3.572 \(\int \frac{(d+e x) \left (1+2 x+x^2\right )^5}{x^6} \, dx\)

Optimal. Leaf size=140 \[ \frac{1}{5} x^5 (d+10 e)+\frac{5}{4} x^4 (2 d+9 e)-\frac{10 d+e}{4 x^4}+5 x^3 (3 d+8 e)-\frac{5 (9 d+2 e)}{3 x^3}+15 x^2 (4 d+7 e)-\frac{15 (8 d+3 e)}{2 x^2}+42 x (5 d+6 e)-\frac{30 (7 d+4 e)}{x}+42 (6 d+5 e) \log (x)-\frac{d}{5 x^5}+\frac{e x^6}{6} \]

[Out]

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

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Rubi [A]  time = 0.217215, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{1}{5} x^5 (d+10 e)+\frac{5}{4} x^4 (2 d+9 e)-\frac{10 d+e}{4 x^4}+5 x^3 (3 d+8 e)-\frac{5 (9 d+2 e)}{3 x^3}+15 x^2 (4 d+7 e)-\frac{15 (8 d+3 e)}{2 x^2}+42 x (5 d+6 e)-\frac{30 (7 d+4 e)}{x}+42 (6 d+5 e) \log (x)-\frac{d}{5 x^5}+\frac{e x^6}{6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{d}{5 x^{5}} + \frac{e x^{6}}{6} + x^{5} \left (\frac{d}{5} + 2 e\right ) + x^{4} \left (\frac{5 d}{2} + \frac{45 e}{4}\right ) + x^{3} \left (15 d + 40 e\right ) + x \left (210 d + 252 e\right ) + \left (120 d + 210 e\right ) \int x\, dx + \left (252 d + 210 e\right ) \log{\left (x \right )} - \frac{210 d + 120 e}{x} - \frac{60 d + \frac{45 e}{2}}{x^{2}} - \frac{15 d + \frac{10 e}{3}}{x^{3}} - \frac{\frac{5 d}{2} + \frac{e}{4}}{x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(x**2+2*x+1)**5/x**6,x)

[Out]

-d/(5*x**5) + e*x**6/6 + x**5*(d/5 + 2*e) + x**4*(5*d/2 + 45*e/4) + x**3*(15*d +
40*e) + x*(210*d + 252*e) + (120*d + 210*e)*Integral(x, x) + (252*d + 210*e)*lo
g(x) - (210*d + 120*e)/x - (60*d + 45*e/2)/x**2 - (15*d + 10*e/3)/x**3 - (5*d/2
+ e/4)/x**4

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Mathematica [A]  time = 0.0758219, size = 142, normalized size = 1.01 \[ \frac{1}{5} x^5 (d+10 e)+\frac{5}{4} x^4 (2 d+9 e)+\frac{-10 d-e}{4 x^4}+5 x^3 (3 d+8 e)-\frac{5 (9 d+2 e)}{3 x^3}+15 x^2 (4 d+7 e)-\frac{15 (8 d+3 e)}{2 x^2}+42 x (5 d+6 e)-\frac{30 (7 d+4 e)}{x}+42 (6 d+5 e) \log (x)-\frac{d}{5 x^5}+\frac{e x^6}{6} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.012, size = 128, normalized size = 0.9 \[{\frac{e{x}^{6}}{6}}+{\frac{d{x}^{5}}{5}}+2\,e{x}^{5}+{\frac{5\,d{x}^{4}}{2}}+{\frac{45\,e{x}^{4}}{4}}+15\,d{x}^{3}+40\,e{x}^{3}+60\,d{x}^{2}+105\,e{x}^{2}+210\,dx+252\,ex+252\,d\ln \left ( x \right ) +210\,e\ln \left ( x \right ) -{\frac{5\,d}{2\,{x}^{4}}}-{\frac{e}{4\,{x}^{4}}}-15\,{\frac{d}{{x}^{3}}}-{\frac{10\,e}{3\,{x}^{3}}}-60\,{\frac{d}{{x}^{2}}}-{\frac{45\,e}{2\,{x}^{2}}}-{\frac{d}{5\,{x}^{5}}}-210\,{\frac{d}{x}}-120\,{\frac{e}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(x^2+2*x+1)^5/x^6,x)

[Out]

1/6*e*x^6+1/5*d*x^5+2*e*x^5+5/2*d*x^4+45/4*e*x^4+15*d*x^3+40*e*x^3+60*d*x^2+105*
e*x^2+210*d*x+252*e*x+252*d*ln(x)+210*e*ln(x)-5/2*d/x^4-1/4*e/x^4-15*d/x^3-10/3*
e/x^3-60*d/x^2-45/2*e/x^2-1/5*d/x^5-210*d/x-120*e/x

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Maxima [A]  time = 0.681915, size = 171, normalized size = 1.22 \[ \frac{1}{6} \, e x^{6} + \frac{1}{5} \,{\left (d + 10 \, e\right )} x^{5} + \frac{5}{4} \,{\left (2 \, d + 9 \, e\right )} x^{4} + 5 \,{\left (3 \, d + 8 \, e\right )} x^{3} + 15 \,{\left (4 \, d + 7 \, e\right )} x^{2} + 42 \,{\left (5 \, d + 6 \, e\right )} x + 42 \,{\left (6 \, d + 5 \, e\right )} \log \left (x\right ) - \frac{1800 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 450 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 100 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 15 \,{\left (10 \, d + e\right )} x + 12 \, d}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5/x^6,x, algorithm="maxima")

[Out]

1/6*e*x^6 + 1/5*(d + 10*e)*x^5 + 5/4*(2*d + 9*e)*x^4 + 5*(3*d + 8*e)*x^3 + 15*(4
*d + 7*e)*x^2 + 42*(5*d + 6*e)*x + 42*(6*d + 5*e)*log(x) - 1/60*(1800*(7*d + 4*e
)*x^4 + 450*(8*d + 3*e)*x^3 + 100*(9*d + 2*e)*x^2 + 15*(10*d + e)*x + 12*d)/x^5

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Fricas [A]  time = 0.278008, size = 177, normalized size = 1.26 \[ \frac{10 \, e x^{11} + 12 \,{\left (d + 10 \, e\right )} x^{10} + 75 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 300 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 900 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 2520 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 2520 \,{\left (6 \, d + 5 \, e\right )} x^{5} \log \left (x\right ) - 1800 \,{\left (7 \, d + 4 \, e\right )} x^{4} - 450 \,{\left (8 \, d + 3 \, e\right )} x^{3} - 100 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 15 \,{\left (10 \, d + e\right )} x - 12 \, d}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5/x^6,x, algorithm="fricas")

[Out]

1/60*(10*e*x^11 + 12*(d + 10*e)*x^10 + 75*(2*d + 9*e)*x^9 + 300*(3*d + 8*e)*x^8
+ 900*(4*d + 7*e)*x^7 + 2520*(5*d + 6*e)*x^6 + 2520*(6*d + 5*e)*x^5*log(x) - 180
0*(7*d + 4*e)*x^4 - 450*(8*d + 3*e)*x^3 - 100*(9*d + 2*e)*x^2 - 15*(10*d + e)*x
- 12*d)/x^5

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Sympy [A]  time = 5.10036, size = 117, normalized size = 0.84 \[ \frac{e x^{6}}{6} + x^{5} \left (\frac{d}{5} + 2 e\right ) + x^{4} \left (\frac{5 d}{2} + \frac{45 e}{4}\right ) + x^{3} \left (15 d + 40 e\right ) + x^{2} \left (60 d + 105 e\right ) + x \left (210 d + 252 e\right ) + 42 \left (6 d + 5 e\right ) \log{\left (x \right )} - \frac{12 d + x^{4} \left (12600 d + 7200 e\right ) + x^{3} \left (3600 d + 1350 e\right ) + x^{2} \left (900 d + 200 e\right ) + x \left (150 d + 15 e\right )}{60 x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(x**2+2*x+1)**5/x**6,x)

[Out]

e*x**6/6 + x**5*(d/5 + 2*e) + x**4*(5*d/2 + 45*e/4) + x**3*(15*d + 40*e) + x**2*
(60*d + 105*e) + x*(210*d + 252*e) + 42*(6*d + 5*e)*log(x) - (12*d + x**4*(12600
*d + 7200*e) + x**3*(3600*d + 1350*e) + x**2*(900*d + 200*e) + x*(150*d + 15*e))
/(60*x**5)

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GIAC/XCAS [A]  time = 0.271743, size = 188, normalized size = 1.34 \[ \frac{1}{6} \, x^{6} e + \frac{1}{5} \, d x^{5} + 2 \, x^{5} e + \frac{5}{2} \, d x^{4} + \frac{45}{4} \, x^{4} e + 15 \, d x^{3} + 40 \, x^{3} e + 60 \, d x^{2} + 105 \, x^{2} e + 210 \, d x + 252 \, x e + 42 \,{\left (6 \, d + 5 \, e\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{1800 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 450 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 100 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 15 \,{\left (10 \, d + e\right )} x + 12 \, d}{60 \, x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5/x^6,x, algorithm="giac")

[Out]

1/6*x^6*e + 1/5*d*x^5 + 2*x^5*e + 5/2*d*x^4 + 45/4*x^4*e + 15*d*x^3 + 40*x^3*e +
60*d*x^2 + 105*x^2*e + 210*d*x + 252*x*e + 42*(6*d + 5*e)*ln(abs(x)) - 1/60*(18
00*(7*d + 4*e)*x^4 + 450*(8*d + 3*e)*x^3 + 100*(9*d + 2*e)*x^2 + 15*(10*d + e)*x
+ 12*d)/x^5