3.569 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^3} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0698625, 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} \[ \frac{1}{8} x^8 (d+10 e)+\frac{5}{7} x^7 (2 d+9 e)+\frac{5}{2} x^6 (3 d+8 e)+6 x^5 (4 d+7 e)+\frac{21}{2} x^4 (5 d+6 e)+14 x^3 (6 d+5 e)+15 x^2 (7 d+4 e)+15 x (8 d+3 e)-\frac{10 d+e}{x}+5 (9 d+2 e) \log (x)-\frac{d}{2 x^2}+\frac{e x^9}{9} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^3,x]

[Out]

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

Mathematica [A]  time = 0.0405465, size = 139, normalized size = 1.01 \[ \frac{1}{8} x^8 (d+10 e)+\frac{5}{7} x^7 (2 d+9 e)+\frac{5}{2} x^6 (3 d+8 e)+6 x^5 (4 d+7 e)+\frac{21}{2} x^4 (5 d+6 e)+14 x^3 (6 d+5 e)+15 x^2 (7 d+4 e)+15 x (8 d+3 e)+\frac{-10 d-e}{x}+5 (9 d+2 e) \log (x)-\frac{d}{2 x^2}+\frac{e x^9}{9} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^3,x]

[Out]

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

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Maple [A]  time = 0.006, size = 128, normalized size = 0.9 \begin{align*}{\frac{e{x}^{9}}{9}}+{\frac{d{x}^{8}}{8}}+{\frac{5\,e{x}^{8}}{4}}+{\frac{10\,d{x}^{7}}{7}}+{\frac{45\,e{x}^{7}}{7}}+{\frac{15\,d{x}^{6}}{2}}+20\,e{x}^{6}+24\,d{x}^{5}+42\,e{x}^{5}+{\frac{105\,d{x}^{4}}{2}}+63\,e{x}^{4}+84\,d{x}^{3}+70\,e{x}^{3}+105\,d{x}^{2}+60\,e{x}^{2}+120\,dx+45\,ex+45\,d\ln \left ( x \right ) +10\,e\ln \left ( x \right ) -{\frac{d}{2\,{x}^{2}}}-10\,{\frac{d}{x}}-{\frac{e}{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^3,x)

[Out]

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

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Maxima [A]  time = 1.0798, size = 169, normalized size = 1.22 \begin{align*} \frac{1}{9} \, e x^{9} + \frac{1}{8} \,{\left (d + 10 \, e\right )} x^{8} + \frac{5}{7} \,{\left (2 \, d + 9 \, e\right )} x^{7} + \frac{5}{2} \,{\left (3 \, d + 8 \, e\right )} x^{6} + 6 \,{\left (4 \, d + 7 \, e\right )} x^{5} + \frac{21}{2} \,{\left (5 \, d + 6 \, e\right )} x^{4} + 14 \,{\left (6 \, d + 5 \, e\right )} x^{3} + 15 \,{\left (7 \, d + 4 \, e\right )} x^{2} + 15 \,{\left (8 \, d + 3 \, e\right )} x + 5 \,{\left (9 \, d + 2 \, e\right )} \log \left (x\right ) - \frac{2 \,{\left (10 \, d + e\right )} x + d}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^3,x, algorithm="maxima")

[Out]

1/9*e*x^9 + 1/8*(d + 10*e)*x^8 + 5/7*(2*d + 9*e)*x^7 + 5/2*(3*d + 8*e)*x^6 + 6*(4*d + 7*e)*x^5 + 21/2*(5*d + 6
*e)*x^4 + 14*(6*d + 5*e)*x^3 + 15*(7*d + 4*e)*x^2 + 15*(8*d + 3*e)*x + 5*(9*d + 2*e)*log(x) - 1/2*(2*(10*d + e
)*x + d)/x^2

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Fricas [A]  time = 1.3029, size = 352, normalized size = 2.55 \begin{align*} \frac{56 \, e x^{11} + 63 \,{\left (d + 10 \, e\right )} x^{10} + 360 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 1260 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 3024 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 5292 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 7056 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 7560 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 7560 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 2520 \,{\left (9 \, d + 2 \, e\right )} x^{2} \log \left (x\right ) - 504 \,{\left (10 \, d + e\right )} x - 252 \, d}{504 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^3,x, algorithm="fricas")

[Out]

1/504*(56*e*x^11 + 63*(d + 10*e)*x^10 + 360*(2*d + 9*e)*x^9 + 1260*(3*d + 8*e)*x^8 + 3024*(4*d + 7*e)*x^7 + 52
92*(5*d + 6*e)*x^6 + 7056*(6*d + 5*e)*x^5 + 7560*(7*d + 4*e)*x^4 + 7560*(8*d + 3*e)*x^3 + 2520*(9*d + 2*e)*x^2
*log(x) - 504*(10*d + e)*x - 252*d)/x^2

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Sympy [A]  time = 0.920426, size = 121, normalized size = 0.88 \begin{align*} \frac{e x^{9}}{9} + x^{8} \left (\frac{d}{8} + \frac{5 e}{4}\right ) + x^{7} \left (\frac{10 d}{7} + \frac{45 e}{7}\right ) + x^{6} \left (\frac{15 d}{2} + 20 e\right ) + x^{5} \left (24 d + 42 e\right ) + x^{4} \left (\frac{105 d}{2} + 63 e\right ) + x^{3} \left (84 d + 70 e\right ) + x^{2} \left (105 d + 60 e\right ) + x \left (120 d + 45 e\right ) + 5 \left (9 d + 2 e\right ) \log{\left (x \right )} - \frac{d + x \left (20 d + 2 e\right )}{2 x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**3,x)

[Out]

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

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Giac [A]  time = 1.15364, size = 185, normalized size = 1.34 \begin{align*} \frac{1}{9} \, x^{9} e + \frac{1}{8} \, d x^{8} + \frac{5}{4} \, x^{8} e + \frac{10}{7} \, d x^{7} + \frac{45}{7} \, x^{7} e + \frac{15}{2} \, d x^{6} + 20 \, x^{6} e + 24 \, d x^{5} + 42 \, x^{5} e + \frac{105}{2} \, d x^{4} + 63 \, x^{4} e + 84 \, d x^{3} + 70 \, x^{3} e + 105 \, d x^{2} + 60 \, x^{2} e + 120 \, d x + 45 \, x e + 5 \,{\left (9 \, d + 2 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac{2 \,{\left (10 \, d + e\right )} x + d}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^3,x, algorithm="giac")

[Out]

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