3.6.10 \(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^{17}} \, dx\)

Optimal. Leaf size=109 \[ \frac {(x+1)^{11} (5 d-16 e)}{240 x^{15}}-\frac {(x+1)^{11} (5 d-16 e)}{840 x^{14}}+\frac {(x+1)^{11} (5 d-16 e)}{3640 x^{13}}-\frac {(x+1)^{11} (5 d-16 e)}{21840 x^{12}}+\frac {(x+1)^{11} (5 d-16 e)}{240240 x^{11}}-\frac {d (x+1)^{11}}{16 x^{16}} \]

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Rubi [A]  time = 0.03, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {27, 78, 45, 37} \begin {gather*} \frac {(x+1)^{11} (5 d-16 e)}{240240 x^{11}}-\frac {(x+1)^{11} (5 d-16 e)}{21840 x^{12}}+\frac {(x+1)^{11} (5 d-16 e)}{3640 x^{13}}-\frac {(x+1)^{11} (5 d-16 e)}{840 x^{14}}+\frac {(x+1)^{11} (5 d-16 e)}{240 x^{15}}-\frac {d (x+1)^{11}}{16 x^{16}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^17,x]

[Out]

-(d*(1 + x)^11)/(16*x^16) + ((5*d - 16*e)*(1 + x)^11)/(240*x^15) - ((5*d - 16*e)*(1 + x)^11)/(840*x^14) + ((5*
d - 16*e)*(1 + x)^11)/(3640*x^13) - ((5*d - 16*e)*(1 + x)^11)/(21840*x^12) + ((5*d - 16*e)*(1 + x)^11)/(240240
*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 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]

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 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]))))

Rubi steps

\begin {align*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{17}} \, dx\\ &=-\frac {d (1+x)^{11}}{16 x^{16}}-\frac {1}{16} (5 d-16 e) \int \frac {(1+x)^{10}}{x^{16}} \, dx\\ &=-\frac {d (1+x)^{11}}{16 x^{16}}+\frac {(5 d-16 e) (1+x)^{11}}{240 x^{15}}-\frac {1}{60} (-5 d+16 e) \int \frac {(1+x)^{10}}{x^{15}} \, dx\\ &=-\frac {d (1+x)^{11}}{16 x^{16}}+\frac {(5 d-16 e) (1+x)^{11}}{240 x^{15}}-\frac {(5 d-16 e) (1+x)^{11}}{840 x^{14}}-\frac {1}{280} (5 d-16 e) \int \frac {(1+x)^{10}}{x^{14}} \, dx\\ &=-\frac {d (1+x)^{11}}{16 x^{16}}+\frac {(5 d-16 e) (1+x)^{11}}{240 x^{15}}-\frac {(5 d-16 e) (1+x)^{11}}{840 x^{14}}+\frac {(5 d-16 e) (1+x)^{11}}{3640 x^{13}}-\frac {(-5 d+16 e) \int \frac {(1+x)^{10}}{x^{13}} \, dx}{1820}\\ &=-\frac {d (1+x)^{11}}{16 x^{16}}+\frac {(5 d-16 e) (1+x)^{11}}{240 x^{15}}-\frac {(5 d-16 e) (1+x)^{11}}{840 x^{14}}+\frac {(5 d-16 e) (1+x)^{11}}{3640 x^{13}}-\frac {(5 d-16 e) (1+x)^{11}}{21840 x^{12}}-\frac {(5 d-16 e) \int \frac {(1+x)^{10}}{x^{12}} \, dx}{21840}\\ &=-\frac {d (1+x)^{11}}{16 x^{16}}+\frac {(5 d-16 e) (1+x)^{11}}{240 x^{15}}-\frac {(5 d-16 e) (1+x)^{11}}{840 x^{14}}+\frac {(5 d-16 e) (1+x)^{11}}{3640 x^{13}}-\frac {(5 d-16 e) (1+x)^{11}}{21840 x^{12}}+\frac {(5 d-16 e) (1+x)^{11}}{240240 x^{11}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 153, normalized size = 1.40 \begin {gather*} -\frac {10 d+e}{15 x^{15}}-\frac {5 (9 d+2 e)}{14 x^{14}}-\frac {15 (8 d+3 e)}{13 x^{13}}-\frac {5 (7 d+4 e)}{2 x^{12}}-\frac {42 (6 d+5 e)}{11 x^{11}}-\frac {21 (5 d+6 e)}{5 x^{10}}-\frac {10 (4 d+7 e)}{3 x^9}-\frac {15 (3 d+8 e)}{8 x^8}-\frac {5 (2 d+9 e)}{7 x^7}-\frac {d+10 e}{6 x^6}-\frac {d}{16 x^{16}}-\frac {e}{5 x^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^17,x]

[Out]

-1/16*d/x^16 - (10*d + e)/(15*x^15) - (5*(9*d + 2*e))/(14*x^14) - (15*(8*d + 3*e))/(13*x^13) - (5*(7*d + 4*e))
/(2*x^12) - (42*(6*d + 5*e))/(11*x^11) - (21*(5*d + 6*e))/(5*x^10) - (10*(4*d + 7*e))/(3*x^9) - (15*(3*d + 8*e
))/(8*x^8) - (5*(2*d + 9*e))/(7*x^7) - (d + 10*e)/(6*x^6) - e/(5*x^5)

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x) \left (1+2 x+x^2\right )^5}{x^{17}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^17,x]

[Out]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^17, x]

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fricas [A]  time = 0.43, size = 129, normalized size = 1.18 \begin {gather*} -\frac {48048 \, e x^{11} + 40040 \, {\left (d + 10 \, e\right )} x^{10} + 171600 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 450450 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 800800 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 1009008 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 917280 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 600600 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 277200 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 85800 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 16016 \, {\left (10 \, d + e\right )} x + 15015 \, d}{240240 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^17,x, algorithm="fricas")

[Out]

-1/240240*(48048*e*x^11 + 40040*(d + 10*e)*x^10 + 171600*(2*d + 9*e)*x^9 + 450450*(3*d + 8*e)*x^8 + 800800*(4*
d + 7*e)*x^7 + 1009008*(5*d + 6*e)*x^6 + 917280*(6*d + 5*e)*x^5 + 600600*(7*d + 4*e)*x^4 + 277200*(8*d + 3*e)*
x^3 + 85800*(9*d + 2*e)*x^2 + 16016*(10*d + e)*x + 15015*d)/x^16

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giac [A]  time = 0.16, size = 142, normalized size = 1.30 \begin {gather*} -\frac {48048 \, x^{11} e + 40040 \, d x^{10} + 400400 \, x^{10} e + 343200 \, d x^{9} + 1544400 \, x^{9} e + 1351350 \, d x^{8} + 3603600 \, x^{8} e + 3203200 \, d x^{7} + 5605600 \, x^{7} e + 5045040 \, d x^{6} + 6054048 \, x^{6} e + 5503680 \, d x^{5} + 4586400 \, x^{5} e + 4204200 \, d x^{4} + 2402400 \, x^{4} e + 2217600 \, d x^{3} + 831600 \, x^{3} e + 772200 \, d x^{2} + 171600 \, x^{2} e + 160160 \, d x + 16016 \, x e + 15015 \, d}{240240 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^17,x, algorithm="giac")

[Out]

-1/240240*(48048*x^11*e + 40040*d*x^10 + 400400*x^10*e + 343200*d*x^9 + 1544400*x^9*e + 1351350*d*x^8 + 360360
0*x^8*e + 3203200*d*x^7 + 5605600*x^7*e + 5045040*d*x^6 + 6054048*x^6*e + 5503680*d*x^5 + 4586400*x^5*e + 4204
200*d*x^4 + 2402400*x^4*e + 2217600*d*x^3 + 831600*x^3*e + 772200*d*x^2 + 171600*x^2*e + 160160*d*x + 16016*x*
e + 15015*d)/x^16

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maple [A]  time = 0.05, size = 130, normalized size = 1.19 \begin {gather*} -\frac {e}{5 x^{5}}-\frac {d +10 e}{6 x^{6}}-\frac {10 d +45 e}{7 x^{7}}-\frac {45 d +120 e}{8 x^{8}}-\frac {120 d +210 e}{9 x^{9}}-\frac {210 d +252 e}{10 x^{10}}-\frac {252 d +210 e}{11 x^{11}}-\frac {210 d +120 e}{12 x^{12}}-\frac {120 d +45 e}{13 x^{13}}-\frac {45 d +10 e}{14 x^{14}}-\frac {d}{16 x^{16}}-\frac {10 d +e}{15 x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^17,x)

[Out]

-1/5*e/x^5-1/12*(210*d+120*e)/x^12-1/16*d/x^16-1/8*(45*d+120*e)/x^8-1/10*(210*d+252*e)/x^10-1/15*(10*d+e)/x^15
-1/13*(120*d+45*e)/x^13-1/14*(45*d+10*e)/x^14-1/9*(120*d+210*e)/x^9-1/7*(10*d+45*e)/x^7-1/6*(d+10*e)/x^6-1/11*
(252*d+210*e)/x^11

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maxima [A]  time = 0.71, size = 129, normalized size = 1.18 \begin {gather*} -\frac {48048 \, e x^{11} + 40040 \, {\left (d + 10 \, e\right )} x^{10} + 171600 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 450450 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 800800 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 1009008 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 917280 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 600600 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 277200 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 85800 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 16016 \, {\left (10 \, d + e\right )} x + 15015 \, d}{240240 \, x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^17,x, algorithm="maxima")

[Out]

-1/240240*(48048*e*x^11 + 40040*(d + 10*e)*x^10 + 171600*(2*d + 9*e)*x^9 + 450450*(3*d + 8*e)*x^8 + 800800*(4*
d + 7*e)*x^7 + 1009008*(5*d + 6*e)*x^6 + 917280*(6*d + 5*e)*x^5 + 600600*(7*d + 4*e)*x^4 + 277200*(8*d + 3*e)*
x^3 + 85800*(9*d + 2*e)*x^2 + 16016*(10*d + e)*x + 15015*d)/x^16

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mupad [B]  time = 0.12, size = 123, normalized size = 1.13 \begin {gather*} -\frac {\frac {e\,x^{11}}{5}+\left (\frac {d}{6}+\frac {5\,e}{3}\right )\,x^{10}+\left (\frac {10\,d}{7}+\frac {45\,e}{7}\right )\,x^9+\left (\frac {45\,d}{8}+15\,e\right )\,x^8+\left (\frac {40\,d}{3}+\frac {70\,e}{3}\right )\,x^7+\left (21\,d+\frac {126\,e}{5}\right )\,x^6+\left (\frac {252\,d}{11}+\frac {210\,e}{11}\right )\,x^5+\left (\frac {35\,d}{2}+10\,e\right )\,x^4+\left (\frac {120\,d}{13}+\frac {45\,e}{13}\right )\,x^3+\left (\frac {45\,d}{14}+\frac {5\,e}{7}\right )\,x^2+\left (\frac {2\,d}{3}+\frac {e}{15}\right )\,x+\frac {d}{16}}{x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x)*(2*x + x^2 + 1)^5)/x^17,x)

[Out]

-(d/16 + x^10*(d/6 + (5*e)/3) + x^4*((35*d)/2 + 10*e) + x^2*((45*d)/14 + (5*e)/7) + x^8*((45*d)/8 + 15*e) + x^
9*((10*d)/7 + (45*e)/7) + x^7*((40*d)/3 + (70*e)/3) + x^6*(21*d + (126*e)/5) + x^3*((120*d)/13 + (45*e)/13) +
x^5*((252*d)/11 + (210*e)/11) + (e*x^11)/5 + x*((2*d)/3 + e/15))/x^16

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sympy [A]  time = 21.70, size = 131, normalized size = 1.20 \begin {gather*} \frac {- 15015 d - 48048 e x^{11} + x^{10} \left (- 40040 d - 400400 e\right ) + x^{9} \left (- 343200 d - 1544400 e\right ) + x^{8} \left (- 1351350 d - 3603600 e\right ) + x^{7} \left (- 3203200 d - 5605600 e\right ) + x^{6} \left (- 5045040 d - 6054048 e\right ) + x^{5} \left (- 5503680 d - 4586400 e\right ) + x^{4} \left (- 4204200 d - 2402400 e\right ) + x^{3} \left (- 2217600 d - 831600 e\right ) + x^{2} \left (- 772200 d - 171600 e\right ) + x \left (- 160160 d - 16016 e\right )}{240240 x^{16}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**17,x)

[Out]

(-15015*d - 48048*e*x**11 + x**10*(-40040*d - 400400*e) + x**9*(-343200*d - 1544400*e) + x**8*(-1351350*d - 36
03600*e) + x**7*(-3203200*d - 5605600*e) + x**6*(-5045040*d - 6054048*e) + x**5*(-5503680*d - 4586400*e) + x**
4*(-4204200*d - 2402400*e) + x**3*(-2217600*d - 831600*e) + x**2*(-772200*d - 171600*e) + x*(-160160*d - 16016
*e))/(240240*x**16)

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