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

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 + (1
20*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 = 0.0536237, 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 \[ \frac{d x^{10}}{10}+\frac{10 d x^9}{9}+\frac{45 d x^8}{8}+\frac{120 d x^7}{7}+35 d x^6+\frac{252 d x^5}{5}+\frac{105 d x^4}{2}+40 d x^3+\frac{45 d x^2}{2}+10 d x+d \log (x)+\frac{1}{11} e (x+1)^{11} \]

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 + (1
20*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 in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{d x^{10}}{10} + \frac{10 d x^{9}}{9} + \frac{45 d x^{8}}{8} + \frac{120 d x^{7}}{7} + 35 d x^{6} + \frac{252 d x^{5}}{5} + \frac{105 d x^{4}}{2} + 40 d x^{3} + 10 d x + d \log{\left (x \right )} + 45 d \int x\, dx + \frac{e \left (x + 1\right )^{11}}{11} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0604192, 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 + (2
52*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 = 0.005, size = 126, normalized size = 1.5 \[{\frac{e{x}^{11}}{11}}+{\frac{d{x}^{10}}{10}}+e{x}^{10}+{\frac{10\,d{x}^{9}}{9}}+5\,e{x}^{9}+{\frac{45\,d{x}^{8}}{8}}+15\,e{x}^{8}+{\frac{120\,d{x}^{7}}{7}}+30\,e{x}^{7}+35\,d{x}^{6}+42\,e{x}^{6}+{\frac{252\,d{x}^{5}}{5}}+42\,e{x}^{5}+{\frac{105\,d{x}^{4}}{2}}+30\,e{x}^{4}+40\,d{x}^{3}+15\,e{x}^{3}+{\frac{45\,d{x}^{2}}{2}}+5\,e{x}^{2}+10\,dx+ex+d\ln \left ( x \right ) \]

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+252/5*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 = 0.6756, size = 167, normalized size = 1.92 \[ \frac{1}{11} \, e x^{11} + \frac{1}{10} \,{\left (d + 10 \, e\right )} x^{10} + \frac{5}{9} \,{\left (2 \, d + 9 \, e\right )} x^{9} + \frac{15}{8} \,{\left (3 \, d + 8 \, e\right )} x^{8} + \frac{30}{7} \,{\left (4 \, d + 7 \, e\right )} x^{7} + 7 \,{\left (5 \, d + 6 \, e\right )} x^{6} + \frac{42}{5} \,{\left (6 \, d + 5 \, e\right )} x^{5} + \frac{15}{2} \,{\left (7 \, d + 4 \, e\right )} x^{4} + 5 \,{\left (8 \, d + 3 \, e\right )} x^{3} + \frac{5}{2} \,{\left (9 \, d + 2 \, e\right )} x^{2} +{\left (10 \, d + e\right )} x + d \log \left (x\right ) \]

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 = 0.273505, size = 167, normalized size = 1.92 \[ \frac{1}{11} \, e x^{11} + \frac{1}{10} \,{\left (d + 10 \, e\right )} x^{10} + \frac{5}{9} \,{\left (2 \, d + 9 \, e\right )} x^{9} + \frac{15}{8} \,{\left (3 \, d + 8 \, e\right )} x^{8} + \frac{30}{7} \,{\left (4 \, d + 7 \, e\right )} x^{7} + 7 \,{\left (5 \, d + 6 \, e\right )} x^{6} + \frac{42}{5} \,{\left (6 \, d + 5 \, e\right )} x^{5} + \frac{15}{2} \,{\left (7 \, d + 4 \, e\right )} x^{4} + 5 \,{\left (8 \, d + 3 \, e\right )} x^{3} + \frac{5}{2} \,{\left (9 \, d + 2 \, e\right )} x^{2} +{\left (10 \, d + e\right )} x + d \log \left (x\right ) \]

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 = 0.971253, size = 117, normalized size = 1.34 \[ d \log{\left (x \right )} + \frac{e x^{11}}{11} + x^{10} \left (\frac{d}{10} + e\right ) + x^{9} \left (\frac{10 d}{9} + 5 e\right ) + x^{8} \left (\frac{45 d}{8} + 15 e\right ) + x^{7} \left (\frac{120 d}{7} + 30 e\right ) + x^{6} \left (35 d + 42 e\right ) + x^{5} \left (\frac{252 d}{5} + 42 e\right ) + x^{4} \left (\frac{105 d}{2} + 30 e\right ) + x^{3} \left (40 d + 15 e\right ) + x^{2} \left (\frac{45 d}{2} + 5 e\right ) + x \left (10 d + e\right ) \]

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/XCAS [A]  time = 0.273581, size = 185, normalized size = 2.13 \[ \frac{1}{11} \, x^{11} e + \frac{1}{10} \, d x^{10} + x^{10} e + \frac{10}{9} \, d x^{9} + 5 \, x^{9} e + \frac{45}{8} \, d x^{8} + 15 \, x^{8} e + \frac{120}{7} \, d x^{7} + 30 \, x^{7} e + 35 \, d x^{6} + 42 \, x^{6} e + \frac{252}{5} \, d x^{5} + 42 \, x^{5} e + \frac{105}{2} \, d x^{4} + 30 \, x^{4} e + 40 \, d x^{3} + 15 \, x^{3} e + \frac{45}{2} \, d x^{2} + 5 \, x^{2} e + 10 \, d x + x e + d{\rm ln}\left ({\left | x \right |}\right ) \]

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 + 35*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*ln(abs(x))