3.573 \(\int \frac{(d+e x) \left (1+2 x+x^2\right )^5}{x^7} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.220094, 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 \[ -\frac{10 d+e}{5 x^5}+\frac{1}{4} x^4 (d+10 e)-\frac{5 (9 d+2 e)}{4 x^4}+\frac{5}{3} x^3 (2 d+9 e)-\frac{5 (8 d+3 e)}{x^3}+\frac{15}{2} x^2 (3 d+8 e)-\frac{15 (7 d+4 e)}{x^2}+30 x (4 d+7 e)-\frac{42 (6 d+5 e)}{x}+42 (5 d+6 e) \log (x)-\frac{d}{6 x^6}+\frac{e x^5}{5} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^7,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{d}{6 x^{6}} + \frac{e x^{5}}{5} + x^{4} \left (\frac{d}{4} + \frac{5 e}{2}\right ) + x^{3} \left (\frac{10 d}{3} + 15 e\right ) + x \left (120 d + 210 e\right ) + \left (45 d + 120 e\right ) \int x\, dx + \left (210 d + 252 e\right ) \log{\left (x \right )} - \frac{252 d + 210 e}{x} - \frac{105 d + 60 e}{x^{2}} - \frac{40 d + 15 e}{x^{3}} - \frac{\frac{45 d}{4} + \frac{5 e}{2}}{x^{4}} - \frac{2 d + \frac{e}{5}}{x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(x**2+2*x+1)**5/x**7,x)

[Out]

-d/(6*x**6) + e*x**5/5 + x**4*(d/4 + 5*e/2) + x**3*(10*d/3 + 15*e) + x*(120*d +
210*e) + (45*d + 120*e)*Integral(x, x) + (210*d + 252*e)*log(x) - (252*d + 210*e
)/x - (105*d + 60*e)/x**2 - (40*d + 15*e)/x**3 - (45*d/4 + 5*e/2)/x**4 - (2*d +
e/5)/x**5

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Mathematica [A]  time = 0.0885239, size = 142, normalized size = 1.01 \[ \frac{-10 d-e}{5 x^5}+\frac{1}{4} x^4 (d+10 e)-\frac{5 (9 d+2 e)}{4 x^4}+\frac{5}{3} x^3 (2 d+9 e)-\frac{5 (8 d+3 e)}{x^3}+\frac{15}{2} x^2 (3 d+8 e)-\frac{15 (7 d+4 e)}{x^2}+30 x (4 d+7 e)-\frac{42 (6 d+5 e)}{x}+42 (5 d+6 e) \log (x)-\frac{d}{6 x^6}+\frac{e x^5}{5} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^7,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(x^2+2*x+1)^5/x^7,x)

[Out]

1/5*e*x^5+1/4*d*x^4+5/2*e*x^4+10/3*d*x^3+15*e*x^3+45/2*d*x^2+60*e*x^2+120*d*x+21
0*e*x+210*d*ln(x)+252*e*ln(x)-1/6*d/x^6-45/4*d/x^4-5/2*e/x^4-40*d/x^3-15*e/x^3-1
05*d/x^2-60*e/x^2-2*d/x^5-1/5*e/x^5-252*d/x-210*e/x

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Maxima [A]  time = 0.673109, size = 171, normalized size = 1.22 \[ \frac{1}{5} \, e x^{5} + \frac{1}{4} \,{\left (d + 10 \, e\right )} x^{4} + \frac{5}{3} \,{\left (2 \, d + 9 \, e\right )} x^{3} + \frac{15}{2} \,{\left (3 \, d + 8 \, e\right )} x^{2} + 30 \,{\left (4 \, d + 7 \, e\right )} x + 42 \,{\left (5 \, d + 6 \, e\right )} \log \left (x\right ) - \frac{2520 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 900 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 300 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 75 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 12 \,{\left (10 \, d + e\right )} x + 10 \, d}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5/x^7,x, algorithm="maxima")

[Out]

1/5*e*x^5 + 1/4*(d + 10*e)*x^4 + 5/3*(2*d + 9*e)*x^3 + 15/2*(3*d + 8*e)*x^2 + 30
*(4*d + 7*e)*x + 42*(5*d + 6*e)*log(x) - 1/60*(2520*(6*d + 5*e)*x^5 + 900*(7*d +
4*e)*x^4 + 300*(8*d + 3*e)*x^3 + 75*(9*d + 2*e)*x^2 + 12*(10*d + e)*x + 10*d)/x
^6

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Fricas [A]  time = 0.280926, size = 177, normalized size = 1.26 \[ \frac{12 \, e x^{11} + 15 \,{\left (d + 10 \, e\right )} x^{10} + 100 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 450 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 1800 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 2520 \,{\left (5 \, d + 6 \, e\right )} x^{6} \log \left (x\right ) - 2520 \,{\left (6 \, d + 5 \, e\right )} x^{5} - 900 \,{\left (7 \, d + 4 \, e\right )} x^{4} - 300 \,{\left (8 \, d + 3 \, e\right )} x^{3} - 75 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 12 \,{\left (10 \, d + e\right )} x - 10 \, d}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5/x^7,x, algorithm="fricas")

[Out]

1/60*(12*e*x^11 + 15*(d + 10*e)*x^10 + 100*(2*d + 9*e)*x^9 + 450*(3*d + 8*e)*x^8
+ 1800*(4*d + 7*e)*x^7 + 2520*(5*d + 6*e)*x^6*log(x) - 2520*(6*d + 5*e)*x^5 - 9
00*(7*d + 4*e)*x^4 - 300*(8*d + 3*e)*x^3 - 75*(9*d + 2*e)*x^2 - 12*(10*d + e)*x
- 10*d)/x^6

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Sympy [A]  time = 7.45828, size = 119, normalized size = 0.85 \[ \frac{e x^{5}}{5} + x^{4} \left (\frac{d}{4} + \frac{5 e}{2}\right ) + x^{3} \left (\frac{10 d}{3} + 15 e\right ) + x^{2} \left (\frac{45 d}{2} + 60 e\right ) + x \left (120 d + 210 e\right ) + 42 \left (5 d + 6 e\right ) \log{\left (x \right )} - \frac{10 d + x^{5} \left (15120 d + 12600 e\right ) + x^{4} \left (6300 d + 3600 e\right ) + x^{3} \left (2400 d + 900 e\right ) + x^{2} \left (675 d + 150 e\right ) + x \left (120 d + 12 e\right )}{60 x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(x**2+2*x+1)**5/x**7,x)

[Out]

e*x**5/5 + x**4*(d/4 + 5*e/2) + x**3*(10*d/3 + 15*e) + x**2*(45*d/2 + 60*e) + x*
(120*d + 210*e) + 42*(5*d + 6*e)*log(x) - (10*d + x**5*(15120*d + 12600*e) + x**
4*(6300*d + 3600*e) + x**3*(2400*d + 900*e) + x**2*(675*d + 150*e) + x*(120*d +
12*e))/(60*x**6)

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GIAC/XCAS [A]  time = 0.271418, size = 188, normalized size = 1.34 \[ \frac{1}{5} \, x^{5} e + \frac{1}{4} \, d x^{4} + \frac{5}{2} \, x^{4} e + \frac{10}{3} \, d x^{3} + 15 \, x^{3} e + \frac{45}{2} \, d x^{2} + 60 \, x^{2} e + 120 \, d x + 210 \, x e + 42 \,{\left (5 \, d + 6 \, e\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2520 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 900 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 300 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 75 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 12 \,{\left (10 \, d + e\right )} x + 10 \, d}{60 \, x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5/x^7,x, algorithm="giac")

[Out]

1/5*x^5*e + 1/4*d*x^4 + 5/2*x^4*e + 10/3*d*x^3 + 15*x^3*e + 45/2*d*x^2 + 60*x^2*
e + 120*d*x + 210*x*e + 42*(5*d + 6*e)*ln(abs(x)) - 1/60*(2520*(6*d + 5*e)*x^5 +
900*(7*d + 4*e)*x^4 + 300*(8*d + 3*e)*x^3 + 75*(9*d + 2*e)*x^2 + 12*(10*d + e)*
x + 10*d)/x^6