3.6.9 \(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^{16}} \, dx\)

Optimal. Leaf size=90 \[ \frac {(x+1)^{11} (4 d-15 e)}{210 x^{14}}-\frac {(x+1)^{11} (4 d-15 e)}{910 x^{13}}+\frac {(x+1)^{11} (4 d-15 e)}{5460 x^{12}}-\frac {(x+1)^{11} (4 d-15 e)}{60060 x^{11}}-\frac {d (x+1)^{11}}{15 x^{15}} \]

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Rubi [A]  time = 0.02, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps used = 6, 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} (4 d-15 e)}{60060 x^{11}}+\frac {(x+1)^{11} (4 d-15 e)}{5460 x^{12}}-\frac {(x+1)^{11} (4 d-15 e)}{910 x^{13}}+\frac {(x+1)^{11} (4 d-15 e)}{210 x^{14}}-\frac {d (x+1)^{11}}{15 x^{15}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^16,x]

[Out]

-(d*(1 + x)^11)/(15*x^15) + ((4*d - 15*e)*(1 + x)^11)/(210*x^14) - ((4*d - 15*e)*(1 + x)^11)/(910*x^13) + ((4*
d - 15*e)*(1 + x)^11)/(5460*x^12) - ((4*d - 15*e)*(1 + x)^11)/(60060*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^{16}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{16}} \, dx\\ &=-\frac {d (1+x)^{11}}{15 x^{15}}-\frac {1}{15} (4 d-15 e) \int \frac {(1+x)^{10}}{x^{15}} \, dx\\ &=-\frac {d (1+x)^{11}}{15 x^{15}}+\frac {(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac {1}{70} (-4 d+15 e) \int \frac {(1+x)^{10}}{x^{14}} \, dx\\ &=-\frac {d (1+x)^{11}}{15 x^{15}}+\frac {(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac {(4 d-15 e) (1+x)^{11}}{910 x^{13}}-\frac {1}{455} (4 d-15 e) \int \frac {(1+x)^{10}}{x^{13}} \, dx\\ &=-\frac {d (1+x)^{11}}{15 x^{15}}+\frac {(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac {(4 d-15 e) (1+x)^{11}}{910 x^{13}}+\frac {(4 d-15 e) (1+x)^{11}}{5460 x^{12}}-\frac {(-4 d+15 e) \int \frac {(1+x)^{10}}{x^{12}} \, dx}{5460}\\ &=-\frac {d (1+x)^{11}}{15 x^{15}}+\frac {(4 d-15 e) (1+x)^{11}}{210 x^{14}}-\frac {(4 d-15 e) (1+x)^{11}}{910 x^{13}}+\frac {(4 d-15 e) (1+x)^{11}}{5460 x^{12}}-\frac {(4 d-15 e) (1+x)^{11}}{60060 x^{11}}\\ \end {align*}

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Mathematica [A]  time = 0.04, size = 153, normalized size = 1.70 \begin {gather*} -\frac {10 d+e}{14 x^{14}}-\frac {5 (9 d+2 e)}{13 x^{13}}-\frac {5 (8 d+3 e)}{4 x^{12}}-\frac {30 (7 d+4 e)}{11 x^{11}}-\frac {21 (6 d+5 e)}{5 x^{10}}-\frac {14 (5 d+6 e)}{3 x^9}-\frac {15 (4 d+7 e)}{4 x^8}-\frac {15 (3 d+8 e)}{7 x^7}-\frac {5 (2 d+9 e)}{6 x^6}-\frac {d+10 e}{5 x^5}-\frac {d}{15 x^{15}}-\frac {e}{4 x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^16,x]

[Out]

-1/15*d/x^15 - (10*d + e)/(14*x^14) - (5*(9*d + 2*e))/(13*x^13) - (5*(8*d + 3*e))/(4*x^12) - (30*(7*d + 4*e))/
(11*x^11) - (21*(6*d + 5*e))/(5*x^10) - (14*(5*d + 6*e))/(3*x^9) - (15*(4*d + 7*e))/(4*x^8) - (15*(3*d + 8*e))
/(7*x^7) - (5*(2*d + 9*e))/(6*x^6) - (d + 10*e)/(5*x^5) - e/(4*x^4)

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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^{16}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^16,x]

[Out]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^16, x]

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fricas [A]  time = 0.41, size = 129, normalized size = 1.43 \begin {gather*} -\frac {15015 \, e x^{11} + 12012 \, {\left (d + 10 \, e\right )} x^{10} + 50050 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 128700 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 225225 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 280280 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 252252 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 163800 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 75075 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 23100 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 4290 \, {\left (10 \, d + e\right )} x + 4004 \, d}{60060 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^16,x, algorithm="fricas")

[Out]

-1/60060*(15015*e*x^11 + 12012*(d + 10*e)*x^10 + 50050*(2*d + 9*e)*x^9 + 128700*(3*d + 8*e)*x^8 + 225225*(4*d
+ 7*e)*x^7 + 280280*(5*d + 6*e)*x^6 + 252252*(6*d + 5*e)*x^5 + 163800*(7*d + 4*e)*x^4 + 75075*(8*d + 3*e)*x^3
+ 23100*(9*d + 2*e)*x^2 + 4290*(10*d + e)*x + 4004*d)/x^15

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giac [A]  time = 0.16, size = 142, normalized size = 1.58 \begin {gather*} -\frac {15015 \, x^{11} e + 12012 \, d x^{10} + 120120 \, x^{10} e + 100100 \, d x^{9} + 450450 \, x^{9} e + 386100 \, d x^{8} + 1029600 \, x^{8} e + 900900 \, d x^{7} + 1576575 \, x^{7} e + 1401400 \, d x^{6} + 1681680 \, x^{6} e + 1513512 \, d x^{5} + 1261260 \, x^{5} e + 1146600 \, d x^{4} + 655200 \, x^{4} e + 600600 \, d x^{3} + 225225 \, x^{3} e + 207900 \, d x^{2} + 46200 \, x^{2} e + 42900 \, d x + 4290 \, x e + 4004 \, d}{60060 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^16,x, algorithm="giac")

[Out]

-1/60060*(15015*x^11*e + 12012*d*x^10 + 120120*x^10*e + 100100*d*x^9 + 450450*x^9*e + 386100*d*x^8 + 1029600*x
^8*e + 900900*d*x^7 + 1576575*x^7*e + 1401400*d*x^6 + 1681680*x^6*e + 1513512*d*x^5 + 1261260*x^5*e + 1146600*
d*x^4 + 655200*x^4*e + 600600*d*x^3 + 225225*x^3*e + 207900*d*x^2 + 46200*x^2*e + 42900*d*x + 4290*x*e + 4004*
d)/x^15

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maple [A]  time = 0.06, size = 130, normalized size = 1.44 \begin {gather*} -\frac {e}{4 x^{4}}-\frac {d +10 e}{5 x^{5}}-\frac {10 d +45 e}{6 x^{6}}-\frac {45 d +120 e}{7 x^{7}}-\frac {120 d +210 e}{8 x^{8}}-\frac {210 d +252 e}{9 x^{9}}-\frac {252 d +210 e}{10 x^{10}}-\frac {210 d +120 e}{11 x^{11}}-\frac {120 d +45 e}{12 x^{12}}-\frac {45 d +10 e}{13 x^{13}}-\frac {d}{15 x^{15}}-\frac {10 d +e}{14 x^{14}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^16,x)

[Out]

-1/5*(d+10*e)/x^5-1/4*e/x^4-1/8*(120*d+210*e)/x^8-1/13*(45*d+10*e)/x^13-1/10*(252*d+210*e)/x^10-1/15*d/x^15-1/
9*(210*d+252*e)/x^9-1/7*(45*d+120*e)/x^7-1/14*(10*d+e)/x^14-1/6*(10*d+45*e)/x^6-1/12*(120*d+45*e)/x^12-1/11*(2
10*d+120*e)/x^11

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maxima [A]  time = 0.71, size = 129, normalized size = 1.43 \begin {gather*} -\frac {15015 \, e x^{11} + 12012 \, {\left (d + 10 \, e\right )} x^{10} + 50050 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 128700 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 225225 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 280280 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 252252 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 163800 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 75075 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 23100 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 4290 \, {\left (10 \, d + e\right )} x + 4004 \, d}{60060 \, x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^16,x, algorithm="maxima")

[Out]

-1/60060*(15015*e*x^11 + 12012*(d + 10*e)*x^10 + 50050*(2*d + 9*e)*x^9 + 128700*(3*d + 8*e)*x^8 + 225225*(4*d
+ 7*e)*x^7 + 280280*(5*d + 6*e)*x^6 + 252252*(6*d + 5*e)*x^5 + 163800*(7*d + 4*e)*x^4 + 75075*(8*d + 3*e)*x^3
+ 23100*(9*d + 2*e)*x^2 + 4290*(10*d + e)*x + 4004*d)/x^15

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mupad [B]  time = 0.12, size = 123, normalized size = 1.37 \begin {gather*} -\frac {\frac {e\,x^{11}}{4}+\left (\frac {d}{5}+2\,e\right )\,x^{10}+\left (\frac {5\,d}{3}+\frac {15\,e}{2}\right )\,x^9+\left (\frac {45\,d}{7}+\frac {120\,e}{7}\right )\,x^8+\left (15\,d+\frac {105\,e}{4}\right )\,x^7+\left (\frac {70\,d}{3}+28\,e\right )\,x^6+\left (\frac {126\,d}{5}+21\,e\right )\,x^5+\left (\frac {210\,d}{11}+\frac {120\,e}{11}\right )\,x^4+\left (10\,d+\frac {15\,e}{4}\right )\,x^3+\left (\frac {45\,d}{13}+\frac {10\,e}{13}\right )\,x^2+\left (\frac {5\,d}{7}+\frac {e}{14}\right )\,x+\frac {d}{15}}{x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x)*(2*x + x^2 + 1)^5)/x^16,x)

[Out]

-(d/15 + x^10*(d/5 + 2*e) + x^3*(10*d + (15*e)/4) + x^9*((5*d)/3 + (15*e)/2) + x^2*((45*d)/13 + (10*e)/13) + x
^6*((70*d)/3 + 28*e) + x^7*(15*d + (105*e)/4) + x^5*((126*d)/5 + 21*e) + x^8*((45*d)/7 + (120*e)/7) + x^4*((21
0*d)/11 + (120*e)/11) + (e*x^11)/4 + x*((5*d)/7 + e/14))/x^15

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sympy [A]  time = 19.11, size = 131, normalized size = 1.46 \begin {gather*} \frac {- 4004 d - 15015 e x^{11} + x^{10} \left (- 12012 d - 120120 e\right ) + x^{9} \left (- 100100 d - 450450 e\right ) + x^{8} \left (- 386100 d - 1029600 e\right ) + x^{7} \left (- 900900 d - 1576575 e\right ) + x^{6} \left (- 1401400 d - 1681680 e\right ) + x^{5} \left (- 1513512 d - 1261260 e\right ) + x^{4} \left (- 1146600 d - 655200 e\right ) + x^{3} \left (- 600600 d - 225225 e\right ) + x^{2} \left (- 207900 d - 46200 e\right ) + x \left (- 42900 d - 4290 e\right )}{60060 x^{15}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**16,x)

[Out]

(-4004*d - 15015*e*x**11 + x**10*(-12012*d - 120120*e) + x**9*(-100100*d - 450450*e) + x**8*(-386100*d - 10296
00*e) + x**7*(-900900*d - 1576575*e) + x**6*(-1401400*d - 1681680*e) + x**5*(-1513512*d - 1261260*e) + x**4*(-
1146600*d - 655200*e) + x**3*(-600600*d - 225225*e) + x**2*(-207900*d - 46200*e) + x*(-42900*d - 4290*e))/(600
60*x**15)

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