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

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

[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]

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Rubi [A]  time = 0.0737919, antiderivative size = 104, normalized size of antiderivative = 1., 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 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 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])

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 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 31

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

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*}

Mathematica [A]  time = 0.0222318, 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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Maple [A]  time = 0.008, size = 70, normalized size = 0.7 \begin{align*} -128\, \left ( 2+x \right ) ^{-4}-{\frac{256}{3\, \left ( 2+x \right ) ^{3}}}-384\, \left ( 2+x \right ) ^{-2}-1024\, \left ( 2+x \right ) ^{-1}+1472\,\ln \left ( 2+x \right ) +{\frac{1}{4\, \left ( 1+x \right ) ^{4}}}-{\frac{14}{3\, \left ( 1+x \right ) ^{3}}}+48\, \left ( 1+x \right ) ^{-2}-434\, \left ( 1+x \right ) ^{-1}-1471\,\ln \left ( 1+x \right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-128/(2+x)^4-256/3/(2+x)^3-384/(2+x)^2-1024/(2+x)+1472*ln(2+x)+1/4/(1+x)^4-14/3/(1+x)^3+48/(1+x)^2-434/(1+x)-1
471*ln(1+x)

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Maxima [A]  time = 0.935198, size = 122, normalized size = 1.17 \begin{align*} -\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 ) \end{align*}

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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Fricas [A]  time = 1.6261, size = 505, normalized size = 4.86 \begin{align*} -\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 )}} \end{align*}

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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Sympy [A]  time = 0.18975, size = 88, normalized size = 0.85 \begin{align*} - \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 )} \end{align*}

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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Giac [A]  time = 1.05595, size = 84, normalized size = 0.81 \begin{align*} -\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 ) \end{align*}

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))