3.577 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^{11}} \, dx\)

Optimal. Leaf size=138 \[ -\frac{15 (3 d+8 e)}{2 x^2}-\frac{10 (4 d+7 e)}{x^3}-\frac{21 (5 d+6 e)}{2 x^4}-\frac{42 (6 d+5 e)}{5 x^5}-\frac{5 (7 d+4 e)}{x^6}-\frac{15 (8 d+3 e)}{7 x^7}-\frac{5 (9 d+2 e)}{8 x^8}-\frac{10 d+e}{9 x^9}-\frac{5 (2 d+9 e)}{x}+(d+10 e) \log (x)-\frac{d}{10 x^{10}}+e x \]

[Out]

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

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Rubi [A]  time = 0.0708308, 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, Rules used = {27, 76} \[ -\frac{15 (3 d+8 e)}{2 x^2}-\frac{10 (4 d+7 e)}{x^3}-\frac{21 (5 d+6 e)}{2 x^4}-\frac{42 (6 d+5 e)}{5 x^5}-\frac{5 (7 d+4 e)}{x^6}-\frac{15 (8 d+3 e)}{7 x^7}-\frac{5 (9 d+2 e)}{8 x^8}-\frac{10 d+e}{9 x^9}-\frac{5 (2 d+9 e)}{x}+(d+10 e) \log (x)-\frac{d}{10 x^{10}}+e x \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^11,x]

[Out]

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

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 76

Int[((d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_))*((e_) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*
x)*(d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, d, e, f, n}, x] && IGtQ[p, 0] && (NeQ[n, -1] || EqQ[p, 1]) && N
eQ[b*e + a*f, 0] && ( !IntegerQ[n] || LtQ[9*p + 5*n, 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && Rational
Q[a, b, d, e, f])) && (NeQ[n + p + 3, 0] || EqQ[p, 1])

Rubi steps

\begin{align*} \int \frac{(d+e x) \left (1+2 x+x^2\right )^5}{x^{11}} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^{11}} \, dx\\ &=\int \left (e+\frac{d}{x^{11}}+\frac{10 d+e}{x^{10}}+\frac{5 (9 d+2 e)}{x^9}+\frac{15 (8 d+3 e)}{x^8}+\frac{30 (7 d+4 e)}{x^7}+\frac{42 (6 d+5 e)}{x^6}+\frac{42 (5 d+6 e)}{x^5}+\frac{30 (4 d+7 e)}{x^4}+\frac{15 (3 d+8 e)}{x^3}+\frac{5 (2 d+9 e)}{x^2}+\frac{d+10 e}{x}\right ) \, dx\\ &=-\frac{d}{10 x^{10}}-\frac{10 d+e}{9 x^9}-\frac{5 (9 d+2 e)}{8 x^8}-\frac{15 (8 d+3 e)}{7 x^7}-\frac{5 (7 d+4 e)}{x^6}-\frac{42 (6 d+5 e)}{5 x^5}-\frac{21 (5 d+6 e)}{2 x^4}-\frac{10 (4 d+7 e)}{x^3}-\frac{15 (3 d+8 e)}{2 x^2}-\frac{5 (2 d+9 e)}{x}+e x+(d+10 e) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0402306, size = 140, normalized size = 1.01 \[ -\frac{15 (3 d+8 e)}{2 x^2}-\frac{10 (4 d+7 e)}{x^3}-\frac{21 (5 d+6 e)}{2 x^4}-\frac{42 (6 d+5 e)}{5 x^5}-\frac{5 (7 d+4 e)}{x^6}-\frac{15 (8 d+3 e)}{7 x^7}-\frac{5 (9 d+2 e)}{8 x^8}+\frac{-10 d-e}{9 x^9}-\frac{5 (2 d+9 e)}{x}+(d+10 e) \log (x)-\frac{d}{10 x^{10}}+e x \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^11,x]

[Out]

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

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Maple [A]  time = 0.007, size = 128, normalized size = 0.9 \begin{align*} ex-{\frac{10\,d}{9\,{x}^{9}}}-{\frac{e}{9\,{x}^{9}}}+d\ln \left ( x \right ) +10\,e\ln \left ( x \right ) -{\frac{252\,d}{5\,{x}^{5}}}-42\,{\frac{e}{{x}^{5}}}-40\,{\frac{d}{{x}^{3}}}-70\,{\frac{e}{{x}^{3}}}-{\frac{105\,d}{2\,{x}^{4}}}-63\,{\frac{e}{{x}^{4}}}-{\frac{45\,d}{2\,{x}^{2}}}-60\,{\frac{e}{{x}^{2}}}-10\,{\frac{d}{x}}-45\,{\frac{e}{x}}-{\frac{d}{10\,{x}^{10}}}-{\frac{120\,d}{7\,{x}^{7}}}-{\frac{45\,e}{7\,{x}^{7}}}-35\,{\frac{d}{{x}^{6}}}-20\,{\frac{e}{{x}^{6}}}-{\frac{45\,d}{8\,{x}^{8}}}-{\frac{5\,e}{4\,{x}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^11,x)

[Out]

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

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Maxima [A]  time = 1.00634, size = 169, normalized size = 1.22 \begin{align*} e x +{\left (d + 10 \, e\right )} \log \left (x\right ) - \frac{12600 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 18900 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 25200 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 26460 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 21168 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 12600 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 5400 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 1575 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 280 \,{\left (10 \, d + e\right )} x + 252 \, d}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^11,x, algorithm="maxima")

[Out]

e*x + (d + 10*e)*log(x) - 1/2520*(12600*(2*d + 9*e)*x^9 + 18900*(3*d + 8*e)*x^8 + 25200*(4*d + 7*e)*x^7 + 2646
0*(5*d + 6*e)*x^6 + 21168*(6*d + 5*e)*x^5 + 12600*(7*d + 4*e)*x^4 + 5400*(8*d + 3*e)*x^3 + 1575*(9*d + 2*e)*x^
2 + 280*(10*d + e)*x + 252*d)/x^10

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Fricas [A]  time = 1.25995, size = 370, normalized size = 2.68 \begin{align*} \frac{2520 \, e x^{11} + 2520 \,{\left (d + 10 \, e\right )} x^{10} \log \left (x\right ) - 12600 \,{\left (2 \, d + 9 \, e\right )} x^{9} - 18900 \,{\left (3 \, d + 8 \, e\right )} x^{8} - 25200 \,{\left (4 \, d + 7 \, e\right )} x^{7} - 26460 \,{\left (5 \, d + 6 \, e\right )} x^{6} - 21168 \,{\left (6 \, d + 5 \, e\right )} x^{5} - 12600 \,{\left (7 \, d + 4 \, e\right )} x^{4} - 5400 \,{\left (8 \, d + 3 \, e\right )} x^{3} - 1575 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 280 \,{\left (10 \, d + e\right )} x - 252 \, d}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^11,x, algorithm="fricas")

[Out]

1/2520*(2520*e*x^11 + 2520*(d + 10*e)*x^10*log(x) - 12600*(2*d + 9*e)*x^9 - 18900*(3*d + 8*e)*x^8 - 25200*(4*d
+ 7*e)*x^7 - 26460*(5*d + 6*e)*x^6 - 21168*(6*d + 5*e)*x^5 - 12600*(7*d + 4*e)*x^4 - 5400*(8*d + 3*e)*x^3 - 1
575*(9*d + 2*e)*x^2 - 280*(10*d + e)*x - 252*d)/x^10

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Sympy [A]  time = 9.63133, size = 109, normalized size = 0.79 \begin{align*} e x + \left (d + 10 e\right ) \log{\left (x \right )} - \frac{252 d + x^{9} \left (25200 d + 113400 e\right ) + x^{8} \left (56700 d + 151200 e\right ) + x^{7} \left (100800 d + 176400 e\right ) + x^{6} \left (132300 d + 158760 e\right ) + x^{5} \left (127008 d + 105840 e\right ) + x^{4} \left (88200 d + 50400 e\right ) + x^{3} \left (43200 d + 16200 e\right ) + x^{2} \left (14175 d + 3150 e\right ) + x \left (2800 d + 280 e\right )}{2520 x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**11,x)

[Out]

e*x + (d + 10*e)*log(x) - (252*d + x**9*(25200*d + 113400*e) + x**8*(56700*d + 151200*e) + x**7*(100800*d + 17
6400*e) + x**6*(132300*d + 158760*e) + x**5*(127008*d + 105840*e) + x**4*(88200*d + 50400*e) + x**3*(43200*d +
16200*e) + x**2*(14175*d + 3150*e) + x*(2800*d + 280*e))/(2520*x**10)

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Giac [A]  time = 1.14779, size = 185, normalized size = 1.34 \begin{align*} x e +{\left (d + 10 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac{12600 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 18900 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 25200 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 26460 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 21168 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 12600 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 5400 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 1575 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 280 \,{\left (10 \, d + e\right )} x + 252 \, d}{2520 \, x^{10}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^11,x, algorithm="giac")

[Out]

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