3.580 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^{14}} \, dx\)

Optimal. Leaf size=52 \[ -\frac{(x+1)^{11} (2 d-13 e)}{1716 x^{11}}+\frac{(x+1)^{11} (2 d-13 e)}{156 x^{12}}-\frac{d (x+1)^{11}}{13 x^{13}} \]

[Out]

-(d*(1 + x)^11)/(13*x^13) + ((2*d - 13*e)*(1 + x)^11)/(156*x^12) - ((2*d - 13*e)*(1 + x)^11)/(1716*x^11)

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Rubi [A]  time = 0.0115211, antiderivative size = 52, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21, Rules used = {27, 78, 45, 37} \[ -\frac{(x+1)^{11} (2 d-13 e)}{1716 x^{11}}+\frac{(x+1)^{11} (2 d-13 e)}{156 x^{12}}-\frac{d (x+1)^{11}}{13 x^{13}} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^14,x]

[Out]

-(d*(1 + x)^11)/(13*x^13) + ((2*d - 13*e)*(1 + x)^11)/(156*x^12) - ((2*d - 13*e)*(1 + x)^11)/(1716*x^11)

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 78

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> -Simp[((b*e - a*f
)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(f*(p + 1)*(c*f - d*e)), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1)
+ c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f,
n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ
[p, n]))))

Rule 45

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n + 1
))/((b*c - a*d)*(m + 1)), x] - Dist[(d*Simplify[m + n + 2])/((b*c - a*d)*(m + 1)), Int[(a + b*x)^Simplify[m +
1]*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && ILtQ[Simplify[m + n + 2], 0] &&
NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (
SumSimplerQ[m, 1] ||  !SumSimplerQ[n, 1])

Rule 37

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^(n +
1))/((b*c - a*d)*(m + 1)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rubi steps

\begin{align*} \int \frac{(d+e x) \left (1+2 x+x^2\right )^5}{x^{14}} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^{14}} \, dx\\ &=-\frac{d (1+x)^{11}}{13 x^{13}}-\frac{1}{13} (2 d-13 e) \int \frac{(1+x)^{10}}{x^{13}} \, dx\\ &=-\frac{d (1+x)^{11}}{13 x^{13}}+\frac{(2 d-13 e) (1+x)^{11}}{156 x^{12}}-\frac{1}{156} (-2 d+13 e) \int \frac{(1+x)^{10}}{x^{12}} \, dx\\ &=-\frac{d (1+x)^{11}}{13 x^{13}}+\frac{(2 d-13 e) (1+x)^{11}}{156 x^{12}}-\frac{(2 d-13 e) (1+x)^{11}}{1716 x^{11}}\\ \end{align*}

Mathematica [B]  time = 0.0239873, size = 115, normalized size = 2.21 \[ -\frac{2 d \left (286 x^{10}+2145 x^9+7722 x^8+17160 x^7+25740 x^6+27027 x^5+20020 x^4+10296 x^3+3510 x^2+715 x+66\right )+13 e x \left (66 x^{10}+440 x^9+1485 x^8+3168 x^7+4620 x^6+4752 x^5+3465 x^4+1760 x^3+594 x^2+120 x+11\right )}{1716 x^{13}} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^14,x]

[Out]

-(13*e*x*(11 + 120*x + 594*x^2 + 1760*x^3 + 3465*x^4 + 4752*x^5 + 4620*x^6 + 3168*x^7 + 1485*x^8 + 440*x^9 + 6
6*x^10) + 2*d*(66 + 715*x + 3510*x^2 + 10296*x^3 + 20020*x^4 + 27027*x^5 + 25740*x^6 + 17160*x^7 + 7722*x^8 +
2145*x^9 + 286*x^10))/(1716*x^13)

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Maple [B]  time = 0.007, size = 130, normalized size = 2.5 \begin{align*} -{\frac{210\,d+120\,e}{9\,{x}^{9}}}-{\frac{d}{13\,{x}^{13}}}-{\frac{120\,d+45\,e}{10\,{x}^{10}}}-{\frac{d+10\,e}{3\,{x}^{3}}}-{\frac{e}{2\,{x}^{2}}}-{\frac{210\,d+252\,e}{7\,{x}^{7}}}-{\frac{45\,d+10\,e}{11\,{x}^{11}}}-{\frac{252\,d+210\,e}{8\,{x}^{8}}}-{\frac{10\,d+e}{12\,{x}^{12}}}-{\frac{120\,d+210\,e}{6\,{x}^{6}}}-{\frac{45\,d+120\,e}{5\,{x}^{5}}}-{\frac{10\,d+45\,e}{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^14,x)

[Out]

-1/9*(210*d+120*e)/x^9-1/13*d/x^13-1/10*(120*d+45*e)/x^10-1/3*(d+10*e)/x^3-1/2*e/x^2-1/7*(210*d+252*e)/x^7-1/1
1*(45*d+10*e)/x^11-1/8*(252*d+210*e)/x^8-1/12*(10*d+e)/x^12-1/6*(120*d+210*e)/x^6-1/5*(45*d+120*e)/x^5-1/4*(10
*d+45*e)/x^4

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Maxima [B]  time = 1.0842, size = 174, normalized size = 3.35 \begin{align*} -\frac{858 \, e x^{11} + 572 \,{\left (d + 10 \, e\right )} x^{10} + 2145 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 5148 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 8580 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 10296 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 9009 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 5720 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 2574 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 780 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 143 \,{\left (10 \, d + e\right )} x + 132 \, d}{1716 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^14,x, algorithm="maxima")

[Out]

-1/1716*(858*e*x^11 + 572*(d + 10*e)*x^10 + 2145*(2*d + 9*e)*x^9 + 5148*(3*d + 8*e)*x^8 + 8580*(4*d + 7*e)*x^7
+ 10296*(5*d + 6*e)*x^6 + 9009*(6*d + 5*e)*x^5 + 5720*(7*d + 4*e)*x^4 + 2574*(8*d + 3*e)*x^3 + 780*(9*d + 2*e
)*x^2 + 143*(10*d + e)*x + 132*d)/x^13

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Fricas [B]  time = 1.20287, size = 351, normalized size = 6.75 \begin{align*} -\frac{858 \, e x^{11} + 572 \,{\left (d + 10 \, e\right )} x^{10} + 2145 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 5148 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 8580 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 10296 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 9009 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 5720 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 2574 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 780 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 143 \,{\left (10 \, d + e\right )} x + 132 \, d}{1716 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^14,x, algorithm="fricas")

[Out]

-1/1716*(858*e*x^11 + 572*(d + 10*e)*x^10 + 2145*(2*d + 9*e)*x^9 + 5148*(3*d + 8*e)*x^8 + 8580*(4*d + 7*e)*x^7
+ 10296*(5*d + 6*e)*x^6 + 9009*(6*d + 5*e)*x^5 + 5720*(7*d + 4*e)*x^4 + 2574*(8*d + 3*e)*x^3 + 780*(9*d + 2*e
)*x^2 + 143*(10*d + e)*x + 132*d)/x^13

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Sympy [B]  time = 14.2426, size = 116, normalized size = 2.23 \begin{align*} - \frac{132 d + 858 e x^{11} + x^{10} \left (572 d + 5720 e\right ) + x^{9} \left (4290 d + 19305 e\right ) + x^{8} \left (15444 d + 41184 e\right ) + x^{7} \left (34320 d + 60060 e\right ) + x^{6} \left (51480 d + 61776 e\right ) + x^{5} \left (54054 d + 45045 e\right ) + x^{4} \left (40040 d + 22880 e\right ) + x^{3} \left (20592 d + 7722 e\right ) + x^{2} \left (7020 d + 1560 e\right ) + x \left (1430 d + 143 e\right )}{1716 x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**14,x)

[Out]

-(132*d + 858*e*x**11 + x**10*(572*d + 5720*e) + x**9*(4290*d + 19305*e) + x**8*(15444*d + 41184*e) + x**7*(34
320*d + 60060*e) + x**6*(51480*d + 61776*e) + x**5*(54054*d + 45045*e) + x**4*(40040*d + 22880*e) + x**3*(2059
2*d + 7722*e) + x**2*(7020*d + 1560*e) + x*(1430*d + 143*e))/(1716*x**13)

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Giac [B]  time = 1.13437, size = 192, normalized size = 3.69 \begin{align*} -\frac{858 \, x^{11} e + 572 \, d x^{10} + 5720 \, x^{10} e + 4290 \, d x^{9} + 19305 \, x^{9} e + 15444 \, d x^{8} + 41184 \, x^{8} e + 34320 \, d x^{7} + 60060 \, x^{7} e + 51480 \, d x^{6} + 61776 \, x^{6} e + 54054 \, d x^{5} + 45045 \, x^{5} e + 40040 \, d x^{4} + 22880 \, x^{4} e + 20592 \, d x^{3} + 7722 \, x^{3} e + 7020 \, d x^{2} + 1560 \, x^{2} e + 1430 \, d x + 143 \, x e + 132 \, d}{1716 \, x^{13}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^14,x, algorithm="giac")

[Out]

-1/1716*(858*x^11*e + 572*d*x^10 + 5720*x^10*e + 4290*d*x^9 + 19305*x^9*e + 15444*d*x^8 + 41184*x^8*e + 34320*
d*x^7 + 60060*x^7*e + 51480*d*x^6 + 61776*x^6*e + 54054*d*x^5 + 45045*x^5*e + 40040*d*x^4 + 22880*x^4*e + 2059
2*d*x^3 + 7722*x^3*e + 7020*d*x^2 + 1560*x^2*e + 1430*d*x + 143*x*e + 132*d)/x^13