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

Optimal. Leaf size=139 \[ \frac{1}{9} x^9 (d+10 e)+\frac{5}{8} x^8 (2 d+9 e)+\frac{15}{7} x^7 (3 d+8 e)+5 x^6 (4 d+7 e)+\frac{42}{5} x^5 (5 d+6 e)+\frac{21}{2} x^4 (6 d+5 e)+10 x^3 (7 d+4 e)+\frac{15}{2} x^2 (8 d+3 e)+5 x (9 d+2 e)+(10 d+e) \log (x)-\frac{d}{x}+\frac{e x^{10}}{10} \]

[Out]

-(d/x) + 5*(9*d + 2*e)*x + (15*(8*d + 3*e)*x^2)/2 + 10*(7*d + 4*e)*x^3 + (21*(6*
d + 5*e)*x^4)/2 + (42*(5*d + 6*e)*x^5)/5 + 5*(4*d + 7*e)*x^6 + (15*(3*d + 8*e)*x
^7)/7 + (5*(2*d + 9*e)*x^8)/8 + ((d + 10*e)*x^9)/9 + (e*x^10)/10 + (10*d + e)*Lo
g[x]

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Rubi [A]  time = 0.221759, antiderivative size = 139, 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 \[ \frac{1}{9} x^9 (d+10 e)+\frac{5}{8} x^8 (2 d+9 e)+\frac{15}{7} x^7 (3 d+8 e)+5 x^6 (4 d+7 e)+\frac{42}{5} x^5 (5 d+6 e)+\frac{21}{2} x^4 (6 d+5 e)+10 x^3 (7 d+4 e)+\frac{15}{2} x^2 (8 d+3 e)+5 x (9 d+2 e)+(10 d+e) \log (x)-\frac{d}{x}+\frac{e x^{10}}{10} \]

Antiderivative was successfully verified.

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

[Out]

-(d/x) + 5*(9*d + 2*e)*x + (15*(8*d + 3*e)*x^2)/2 + 10*(7*d + 4*e)*x^3 + (21*(6*
d + 5*e)*x^4)/2 + (42*(5*d + 6*e)*x^5)/5 + 5*(4*d + 7*e)*x^6 + (15*(3*d + 8*e)*x
^7)/7 + (5*(2*d + 9*e)*x^8)/8 + ((d + 10*e)*x^9)/9 + (e*x^10)/10 + (10*d + e)*Lo
g[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{d}{x} + \frac{e x^{10}}{10} + x^{9} \left (\frac{d}{9} + \frac{10 e}{9}\right ) + x^{8} \left (\frac{5 d}{4} + \frac{45 e}{8}\right ) + x^{7} \left (\frac{45 d}{7} + \frac{120 e}{7}\right ) + x^{6} \left (20 d + 35 e\right ) + x^{5} \left (42 d + \frac{252 e}{5}\right ) + x^{4} \left (63 d + \frac{105 e}{2}\right ) + x^{3} \left (70 d + 40 e\right ) + x \left (45 d + 10 e\right ) + \left (10 d + e\right ) \log{\left (x \right )} + \left (120 d + 45 e\right ) \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/x + e*x**10/10 + x**9*(d/9 + 10*e/9) + x**8*(5*d/4 + 45*e/8) + x**7*(45*d/7 +
120*e/7) + x**6*(20*d + 35*e) + x**5*(42*d + 252*e/5) + x**4*(63*d + 105*e/2) +
x**3*(70*d + 40*e) + x*(45*d + 10*e) + (10*d + e)*log(x) + (120*d + 45*e)*Integ
ral(x, x)

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Mathematica [A]  time = 0.0880862, size = 139, normalized size = 1. \[ \frac{1}{9} x^9 (d+10 e)+\frac{5}{8} x^8 (2 d+9 e)+\frac{15}{7} x^7 (3 d+8 e)+5 x^6 (4 d+7 e)+\frac{42}{5} x^5 (5 d+6 e)+\frac{21}{2} x^4 (6 d+5 e)+10 x^3 (7 d+4 e)+\frac{15}{2} x^2 (8 d+3 e)+5 x (9 d+2 e)+(10 d+e) \log (x)-\frac{d}{x}+\frac{e x^{10}}{10} \]

Antiderivative was successfully verified.

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

[Out]

-(d/x) + 5*(9*d + 2*e)*x + (15*(8*d + 3*e)*x^2)/2 + 10*(7*d + 4*e)*x^3 + (21*(6*
d + 5*e)*x^4)/2 + (42*(5*d + 6*e)*x^5)/5 + 5*(4*d + 7*e)*x^6 + (15*(3*d + 8*e)*x
^7)/7 + (5*(2*d + 9*e)*x^8)/8 + ((d + 10*e)*x^9)/9 + (e*x^10)/10 + (10*d + e)*Lo
g[x]

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Maple [A]  time = 0.01, size = 127, normalized size = 0.9 \[{\frac{e{x}^{10}}{10}}+{\frac{d{x}^{9}}{9}}+{\frac{10\,e{x}^{9}}{9}}+{\frac{5\,d{x}^{8}}{4}}+{\frac{45\,e{x}^{8}}{8}}+{\frac{45\,d{x}^{7}}{7}}+{\frac{120\,e{x}^{7}}{7}}+20\,d{x}^{6}+35\,e{x}^{6}+42\,d{x}^{5}+{\frac{252\,e{x}^{5}}{5}}+63\,d{x}^{4}+{\frac{105\,e{x}^{4}}{2}}+70\,d{x}^{3}+40\,e{x}^{3}+60\,d{x}^{2}+{\frac{45\,e{x}^{2}}{2}}+45\,dx+10\,ex+10\,d\ln \left ( x \right ) +e\ln \left ( x \right ) -{\frac{d}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*e*x^10+1/9*d*x^9+10/9*e*x^9+5/4*d*x^8+45/8*e*x^8+45/7*d*x^7+120/7*e*x^7+20*
d*x^6+35*e*x^6+42*d*x^5+252/5*e*x^5+63*d*x^4+105/2*e*x^4+70*d*x^3+40*e*x^3+60*d*
x^2+45/2*e*x^2+45*d*x+10*e*x+10*d*ln(x)+e*ln(x)-d/x

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Maxima [A]  time = 0.68096, size = 169, normalized size = 1.22 \[ \frac{1}{10} \, e x^{10} + \frac{1}{9} \,{\left (d + 10 \, e\right )} x^{9} + \frac{5}{8} \,{\left (2 \, d + 9 \, e\right )} x^{8} + \frac{15}{7} \,{\left (3 \, d + 8 \, e\right )} x^{7} + 5 \,{\left (4 \, d + 7 \, e\right )} x^{6} + \frac{42}{5} \,{\left (5 \, d + 6 \, e\right )} x^{5} + \frac{21}{2} \,{\left (6 \, d + 5 \, e\right )} x^{4} + 10 \,{\left (7 \, d + 4 \, e\right )} x^{3} + \frac{15}{2} \,{\left (8 \, d + 3 \, e\right )} x^{2} + 5 \,{\left (9 \, d + 2 \, e\right )} x +{\left (10 \, d + e\right )} \log \left (x\right ) - \frac{d}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*e*x^10 + 1/9*(d + 10*e)*x^9 + 5/8*(2*d + 9*e)*x^8 + 15/7*(3*d + 8*e)*x^7 +
5*(4*d + 7*e)*x^6 + 42/5*(5*d + 6*e)*x^5 + 21/2*(6*d + 5*e)*x^4 + 10*(7*d + 4*e)
*x^3 + 15/2*(8*d + 3*e)*x^2 + 5*(9*d + 2*e)*x + (10*d + e)*log(x) - d/x

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Fricas [A]  time = 0.282522, size = 177, normalized size = 1.27 \[ \frac{252 \, e x^{11} + 280 \,{\left (d + 10 \, e\right )} x^{10} + 1575 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 5400 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 12600 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 21168 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 26460 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 25200 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 18900 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 12600 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 2520 \,{\left (10 \, d + e\right )} x \log \left (x\right ) - 2520 \, d}{2520 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2520*(252*e*x^11 + 280*(d + 10*e)*x^10 + 1575*(2*d + 9*e)*x^9 + 5400*(3*d + 8*
e)*x^8 + 12600*(4*d + 7*e)*x^7 + 21168*(5*d + 6*e)*x^6 + 26460*(6*d + 5*e)*x^5 +
25200*(7*d + 4*e)*x^4 + 18900*(8*d + 3*e)*x^3 + 12600*(9*d + 2*e)*x^2 + 2520*(1
0*d + e)*x*log(x) - 2520*d)/x

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Sympy [A]  time = 1.88031, size = 121, normalized size = 0.87 \[ - \frac{d}{x} + \frac{e x^{10}}{10} + x^{9} \left (\frac{d}{9} + \frac{10 e}{9}\right ) + x^{8} \left (\frac{5 d}{4} + \frac{45 e}{8}\right ) + x^{7} \left (\frac{45 d}{7} + \frac{120 e}{7}\right ) + x^{6} \left (20 d + 35 e\right ) + x^{5} \left (42 d + \frac{252 e}{5}\right ) + x^{4} \left (63 d + \frac{105 e}{2}\right ) + x^{3} \left (70 d + 40 e\right ) + x^{2} \left (60 d + \frac{45 e}{2}\right ) + x \left (45 d + 10 e\right ) + \left (10 d + e\right ) \log{\left (x \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d/x + e*x**10/10 + x**9*(d/9 + 10*e/9) + x**8*(5*d/4 + 45*e/8) + x**7*(45*d/7 +
120*e/7) + x**6*(20*d + 35*e) + x**5*(42*d + 252*e/5) + x**4*(63*d + 105*e/2) +
x**3*(70*d + 40*e) + x**2*(60*d + 45*e/2) + x*(45*d + 10*e) + (10*d + e)*log(x)

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GIAC/XCAS [A]  time = 0.271083, size = 185, normalized size = 1.33 \[ \frac{1}{10} \, x^{10} e + \frac{1}{9} \, d x^{9} + \frac{10}{9} \, x^{9} e + \frac{5}{4} \, d x^{8} + \frac{45}{8} \, x^{8} e + \frac{45}{7} \, d x^{7} + \frac{120}{7} \, x^{7} e + 20 \, d x^{6} + 35 \, x^{6} e + 42 \, d x^{5} + \frac{252}{5} \, x^{5} e + 63 \, d x^{4} + \frac{105}{2} \, x^{4} e + 70 \, d x^{3} + 40 \, x^{3} e + 60 \, d x^{2} + \frac{45}{2} \, x^{2} e + 45 \, d x + 10 \, x e +{\left (10 \, d + e\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{d}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/10*x^10*e + 1/9*d*x^9 + 10/9*x^9*e + 5/4*d*x^8 + 45/8*x^8*e + 45/7*d*x^7 + 120
/7*x^7*e + 20*d*x^6 + 35*x^6*e + 42*d*x^5 + 252/5*x^5*e + 63*d*x^4 + 105/2*x^4*e
+ 70*d*x^3 + 40*x^3*e + 60*d*x^2 + 45/2*x^2*e + 45*d*x + 10*x*e + (10*d + e)*ln
(abs(x)) - d/x