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

Optimal. Leaf size=137 \[ \frac{1}{6} x^6 (d+10 e)+x^5 (2 d+9 e)+\frac{15}{4} x^4 (3 d+8 e)+10 x^3 (4 d+7 e)+21 x^2 (5 d+6 e)-\frac{5 (9 d+2 e)}{2 x^2}-\frac{10 d+e}{3 x^3}+42 x (6 d+5 e)-\frac{15 (8 d+3 e)}{x}+30 (7 d+4 e) \log (x)-\frac{d}{4 x^4}+\frac{e x^7}{7} \]

[Out]

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

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Rubi [A]  time = 0.0683002, 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} \[ \frac{1}{6} x^6 (d+10 e)+x^5 (2 d+9 e)+\frac{15}{4} x^4 (3 d+8 e)+10 x^3 (4 d+7 e)+21 x^2 (5 d+6 e)-\frac{5 (9 d+2 e)}{2 x^2}-\frac{10 d+e}{3 x^3}+42 x (6 d+5 e)-\frac{15 (8 d+3 e)}{x}+30 (7 d+4 e) \log (x)-\frac{d}{4 x^4}+\frac{e x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0345277, size = 139, normalized size = 1.01 \[ \frac{1}{6} x^6 (d+10 e)+x^5 (2 d+9 e)+\frac{15}{4} x^4 (3 d+8 e)+10 x^3 (4 d+7 e)+21 x^2 (5 d+6 e)-\frac{5 (9 d+2 e)}{2 x^2}+\frac{-10 d-e}{3 x^3}+42 x (6 d+5 e)-\frac{15 (8 d+3 e)}{x}+30 (7 d+4 e) \log (x)-\frac{d}{4 x^4}+\frac{e x^7}{7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-d/(4*x^4) + (-10*d - e)/(3*x^3) - (5*(9*d + 2*e))/(2*x^2) - (15*(8*d + 3*e))/x + 42*(6*d + 5*e)*x + 21*(5*d +
6*e)*x^2 + 10*(4*d + 7*e)*x^3 + (15*(3*d + 8*e)*x^4)/4 + (2*d + 9*e)*x^5 + ((d + 10*e)*x^6)/6 + (e*x^7)/7 + 3
0*(7*d + 4*e)*Log[x]

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Maple [A]  time = 0.007, size = 128, normalized size = 0.9 \begin{align*}{\frac{e{x}^{7}}{7}}+{\frac{d{x}^{6}}{6}}+{\frac{5\,e{x}^{6}}{3}}+2\,d{x}^{5}+9\,e{x}^{5}+{\frac{45\,d{x}^{4}}{4}}+30\,e{x}^{4}+40\,d{x}^{3}+70\,e{x}^{3}+105\,d{x}^{2}+126\,e{x}^{2}+252\,dx+210\,ex+210\,d\ln \left ( x \right ) +120\,e\ln \left ( x \right ) -{\frac{10\,d}{3\,{x}^{3}}}-{\frac{e}{3\,{x}^{3}}}-{\frac{45\,d}{2\,{x}^{2}}}-5\,{\frac{e}{{x}^{2}}}-120\,{\frac{d}{x}}-45\,{\frac{e}{x}}-{\frac{d}{4\,{x}^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.01446, size = 170, normalized size = 1.24 \begin{align*} \frac{1}{7} \, e x^{7} + \frac{1}{6} \,{\left (d + 10 \, e\right )} x^{6} +{\left (2 \, d + 9 \, e\right )} x^{5} + \frac{15}{4} \,{\left (3 \, d + 8 \, e\right )} x^{4} + 10 \,{\left (4 \, d + 7 \, e\right )} x^{3} + 21 \,{\left (5 \, d + 6 \, e\right )} x^{2} + 42 \,{\left (6 \, d + 5 \, e\right )} x + 30 \,{\left (7 \, d + 4 \, e\right )} \log \left (x\right ) - \frac{180 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 30 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 4 \,{\left (10 \, d + e\right )} x + 3 \, d}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*e*x^7 + 1/6*(d + 10*e)*x^6 + (2*d + 9*e)*x^5 + 15/4*(3*d + 8*e)*x^4 + 10*(4*d + 7*e)*x^3 + 21*(5*d + 6*e)*
x^2 + 42*(6*d + 5*e)*x + 30*(7*d + 4*e)*log(x) - 1/12*(180*(8*d + 3*e)*x^3 + 30*(9*d + 2*e)*x^2 + 4*(10*d + e)
*x + 3*d)/x^4

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Fricas [A]  time = 1.41654, size = 343, normalized size = 2.5 \begin{align*} \frac{12 \, e x^{11} + 14 \,{\left (d + 10 \, e\right )} x^{10} + 84 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 315 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 840 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 1764 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 3528 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 2520 \,{\left (7 \, d + 4 \, e\right )} x^{4} \log \left (x\right ) - 1260 \,{\left (8 \, d + 3 \, e\right )} x^{3} - 210 \,{\left (9 \, d + 2 \, e\right )} x^{2} - 28 \,{\left (10 \, d + e\right )} x - 21 \, d}{84 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]  time = 1.96907, size = 117, normalized size = 0.85 \begin{align*} \frac{e x^{7}}{7} + x^{6} \left (\frac{d}{6} + \frac{5 e}{3}\right ) + x^{5} \left (2 d + 9 e\right ) + x^{4} \left (\frac{45 d}{4} + 30 e\right ) + x^{3} \left (40 d + 70 e\right ) + x^{2} \left (105 d + 126 e\right ) + x \left (252 d + 210 e\right ) + 30 \left (7 d + 4 e\right ) \log{\left (x \right )} - \frac{3 d + x^{3} \left (1440 d + 540 e\right ) + x^{2} \left (270 d + 60 e\right ) + x \left (40 d + 4 e\right )}{12 x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e*x**7/7 + x**6*(d/6 + 5*e/3) + x**5*(2*d + 9*e) + x**4*(45*d/4 + 30*e) + x**3*(40*d + 70*e) + x**2*(105*d + 1
26*e) + x*(252*d + 210*e) + 30*(7*d + 4*e)*log(x) - (3*d + x**3*(1440*d + 540*e) + x**2*(270*d + 60*e) + x*(40
*d + 4*e))/(12*x**4)

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Giac [A]  time = 1.17282, size = 188, normalized size = 1.37 \begin{align*} \frac{1}{7} \, x^{7} e + \frac{1}{6} \, d x^{6} + \frac{5}{3} \, x^{6} e + 2 \, d x^{5} + 9 \, x^{5} e + \frac{45}{4} \, d x^{4} + 30 \, x^{4} e + 40 \, d x^{3} + 70 \, x^{3} e + 105 \, d x^{2} + 126 \, x^{2} e + 252 \, d x + 210 \, x e + 30 \,{\left (7 \, d + 4 \, e\right )} \log \left ({\left | x \right |}\right ) - \frac{180 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 30 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 4 \,{\left (10 \, d + e\right )} x + 3 \, d}{12 \, x^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/7*x^7*e + 1/6*d*x^6 + 5/3*x^6*e + 2*d*x^5 + 9*x^5*e + 45/4*d*x^4 + 30*x^4*e + 40*d*x^3 + 70*x^3*e + 105*d*x^
2 + 126*x^2*e + 252*d*x + 210*x*e + 30*(7*d + 4*e)*log(abs(x)) - 1/12*(180*(8*d + 3*e)*x^3 + 30*(9*d + 2*e)*x^
2 + 4*(10*d + e)*x + 3*d)/x^4