3.572 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^6} \, dx\)

Optimal. Leaf size=140 \[ \frac{1}{5} x^5 (d+10 e)+\frac{5}{4} x^4 (2 d+9 e)+5 x^3 (3 d+8 e)+15 x^2 (4 d+7 e)-\frac{15 (8 d+3 e)}{2 x^2}-\frac{5 (9 d+2 e)}{3 x^3}-\frac{10 d+e}{4 x^4}+42 x (5 d+6 e)-\frac{30 (7 d+4 e)}{x}+42 (6 d+5 e) \log (x)-\frac{d}{5 x^5}+\frac{e x^6}{6} \]

[Out]

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

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Rubi [A]  time = 0.0706684, antiderivative size = 140, 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} \[ \frac{1}{5} x^5 (d+10 e)+\frac{5}{4} x^4 (2 d+9 e)+5 x^3 (3 d+8 e)+15 x^2 (4 d+7 e)-\frac{15 (8 d+3 e)}{2 x^2}-\frac{5 (9 d+2 e)}{3 x^3}-\frac{10 d+e}{4 x^4}+42 x (5 d+6 e)-\frac{30 (7 d+4 e)}{x}+42 (6 d+5 e) \log (x)-\frac{d}{5 x^5}+\frac{e x^6}{6} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^6,x]

[Out]

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

Mathematica [A]  time = 0.0348496, size = 142, normalized size = 1.01 \[ \frac{1}{5} x^5 (d+10 e)+\frac{5}{4} x^4 (2 d+9 e)+5 x^3 (3 d+8 e)+15 x^2 (4 d+7 e)-\frac{15 (8 d+3 e)}{2 x^2}-\frac{5 (9 d+2 e)}{3 x^3}+\frac{-10 d-e}{4 x^4}+42 x (5 d+6 e)-\frac{30 (7 d+4 e)}{x}+42 (6 d+5 e) \log (x)-\frac{d}{5 x^5}+\frac{e x^6}{6} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^6,x]

[Out]

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

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Maple [A]  time = 0.007, size = 128, normalized size = 0.9 \begin{align*}{\frac{e{x}^{6}}{6}}+{\frac{d{x}^{5}}{5}}+2\,e{x}^{5}+{\frac{5\,d{x}^{4}}{2}}+{\frac{45\,e{x}^{4}}{4}}+15\,d{x}^{3}+40\,e{x}^{3}+60\,d{x}^{2}+105\,e{x}^{2}+210\,dx+252\,ex+252\,d\ln \left ( x \right ) +210\,e\ln \left ( x \right ) -15\,{\frac{d}{{x}^{3}}}-{\frac{10\,e}{3\,{x}^{3}}}-{\frac{5\,d}{2\,{x}^{4}}}-{\frac{e}{4\,{x}^{4}}}-60\,{\frac{d}{{x}^{2}}}-{\frac{45\,e}{2\,{x}^{2}}}-210\,{\frac{d}{x}}-120\,{\frac{e}{x}}-{\frac{d}{5\,{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^6,x)

[Out]

1/6*e*x^6+1/5*d*x^5+2*e*x^5+5/2*d*x^4+45/4*e*x^4+15*d*x^3+40*e*x^3+60*d*x^2+105*e*x^2+210*d*x+252*e*x+252*d*ln
(x)+210*e*ln(x)-15*d/x^3-10/3*e/x^3-5/2*d/x^4-1/4*e/x^4-60*d/x^2-45/2*e/x^2-210*d/x-120*e/x-1/5*d/x^5

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Maxima [A]  time = 0.962507, size = 171, normalized size = 1.22 \begin{align*} \frac{1}{6} \, e x^{6} + \frac{1}{5} \,{\left (d + 10 \, e\right )} x^{5} + \frac{5}{4} \,{\left (2 \, d + 9 \, e\right )} x^{4} + 5 \,{\left (3 \, d + 8 \, e\right )} x^{3} + 15 \,{\left (4 \, d + 7 \, e\right )} x^{2} + 42 \,{\left (5 \, d + 6 \, e\right )} x + 42 \,{\left (6 \, d + 5 \, e\right )} \log \left (x\right ) - \frac{1800 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 450 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 100 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 15 \,{\left (10 \, d + e\right )} x + 12 \, d}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^6,x, algorithm="maxima")

[Out]

1/6*e*x^6 + 1/5*(d + 10*e)*x^5 + 5/4*(2*d + 9*e)*x^4 + 5*(3*d + 8*e)*x^3 + 15*(4*d + 7*e)*x^2 + 42*(5*d + 6*e)
*x + 42*(6*d + 5*e)*log(x) - 1/60*(1800*(7*d + 4*e)*x^4 + 450*(8*d + 3*e)*x^3 + 100*(9*d + 2*e)*x^2 + 15*(10*d
+ e)*x + 12*d)/x^5

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Fricas [A]  time = 1.21954, size = 342, normalized size = 2.44 \begin{align*} \frac{10 \, e x^{11} + 12 \,{\left (d + 10 \, e\right )} x^{10} + 75 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 300 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 900 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 2520 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 2520 \,{\left (6 \, d + 5 \, e\right )} x^{5} \log \left (x\right ) - 1800 \,{\left (7 \, d + 4 \, e\right )} x^{4} - 450 \,{\left (8 \, d + 3 \, e\right )} x^{3} - 100 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 15 \,{\left (10 \, d + e\right )} x - 12 \, d}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^6,x, algorithm="fricas")

[Out]

1/60*(10*e*x^11 + 12*(d + 10*e)*x^10 + 75*(2*d + 9*e)*x^9 + 300*(3*d + 8*e)*x^8 + 900*(4*d + 7*e)*x^7 + 2520*(
5*d + 6*e)*x^6 + 2520*(6*d + 5*e)*x^5*log(x) - 1800*(7*d + 4*e)*x^4 - 450*(8*d + 3*e)*x^3 - 100*(9*d + 2*e)*x^
2 - 15*(10*d + e)*x - 12*d)/x^5

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Sympy [A]  time = 2.17201, size = 117, normalized size = 0.84 \begin{align*} \frac{e x^{6}}{6} + x^{5} \left (\frac{d}{5} + 2 e\right ) + x^{4} \left (\frac{5 d}{2} + \frac{45 e}{4}\right ) + x^{3} \left (15 d + 40 e\right ) + x^{2} \left (60 d + 105 e\right ) + x \left (210 d + 252 e\right ) + 42 \left (6 d + 5 e\right ) \log{\left (x \right )} - \frac{12 d + x^{4} \left (12600 d + 7200 e\right ) + x^{3} \left (3600 d + 1350 e\right ) + x^{2} \left (900 d + 200 e\right ) + x \left (150 d + 15 e\right )}{60 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**6,x)

[Out]

e*x**6/6 + x**5*(d/5 + 2*e) + x**4*(5*d/2 + 45*e/4) + x**3*(15*d + 40*e) + x**2*(60*d + 105*e) + x*(210*d + 25
2*e) + 42*(6*d + 5*e)*log(x) - (12*d + x**4*(12600*d + 7200*e) + x**3*(3600*d + 1350*e) + x**2*(900*d + 200*e)
+ x*(150*d + 15*e))/(60*x**5)

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Giac [A]  time = 1.20666, size = 188, normalized size = 1.34 \begin{align*} \frac{1}{6} \, x^{6} e + \frac{1}{5} \, d x^{5} + 2 \, x^{5} e + \frac{5}{2} \, d x^{4} + \frac{45}{4} \, x^{4} e + 15 \, d x^{3} + 40 \, x^{3} e + 60 \, d x^{2} + 105 \, x^{2} e + 210 \, d x + 252 \, x e + 42 \,{\left (6 \, d + 5 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac{1800 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 450 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 100 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 15 \,{\left (10 \, d + e\right )} x + 12 \, d}{60 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^6,x, algorithm="giac")

[Out]

1/6*x^6*e + 1/5*d*x^5 + 2*x^5*e + 5/2*d*x^4 + 45/4*x^4*e + 15*d*x^3 + 40*x^3*e + 60*d*x^2 + 105*x^2*e + 210*d*
x + 252*x*e + 42*(6*d + 5*e)*log(abs(x)) - 1/60*(1800*(7*d + 4*e)*x^4 + 450*(8*d + 3*e)*x^3 + 100*(9*d + 2*e)*
x^2 + 15*(10*d + e)*x + 12*d)/x^5