3.6.85 \(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^{19}} \, dx\) [585]

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

[Out]

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

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Rubi [A]
time = 0.05, antiderivative size = 151, normalized size of antiderivative = 1.00, 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, 77} \begin {gather*} -\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 {d}{18 x^{18}}-\frac {e}{7 x^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/18*d/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 77

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*}

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Mathematica [A]
time = 0.03, size = 151, normalized size = 1.00 \begin {gather*} -\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 {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/18*d/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 = 0.68, size = 130, normalized size = 0.86

method	result	size
norman	\(\frac {-\frac {d}{18}+\left (-\frac {10 d}{17}-\frac {e}{17}\right ) x +\left (-\frac {45 d}{16}-\frac {5 e}{8}\right ) x^{2}+\left (-8 d -3 e \right ) x^{3}+\left (-15 d -\frac {60 e}{7}\right ) x^{4}+\left (-\frac {252 d}{13}-\frac {210 e}{13}\right ) x^{5}+\left (-\frac {35 d}{2}-21 e \right ) x^{6}+\left (-\frac {120 d}{11}-\frac {210 e}{11}\right ) x^{7}+\left (-\frac {9 d}{2}-12 e \right ) x^{8}+\left (-\frac {10 d}{9}-5 e \right ) x^{9}+\left (-\frac {d}{8}-\frac {5 e}{4}\right ) x^{10}-\frac {e \,x^{11}}{7}}{x^{18}}\)	\(123\)
risch	\(\frac {-\frac {d}{18}+\left (-\frac {10 d}{17}-\frac {e}{17}\right ) x +\left (-\frac {45 d}{16}-\frac {5 e}{8}\right ) x^{2}+\left (-8 d -3 e \right ) x^{3}+\left (-15 d -\frac {60 e}{7}\right ) x^{4}+\left (-\frac {252 d}{13}-\frac {210 e}{13}\right ) x^{5}+\left (-\frac {35 d}{2}-21 e \right ) x^{6}+\left (-\frac {120 d}{11}-\frac {210 e}{11}\right ) x^{7}+\left (-\frac {9 d}{2}-12 e \right ) x^{8}+\left (-\frac {10 d}{9}-5 e \right ) x^{9}+\left (-\frac {d}{8}-\frac {5 e}{4}\right ) x^{10}-\frac {e \,x^{11}}{7}}{x^{18}}\)	\(123\)
default	\(-\frac {210 d +120 e}{14 x^{14}}-\frac {120 d +210 e}{11 x^{11}}-\frac {120 d +45 e}{15 x^{15}}-\frac {10 d +45 e}{9 x^{9}}-\frac {210 d +252 e}{12 x^{12}}-\frac {252 d +210 e}{13 x^{13}}-\frac {d +10 e}{8 x^{8}}-\frac {e}{7 x^{7}}-\frac {45 d +120 e}{10 x^{10}}-\frac {10 d +e}{17 x^{17}}-\frac {45 d +10 e}{16 x^{16}}-\frac {d}{18 x^{18}}\)	\(130\)
gosper	\(-\frac {350064 e \,x^{11}+306306 d \,x^{10}+3063060 e \,x^{10}+2722720 d \,x^{9}[phone] e \,x^{9}[phone] d \,x^{8}[phone] e \,x^{8}[phone] d \,x^{7}[phone] e \,x^{7}[phone] d \,x^{6}[phone] x^{6} e [phone] d \,x^{5}[phone] e \,x^{5}[phone] d \,x^{4}[phone] x^{4} e [phone] d \,x^{3}+7351344 e \,x^{3}+6891885 d \,x^{2}+1531530 e \,x^{2}+1441440 d x +144144 e x +136136 d}{2450448 x^{18}}\)	\(132\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 140, normalized size = 0.93 \begin {gather*} -\frac {350064 \, x^{11} e + 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 {gather*}

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*x^11*e + 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 = 2.81, size = 125, normalized size = 0.83 \begin {gather*} -\frac {306306 \, d x^{10} + 2722720 \, d x^{9} + 11027016 \, d x^{8} + 26732160 \, d x^{7} + 42882840 \, d x^{6} + 47500992 \, d x^{5} + 36756720 \, d x^{4} + 19603584 \, d x^{3} + 6891885 \, d x^{2} + 1441440 \, d x + 18 \, {\left (19448 \, x^{11} + 170170 \, x^{10} + 680680 \, x^{9} + 1633632 \, x^{8} + 2598960 \, x^{7} + 2858856 \, x^{6} + 2199120 \, x^{5} + 1166880 \, x^{4} + 408408 \, x^{3} + 85085 \, x^{2} + 8008 \, x\right )} e + 136136 \, d}{2450448 \, x^{18}} \end {gather*}

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*(306306*d*x^10 + 2722720*d*x^9 + 11027016*d*x^8 + 26732160*d*x^7 + 42882840*d*x^6 + 47500992*d*x^5
+ 36756720*d*x^4 + 19603584*d*x^3 + 6891885*d*x^2 + 1441440*d*x + 18*(19448*x^11 + 170170*x^10 + 680680*x^9 +
1633632*x^8 + 2598960*x^7 + 2858856*x^6 + 2199120*x^5 + 1166880*x^4 + 408408*x^3 + 85085*x^2 + 8008*x)*e + 136
136*d)/x^18

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Sympy [A]
time = 17.77, size = 131, normalized size = 0.87 \begin {gather*} \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 {gather*}

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 - 395
84160*e) + x**4*(-36756720*d - 21003840*e) + x**3*(-19603584*d - 7351344*e) + x**2*(-6891885*d - 1531530*e) +
x*(-1441440*d - 144144*e))/(2450448*x**18)

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Giac [A]
time = 0.98, size = 142, normalized size = 0.94 \begin {gather*} -\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 {gather*}

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

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Mupad [B]
time = 0.12, size = 123, normalized size = 0.81 \begin {gather*} -\frac {\frac {e\,x^{11}}{7}+\left (\frac {d}{8}+\frac {5\,e}{4}\right )\,x^{10}+\left (\frac {10\,d}{9}+5\,e\right )\,x^9+\left (\frac {9\,d}{2}+12\,e\right )\,x^8+\left (\frac {120\,d}{11}+\frac {210\,e}{11}\right )\,x^7+\left (\frac {35\,d}{2}+21\,e\right )\,x^6+\left (\frac {252\,d}{13}+\frac {210\,e}{13}\right )\,x^5+\left (15\,d+\frac {60\,e}{7}\right )\,x^4+\left (8\,d+3\,e\right )\,x^3+\left (\frac {45\,d}{16}+\frac {5\,e}{8}\right )\,x^2+\left (\frac {10\,d}{17}+\frac {e}{17}\right )\,x+\frac {d}{18}}{x^{18}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(d/18 + x^3*(8*d + 3*e) + x^10*(d/8 + (5*e)/4) + x^8*((9*d)/2 + 12*e) + x^9*((10*d)/9 + 5*e) + x^6*((35*d)/2
+ 21*e) + x^2*((45*d)/16 + (5*e)/8) + x^4*(15*d + (60*e)/7) + x^7*((120*d)/11 + (210*e)/11) + x^5*((252*d)/13
+ (210*e)/13) + (e*x^11)/7 + x*((10*d)/17 + e/17))/x^18

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