3.585 \(\int \frac{(d+e x) (1+2 x+x^2)^5}{x^{19}} \, dx\)

Optimal. Leaf size=151 \[ -\frac{d+10 e}{8 x^8}-\frac{5 (2 d+9 e)}{9 x^9}-\frac{3 (3 d+8 e)}{2 x^{10}}-\frac{30 (4 d+7 e)}{11 x^{11}}-\frac{7 (5 d+6 e)}{2 x^{12}}-\frac{42 (6 d+5 e)}{13 x^{13}}-\frac{15 (7 d+4 e)}{7 x^{14}}-\frac{8 d+3 e}{x^{15}}-\frac{5 (9 d+2 e)}{16 x^{16}}-\frac{10 d+e}{17 x^{17}}-\frac{d}{18 x^{18}}-\frac{e}{7 x^7} \]

[Out]

-d/(18*x^18) - (10*d + e)/(17*x^17) - (5*(9*d + 2*e))/(16*x^16) - (8*d + 3*e)/x^15 - (15*(7*d + 4*e))/(7*x^14)
- (42*(6*d + 5*e))/(13*x^13) - (7*(5*d + 6*e))/(2*x^12) - (30*(4*d + 7*e))/(11*x^11) - (3*(3*d + 8*e))/(2*x^1
0) - (5*(2*d + 9*e))/(9*x^9) - (d + 10*e)/(8*x^8) - e/(7*x^7)

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Rubi [A]  time = 0.0745416, antiderivative size = 151, 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} \[ -\frac{d+10 e}{8 x^8}-\frac{5 (2 d+9 e)}{9 x^9}-\frac{3 (3 d+8 e)}{2 x^{10}}-\frac{30 (4 d+7 e)}{11 x^{11}}-\frac{7 (5 d+6 e)}{2 x^{12}}-\frac{42 (6 d+5 e)}{13 x^{13}}-\frac{15 (7 d+4 e)}{7 x^{14}}-\frac{8 d+3 e}{x^{15}}-\frac{5 (9 d+2 e)}{16 x^{16}}-\frac{10 d+e}{17 x^{17}}-\frac{d}{18 x^{18}}-\frac{e}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^19,x]

[Out]

-d/(18*x^18) - (10*d + e)/(17*x^17) - (5*(9*d + 2*e))/(16*x^16) - (8*d + 3*e)/x^15 - (15*(7*d + 4*e))/(7*x^14)
- (42*(6*d + 5*e))/(13*x^13) - (7*(5*d + 6*e))/(2*x^12) - (30*(4*d + 7*e))/(11*x^11) - (3*(3*d + 8*e))/(2*x^1
0) - (5*(2*d + 9*e))/(9*x^9) - (d + 10*e)/(8*x^8) - e/(7*x^7)

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^{19}} \, dx &=\int \frac{(1+x)^{10} (d+e x)}{x^{19}} \, dx\\ &=\int \left (\frac{d}{x^{19}}+\frac{10 d+e}{x^{18}}+\frac{5 (9 d+2 e)}{x^{17}}+\frac{15 (8 d+3 e)}{x^{16}}+\frac{30 (7 d+4 e)}{x^{15}}+\frac{42 (6 d+5 e)}{x^{14}}+\frac{42 (5 d+6 e)}{x^{13}}+\frac{30 (4 d+7 e)}{x^{12}}+\frac{15 (3 d+8 e)}{x^{11}}+\frac{5 (2 d+9 e)}{x^{10}}+\frac{d+10 e}{x^9}+\frac{e}{x^8}\right ) \, dx\\ &=-\frac{d}{18 x^{18}}-\frac{10 d+e}{17 x^{17}}-\frac{5 (9 d+2 e)}{16 x^{16}}-\frac{8 d+3 e}{x^{15}}-\frac{15 (7 d+4 e)}{7 x^{14}}-\frac{42 (6 d+5 e)}{13 x^{13}}-\frac{7 (5 d+6 e)}{2 x^{12}}-\frac{30 (4 d+7 e)}{11 x^{11}}-\frac{3 (3 d+8 e)}{2 x^{10}}-\frac{5 (2 d+9 e)}{9 x^9}-\frac{d+10 e}{8 x^8}-\frac{e}{7 x^7}\\ \end{align*}

Mathematica [A]  time = 0.0445914, size = 151, normalized size = 1. \[ -\frac{d+10 e}{8 x^8}-\frac{5 (2 d+9 e)}{9 x^9}-\frac{3 (3 d+8 e)}{2 x^{10}}-\frac{30 (4 d+7 e)}{11 x^{11}}-\frac{7 (5 d+6 e)}{2 x^{12}}-\frac{42 (6 d+5 e)}{13 x^{13}}-\frac{15 (7 d+4 e)}{7 x^{14}}-\frac{8 d+3 e}{x^{15}}-\frac{5 (9 d+2 e)}{16 x^{16}}-\frac{10 d+e}{17 x^{17}}-\frac{d}{18 x^{18}}-\frac{e}{7 x^7} \]

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^19,x]

[Out]

-d/(18*x^18) - (10*d + e)/(17*x^17) - (5*(9*d + 2*e))/(16*x^16) - (8*d + 3*e)/x^15 - (15*(7*d + 4*e))/(7*x^14)
- (42*(6*d + 5*e))/(13*x^13) - (7*(5*d + 6*e))/(2*x^12) - (30*(4*d + 7*e))/(11*x^11) - (3*(3*d + 8*e))/(2*x^1
0) - (5*(2*d + 9*e))/(9*x^9) - (d + 10*e)/(8*x^8) - e/(7*x^7)

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Maple [A]  time = 0.006, size = 130, normalized size = 0.9 \begin{align*} -{\frac{10\,d+45\,e}{9\,{x}^{9}}}-{\frac{45\,d+120\,e}{10\,{x}^{10}}}-{\frac{120\,d+45\,e}{15\,{x}^{15}}}-{\frac{d}{18\,{x}^{18}}}-{\frac{210\,d+252\,e}{12\,{x}^{12}}}-{\frac{120\,d+210\,e}{11\,{x}^{11}}}-{\frac{e}{7\,{x}^{7}}}-{\frac{10\,d+e}{17\,{x}^{17}}}-{\frac{d+10\,e}{8\,{x}^{8}}}-{\frac{45\,d+10\,e}{16\,{x}^{16}}}-{\frac{210\,d+120\,e}{14\,{x}^{14}}}-{\frac{252\,d+210\,e}{13\,{x}^{13}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^19,x)

[Out]

-1/9*(10*d+45*e)/x^9-1/10*(45*d+120*e)/x^10-1/15*(120*d+45*e)/x^15-1/18*d/x^18-1/12*(210*d+252*e)/x^12-1/11*(1
20*d+210*e)/x^11-1/7*e/x^7-1/17*(10*d+e)/x^17-1/8*(d+10*e)/x^8-1/16*(45*d+10*e)/x^16-1/14*(210*d+120*e)/x^14-1
/13*(252*d+210*e)/x^13

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Maxima [A]  time = 1.00285, size = 174, normalized size = 1.15 \begin{align*} -\frac{350064 \, e x^{11} + 306306 \,{\left (d + 10 \, e\right )} x^{10} + 1361360 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 3675672 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 6683040 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 8576568 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 7916832 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 5250960 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 2450448 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 765765 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 144144 \,{\left (10 \, d + e\right )} x + 136136 \, d}{2450448 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^19,x, algorithm="maxima")

[Out]

-1/2450448*(350064*e*x^11 + 306306*(d + 10*e)*x^10 + 1361360*(2*d + 9*e)*x^9 + 3675672*(3*d + 8*e)*x^8 + 66830
40*(4*d + 7*e)*x^7 + 8576568*(5*d + 6*e)*x^6 + 7916832*(6*d + 5*e)*x^5 + 5250960*(7*d + 4*e)*x^4 + 2450448*(8*
d + 3*e)*x^3 + 765765*(9*d + 2*e)*x^2 + 144144*(10*d + e)*x + 136136*d)/x^18

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Fricas [A]  time = 1.21474, size = 402, normalized size = 2.66 \begin{align*} -\frac{350064 \, e x^{11} + 306306 \,{\left (d + 10 \, e\right )} x^{10} + 1361360 \,{\left (2 \, d + 9 \, e\right )} x^{9} + 3675672 \,{\left (3 \, d + 8 \, e\right )} x^{8} + 6683040 \,{\left (4 \, d + 7 \, e\right )} x^{7} + 8576568 \,{\left (5 \, d + 6 \, e\right )} x^{6} + 7916832 \,{\left (6 \, d + 5 \, e\right )} x^{5} + 5250960 \,{\left (7 \, d + 4 \, e\right )} x^{4} + 2450448 \,{\left (8 \, d + 3 \, e\right )} x^{3} + 765765 \,{\left (9 \, d + 2 \, e\right )} x^{2} + 144144 \,{\left (10 \, d + e\right )} x + 136136 \, d}{2450448 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^19,x, algorithm="fricas")

[Out]

-1/2450448*(350064*e*x^11 + 306306*(d + 10*e)*x^10 + 1361360*(2*d + 9*e)*x^9 + 3675672*(3*d + 8*e)*x^8 + 66830
40*(4*d + 7*e)*x^7 + 8576568*(5*d + 6*e)*x^6 + 7916832*(6*d + 5*e)*x^5 + 5250960*(7*d + 4*e)*x^4 + 2450448*(8*
d + 3*e)*x^3 + 765765*(9*d + 2*e)*x^2 + 144144*(10*d + e)*x + 136136*d)/x^18

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Sympy [A]  time = 25.6633, size = 116, normalized size = 0.77 \begin{align*} - \frac{136136 d + 350064 e x^{11} + x^{10} \left (306306 d + 3063060 e\right ) + x^{9} \left (2722720 d + 12252240 e\right ) + x^{8} \left (11027016 d + 29405376 e\right ) + x^{7} \left (26732160 d + 46781280 e\right ) + x^{6} \left (42882840 d + 51459408 e\right ) + x^{5} \left (47500992 d + 39584160 e\right ) + x^{4} \left (36756720 d + 21003840 e\right ) + x^{3} \left (19603584 d + 7351344 e\right ) + x^{2} \left (6891885 d + 1531530 e\right ) + x \left (1441440 d + 144144 e\right )}{2450448 x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**19,x)

[Out]

-(136136*d + 350064*e*x**11 + x**10*(306306*d + 3063060*e) + x**9*(2722720*d + 12252240*e) + x**8*(11027016*d
+ 29405376*e) + x**7*(26732160*d + 46781280*e) + x**6*(42882840*d + 51459408*e) + x**5*(47500992*d + 39584160*
e) + x**4*(36756720*d + 21003840*e) + x**3*(19603584*d + 7351344*e) + x**2*(6891885*d + 1531530*e) + x*(144144
0*d + 144144*e))/(2450448*x**18)

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Giac [A]  time = 1.14552, size = 192, normalized size = 1.27 \begin{align*} -\frac{350064 \, x^{11} e + 306306 \, d x^{10} + 3063060 \, x^{10} e + 2722720 \, d x^{9} + 12252240 \, x^{9} e + 11027016 \, d x^{8} + 29405376 \, x^{8} e + 26732160 \, d x^{7} + 46781280 \, x^{7} e + 42882840 \, d x^{6} + 51459408 \, x^{6} e + 47500992 \, d x^{5} + 39584160 \, x^{5} e + 36756720 \, d x^{4} + 21003840 \, x^{4} e + 19603584 \, d x^{3} + 7351344 \, x^{3} e + 6891885 \, d x^{2} + 1531530 \, x^{2} e + 1441440 \, d x + 144144 \, x e + 136136 \, d}{2450448 \, x^{18}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^19,x, algorithm="giac")

[Out]

-1/2450448*(350064*x^11*e + 306306*d*x^10 + 3063060*x^10*e + 2722720*d*x^9 + 12252240*x^9*e + 11027016*d*x^8 +
29405376*x^8*e + 26732160*d*x^7 + 46781280*x^7*e + 42882840*d*x^6 + 51459408*x^6*e + 47500992*d*x^5 + 3958416
0*x^5*e + 36756720*d*x^4 + 21003840*x^4*e + 19603584*d*x^3 + 7351344*x^3*e + 6891885*d*x^2 + 1531530*x^2*e + 1
441440*d*x + 144144*x*e + 136136*d)/x^18