3.6.5 \(\int \frac {(d+e x) (1+2 x+x^2)^5}{x^{12}} \, dx\)

Optimal. Leaf size=92 \[ -\frac {d (x+1)^{11}}{11 x^{11}}-\frac {e}{10 x^{10}}-\frac {10 e}{9 x^9}-\frac {45 e}{8 x^8}-\frac {120 e}{7 x^7}-\frac {35 e}{x^6}-\frac {252 e}{5 x^5}-\frac {105 e}{2 x^4}-\frac {40 e}{x^3}-\frac {45 e}{2 x^2}-\frac {10 e}{x}+e \log (x) \]

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Rubi [A]  time = 0.03, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {27, 78, 43} \begin {gather*} -\frac {d (x+1)^{11}}{11 x^{11}}-\frac {45 e}{2 x^2}-\frac {40 e}{x^3}-\frac {105 e}{2 x^4}-\frac {252 e}{5 x^5}-\frac {35 e}{x^6}-\frac {120 e}{7 x^7}-\frac {45 e}{8 x^8}-\frac {10 e}{9 x^9}-\frac {e}{10 x^{10}}-\frac {10 e}{x}+e \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((d + e*x)*(1 + 2*x + x^2)^5)/x^12,x]

[Out]

-e/(10*x^10) - (10*e)/(9*x^9) - (45*e)/(8*x^8) - (120*e)/(7*x^7) - (35*e)/x^6 - (252*e)/(5*x^5) - (105*e)/(2*x
^4) - (40*e)/x^3 - (45*e)/(2*x^2) - (10*e)/x - (d*(1 + x)^11)/(11*x^11) + e*Log[x]

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 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

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^{12}} \, dx &=\int \frac {(1+x)^{10} (d+e x)}{x^{12}} \, dx\\ &=-\frac {d (1+x)^{11}}{11 x^{11}}+e \int \frac {(1+x)^{10}}{x^{11}} \, dx\\ &=-\frac {d (1+x)^{11}}{11 x^{11}}+e \int \left (\frac {1}{x^{11}}+\frac {10}{x^{10}}+\frac {45}{x^9}+\frac {120}{x^8}+\frac {210}{x^7}+\frac {252}{x^6}+\frac {210}{x^5}+\frac {120}{x^4}+\frac {45}{x^3}+\frac {10}{x^2}+\frac {1}{x}\right ) \, dx\\ &=-\frac {e}{10 x^{10}}-\frac {10 e}{9 x^9}-\frac {45 e}{8 x^8}-\frac {120 e}{7 x^7}-\frac {35 e}{x^6}-\frac {252 e}{5 x^5}-\frac {105 e}{2 x^4}-\frac {40 e}{x^3}-\frac {45 e}{2 x^2}-\frac {10 e}{x}-\frac {d (1+x)^{11}}{11 x^{11}}+e \log (x)\\ \end {align*}

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Mathematica [A]  time = 0.05, size = 143, normalized size = 1.55 \begin {gather*} -\frac {10 d+e}{10 x^{10}}-\frac {5 (9 d+2 e)}{9 x^9}-\frac {15 (8 d+3 e)}{8 x^8}-\frac {30 (7 d+4 e)}{7 x^7}-\frac {7 (6 d+5 e)}{x^6}-\frac {42 (5 d+6 e)}{5 x^5}-\frac {15 (4 d+7 e)}{2 x^4}-\frac {5 (3 d+8 e)}{x^3}-\frac {5 (2 d+9 e)}{2 x^2}-\frac {d+10 e}{x}-\frac {d}{11 x^{11}}+e \log (x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((d + e*x)*(1 + 2*x + x^2)^5)/x^12,x]

[Out]

-1/11*d/x^11 - (10*d + e)/(10*x^10) - (5*(9*d + 2*e))/(9*x^9) - (15*(8*d + 3*e))/(8*x^8) - (30*(7*d + 4*e))/(7
*x^7) - (7*(6*d + 5*e))/x^6 - (42*(5*d + 6*e))/(5*x^5) - (15*(4*d + 7*e))/(2*x^4) - (5*(3*d + 8*e))/x^3 - (5*(
2*d + 9*e))/(2*x^2) - (d + 10*e)/x + e*Log[x]

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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^{12}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^12,x]

[Out]

IntegrateAlgebraic[((d + e*x)*(1 + 2*x + x^2)^5)/x^12, x]

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fricas [A]  time = 0.40, size = 131, normalized size = 1.42 \begin {gather*} \frac {27720 \, e x^{11} \log \relax (x) - 27720 \, {\left (d + 10 \, e\right )} x^{10} - 69300 \, {\left (2 \, d + 9 \, e\right )} x^{9} - 138600 \, {\left (3 \, d + 8 \, e\right )} x^{8} - 207900 \, {\left (4 \, d + 7 \, e\right )} x^{7} - 232848 \, {\left (5 \, d + 6 \, e\right )} x^{6} - 194040 \, {\left (6 \, d + 5 \, e\right )} x^{5} - 118800 \, {\left (7 \, d + 4 \, e\right )} x^{4} - 51975 \, {\left (8 \, d + 3 \, e\right )} x^{3} - 15400 \, {\left (9 \, d + 2 \, e\right )} x^{2} - 2772 \, {\left (10 \, d + e\right )} x - 2520 \, d}{27720 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^12,x, algorithm="fricas")

[Out]

1/27720*(27720*e*x^11*log(x) - 27720*(d + 10*e)*x^10 - 69300*(2*d + 9*e)*x^9 - 138600*(3*d + 8*e)*x^8 - 207900
*(4*d + 7*e)*x^7 - 232848*(5*d + 6*e)*x^6 - 194040*(6*d + 5*e)*x^5 - 118800*(7*d + 4*e)*x^4 - 51975*(8*d + 3*e
)*x^3 - 15400*(9*d + 2*e)*x^2 - 2772*(10*d + e)*x - 2520*d)/x^11

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giac [A]  time = 0.18, size = 140, normalized size = 1.52 \begin {gather*} e \log \left ({\left | x \right |}\right ) - \frac {27720 \, {\left (d + 10 \, e\right )} x^{10} + 69300 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 138600 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 207900 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 232848 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 194040 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 118800 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 51975 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 15400 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 2772 \, {\left (10 \, d + e\right )} x + 2520 \, d}{27720 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^12,x, algorithm="giac")

[Out]

e*log(abs(x)) - 1/27720*(27720*(d + 10*e)*x^10 + 69300*(2*d + 9*e)*x^9 + 138600*(3*d + 8*e)*x^8 + 207900*(4*d
+ 7*e)*x^7 + 232848*(5*d + 6*e)*x^6 + 194040*(6*d + 5*e)*x^5 + 118800*(7*d + 4*e)*x^4 + 51975*(8*d + 3*e)*x^3
+ 15400*(9*d + 2*e)*x^2 + 2772*(10*d + e)*x + 2520*d)/x^11

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maple [A]  time = 0.05, size = 132, normalized size = 1.43 \begin {gather*} e \ln \relax (x )-\frac {d}{x}-\frac {10 e}{x}-\frac {5 d}{x^{2}}-\frac {45 e}{2 x^{2}}-\frac {15 d}{x^{3}}-\frac {40 e}{x^{3}}-\frac {30 d}{x^{4}}-\frac {105 e}{2 x^{4}}-\frac {42 d}{x^{5}}-\frac {252 e}{5 x^{5}}-\frac {42 d}{x^{6}}-\frac {35 e}{x^{6}}-\frac {30 d}{x^{7}}-\frac {120 e}{7 x^{7}}-\frac {15 d}{x^{8}}-\frac {45 e}{8 x^{8}}-\frac {5 d}{x^{9}}-\frac {10 e}{9 x^{9}}-\frac {d}{x^{10}}-\frac {e}{10 x^{10}}-\frac {d}{11 x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)*(x^2+2*x+1)^5/x^12,x)

[Out]

-42*d/x^5-252/5*e/x^5-30*d/x^4-105/2*e/x^4-15*d/x^3-40*e/x^3-15*d/x^8-45/8*e/x^8-d/x^10-1/10*e/x^10-5*d/x^2-45
/2*e/x^2-5*d/x^9-10/9*e/x^9-30*d/x^7-120/7*e/x^7-42*d/x^6-35*e/x^6-d/x-10*e/x-1/11*d/x^11+e*ln(x)

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maxima [A]  time = 0.58, size = 128, normalized size = 1.39 \begin {gather*} e \log \relax (x) - \frac {27720 \, {\left (d + 10 \, e\right )} x^{10} + 69300 \, {\left (2 \, d + 9 \, e\right )} x^{9} + 138600 \, {\left (3 \, d + 8 \, e\right )} x^{8} + 207900 \, {\left (4 \, d + 7 \, e\right )} x^{7} + 232848 \, {\left (5 \, d + 6 \, e\right )} x^{6} + 194040 \, {\left (6 \, d + 5 \, e\right )} x^{5} + 118800 \, {\left (7 \, d + 4 \, e\right )} x^{4} + 51975 \, {\left (8 \, d + 3 \, e\right )} x^{3} + 15400 \, {\left (9 \, d + 2 \, e\right )} x^{2} + 2772 \, {\left (10 \, d + e\right )} x + 2520 \, d}{27720 \, x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x^2+2*x+1)^5/x^12,x, algorithm="maxima")

[Out]

e*log(x) - 1/27720*(27720*(d + 10*e)*x^10 + 69300*(2*d + 9*e)*x^9 + 138600*(3*d + 8*e)*x^8 + 207900*(4*d + 7*e
)*x^7 + 232848*(5*d + 6*e)*x^6 + 194040*(6*d + 5*e)*x^5 + 118800*(7*d + 4*e)*x^4 + 51975*(8*d + 3*e)*x^3 + 154
00*(9*d + 2*e)*x^2 + 2772*(10*d + e)*x + 2520*d)/x^11

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mupad [B]  time = 1.10, size = 118, normalized size = 1.28 \begin {gather*} e\,\ln \relax (x)-\frac {\left (d+10\,e\right )\,x^{10}+\left (5\,d+\frac {45\,e}{2}\right )\,x^9+\left (15\,d+40\,e\right )\,x^8+\left (30\,d+\frac {105\,e}{2}\right )\,x^7+\left (42\,d+\frac {252\,e}{5}\right )\,x^6+\left (42\,d+35\,e\right )\,x^5+\left (30\,d+\frac {120\,e}{7}\right )\,x^4+\left (15\,d+\frac {45\,e}{8}\right )\,x^3+\left (5\,d+\frac {10\,e}{9}\right )\,x^2+\left (d+\frac {e}{10}\right )\,x+\frac {d}{11}}{x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((d + e*x)*(2*x + x^2 + 1)^5)/x^12,x)

[Out]

e*log(x) - (d/11 + x^2*(5*d + (10*e)/9) + x^9*(5*d + (45*e)/2) + x^8*(15*d + 40*e) + x^3*(15*d + (45*e)/8) + x
^5*(42*d + 35*e) + x^7*(30*d + (105*e)/2) + x^4*(30*d + (120*e)/7) + x^6*(42*d + (252*e)/5) + x*(d + e/10) + x
^10*(d + 10*e))/x^11

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sympy [A]  time = 9.30, size = 129, normalized size = 1.40 \begin {gather*} e \log {\relax (x )} + \frac {- 2520 d + x^{10} \left (- 27720 d - 277200 e\right ) + x^{9} \left (- 138600 d - 623700 e\right ) + x^{8} \left (- 415800 d - 1108800 e\right ) + x^{7} \left (- 831600 d - 1455300 e\right ) + x^{6} \left (- 1164240 d - 1397088 e\right ) + x^{5} \left (- 1164240 d - 970200 e\right ) + x^{4} \left (- 831600 d - 475200 e\right ) + x^{3} \left (- 415800 d - 155925 e\right ) + x^{2} \left (- 138600 d - 30800 e\right ) + x \left (- 27720 d - 2772 e\right )}{27720 x^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)*(x**2+2*x+1)**5/x**12,x)

[Out]

e*log(x) + (-2520*d + x**10*(-27720*d - 277200*e) + x**9*(-138600*d - 623700*e) + x**8*(-415800*d - 1108800*e)
+ x**7*(-831600*d - 1455300*e) + x**6*(-1164240*d - 1397088*e) + x**5*(-1164240*d - 970200*e) + x**4*(-831600
*d - 475200*e) + x**3*(-415800*d - 155925*e) + x**2*(-138600*d - 30800*e) + x*(-27720*d - 2772*e))/(27720*x**1
1)

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