3.204 \(\int \frac {x^9}{(2+3 x+x^2)^5} \, dx\)

Optimal. Leaf size=104 \[ -\frac {(1593 x+2206) x^2}{2 \left (x^2+3 x+2\right )}+\frac {(3 x+4) x^8}{4 \left (x^2+3 x+2\right )^4}-\frac {(81 x+110) x^6}{12 \left (x^2+3 x+2\right )^3}+\frac {(135 x+184) x^4}{2 \left (x^2+3 x+2\right )^2}+735 x-1471 \log (x+1)+1472 \log (x+2) \]

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Rubi [A]  time = 0.07, antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {738, 818, 773, 632, 31} \[ \frac {(3 x+4) x^8}{4 \left (x^2+3 x+2\right )^4}-\frac {(81 x+110) x^6}{12 \left (x^2+3 x+2\right )^3}+\frac {(135 x+184) x^4}{2 \left (x^2+3 x+2\right )^2}-\frac {(1593 x+2206) x^2}{2 \left (x^2+3 x+2\right )}+735 x-1471 \log (x+1)+1472 \log (x+2) \]

Antiderivative was successfully verified.

[In]

Int[x^9/(2 + 3*x + x^2)^5,x]

[Out]

735*x + (x^8*(4 + 3*x))/(4*(2 + 3*x + x^2)^4) - (x^6*(110 + 81*x))/(12*(2 + 3*x + x^2)^3) + (x^4*(184 + 135*x)
)/(2*(2 + 3*x + x^2)^2) - (x^2*(2206 + 1593*x))/(2*(2 + 3*x + x^2)) - 1471*Log[1 + x] + 1472*Log[2 + x]

Rule 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rule 632

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Dist[
(c*d - e*(b/2 - q/2))/q, Int[1/(b/2 - q/2 + c*x), x], x] - Dist[(c*d - e*(b/2 + q/2))/q, Int[1/(b/2 + q/2 + c*
x), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] && NiceSqrtQ[b^2 - 4*a*
c]

Rule 738

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]

Rule 773

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rubi steps

\begin {align*} \int \frac {x^9}{\left (2+3 x+x^2\right )^5} \, dx &=\frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {1}{4} \int \frac {x^7 (32+3 x)}{\left (2+3 x+x^2\right )^4} \, dx\\ &=\frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}-\frac {1}{12} \int \frac {(-660-72 x) x^5}{\left (2+3 x+x^2\right )^3} \, dx\\ &=\frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}+\frac {x^4 (184+135 x)}{2 \left (2+3 x+x^2\right )^2}-\frac {1}{24} \int \frac {x^3 (8832+1476 x)}{\left (2+3 x+x^2\right )^2} \, dx\\ &=\frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}+\frac {x^4 (184+135 x)}{2 \left (2+3 x+x^2\right )^2}-\frac {x^2 (2206+1593 x)}{2 \left (2+3 x+x^2\right )}-\frac {1}{24} \int \frac {(-52944-17640 x) x}{2+3 x+x^2} \, dx\\ &=735 x+\frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}+\frac {x^4 (184+135 x)}{2 \left (2+3 x+x^2\right )^2}-\frac {x^2 (2206+1593 x)}{2 \left (2+3 x+x^2\right )}-\frac {1}{24} \int \frac {35280-24 x}{2+3 x+x^2} \, dx\\ &=735 x+\frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}+\frac {x^4 (184+135 x)}{2 \left (2+3 x+x^2\right )^2}-\frac {x^2 (2206+1593 x)}{2 \left (2+3 x+x^2\right )}-1471 \int \frac {1}{1+x} \, dx+1472 \int \frac {1}{2+x} \, dx\\ &=735 x+\frac {x^8 (4+3 x)}{4 \left (2+3 x+x^2\right )^4}-\frac {x^6 (110+81 x)}{12 \left (2+3 x+x^2\right )^3}+\frac {x^4 (184+135 x)}{2 \left (2+3 x+x^2\right )^2}-\frac {x^2 (2206+1593 x)}{2 \left (2+3 x+x^2\right )}-1471 \log (1+x)+1472 \log (2+x)\\ \end {align*}

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Mathematica [A]  time = 0.03, size = 87, normalized size = 0.84 \[ \frac {3 (456 x+451)}{4 \left (x^2+3 x+2\right )^2}-\frac {2 (729 x+1114)}{x^2+3 x+2}+\frac {1998 x+415}{12 \left (x^2+3 x+2\right )^3}+\frac {513 x+514}{4 \left (x^2+3 x+2\right )^4}-1471 \log (x+1)+1472 \log (x+2) \]

Antiderivative was successfully verified.

[In]

Integrate[x^9/(2 + 3*x + x^2)^5,x]

[Out]

(514 + 513*x)/(4*(2 + 3*x + x^2)^4) + (415 + 1998*x)/(12*(2 + 3*x + x^2)^3) + (3*(451 + 456*x))/(4*(2 + 3*x +
x^2)^2) - (2*(1114 + 729*x))/(2 + 3*x + x^2) - 1471*Log[1 + x] + 1472*Log[2 + x]

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IntegrateAlgebraic [A]  time = 0.04, size = 62, normalized size = 0.60 \[ \frac {-17496 x^7-184200 x^6-813888 x^5-1955853 x^4-2759400 x^3-2286008 x^2-1030560 x-195280}{12 \left (x^2+3 x+2\right )^4}-1471 \log (x+1)+1472 \log (x+2) \]

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^9/(2 + 3*x + x^2)^5,x]

[Out]

(-195280 - 1030560*x - 2286008*x^2 - 2759400*x^3 - 1955853*x^4 - 813888*x^5 - 184200*x^6 - 17496*x^7)/(12*(2 +
3*x + x^2)^4) - 1471*Log[1 + x] + 1472*Log[2 + x]

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fricas [A]  time = 1.23, size = 165, normalized size = 1.59 \[ -\frac {17496 \, x^{7} + 184200 \, x^{6} + 813888 \, x^{5} + 1955853 \, x^{4} + 2759400 \, x^{3} + 2286008 \, x^{2} - 17664 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )} \log \left (x + 2\right ) + 17652 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )} \log \left (x + 1\right ) + 1030560 \, x + 195280}{12 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(x^2+3*x+2)^5,x, algorithm="fricas")

[Out]

-1/12*(17496*x^7 + 184200*x^6 + 813888*x^5 + 1955853*x^4 + 2759400*x^3 + 2286008*x^2 - 17664*(x^8 + 12*x^7 + 6
2*x^6 + 180*x^5 + 321*x^4 + 360*x^3 + 248*x^2 + 96*x + 16)*log(x + 2) + 17652*(x^8 + 12*x^7 + 62*x^6 + 180*x^5
+ 321*x^4 + 360*x^3 + 248*x^2 + 96*x + 16)*log(x + 1) + 1030560*x + 195280)/(x^8 + 12*x^7 + 62*x^6 + 180*x^5
+ 321*x^4 + 360*x^3 + 248*x^2 + 96*x + 16)

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giac [A]  time = 0.98, size = 62, normalized size = 0.60 \[ -\frac {17496 \, x^{7} + 184200 \, x^{6} + 813888 \, x^{5} + 1955853 \, x^{4} + 2759400 \, x^{3} + 2286008 \, x^{2} + 1030560 \, x + 195280}{12 \, {\left (x + 2\right )}^{4} {\left (x + 1\right )}^{4}} + 1472 \, \log \left ({\left | x + 2 \right |}\right ) - 1471 \, \log \left ({\left | x + 1 \right |}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(x^2+3*x+2)^5,x, algorithm="giac")

[Out]

-1/12*(17496*x^7 + 184200*x^6 + 813888*x^5 + 1955853*x^4 + 2759400*x^3 + 2286008*x^2 + 1030560*x + 195280)/((x
+ 2)^4*(x + 1)^4) + 1472*log(abs(x + 2)) - 1471*log(abs(x + 1))

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maple [A]  time = 0.29, size = 60, normalized size = 0.58

method	result	size
norman	\(\frac {-229950 x^{3}-85880 x -67824 x^{5}-15350 x^{6}-1458 x^{7}-\frac {651951}{4} x^{4}-\frac {571502}{3} x^{2}-\frac {48820}{3}}{\left (x^{2}+3 x +2\right )^{4}}-1471 \ln \left (1+x \right )+1472 \ln \left (2+x \right )\)	\(60\)
risch	\(\frac {-229950 x^{3}-85880 x -67824 x^{5}-15350 x^{6}-1458 x^{7}-\frac {651951}{4} x^{4}-\frac {571502}{3} x^{2}-\frac {48820}{3}}{\left (x^{2}+3 x +2\right )^{4}}-1471 \ln \left (1+x \right )+1472 \ln \left (2+x \right )\)	\(60\)
default	\(-\frac {128}{\left (2+x \right )^{4}}-\frac {256}{3 \left (2+x \right )^{3}}-\frac {384}{\left (2+x \right )^{2}}-\frac {1024}{2+x}+1472 \ln \left (2+x \right )+\frac {1}{4 \left (1+x \right )^{4}}-\frac {14}{3 \left (1+x \right )^{3}}+\frac {48}{\left (1+x \right )^{2}}-\frac {434}{1+x}-1471 \ln \left (1+x \right )\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(x^2+3*x+2)^5,x,method=_RETURNVERBOSE)

[Out]

(-229950*x^3-85880*x-67824*x^5-15350*x^6-1458*x^7-651951/4*x^4-571502/3*x^2-48820/3)/(x^2+3*x+2)^4-1471*ln(1+x
)+1472*ln(2+x)

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maxima [A]  time = 0.55, size = 90, normalized size = 0.87 \[ -\frac {17496 \, x^{7} + 184200 \, x^{6} + 813888 \, x^{5} + 1955853 \, x^{4} + 2759400 \, x^{3} + 2286008 \, x^{2} + 1030560 \, x + 195280}{12 \, {\left (x^{8} + 12 \, x^{7} + 62 \, x^{6} + 180 \, x^{5} + 321 \, x^{4} + 360 \, x^{3} + 248 \, x^{2} + 96 \, x + 16\right )}} + 1472 \, \log \left (x + 2\right ) - 1471 \, \log \left (x + 1\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^9/(x^2+3*x+2)^5,x, algorithm="maxima")

[Out]

-1/12*(17496*x^7 + 184200*x^6 + 813888*x^5 + 1955853*x^4 + 2759400*x^3 + 2286008*x^2 + 1030560*x + 195280)/(x^
8 + 12*x^7 + 62*x^6 + 180*x^5 + 321*x^4 + 360*x^3 + 248*x^2 + 96*x + 16) + 1472*log(x + 2) - 1471*log(x + 1)

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mupad [B]  time = 0.21, size = 90, normalized size = 0.87 \[ 1472\,\ln \left (x+2\right )-1471\,\ln \left (x+1\right )-\frac {1458\,x^7+15350\,x^6+67824\,x^5+\frac {651951\,x^4}{4}+229950\,x^3+\frac {571502\,x^2}{3}+85880\,x+\frac {48820}{3}}{x^8+12\,x^7+62\,x^6+180\,x^5+321\,x^4+360\,x^3+248\,x^2+96\,x+16} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^9/(3*x + x^2 + 2)^5,x)

[Out]

1472*log(x + 2) - 1471*log(x + 1) - (85880*x + (571502*x^2)/3 + 229950*x^3 + (651951*x^4)/4 + 67824*x^5 + 1535
0*x^6 + 1458*x^7 + 48820/3)/(96*x + 248*x^2 + 360*x^3 + 321*x^4 + 180*x^5 + 62*x^6 + 12*x^7 + x^8 + 16)

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sympy [A]  time = 0.21, size = 90, normalized size = 0.87 \[ \frac {- 17496 x^{7} - 184200 x^{6} - 813888 x^{5} - 1955853 x^{4} - 2759400 x^{3} - 2286008 x^{2} - 1030560 x - 195280}{12 x^{8} + 144 x^{7} + 744 x^{6} + 2160 x^{5} + 3852 x^{4} + 4320 x^{3} + 2976 x^{2} + 1152 x + 192} - 1471 \log {\left (x + 1 \right )} + 1472 \log {\left (x + 2 \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**9/(x**2+3*x+2)**5,x)

[Out]

(-17496*x**7 - 184200*x**6 - 813888*x**5 - 1955853*x**4 - 2759400*x**3 - 2286008*x**2 - 1030560*x - 195280)/(1
2*x**8 + 144*x**7 + 744*x**6 + 2160*x**5 + 3852*x**4 + 4320*x**3 + 2976*x**2 + 1152*x + 192) - 1471*log(x + 1)
+ 1472*log(x + 2)

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