3.570 \(\int \frac{(d+e x) \left (1+2 x+x^2\right )^5}{x^4} \, dx\)

Optimal. Leaf size=138 \[ \frac{1}{7} x^7 (d+10 e)+\frac{5}{6} x^6 (2 d+9 e)+3 x^5 (3 d+8 e)+\frac{15}{2} x^4 (4 d+7 e)+14 x^3 (5 d+6 e)+21 x^2 (6 d+5 e)-\frac{10 d+e}{2 x^2}+30 x (7 d+4 e)-\frac{5 (9 d+2 e)}{x}+15 (8 d+3 e) \log (x)-\frac{d}{3 x^3}+\frac{e x^8}{8} \]

[Out]

-d/(3*x^3) - (10*d + e)/(2*x^2) - (5*(9*d + 2*e))/x + 30*(7*d + 4*e)*x + 21*(6*d
+ 5*e)*x^2 + 14*(5*d + 6*e)*x^3 + (15*(4*d + 7*e)*x^4)/2 + 3*(3*d + 8*e)*x^5 +
(5*(2*d + 9*e)*x^6)/6 + ((d + 10*e)*x^7)/7 + (e*x^8)/8 + 15*(8*d + 3*e)*Log[x]

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Rubi [A]  time = 0.223797, antiderivative size = 138, 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 \[ \frac{1}{7} x^7 (d+10 e)+\frac{5}{6} x^6 (2 d+9 e)+3 x^5 (3 d+8 e)+\frac{15}{2} x^4 (4 d+7 e)+14 x^3 (5 d+6 e)+21 x^2 (6 d+5 e)-\frac{10 d+e}{2 x^2}+30 x (7 d+4 e)-\frac{5 (9 d+2 e)}{x}+15 (8 d+3 e) \log (x)-\frac{d}{3 x^3}+\frac{e x^8}{8} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^4,x]

[Out]

-d/(3*x^3) - (10*d + e)/(2*x^2) - (5*(9*d + 2*e))/x + 30*(7*d + 4*e)*x + 21*(6*d
+ 5*e)*x^2 + 14*(5*d + 6*e)*x^3 + (15*(4*d + 7*e)*x^4)/2 + 3*(3*d + 8*e)*x^5 +
(5*(2*d + 9*e)*x^6)/6 + ((d + 10*e)*x^7)/7 + (e*x^8)/8 + 15*(8*d + 3*e)*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{d}{3 x^{3}} + \frac{e x^{8}}{8} + x^{7} \left (\frac{d}{7} + \frac{10 e}{7}\right ) + x^{6} \left (\frac{5 d}{3} + \frac{15 e}{2}\right ) + x^{5} \left (9 d + 24 e\right ) + x^{4} \left (30 d + \frac{105 e}{2}\right ) + x^{3} \left (70 d + 84 e\right ) + x \left (210 d + 120 e\right ) + \left (120 d + 45 e\right ) \log{\left (x \right )} + \left (252 d + 210 e\right ) \int x\, dx - \frac{45 d + 10 e}{x} - \frac{5 d + \frac{e}{2}}{x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(x**2+2*x+1)**5/x**4,x)

[Out]

-d/(3*x**3) + e*x**8/8 + x**7*(d/7 + 10*e/7) + x**6*(5*d/3 + 15*e/2) + x**5*(9*d
+ 24*e) + x**4*(30*d + 105*e/2) + x**3*(70*d + 84*e) + x*(210*d + 120*e) + (120
*d + 45*e)*log(x) + (252*d + 210*e)*Integral(x, x) - (45*d + 10*e)/x - (5*d + e/
2)/x**2

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Mathematica [A]  time = 0.0767537, size = 140, normalized size = 1.01 \[ \frac{1}{7} x^7 (d+10 e)+\frac{5}{6} x^6 (2 d+9 e)+3 x^5 (3 d+8 e)+\frac{15}{2} x^4 (4 d+7 e)+14 x^3 (5 d+6 e)+21 x^2 (6 d+5 e)+\frac{-10 d-e}{2 x^2}+30 x (7 d+4 e)-\frac{5 (9 d+2 e)}{x}+15 (8 d+3 e) \log (x)-\frac{d}{3 x^3}+\frac{e x^8}{8} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^4,x]

[Out]

-d/(3*x^3) + (-10*d - e)/(2*x^2) - (5*(9*d + 2*e))/x + 30*(7*d + 4*e)*x + 21*(6*
d + 5*e)*x^2 + 14*(5*d + 6*e)*x^3 + (15*(4*d + 7*e)*x^4)/2 + 3*(3*d + 8*e)*x^5 +
(5*(2*d + 9*e)*x^6)/6 + ((d + 10*e)*x^7)/7 + (e*x^8)/8 + 15*(8*d + 3*e)*Log[x]

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Maple [A]  time = 0.01, size = 128, normalized size = 0.9 \[{\frac{e{x}^{8}}{8}}+{\frac{d{x}^{7}}{7}}+{\frac{10\,e{x}^{7}}{7}}+{\frac{5\,d{x}^{6}}{3}}+{\frac{15\,e{x}^{6}}{2}}+9\,d{x}^{5}+24\,e{x}^{5}+30\,d{x}^{4}+{\frac{105\,e{x}^{4}}{2}}+70\,d{x}^{3}+84\,e{x}^{3}+126\,d{x}^{2}+105\,e{x}^{2}+210\,dx+120\,ex+120\,d\ln \left ( x \right ) +45\,e\ln \left ( x \right ) -{\frac{d}{3\,{x}^{3}}}-5\,{\frac{d}{{x}^{2}}}-{\frac{e}{2\,{x}^{2}}}-45\,{\frac{d}{x}}-10\,{\frac{e}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(x^2+2*x+1)^5/x^4,x)

[Out]

1/8*e*x^8+1/7*d*x^7+10/7*e*x^7+5/3*d*x^6+15/2*e*x^6+9*d*x^5+24*e*x^5+30*d*x^4+10
5/2*e*x^4+70*d*x^3+84*e*x^3+126*d*x^2+105*e*x^2+210*d*x+120*e*x+120*d*ln(x)+45*e
*ln(x)-1/3*d/x^3-5*d/x^2-1/2*e/x^2-45*d/x-10*e/x

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Maxima [A]  time = 0.679481, size = 171, normalized size = 1.24 \[ \frac{1}{8} \, e x^{8} + \frac{1}{7} \,{\left (d + 10 \, e\right )} x^{7} + \frac{5}{6} \,{\left (2 \, d + 9 \, e\right )} x^{6} + 3 \,{\left (3 \, d + 8 \, e\right )} x^{5} + \frac{15}{2} \,{\left (4 \, d + 7 \, e\right )} x^{4} + 14 \,{\left (5 \, d + 6 \, e\right )} x^{3} + 21 \,{\left (6 \, d + 5 \, e\right )} x^{2} + 30 \,{\left (7 \, d + 4 \, e\right )} x + 15 \,{\left (8 \, d + 3 \, e\right )} \log \left (x\right ) - \frac{30 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 3 \,{\left (10 \, d + e\right )} x + 2 \, d}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5/x^4,x, algorithm="maxima")

[Out]

1/8*e*x^8 + 1/7*(d + 10*e)*x^7 + 5/6*(2*d + 9*e)*x^6 + 3*(3*d + 8*e)*x^5 + 15/2*
(4*d + 7*e)*x^4 + 14*(5*d + 6*e)*x^3 + 21*(6*d + 5*e)*x^2 + 30*(7*d + 4*e)*x + 1
5*(8*d + 3*e)*log(x) - 1/6*(30*(9*d + 2*e)*x^2 + 3*(10*d + e)*x + 2*d)/x^3

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Fricas [A]  time = 0.279504, size = 177, normalized size = 1.28 \[ \frac{21 \, e x^{11} + 24 \,{\left (d + 10 \, e\right )} x^{10} + 140 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 504 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 1260 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 2352 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 3528 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 5040 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 2520 \,{\left (8 \, d + 3 \, e\right )} x^{3} \log \left (x\right ) - 840 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 84 \,{\left (10 \, d + e\right )} x - 56 \, d}{168 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5/x^4,x, algorithm="fricas")

[Out]

1/168*(21*e*x^11 + 24*(d + 10*e)*x^10 + 140*(2*d + 9*e)*x^9 + 504*(3*d + 8*e)*x^
8 + 1260*(4*d + 7*e)*x^7 + 2352*(5*d + 6*e)*x^6 + 3528*(6*d + 5*e)*x^5 + 5040*(7
*d + 4*e)*x^4 + 2520*(8*d + 3*e)*x^3*log(x) - 840*(9*d + 2*e)*x^2 - 84*(10*d + e
)*x - 56*d)/x^3

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Sympy [A]  time = 2.86563, size = 121, normalized size = 0.88 \[ \frac{e x^{8}}{8} + x^{7} \left (\frac{d}{7} + \frac{10 e}{7}\right ) + x^{6} \left (\frac{5 d}{3} + \frac{15 e}{2}\right ) + x^{5} \left (9 d + 24 e\right ) + x^{4} \left (30 d + \frac{105 e}{2}\right ) + x^{3} \left (70 d + 84 e\right ) + x^{2} \left (126 d + 105 e\right ) + x \left (210 d + 120 e\right ) + 15 \left (8 d + 3 e\right ) \log{\left (x \right )} - \frac{2 d + x^{2} \left (270 d + 60 e\right ) + x \left (30 d + 3 e\right )}{6 x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(x**2+2*x+1)**5/x**4,x)

[Out]

e*x**8/8 + x**7*(d/7 + 10*e/7) + x**6*(5*d/3 + 15*e/2) + x**5*(9*d + 24*e) + x**
4*(30*d + 105*e/2) + x**3*(70*d + 84*e) + x**2*(126*d + 105*e) + x*(210*d + 120*
e) + 15*(8*d + 3*e)*log(x) - (2*d + x**2*(270*d + 60*e) + x*(30*d + 3*e))/(6*x**
3)

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GIAC/XCAS [A]  time = 0.270894, size = 188, normalized size = 1.36 \[ \frac{1}{8} \, x^{8} e + \frac{1}{7} \, d x^{7} + \frac{10}{7} \, x^{7} e + \frac{5}{3} \, d x^{6} + \frac{15}{2} \, x^{6} e + 9 \, d x^{5} + 24 \, x^{5} e + 30 \, d x^{4} + \frac{105}{2} \, x^{4} e + 70 \, d x^{3} + 84 \, x^{3} e + 126 \, d x^{2} + 105 \, x^{2} e + 210 \, d x + 120 \, x e + 15 \,{\left (8 \, d + 3 \, e\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{30 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 3 \,{\left (10 \, d + e\right )} x + 2 \, d}{6 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5/x^4,x, algorithm="giac")

[Out]

1/8*x^8*e + 1/7*d*x^7 + 10/7*x^7*e + 5/3*d*x^6 + 15/2*x^6*e + 9*d*x^5 + 24*x^5*e
+ 30*d*x^4 + 105/2*x^4*e + 70*d*x^3 + 84*x^3*e + 126*d*x^2 + 105*x^2*e + 210*d*
x + 120*x*e + 15*(8*d + 3*e)*ln(abs(x)) - 1/6*(30*(9*d + 2*e)*x^2 + 3*(10*d + e)
*x + 2*d)/x^3