3.576 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^{10}} \, dx\)

Optimal. Leaf size=137 \[ -\frac{15 (4 d+7 e)}{x^2}-\frac{14 (5 d+6 e)}{x^3}-\frac{21 (6 d+5 e)}{2 x^4}-\frac{6 (7 d+4 e)}{x^5}-\frac{5 (8 d+3 e)}{2 x^6}-\frac{5 (9 d+2 e)}{7 x^7}-\frac{10 d+e}{8 x^8}+x (d+10 e)-\frac{15 (3 d+8 e)}{x}+5 (2 d+9 e) \log (x)-\frac{d}{9 x^9}+\frac{e x^2}{2} \]

[Out]

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

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Rubi [A]  time = 0.079851, antiderivative size = 137, 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} \[ -\frac{15 (4 d+7 e)}{x^2}-\frac{14 (5 d+6 e)}{x^3}-\frac{21 (6 d+5 e)}{2 x^4}-\frac{6 (7 d+4 e)}{x^5}-\frac{5 (8 d+3 e)}{2 x^6}-\frac{5 (9 d+2 e)}{7 x^7}-\frac{10 d+e}{8 x^8}+x (d+10 e)-\frac{15 (3 d+8 e)}{x}+5 (2 d+9 e) \log (x)-\frac{d}{9 x^9}+\frac{e x^2}{2} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^10,x]

[Out]

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

Mathematica [A]  time = 0.0411904, size = 139, normalized size = 1.01 \[ -\frac{15 (4 d+7 e)}{x^2}-\frac{14 (5 d+6 e)}{x^3}-\frac{21 (6 d+5 e)}{2 x^4}-\frac{6 (7 d+4 e)}{x^5}-\frac{5 (8 d+3 e)}{2 x^6}-\frac{5 (9 d+2 e)}{7 x^7}+\frac{-10 d-e}{8 x^8}+x (d+10 e)-\frac{15 (3 d+8 e)}{x}+5 (2 d+9 e) \log (x)-\frac{d}{9 x^9}+\frac{e x^2}{2} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^10,x]

[Out]

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

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Maple [A]  time = 0.009, size = 127, normalized size = 0.9 \begin{align*}{\frac{e{x}^{2}}{2}}+dx+10\,ex-{\frac{d}{9\,{x}^{9}}}+10\,d\ln \left ( x \right ) +45\,e\ln \left ( x \right ) -63\,{\frac{d}{{x}^{4}}}-{\frac{105\,e}{2\,{x}^{4}}}-70\,{\frac{d}{{x}^{3}}}-84\,{\frac{e}{{x}^{3}}}-60\,{\frac{d}{{x}^{2}}}-105\,{\frac{e}{{x}^{2}}}-45\,{\frac{d}{x}}-120\,{\frac{e}{x}}-{\frac{45\,d}{7\,{x}^{7}}}-{\frac{10\,e}{7\,{x}^{7}}}-{\frac{5\,d}{4\,{x}^{8}}}-{\frac{e}{8\,{x}^{8}}}-20\,{\frac{d}{{x}^{6}}}-{\frac{15\,e}{2\,{x}^{6}}}-42\,{\frac{d}{{x}^{5}}}-24\,{\frac{e}{{x}^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^10,x)

[Out]

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

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Maxima [A]  time = 1.05922, size = 170, normalized size = 1.24 \begin{align*} \frac{1}{2} \, e x^{2} +{\left (d + 10 \, e\right )} x + 5 \,{\left (2 \, d + 9 \, e\right )} \log \left (x\right ) - \frac{7560 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 7560 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 7056 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 5292 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 3024 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 1260 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 360 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 63 \,{\left (10 \, d + e\right )} x + 56 \, d}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^10,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.34286, size = 352, normalized size = 2.57 \begin{align*} \frac{252 \, e x^{11} + 504 \,{\left (d + 10 \, e\right )} x^{10} + 2520 \,{\left (2 \, d + 9 \, e\right )} x^{9} \log \left (x\right ) - 7560 \,{\left (3 \, d + 8 \, e\right )} x^{8} - 7560 \,{\left (4 \, d + 7 \, e\right )} x^{7} - 7056 \,{\left (5 \, d + 6 \, e\right )} x^{6} - 5292 \,{\left (6 \, d + 5 \, e\right )} x^{5} - 3024 \,{\left (7 \, d + 4 \, e\right )} x^{4} - 1260 \,{\left (8 \, d + 3 \, e\right )} x^{3} - 360 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 63 \,{\left (10 \, d + e\right )} x - 56 \, d}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^10,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.26872, size = 112, normalized size = 0.82 \begin{align*} \frac{e x^{2}}{2} + x \left (d + 10 e\right ) + 5 \left (2 d + 9 e\right ) \log{\left (x \right )} - \frac{56 d + x^{8} \left (22680 d + 60480 e\right ) + x^{7} \left (30240 d + 52920 e\right ) + x^{6} \left (35280 d + 42336 e\right ) + x^{5} \left (31752 d + 26460 e\right ) + x^{4} \left (21168 d + 12096 e\right ) + x^{3} \left (10080 d + 3780 e\right ) + x^{2} \left (3240 d + 720 e\right ) + x \left (630 d + 63 e\right )}{504 x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**10,x)

[Out]

e*x**2/2 + x*(d + 10*e) + 5*(2*d + 9*e)*log(x) - (56*d + x**8*(22680*d + 60480*e) + x**7*(30240*d + 52920*e) +
x**6*(35280*d + 42336*e) + x**5*(31752*d + 26460*e) + x**4*(21168*d + 12096*e) + x**3*(10080*d + 3780*e) + x*
*2*(3240*d + 720*e) + x*(630*d + 63*e))/(504*x**9)

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Giac [A]  time = 1.16276, size = 186, normalized size = 1.36 \begin{align*} \frac{1}{2} \, x^{2} e + d x + 10 \, x e + 5 \,{\left (2 \, d + 9 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac{7560 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 7560 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 7056 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 5292 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 3024 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 1260 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 360 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 63 \,{\left (10 \, d + e\right )} x + 56 \, d}{504 \, x^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^10,x, algorithm="giac")

[Out]

1/2*x^2*e + d*x + 10*x*e + 5*(2*d + 9*e)*log(abs(x)) - 1/504*(7560*(3*d + 8*e)*x^8 + 7560*(4*d + 7*e)*x^7 + 70
56*(5*d + 6*e)*x^6 + 5292*(6*d + 5*e)*x^5 + 3024*(7*d + 4*e)*x^4 + 1260*(8*d + 3*e)*x^3 + 360*(9*d + 2*e)*x^2
+ 63*(10*d + e)*x + 56*d)/x^9