/x^8-45/8*e/x^8-d/x^10-1/10*e/x^10-42*d/x^6-35*e/x^6

________________________________________________________________________________________

Maxima [A]  time = 0.961388, size = 173, normalized size = 1.88 \begin{align*} e \log \left (x\right ) - \frac{27720 \,{\left (d + 10 \, e\right )} x^{10} + 69300 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 138600 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 207900 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 232848 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 194040 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 118800 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 51975 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 15400 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 2772 \,{\left (10 \, d + e\right )} x + 2520 \, d}{27720 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^12,x, algorithm="maxima")

[Out]

e*log(x) - 1/27720*(27720*(d + 10*e)*x^10 + 69300*(2*d + 9*e)*x^9 + 138600*(3*d + 8*e)*x^8 + 207900*(4*d + 7*e
)*x^7 + 232848*(5*d + 6*e)*x^6 + 194040*(6*d + 5*e)*x^5 + 118800*(7*d + 4*e)*x^4 + 51975*(8*d + 3*e)*x^3 + 154
00*(9*d + 2*e)*x^2 + 2772*(10*d + e)*x + 2520*d)/x^11

________________________________________________________________________________________

Fricas [A]  time = 1.3867, size = 386, normalized size = 4.2 \begin{align*} \frac{27720 \, e x^{11} \log \left (x\right ) - 27720 \,{\left (d + 10 \, e\right )} x^{10} - 69300 \,{\left (2 \, d + 9 \, e\right )} x^{9} - 138600 \,{\left (3 \, d + 8 \, e\right )} x^{8} - 207900 \,{\left (4 \, d + 7 \, e\right )} x^{7} - 232848 \,{\left (5 \, d + 6 \, e\right )} x^{6} - 194040 \,{\left (6 \, d + 5 \, e\right )} x^{5} - 118800 \,{\left (7 \, d + 4 \, e\right )} x^{4} - 51975 \,{\left (8 \, d + 3 \, e\right )} x^{3} - 15400 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 2772 \,{\left (10 \, d + e\right )} x - 2520 \, d}{27720 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^12,x, algorithm="fricas")

[Out]

1/27720*(27720*e*x^11*log(x) - 27720*(d + 10*e)*x^10 - 69300*(2*d + 9*e)*x^9 - 138600*(3*d + 8*e)*x^8 - 207900
*(4*d + 7*e)*x^7 - 232848*(5*d + 6*e)*x^6 - 194040*(6*d + 5*e)*x^5 - 118800*(7*d + 4*e)*x^4 - 51975*(8*d + 3*e
)*x^3 - 15400*(9*d + 2*e)*x^2 - 2772*(10*d + e)*x - 2520*d)/x^11

________________________________________________________________________________________

Sympy [A]  time = 9.91297, size = 112, normalized size = 1.22 \begin{align*} e \log{\left (x \right )} - \frac{2520 d + x^{10} \left (27720 d + 277200 e\right ) + x^{9} \left (138600 d + 623700 e\right ) + x^{8} \left (415800 d + 1108800 e\right ) + x^{7} \left (831600 d + 1455300 e\right ) + x^{6} \left (1164240 d + 1397088 e\right ) + x^{5} \left (1164240 d + 970200 e\right ) + x^{4} \left (831600 d + 475200 e\right ) + x^{3} \left (415800 d + 155925 e\right ) + x^{2} \left (138600 d + 30800 e\right ) + x \left (27720 d + 2772 e\right )}{27720 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**12,x)

[Out]

e*log(x) - (2520*d + x**10*(27720*d + 277200*e) + x**9*(138600*d + 623700*e) + x**8*(415800*d + 1108800*e) + x
**7*(831600*d + 1455300*e) + x**6*(1164240*d + 1397088*e) + x**5*(1164240*d + 970200*e) + x**4*(831600*d + 475
200*e) + x**3*(415800*d + 155925*e) + x**2*(138600*d + 30800*e) + x*(27720*d + 2772*e))/(27720*x**11)

________________________________________________________________________________________

Giac [A]  time = 1.14247, size = 189, normalized size = 2.05 \begin{align*} e \log \left ({\left | x \right |}\right ) - \frac{27720 \,{\left (d + 10 \, e\right )} x^{10} + 69300 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 138600 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 207900 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 232848 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 194040 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 118800 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 51975 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 15400 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 2772 \,{\left (10 \, d + e\right )} x + 2520 \, d}{27720 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^12,x, algorithm="giac")

[Out]

e*log(abs(x)) - 1/27720*(27720*(d + 10*e)*x^10 + 69300*(2*d + 9*e)*x^9 + 138600*(3*d + 8*e)*x^8 + 207900*(4*d
+ 7*e)*x^7 + 232848*(5*d + 6*e)*x^6 + 194040*(6*d + 5*e)*x^5 + 118800*(7*d + 4*e)*x^4 + 51975*(8*d + 3*e)*x^3
+ 15400*(9*d + 2*e)*x^2 + 2772*(10*d + e)*x + 2520*d)/x^11