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

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

[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 = 0.0746818, 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 \[ -\frac{d (x+1)^{11}}{11 x^{11}}-\frac{e}{10 x^{10}}-\frac{10 e}{9 x^9}-\frac{45 e}{8 x^8}-\frac{120 e}{7 x^7}-\frac{35 e}{x^6}-\frac{252 e}{5 x^5}-\frac{105 e}{2 x^4}-\frac{40 e}{x^3}-\frac{45 e}{2 x^2}-\frac{10 e}{x}+e \log (x) \]

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]

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.5436, size = 92, normalized size = 1. \[ - \frac{d \left (x + 1\right )^{11}}{11 x^{11}} + e \log{\left (x \right )} - \frac{10 e}{x} - \frac{45 e}{2 x^{2}} - \frac{40 e}{x^{3}} - \frac{105 e}{2 x^{4}} - \frac{252 e}{5 x^{5}} - \frac{35 e}{x^{6}} - \frac{120 e}{7 x^{7}} - \frac{45 e}{8 x^{8}} - \frac{10 e}{9 x^{9}} - \frac{e}{10 x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(x**2+2*x+1)**5/x**12,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.110622, size = 143, normalized size = 1.55 \[ -\frac{10 d+e}{10 x^{10}}-\frac{5 (9 d+2 e)}{9 x^9}-\frac{15 (8 d+3 e)}{8 x^8}-\frac{30 (7 d+4 e)}{7 x^7}-\frac{7 (6 d+5 e)}{x^6}-\frac{42 (5 d+6 e)}{5 x^5}-\frac{15 (4 d+7 e)}{2 x^4}-\frac{5 (3 d+8 e)}{x^3}-\frac{5 (2 d+9 e)}{2 x^2}-\frac{d+10 e}{x}-\frac{d}{11 x^{11}}+e \log (x) \]

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 = 0.01, size = 132, normalized size = 1.4 \[ e\ln \left ( x \right ) -42\,{\frac{d}{{x}^{6}}}-35\,{\frac{e}{{x}^{6}}}-30\,{\frac{d}{{x}^{4}}}-{\frac{105\,e}{2\,{x}^{4}}}-{\frac{d}{{x}^{10}}}-{\frac{e}{10\,{x}^{10}}}-15\,{\frac{d}{{x}^{8}}}-{\frac{45\,e}{8\,{x}^{8}}}-{\frac{d}{11\,{x}^{11}}}-5\,{\frac{d}{{x}^{9}}}-{\frac{10\,e}{9\,{x}^{9}}}-15\,{\frac{d}{{x}^{3}}}-40\,{\frac{e}{{x}^{3}}}-5\,{\frac{d}{{x}^{2}}}-{\frac{45\,e}{2\,{x}^{2}}}-42\,{\frac{d}{{x}^{5}}}-{\frac{252\,e}{5\,{x}^{5}}}-{\frac{d}{x}}-10\,{\frac{e}{x}}-30\,{\frac{d}{{x}^{7}}}-{\frac{120\,e}{7\,{x}^{7}}} \]

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]

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

_______________________________________________________________________________________

Maxima [A]  time = 0.695528, size = 173, normalized size = 1.88 \[ e \log \left (x\right ) - \frac{27720 \,{\left (d + 10 \, e\right )} x^{10} + 69300 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 138600 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 207900 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 232848 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 194040 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 118800 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 51975 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 15400 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 2772 \,{\left (10 \, d + e\right )} x + 2520 \, d}{27720 \, x^{11}} \]

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 + 15400*(9*d + 2*e)*x^2 +
2772*(10*d + e)*x + 2520*d)/x^11

_______________________________________________________________________________________

Fricas [A]  time = 0.283991, size = 177, normalized size = 1.92 \[ \frac{27720 \, e x^{11} \log \left (x\right ) - 27720 \,{\left (d + 10 \, e\right )} x^{10} - 69300 \,{\left (2 \, d + 9 \, e\right )} x^{9} - 138600 \,{\left (3 \, d + 8 \, e\right )} x^{8} - 207900 \,{\left (4 \, d + 7 \, e\right )} x^{7} - 232848 \,{\left (5 \, d + 6 \, e\right )} x^{6} - 194040 \,{\left (6 \, d + 5 \, e\right )} x^{5} - 118800 \,{\left (7 \, d + 4 \, e\right )} x^{4} - 51975 \,{\left (8 \, d + 3 \, e\right )} x^{3} - 15400 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 2772 \,{\left (10 \, d + e\right )} x - 2520 \, d}{27720 \, x^{11}} \]

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 - 1
38600*(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 = 26.07, size = 112, normalized size = 1.22 \[ e \log{\left (x \right )} - \frac{2520 d + x^{10} \left (27720 d + 277200 e\right ) + x^{9} \left (138600 d + 623700 e\right ) + x^{8} \left (415800 d + 1108800 e\right ) + x^{7} \left (831600 d + 1455300 e\right ) + x^{6} \left (1164240 d + 1397088 e\right ) + x^{5} \left (1164240 d + 970200 e\right ) + x^{4} \left (831600 d + 475200 e\right ) + x^{3} \left (415800 d + 155925 e\right ) + x^{2} \left (138600 d + 30800 e\right ) + x \left (27720 d + 2772 e\right )}{27720 x^{11}} \]

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 + 139
7088*e) + x**5*(1164240*d + 970200*e) + x**4*(831600*d + 475200*e) + x**3*(41580
0*d + 155925*e) + x**2*(138600*d + 30800*e) + x*(27720*d + 2772*e))/(27720*x**11
)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.273567, size = 189, normalized size = 2.05 \[ e{\rm ln}\left ({\left | x \right |}\right ) - \frac{27720 \,{\left (d + 10 \, e\right )} x^{10} + 69300 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 138600 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 207900 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 232848 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 194040 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 118800 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 51975 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 15400 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 2772 \,{\left (10 \, d + e\right )} x + 2520 \, d}{27720 \, x^{11}} \]

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*ln(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