3.3.2 \(\int \frac {1}{(2+3 x+x^2)^5} \, dx\) [202]

Optimal. Leaf size=87 \[ \frac {-3-2 x}{4 \left (2+3 x+x^2\right )^4}+\frac {7 (3+2 x)}{6 \left (2+3 x+x^2\right )^3}-\frac {35 (3+2 x)}{6 \left (2+3 x+x^2\right )^2}+\frac {35 (3+2 x)}{2+3 x+x^2}+70 \log (1+x)-70 \log (2+x) \]

[Out]

1/4*(-3-2*x)/(x^2+3*x+2)^4+7/6*(3+2*x)/(x^2+3*x+2)^3-35/6*(3+2*x)/(x^2+3*x+2)^2+35*(3+2*x)/(x^2+3*x+2)+70*ln(1
+x)-70*ln(2+x)

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Rubi [A]
time = 0.02, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 3, integrand size = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {628, 630, 31} \begin {gather*} \frac {35 (2 x+3)}{x^2+3 x+2}-\frac {35 (2 x+3)}{6 \left (x^2+3 x+2\right )^2}+\frac {7 (2 x+3)}{6 \left (x^2+3 x+2\right )^3}-\frac {2 x+3}{4 \left (x^2+3 x+2\right )^4}+70 \log (x+1)-70 \log (x+2) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x + x^2)^(-5),x]

[Out]

-1/4*(3 + 2*x)/(2 + 3*x + x^2)^4 + (7*(3 + 2*x))/(6*(2 + 3*x + x^2)^3) - (35*(3 + 2*x))/(6*(2 + 3*x + x^2)^2)
+ (35*(3 + 2*x))/(2 + 3*x + x^2) + 70*Log[1 + x] - 70*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 628

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

Rule 630

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

Rubi steps

\begin {align*} \int \frac {1}{\left (2+3 x+x^2\right )^5} \, dx &=-\frac {3+2 x}{4 \left (2+3 x+x^2\right )^4}-\frac {7}{2} \int \frac {1}{\left (2+3 x+x^2\right )^4} \, dx\\ &=-\frac {3+2 x}{4 \left (2+3 x+x^2\right )^4}+\frac {7 (3+2 x)}{6 \left (2+3 x+x^2\right )^3}+\frac {35}{3} \int \frac {1}{\left (2+3 x+x^2\right )^3} \, dx\\ &=-\frac {3+2 x}{4 \left (2+3 x+x^2\right )^4}+\frac {7 (3+2 x)}{6 \left (2+3 x+x^2\right )^3}-\frac {35 (3+2 x)}{6 \left (2+3 x+x^2\right )^2}-35 \int \frac {1}{\left (2+3 x+x^2\right )^2} \, dx\\ &=-\frac {3+2 x}{4 \left (2+3 x+x^2\right )^4}+\frac {7 (3+2 x)}{6 \left (2+3 x+x^2\right )^3}-\frac {35 (3+2 x)}{6 \left (2+3 x+x^2\right )^2}+\frac {35 (3+2 x)}{2+3 x+x^2}+70 \int \frac {1}{2+3 x+x^2} \, dx\\ &=-\frac {3+2 x}{4 \left (2+3 x+x^2\right )^4}+\frac {7 (3+2 x)}{6 \left (2+3 x+x^2\right )^3}-\frac {35 (3+2 x)}{6 \left (2+3 x+x^2\right )^2}+\frac {35 (3+2 x)}{2+3 x+x^2}+70 \int \frac {1}{1+x} \, dx-70 \int \frac {1}{2+x} \, dx\\ &=-\frac {3+2 x}{4 \left (2+3 x+x^2\right )^4}+\frac {7 (3+2 x)}{6 \left (2+3 x+x^2\right )^3}-\frac {35 (3+2 x)}{6 \left (2+3 x+x^2\right )^2}+\frac {35 (3+2 x)}{2+3 x+x^2}+70 \log (1+x)-70 \log (2+x)\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 87, normalized size = 1.00 \begin {gather*} \frac {-3-2 x}{4 \left (2+3 x+x^2\right )^4}+\frac {7 (3+2 x)}{6 \left (2+3 x+x^2\right )^3}-\frac {35 (3+2 x)}{6 \left (2+3 x+x^2\right )^2}+\frac {35 (3+2 x)}{2+3 x+x^2}+70 \log (1+x)-70 \log (2+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x + x^2)^(-5),x]

[Out]

(-3 - 2*x)/(4*(2 + 3*x + x^2)^4) + (7*(3 + 2*x))/(6*(2 + 3*x + x^2)^3) - (35*(3 + 2*x))/(6*(2 + 3*x + x^2)^2)
+ (35*(3 + 2*x))/(2 + 3*x + x^2) + 70*Log[1 + x] - 70*Log[2 + x]

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Maple [A]
time = 0.07, size = 70, normalized size = 0.80

method	result	size
norman	\(\frac {4098 x +70 x^{7}+735 x^{6}+\frac {9730}{3} x^{5}+9093 x^{2}+\frac {15575}{2} x^{4}+\frac {32942}{3} x^{3}+\frac {3105}{4}}{\left (x^{2}+3 x +2\right )^{4}}+70 \ln \left (1+x \right )-70 \ln \left (2+x \right )\)	\(60\)
risch	\(\frac {4098 x +70 x^{7}+735 x^{6}+\frac {9730}{3} x^{5}+9093 x^{2}+\frac {15575}{2} x^{4}+\frac {32942}{3} x^{3}+\frac {3105}{4}}{\left (x^{2}+3 x +2\right )^{4}}+70 \ln \left (1+x \right )-70 \ln \left (2+x \right )\)	\(60\)
default	\(\frac {1}{4 \left (2+x \right )^{4}}+\frac {5}{3 \left (2+x \right )^{3}}+\frac {15}{2 \left (2+x \right )^{2}}+\frac {35}{2+x}-70 \ln \left (2+x \right )-\frac {1}{4 \left (1+x \right )^{4}}+\frac {5}{3 \left (1+x \right )^{3}}-\frac {15}{2 \left (1+x \right )^{2}}+\frac {35}{1+x}+70 \ln \left (1+x \right )\)	\(70\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^2+3*x+2)^5,x,method=_RETURNVERBOSE)

[Out]

1/4/(2+x)^4+5/3/(2+x)^3+15/2/(2+x)^2+35/(2+x)-70*ln(2+x)-1/4/(1+x)^4+5/3/(1+x)^3-15/2/(1+x)^2+35/(1+x)+70*ln(1
+x)

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Maxima [A]
time = 3.23, size = 90, normalized size = 1.03 \begin {gather*} \frac {840 \, x^{7} + 8820 \, x^{6} + 38920 \, x^{5} + 93450 \, x^{4} + 131768 \, x^{3} + 109116 \, x^{2} + 49176 \, x + 9315}{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 )}} - 70 \, \log \left (x + 2\right ) + 70 \, \log \left (x + 1\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+3*x+2)^5,x, algorithm="maxima")

[Out]

1/12*(840*x^7 + 8820*x^6 + 38920*x^5 + 93450*x^4 + 131768*x^3 + 109116*x^2 + 49176*x + 9315)/(x^8 + 12*x^7 + 6
2*x^6 + 180*x^5 + 321*x^4 + 360*x^3 + 248*x^2 + 96*x + 16) - 70*log(x + 2) + 70*log(x + 1)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 165 vs. \(2 (81) = 162\).
time = 0.39, size = 165, normalized size = 1.90 \begin {gather*} \frac {840 \, x^{7} + 8820 \, x^{6} + 38920 \, x^{5} + 93450 \, x^{4} + 131768 \, x^{3} + 109116 \, x^{2} - 840 \, {\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 ) + 840 \, {\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 ) + 49176 \, x + 9315}{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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+3*x+2)^5,x, algorithm="fricas")

[Out]

1/12*(840*x^7 + 8820*x^6 + 38920*x^5 + 93450*x^4 + 131768*x^3 + 109116*x^2 - 840*(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 + 2) + 840*(x^8 + 12*x^7 + 62*x^6 + 180*x^5 + 321*x^4 + 3
60*x^3 + 248*x^2 + 96*x + 16)*log(x + 1) + 49176*x + 9315)/(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 = 0.08, size = 88, normalized size = 1.01 \begin {gather*} \frac {840 x^{7} + 8820 x^{6} + 38920 x^{5} + 93450 x^{4} + 131768 x^{3} + 109116 x^{2} + 49176 x + 9315}{12 x^{8} + 144 x^{7} + 744 x^{6} + 2160 x^{5} + 3852 x^{4} + 4320 x^{3} + 2976 x^{2} + 1152 x + 192} + 70 \log {\left (x + 1 \right )} - 70 \log {\left (x + 2 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x**2+3*x+2)**5,x)

[Out]

(840*x**7 + 8820*x**6 + 38920*x**5 + 93450*x**4 + 131768*x**3 + 109116*x**2 + 49176*x + 9315)/(12*x**8 + 144*x
**7 + 744*x**6 + 2160*x**5 + 3852*x**4 + 4320*x**3 + 2976*x**2 + 1152*x + 192) + 70*log(x + 1) - 70*log(x + 2)

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Giac [A]
time = 1.03, size = 62, normalized size = 0.71 \begin {gather*} \frac {840 \, x^{7} + 8820 \, x^{6} + 38920 \, x^{5} + 93450 \, x^{4} + 131768 \, x^{3} + 109116 \, x^{2} + 49176 \, x + 9315}{12 \, {\left (x^{2} + 3 \, x + 2\right )}^{4}} - 70 \, \log \left ({\left | x + 2 \right |}\right ) + 70 \, \log \left ({\left | x + 1 \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(x^2+3*x+2)^5,x, algorithm="giac")

[Out]

1/12*(840*x^7 + 8820*x^6 + 38920*x^5 + 93450*x^4 + 131768*x^3 + 109116*x^2 + 49176*x + 9315)/(x^2 + 3*x + 2)^4
- 70*log(abs(x + 2)) + 70*log(abs(x + 1))

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Mupad [B]
time = 0.09, size = 65, normalized size = 0.75 \begin {gather*} 70\,\ln \left (\frac {x+1}{x+2}\right )+70\,\left (x+\frac {3}{2}\right )\,\left (\frac {1}{x^2+3\,x+2}-\frac {1}{6\,{\left (x^2+3\,x+2\right )}^2}+\frac {1}{30\,{\left (x^2+3\,x+2\right )}^3}-\frac {1}{140\,{\left (x^2+3\,x+2\right )}^4}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(3*x + x^2 + 2)^5,x)

[Out]

70*log((x + 1)/(x + 2)) + 70*(x + 3/2)*(1/(3*x + x^2 + 2) - 1/(6*(3*x + x^2 + 2)^2) + 1/(30*(3*x + x^2 + 2)^3)
- 1/(140*(3*x + x^2 + 2)^4))

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