3.575 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^9} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0738579, 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} \[ \frac{1}{2} x^2 (d+10 e)-\frac{21 (5 d+6 e)}{x^2}-\frac{14 (6 d+5 e)}{x^3}-\frac{15 (7 d+4 e)}{2 x^4}-\frac{3 (8 d+3 e)}{x^5}-\frac{5 (9 d+2 e)}{6 x^6}-\frac{10 d+e}{7 x^7}+5 x (2 d+9 e)-\frac{30 (4 d+7 e)}{x}+15 (3 d+8 e) \log (x)-\frac{d}{8 x^8}+\frac{e x^3}{3} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^9,x]

[Out]

-d/(8*x^8) - (10*d + e)/(7*x^7) - (5*(9*d + 2*e))/(6*x^6) - (3*(8*d + 3*e))/x^5 - (15*(7*d + 4*e))/(2*x^4) - (
14*(6*d + 5*e))/x^3 - (21*(5*d + 6*e))/x^2 - (30*(4*d + 7*e))/x + 5*(2*d + 9*e)*x + ((d + 10*e)*x^2)/2 + (e*x^
3)/3 + 15*(3*d + 8*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^9} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^9} \, dx\\ &=\int \left (5 (2 d+9 e)+\frac{d}{x^9}+\frac{10 d+e}{x^8}+\frac{5 (9 d+2 e)}{x^7}+\frac{15 (8 d+3 e)}{x^6}+\frac{30 (7 d+4 e)}{x^5}+\frac{42 (6 d+5 e)}{x^4}+\frac{42 (5 d+6 e)}{x^3}+\frac{30 (4 d+7 e)}{x^2}+\frac{15 (3 d+8 e)}{x}+(d+10 e) x+e x^2\right ) \, dx\\ &=-\frac{d}{8 x^8}-\frac{10 d+e}{7 x^7}-\frac{5 (9 d+2 e)}{6 x^6}-\frac{3 (8 d+3 e)}{x^5}-\frac{15 (7 d+4 e)}{2 x^4}-\frac{14 (6 d+5 e)}{x^3}-\frac{21 (5 d+6 e)}{x^2}-\frac{30 (4 d+7 e)}{x}+5 (2 d+9 e) x+\frac{1}{2} (d+10 e) x^2+\frac{e x^3}{3}+15 (3 d+8 e) \log (x)\\ \end{align*}

Mathematica [A]  time = 0.0388898, size = 140, normalized size = 1.01 \[ \frac{1}{2} x^2 (d+10 e)-\frac{21 (5 d+6 e)}{x^2}-\frac{14 (6 d+5 e)}{x^3}-\frac{15 (7 d+4 e)}{2 x^4}-\frac{3 (8 d+3 e)}{x^5}-\frac{5 (9 d+2 e)}{6 x^6}+\frac{-10 d-e}{7 x^7}+5 x (2 d+9 e)-\frac{30 (4 d+7 e)}{x}+15 (3 d+8 e) \log (x)-\frac{d}{8 x^8}+\frac{e x^3}{3} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^9,x]

[Out]

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

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Maple [A]  time = 0.009, size = 128, normalized size = 0.9 \begin{align*}{\frac{e{x}^{3}}{3}}+{\frac{d{x}^{2}}{2}}+5\,e{x}^{2}+10\,dx+45\,ex-24\,{\frac{d}{{x}^{5}}}-9\,{\frac{e}{{x}^{5}}}+45\,d\ln \left ( x \right ) +120\,e\ln \left ( x \right ) -{\frac{15\,d}{2\,{x}^{6}}}-{\frac{5\,e}{3\,{x}^{6}}}-84\,{\frac{d}{{x}^{3}}}-70\,{\frac{e}{{x}^{3}}}-{\frac{105\,d}{2\,{x}^{4}}}-30\,{\frac{e}{{x}^{4}}}-105\,{\frac{d}{{x}^{2}}}-126\,{\frac{e}{{x}^{2}}}-120\,{\frac{d}{x}}-210\,{\frac{e}{x}}-{\frac{10\,d}{7\,{x}^{7}}}-{\frac{e}{7\,{x}^{7}}}-{\frac{d}{8\,{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^9,x)

[Out]

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

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Maxima [A]  time = 0.991006, size = 171, normalized size = 1.24 \begin{align*} \frac{1}{3} \, e x^{3} + \frac{1}{2} \,{\left (d + 10 \, e\right )} x^{2} + 5 \,{\left (2 \, d + 9 \, e\right )} x + 15 \,{\left (3 \, d + 8 \, e\right )} \log \left (x\right ) - \frac{5040 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 3528 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 2352 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 1260 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 504 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 140 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 24 \,{\left (10 \, d + e\right )} x + 21 \, d}{168 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^9,x, algorithm="maxima")

[Out]

1/3*e*x^3 + 1/2*(d + 10*e)*x^2 + 5*(2*d + 9*e)*x + 15*(3*d + 8*e)*log(x) - 1/168*(5040*(4*d + 7*e)*x^7 + 3528*
(5*d + 6*e)*x^6 + 2352*(6*d + 5*e)*x^5 + 1260*(7*d + 4*e)*x^4 + 504*(8*d + 3*e)*x^3 + 140*(9*d + 2*e)*x^2 + 24
*(10*d + e)*x + 21*d)/x^8

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Fricas [A]  time = 1.26131, size = 347, normalized size = 2.51 \begin{align*} \frac{56 \, e x^{11} + 84 \,{\left (d + 10 \, e\right )} x^{10} + 840 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 2520 \,{\left (3 \, d + 8 \, e\right )} x^{8} \log \left (x\right ) - 5040 \,{\left (4 \, d + 7 \, e\right )} x^{7} - 3528 \,{\left (5 \, d + 6 \, e\right )} x^{6} - 2352 \,{\left (6 \, d + 5 \, e\right )} x^{5} - 1260 \,{\left (7 \, d + 4 \, e\right )} x^{4} - 504 \,{\left (8 \, d + 3 \, e\right )} x^{3} - 140 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 24 \,{\left (10 \, d + e\right )} x - 21 \, d}{168 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^9,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 8.29379, size = 114, normalized size = 0.83 \begin{align*} \frac{e x^{3}}{3} + x^{2} \left (\frac{d}{2} + 5 e\right ) + x \left (10 d + 45 e\right ) + 15 \left (3 d + 8 e\right ) \log{\left (x \right )} - \frac{21 d + x^{7} \left (20160 d + 35280 e\right ) + x^{6} \left (17640 d + 21168 e\right ) + x^{5} \left (14112 d + 11760 e\right ) + x^{4} \left (8820 d + 5040 e\right ) + x^{3} \left (4032 d + 1512 e\right ) + x^{2} \left (1260 d + 280 e\right ) + x \left (240 d + 24 e\right )}{168 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**9,x)

[Out]

e*x**3/3 + x**2*(d/2 + 5*e) + x*(10*d + 45*e) + 15*(3*d + 8*e)*log(x) - (21*d + x**7*(20160*d + 35280*e) + x**
6*(17640*d + 21168*e) + x**5*(14112*d + 11760*e) + x**4*(8820*d + 5040*e) + x**3*(4032*d + 1512*e) + x**2*(126
0*d + 280*e) + x*(240*d + 24*e))/(168*x**8)

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Giac [A]  time = 1.11096, size = 188, normalized size = 1.36 \begin{align*} \frac{1}{3} \, x^{3} e + \frac{1}{2} \, d x^{2} + 5 \, x^{2} e + 10 \, d x + 45 \, x e + 15 \,{\left (3 \, d + 8 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac{5040 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 3528 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 2352 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 1260 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 504 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 140 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 24 \,{\left (10 \, d + e\right )} x + 21 \, d}{168 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^9,x, algorithm="giac")

[Out]

1/3*x^3*e + 1/2*d*x^2 + 5*x^2*e + 10*d*x + 45*x*e + 15*(3*d + 8*e)*log(abs(x)) - 1/168*(5040*(4*d + 7*e)*x^7 +
3528*(5*d + 6*e)*x^6 + 2352*(6*d + 5*e)*x^5 + 1260*(7*d + 4*e)*x^4 + 504*(8*d + 3*e)*x^3 + 140*(9*d + 2*e)*x^
2 + 24*(10*d + e)*x + 21*d)/x^8