3.574 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^8} \, dx\)

Optimal. Leaf size=138 \[ \frac{1}{3} x^3 (d+10 e)+\frac{5}{2} x^2 (2 d+9 e)-\frac{21 (6 d+5 e)}{x^2}-\frac{10 (7 d+4 e)}{x^3}-\frac{15 (8 d+3 e)}{4 x^4}-\frac{9 d+2 e}{x^5}-\frac{10 d+e}{6 x^6}+15 x (3 d+8 e)-\frac{42 (5 d+6 e)}{x}+30 (4 d+7 e) \log (x)-\frac{d}{7 x^7}+\frac{e x^4}{4} \]

[Out]

-d/(7*x^7) - (10*d + e)/(6*x^6) - (9*d + 2*e)/x^5 - (15*(8*d + 3*e))/(4*x^4) - (10*(7*d + 4*e))/x^3 - (21*(6*d
+ 5*e))/x^2 - (42*(5*d + 6*e))/x + 15*(3*d + 8*e)*x + (5*(2*d + 9*e)*x^2)/2 + ((d + 10*e)*x^3)/3 + (e*x^4)/4
+ 30*(4*d + 7*e)*Log[x]

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Rubi [A]  time = 0.0734573, 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} \[ \frac{1}{3} x^3 (d+10 e)+\frac{5}{2} x^2 (2 d+9 e)-\frac{21 (6 d+5 e)}{x^2}-\frac{10 (7 d+4 e)}{x^3}-\frac{15 (8 d+3 e)}{4 x^4}-\frac{9 d+2 e}{x^5}-\frac{10 d+e}{6 x^6}+15 x (3 d+8 e)-\frac{42 (5 d+6 e)}{x}+30 (4 d+7 e) \log (x)-\frac{d}{7 x^7}+\frac{e x^4}{4} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^8,x]

[Out]

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

Mathematica [A]  time = 0.0367221, size = 139, normalized size = 1.01 \[ \frac{1}{3} x^3 (d+10 e)+\frac{5}{2} x^2 (2 d+9 e)-\frac{21 (6 d+5 e)}{x^2}-\frac{10 (7 d+4 e)}{x^3}-\frac{15 (8 d+3 e)}{4 x^4}+\frac{-9 d-2 e}{x^5}+\frac{-10 d-e}{6 x^6}+15 x (3 d+8 e)-\frac{42 (5 d+6 e)}{x}+30 (4 d+7 e) \log (x)-\frac{d}{7 x^7}+\frac{e x^4}{4} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^8,x]

[Out]

-d/(7*x^7) + (-10*d - e)/(6*x^6) + (-9*d - 2*e)/x^5 - (15*(8*d + 3*e))/(4*x^4) - (10*(7*d + 4*e))/x^3 - (21*(6
*d + 5*e))/x^2 - (42*(5*d + 6*e))/x + 15*(3*d + 8*e)*x + (5*(2*d + 9*e)*x^2)/2 + ((d + 10*e)*x^3)/3 + (e*x^4)/
4 + 30*(4*d + 7*e)*Log[x]

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Maple [A]  time = 0.008, size = 128, normalized size = 0.9 \begin{align*}{\frac{e{x}^{4}}{4}}+{\frac{d{x}^{3}}{3}}+{\frac{10\,e{x}^{3}}{3}}+5\,d{x}^{2}+{\frac{45\,e{x}^{2}}{2}}+45\,dx+120\,ex+120\,d\ln \left ( x \right ) +210\,e\ln \left ( x \right ) -9\,{\frac{d}{{x}^{5}}}-2\,{\frac{e}{{x}^{5}}}-70\,{\frac{d}{{x}^{3}}}-40\,{\frac{e}{{x}^{3}}}-30\,{\frac{d}{{x}^{4}}}-{\frac{45\,e}{4\,{x}^{4}}}-126\,{\frac{d}{{x}^{2}}}-105\,{\frac{e}{{x}^{2}}}-210\,{\frac{d}{x}}-252\,{\frac{e}{x}}-{\frac{5\,d}{3\,{x}^{6}}}-{\frac{e}{6\,{x}^{6}}}-{\frac{d}{7\,{x}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^8,x)

[Out]

1/4*e*x^4+1/3*d*x^3+10/3*e*x^3+5*d*x^2+45/2*e*x^2+45*d*x+120*e*x+120*d*ln(x)+210*e*ln(x)-9*d/x^5-2*e/x^5-70*d/
x^3-40*e/x^3-30*d/x^4-45/4*e/x^4-126*d/x^2-105*e/x^2-210*d/x-252*e/x-5/3*d/x^6-1/6*e/x^6-1/7*d/x^7

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Maxima [A]  time = 1.03743, size = 171, normalized size = 1.24 \begin{align*} \frac{1}{4} \, e x^{4} + \frac{1}{3} \,{\left (d + 10 \, e\right )} x^{3} + \frac{5}{2} \,{\left (2 \, d + 9 \, e\right )} x^{2} + 15 \,{\left (3 \, d + 8 \, e\right )} x + 30 \,{\left (4 \, d + 7 \, e\right )} \log \left (x\right ) - \frac{3528 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 1764 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 840 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 315 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 84 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 14 \,{\left (10 \, d + e\right )} x + 12 \, d}{84 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^8,x, algorithm="maxima")

[Out]

1/4*e*x^4 + 1/3*(d + 10*e)*x^3 + 5/2*(2*d + 9*e)*x^2 + 15*(3*d + 8*e)*x + 30*(4*d + 7*e)*log(x) - 1/84*(3528*(
5*d + 6*e)*x^6 + 1764*(6*d + 5*e)*x^5 + 840*(7*d + 4*e)*x^4 + 315*(8*d + 3*e)*x^3 + 84*(9*d + 2*e)*x^2 + 14*(1
0*d + e)*x + 12*d)/x^7

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Fricas [A]  time = 1.38635, size = 343, normalized size = 2.49 \begin{align*} \frac{21 \, e x^{11} + 28 \,{\left (d + 10 \, e\right )} x^{10} + 210 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 1260 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 2520 \,{\left (4 \, d + 7 \, e\right )} x^{7} \log \left (x\right ) - 3528 \,{\left (5 \, d + 6 \, e\right )} x^{6} - 1764 \,{\left (6 \, d + 5 \, e\right )} x^{5} - 840 \,{\left (7 \, d + 4 \, e\right )} x^{4} - 315 \,{\left (8 \, d + 3 \, e\right )} x^{3} - 84 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 14 \,{\left (10 \, d + e\right )} x - 12 \, d}{84 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^8,x, algorithm="fricas")

[Out]

1/84*(21*e*x^11 + 28*(d + 10*e)*x^10 + 210*(2*d + 9*e)*x^9 + 1260*(3*d + 8*e)*x^8 + 2520*(4*d + 7*e)*x^7*log(x
) - 3528*(5*d + 6*e)*x^6 - 1764*(6*d + 5*e)*x^5 - 840*(7*d + 4*e)*x^4 - 315*(8*d + 3*e)*x^3 - 84*(9*d + 2*e)*x
^2 - 14*(10*d + e)*x - 12*d)/x^7

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Sympy [A]  time = 3.89584, size = 117, normalized size = 0.85 \begin{align*} \frac{e x^{4}}{4} + x^{3} \left (\frac{d}{3} + \frac{10 e}{3}\right ) + x^{2} \left (5 d + \frac{45 e}{2}\right ) + x \left (45 d + 120 e\right ) + 30 \left (4 d + 7 e\right ) \log{\left (x \right )} - \frac{12 d + x^{6} \left (17640 d + 21168 e\right ) + x^{5} \left (10584 d + 8820 e\right ) + x^{4} \left (5880 d + 3360 e\right ) + x^{3} \left (2520 d + 945 e\right ) + x^{2} \left (756 d + 168 e\right ) + x \left (140 d + 14 e\right )}{84 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**8,x)

[Out]

e*x**4/4 + x**3*(d/3 + 10*e/3) + x**2*(5*d + 45*e/2) + x*(45*d + 120*e) + 30*(4*d + 7*e)*log(x) - (12*d + x**6
*(17640*d + 21168*e) + x**5*(10584*d + 8820*e) + x**4*(5880*d + 3360*e) + x**3*(2520*d + 945*e) + x**2*(756*d
+ 168*e) + x*(140*d + 14*e))/(84*x**7)

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Giac [A]  time = 1.14938, size = 188, normalized size = 1.36 \begin{align*} \frac{1}{4} \, x^{4} e + \frac{1}{3} \, d x^{3} + \frac{10}{3} \, x^{3} e + 5 \, d x^{2} + \frac{45}{2} \, x^{2} e + 45 \, d x + 120 \, x e + 30 \,{\left (4 \, d + 7 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac{3528 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 1764 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 840 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 315 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 84 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 14 \,{\left (10 \, d + e\right )} x + 12 \, d}{84 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^8,x, algorithm="giac")

[Out]

1/4*x^4*e + 1/3*d*x^3 + 10/3*x^3*e + 5*d*x^2 + 45/2*x^2*e + 45*d*x + 120*x*e + 30*(4*d + 7*e)*log(abs(x)) - 1/
84*(3528*(5*d + 6*e)*x^6 + 1764*(6*d + 5*e)*x^5 + 840*(7*d + 4*e)*x^4 + 315*(8*d + 3*e)*x^3 + 84*(9*d + 2*e)*x
^2 + 14*(10*d + e)*x + 12*d)/x^7