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

Optimal. Leaf size=139 \[ \frac{1}{9} x^9 (d+10 e)+\frac{5}{8} x^8 (2 d+9 e)+\frac{15}{7} x^7 (3 d+8 e)+5 x^6 (4 d+7 e)+\frac{42}{5} x^5 (5 d+6 e)+\frac{21}{2} x^4 (6 d+5 e)+10 x^3 (7 d+4 e)+\frac{15}{2} x^2 (8 d+3 e)+5 x (9 d+2 e)+(10 d+e) \log (x)-\frac{d}{x}+\frac{e x^{10}}{10} \]

[Out]

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

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Rubi [A]  time = 0.0677171, antiderivative size = 139, 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} \[ \frac{1}{9} x^9 (d+10 e)+\frac{5}{8} x^8 (2 d+9 e)+\frac{15}{7} x^7 (3 d+8 e)+5 x^6 (4 d+7 e)+\frac{42}{5} x^5 (5 d+6 e)+\frac{21}{2} x^4 (6 d+5 e)+10 x^3 (7 d+4 e)+\frac{15}{2} x^2 (8 d+3 e)+5 x (9 d+2 e)+(10 d+e) \log (x)-\frac{d}{x}+\frac{e x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0433868, size = 139, normalized size = 1. \[ \frac{1}{9} x^9 (d+10 e)+\frac{5}{8} x^8 (2 d+9 e)+\frac{15}{7} x^7 (3 d+8 e)+5 x^6 (4 d+7 e)+\frac{42}{5} x^5 (5 d+6 e)+\frac{21}{2} x^4 (6 d+5 e)+10 x^3 (7 d+4 e)+\frac{15}{2} x^2 (8 d+3 e)+5 x (9 d+2 e)+(10 d+e) \log (x)-\frac{d}{x}+\frac{e x^{10}}{10} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.007, size = 127, normalized size = 0.9 \begin{align*}{\frac{e{x}^{10}}{10}}+{\frac{d{x}^{9}}{9}}+{\frac{10\,e{x}^{9}}{9}}+{\frac{5\,d{x}^{8}}{4}}+{\frac{45\,e{x}^{8}}{8}}+{\frac{45\,d{x}^{7}}{7}}+{\frac{120\,e{x}^{7}}{7}}+20\,d{x}^{6}+35\,e{x}^{6}+42\,d{x}^{5}+{\frac{252\,e{x}^{5}}{5}}+63\,d{x}^{4}+{\frac{105\,e{x}^{4}}{2}}+70\,d{x}^{3}+40\,e{x}^{3}+60\,d{x}^{2}+{\frac{45\,e{x}^{2}}{2}}+45\,dx+10\,ex+10\,d\ln \left ( x \right ) +e\ln \left ( x \right ) -{\frac{d}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.965181, size = 169, normalized size = 1.22 \begin{align*} \frac{1}{10} \, e x^{10} + \frac{1}{9} \,{\left (d + 10 \, e\right )} x^{9} + \frac{5}{8} \,{\left (2 \, d + 9 \, e\right )} x^{8} + \frac{15}{7} \,{\left (3 \, d + 8 \, e\right )} x^{7} + 5 \,{\left (4 \, d + 7 \, e\right )} x^{6} + \frac{42}{5} \,{\left (5 \, d + 6 \, e\right )} x^{5} + \frac{21}{2} \,{\left (6 \, d + 5 \, e\right )} x^{4} + 10 \,{\left (7 \, d + 4 \, e\right )} x^{3} + \frac{15}{2} \,{\left (8 \, d + 3 \, e\right )} x^{2} + 5 \,{\left (9 \, d + 2 \, e\right )} x +{\left (10 \, d + e\right )} \log \left (x\right ) - \frac{d}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*e*x^10 + 1/9*(d + 10*e)*x^9 + 5/8*(2*d + 9*e)*x^8 + 15/7*(3*d + 8*e)*x^7 + 5*(4*d + 7*e)*x^6 + 42/5*(5*d
+ 6*e)*x^5 + 21/2*(6*d + 5*e)*x^4 + 10*(7*d + 4*e)*x^3 + 15/2*(8*d + 3*e)*x^2 + 5*(9*d + 2*e)*x + (10*d + e)*l
og(x) - d/x

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Fricas [A]  time = 1.27784, size = 366, normalized size = 2.63 \begin{align*} \frac{252 \, e x^{11} + 280 \,{\left (d + 10 \, e\right )} x^{10} + 1575 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 5400 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 12600 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 21168 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 26460 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 25200 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 18900 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 12600 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 2520 \,{\left (10 \, d + e\right )} x \log \left (x\right ) - 2520 \, d}{2520 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2520*(252*e*x^11 + 280*(d + 10*e)*x^10 + 1575*(2*d + 9*e)*x^9 + 5400*(3*d + 8*e)*x^8 + 12600*(4*d + 7*e)*x^7
+ 21168*(5*d + 6*e)*x^6 + 26460*(6*d + 5*e)*x^5 + 25200*(7*d + 4*e)*x^4 + 18900*(8*d + 3*e)*x^3 + 12600*(9*d
+ 2*e)*x^2 + 2520*(10*d + e)*x*log(x) - 2520*d)/x

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Sympy [A]  time = 0.748495, size = 121, normalized size = 0.87 \begin{align*} - \frac{d}{x} + \frac{e x^{10}}{10} + x^{9} \left (\frac{d}{9} + \frac{10 e}{9}\right ) + x^{8} \left (\frac{5 d}{4} + \frac{45 e}{8}\right ) + x^{7} \left (\frac{45 d}{7} + \frac{120 e}{7}\right ) + x^{6} \left (20 d + 35 e\right ) + x^{5} \left (42 d + \frac{252 e}{5}\right ) + x^{4} \left (63 d + \frac{105 e}{2}\right ) + x^{3} \left (70 d + 40 e\right ) + x^{2} \left (60 d + \frac{45 e}{2}\right ) + x \left (45 d + 10 e\right ) + \left (10 d + e\right ) \log{\left (x \right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.16534, size = 185, normalized size = 1.33 \begin{align*} \frac{1}{10} \, x^{10} e + \frac{1}{9} \, d x^{9} + \frac{10}{9} \, x^{9} e + \frac{5}{4} \, d x^{8} + \frac{45}{8} \, x^{8} e + \frac{45}{7} \, d x^{7} + \frac{120}{7} \, x^{7} e + 20 \, d x^{6} + 35 \, x^{6} e + 42 \, d x^{5} + \frac{252}{5} \, x^{5} e + 63 \, d x^{4} + \frac{105}{2} \, x^{4} e + 70 \, d x^{3} + 40 \, x^{3} e + 60 \, d x^{2} + \frac{45}{2} \, x^{2} e + 45 \, d x + 10 \, x e +{\left (10 \, d + e\right )} \log \left ({\left | x \right |}\right ) - \frac{d}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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