3.569 \(\int \frac{(d+e x) \left (1+2 x+x^2\right )^5}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.221972, 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 \[ \frac{1}{8} x^8 (d+10 e)+\frac{5}{7} x^7 (2 d+9 e)+\frac{5}{2} x^6 (3 d+8 e)+6 x^5 (4 d+7 e)+\frac{21}{2} x^4 (5 d+6 e)+14 x^3 (6 d+5 e)+15 x^2 (7 d+4 e)+15 x (8 d+3 e)-\frac{10 d+e}{x}+5 (9 d+2 e) \log (x)-\frac{d}{2 x^2}+\frac{e x^9}{9} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^3,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{d}{2 x^{2}} + \frac{e x^{9}}{9} + x^{8} \left (\frac{d}{8} + \frac{5 e}{4}\right ) + x^{7} \left (\frac{10 d}{7} + \frac{45 e}{7}\right ) + x^{6} \left (\frac{15 d}{2} + 20 e\right ) + x^{5} \left (24 d + 42 e\right ) + x^{4} \left (\frac{105 d}{2} + 63 e\right ) + x^{3} \left (84 d + 70 e\right ) + x \left (120 d + 45 e\right ) + \left (45 d + 10 e\right ) \log{\left (x \right )} + \left (210 d + 120 e\right ) \int x\, dx - \frac{10 d + e}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)*(x**2+2*x+1)**5/x**3,x)

[Out]

-d/(2*x**2) + e*x**9/9 + x**8*(d/8 + 5*e/4) + x**7*(10*d/7 + 45*e/7) + x**6*(15*
d/2 + 20*e) + x**5*(24*d + 42*e) + x**4*(105*d/2 + 63*e) + x**3*(84*d + 70*e) +
x*(120*d + 45*e) + (45*d + 10*e)*log(x) + (210*d + 120*e)*Integral(x, x) - (10*d
+ e)/x

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Mathematica [A]  time = 0.0852396, size = 139, normalized size = 1.01 \[ \frac{1}{8} x^8 (d+10 e)+\frac{5}{7} x^7 (2 d+9 e)+\frac{5}{2} x^6 (3 d+8 e)+6 x^5 (4 d+7 e)+\frac{21}{2} x^4 (5 d+6 e)+14 x^3 (6 d+5 e)+15 x^2 (7 d+4 e)+15 x (8 d+3 e)+\frac{-10 d-e}{x}+5 (9 d+2 e) \log (x)-\frac{d}{2 x^2}+\frac{e x^9}{9} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^3,x]

[Out]

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

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Maple [A]  time = 0.01, size = 128, normalized size = 0.9 \[{\frac{e{x}^{9}}{9}}+{\frac{d{x}^{8}}{8}}+{\frac{5\,e{x}^{8}}{4}}+{\frac{10\,d{x}^{7}}{7}}+{\frac{45\,e{x}^{7}}{7}}+{\frac{15\,d{x}^{6}}{2}}+20\,e{x}^{6}+24\,d{x}^{5}+42\,e{x}^{5}+{\frac{105\,d{x}^{4}}{2}}+63\,e{x}^{4}+84\,d{x}^{3}+70\,e{x}^{3}+105\,d{x}^{2}+60\,e{x}^{2}+120\,dx+45\,ex+45\,d\ln \left ( x \right ) +10\,e\ln \left ( x \right ) -{\frac{d}{2\,{x}^{2}}}-10\,{\frac{d}{x}}-{\frac{e}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)*(x^2+2*x+1)^5/x^3,x)

[Out]

1/9*e*x^9+1/8*d*x^8+5/4*e*x^8+10/7*d*x^7+45/7*e*x^7+15/2*d*x^6+20*e*x^6+24*d*x^5
+42*e*x^5+105/2*d*x^4+63*e*x^4+84*d*x^3+70*e*x^3+105*d*x^2+60*e*x^2+120*d*x+45*e
*x+45*d*ln(x)+10*e*ln(x)-1/2*d/x^2-10*d/x-e/x

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Maxima [A]  time = 0.686732, size = 169, normalized size = 1.22 \[ \frac{1}{9} \, e x^{9} + \frac{1}{8} \,{\left (d + 10 \, e\right )} x^{8} + \frac{5}{7} \,{\left (2 \, d + 9 \, e\right )} x^{7} + \frac{5}{2} \,{\left (3 \, d + 8 \, e\right )} x^{6} + 6 \,{\left (4 \, d + 7 \, e\right )} x^{5} + \frac{21}{2} \,{\left (5 \, d + 6 \, e\right )} x^{4} + 14 \,{\left (6 \, d + 5 \, e\right )} x^{3} + 15 \,{\left (7 \, d + 4 \, e\right )} x^{2} + 15 \,{\left (8 \, d + 3 \, e\right )} x + 5 \,{\left (9 \, d + 2 \, e\right )} \log \left (x\right ) - \frac{2 \,{\left (10 \, d + e\right )} x + d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5/x^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.280627, size = 177, normalized size = 1.28 \[ \frac{56 \, e x^{11} + 63 \,{\left (d + 10 \, e\right )} x^{10} + 360 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 1260 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 3024 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 5292 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 7056 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 7560 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 7560 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 2520 \,{\left (9 \, d + 2 \, e\right )} x^{2} \log \left (x\right ) - 504 \,{\left (10 \, d + e\right )} x - 252 \, d}{504 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5/x^3,x, algorithm="fricas")

[Out]

1/504*(56*e*x^11 + 63*(d + 10*e)*x^10 + 360*(2*d + 9*e)*x^9 + 1260*(3*d + 8*e)*x
^8 + 3024*(4*d + 7*e)*x^7 + 5292*(5*d + 6*e)*x^6 + 7056*(6*d + 5*e)*x^5 + 7560*(
7*d + 4*e)*x^4 + 7560*(8*d + 3*e)*x^3 + 2520*(9*d + 2*e)*x^2*log(x) - 504*(10*d
+ e)*x - 252*d)/x^2

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Sympy [A]  time = 2.23155, size = 121, normalized size = 0.88 \[ \frac{e x^{9}}{9} + x^{8} \left (\frac{d}{8} + \frac{5 e}{4}\right ) + x^{7} \left (\frac{10 d}{7} + \frac{45 e}{7}\right ) + x^{6} \left (\frac{15 d}{2} + 20 e\right ) + x^{5} \left (24 d + 42 e\right ) + x^{4} \left (\frac{105 d}{2} + 63 e\right ) + x^{3} \left (84 d + 70 e\right ) + x^{2} \left (105 d + 60 e\right ) + x \left (120 d + 45 e\right ) + 5 \left (9 d + 2 e\right ) \log{\left (x \right )} - \frac{d + x \left (20 d + 2 e\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)*(x**2+2*x+1)**5/x**3,x)

[Out]

e*x**9/9 + x**8*(d/8 + 5*e/4) + x**7*(10*d/7 + 45*e/7) + x**6*(15*d/2 + 20*e) +
x**5*(24*d + 42*e) + x**4*(105*d/2 + 63*e) + x**3*(84*d + 70*e) + x**2*(105*d +
60*e) + x*(120*d + 45*e) + 5*(9*d + 2*e)*log(x) - (d + x*(20*d + 2*e))/(2*x**2)

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GIAC/XCAS [A]  time = 0.271055, size = 185, normalized size = 1.34 \[ \frac{1}{9} \, x^{9} e + \frac{1}{8} \, d x^{8} + \frac{5}{4} \, x^{8} e + \frac{10}{7} \, d x^{7} + \frac{45}{7} \, x^{7} e + \frac{15}{2} \, d x^{6} + 20 \, x^{6} e + 24 \, d x^{5} + 42 \, x^{5} e + \frac{105}{2} \, d x^{4} + 63 \, x^{4} e + 84 \, d x^{3} + 70 \, x^{3} e + 105 \, d x^{2} + 60 \, x^{2} e + 120 \, d x + 45 \, x e + 5 \,{\left (9 \, d + 2 \, e\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{2 \,{\left (10 \, d + e\right )} x + d}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)*(x^2 + 2*x + 1)^5/x^3,x, algorithm="giac")

[Out]

1/9*x^9*e + 1/8*d*x^8 + 5/4*x^8*e + 10/7*d*x^7 + 45/7*x^7*e + 15/2*d*x^6 + 20*x^
6*e + 24*d*x^5 + 42*x^5*e + 105/2*d*x^4 + 63*x^4*e + 84*d*x^3 + 70*x^3*e + 105*d
*x^2 + 60*x^2*e + 120*d*x + 45*x*e + 5*(9*d + 2*e)*ln(abs(x)) - 1/2*(2*(10*d + e
)*x + d)/x^2