Integrand size = 12, antiderivative size = 64 \[ \int \frac {1}{\left (2+4 x-3 x^2\right )^3} \, dx=-\frac {2-3 x}{40 \left (2+4 x-3 x^2\right )^2}-\frac {9 (2-3 x)}{800 \left (2+4 x-3 x^2\right )}-\frac {27 \text {arctanh}\left (\frac {2-3 x}{\sqrt {10}}\right )}{800 \sqrt {10}} \] Output:
-1/40*(2-3*x)/(-3*x^2+4*x+2)^2-9*(2-3*x)/(-2400*x^2+3200*x+1600)-27/8000*a rctanh(1/10*(2-3*x)*10^(1/2))*10^(1/2)
Time = 0.04 (sec) , antiderivative size = 70, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (2+4 x-3 x^2\right )^3} \, dx=\frac {-\frac {20 \left (76-42 x-162 x^2+81 x^3\right )}{\left (2+4 x-3 x^2\right )^2}-27 \sqrt {10} \log \left (2+\sqrt {10}-3 x\right )+27 \sqrt {10} \log \left (-2+\sqrt {10}+3 x\right )}{16000} \] Input:
Integrate[(2 + 4*x - 3*x^2)^(-3),x]
Output:
((-20*(76 - 42*x - 162*x^2 + 81*x^3))/(2 + 4*x - 3*x^2)^2 - 27*Sqrt[10]*Lo g[2 + Sqrt[10] - 3*x] + 27*Sqrt[10]*Log[-2 + Sqrt[10] + 3*x])/16000
Time = 0.39 (sec) , antiderivative size = 95, normalized size of antiderivative = 1.48, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1086, 1086, 1081, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {1}{\left (-3 x^2+4 x+2\right )^3} \, dx\) |
\(\Big \downarrow \) 1086 |
\(\displaystyle \frac {9}{40} \int \frac {1}{\left (-3 x^2+4 x+2\right )^2}dx-\frac {2-3 x}{40 \left (-3 x^2+4 x+2\right )^2}\) |
\(\Big \downarrow \) 1086 |
\(\displaystyle \frac {9}{40} \left (\frac {3}{20} \int \frac {1}{-3 x^2+4 x+2}dx-\frac {2-3 x}{20 \left (-3 x^2+4 x+2\right )}\right )-\frac {2-3 x}{40 \left (-3 x^2+4 x+2\right )^2}\) |
\(\Big \downarrow \) 1081 |
\(\displaystyle \frac {9}{40} \left (-\frac {9}{20} \int \left (\frac {1}{2 \sqrt {10} \left (-3 x-\sqrt {10}+2\right )}-\frac {1}{2 \sqrt {10} \left (-3 x+\sqrt {10}+2\right )}\right )dx-\frac {2-3 x}{20 \left (-3 x^2+4 x+2\right )}\right )-\frac {2-3 x}{40 \left (-3 x^2+4 x+2\right )^2}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {9}{40} \left (-\frac {2-3 x}{20 \left (-3 x^2+4 x+2\right )}-\frac {9}{20} \left (\frac {\log \left (-3 x+\sqrt {10}+2\right )}{6 \sqrt {10}}-\frac {\log \left (-3 x-\sqrt {10}+2\right )}{6 \sqrt {10}}\right )\right )-\frac {2-3 x}{40 \left (-3 x^2+4 x+2\right )^2}\) |
Input:
Int[(2 + 4*x - 3*x^2)^(-3),x]
Output:
-1/40*(2 - 3*x)/(2 + 4*x - 3*x^2)^2 + (9*(-1/20*(2 - 3*x)/(2 + 4*x - 3*x^2 ) - (9*(-1/6*Log[2 - Sqrt[10] - 3*x]/Sqrt[10] + Log[2 + Sqrt[10] - 3*x]/(6 *Sqrt[10])))/20))/40
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[c Int[ExpandIntegrand[1/((b/2 - q/2 + c*x)*(b/2 + q/2 + c*x)), x], x], x]] /; FreeQ[{a, b, c}, x] && NiceSqrtQ[b^2 - 4*a*c]
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] - Simp[2*c*((2*p + 3)/((p + 1)*(b^2 - 4*a*c))) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; Fre eQ[{a, b, c}, x] && ILtQ[p, -1]
Time = 0.58 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.88
method | result | size |
default | \(\frac {6 x -4}{80 \left (3 x^{2}-4 x -2\right )^{2}}-\frac {9 \left (6 x -4\right )}{1600 \left (3 x^{2}-4 x -2\right )}+\frac {27 \sqrt {10}\, \operatorname {arctanh}\left (\frac {\left (6 x -4\right ) \sqrt {10}}{20}\right )}{8000}\) | \(56\) |
risch | \(\frac {-\frac {81}{800} x^{3}+\frac {81}{400} x^{2}+\frac {21}{400} x -\frac {19}{200}}{\left (3 x^{2}-4 x -2\right )^{2}}+\frac {27 \sqrt {10}\, \ln \left (3 x -2+\sqrt {10}\right )}{16000}-\frac {27 \sqrt {10}\, \ln \left (3 x -2-\sqrt {10}\right )}{16000}\) | \(61\) |
Input:
int(1/(-3*x^2+4*x+2)^3,x,method=_RETURNVERBOSE)
Output:
1/80*(6*x-4)/(3*x^2-4*x-2)^2-9/1600*(6*x-4)/(3*x^2-4*x-2)+27/8000*10^(1/2) *arctanh(1/20*(6*x-4)*10^(1/2))
Time = 0.08 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.53 \[ \int \frac {1}{\left (2+4 x-3 x^2\right )^3} \, dx=-\frac {1620 \, x^{3} - 27 \, \sqrt {10} {\left (9 \, x^{4} - 24 \, x^{3} + 4 \, x^{2} + 16 \, x + 4\right )} \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {10} {\left (3 \, x - 2\right )} - 12 \, x + 14}{3 \, x^{2} - 4 \, x - 2}\right ) - 3240 \, x^{2} - 840 \, x + 1520}{16000 \, {\left (9 \, x^{4} - 24 \, x^{3} + 4 \, x^{2} + 16 \, x + 4\right )}} \] Input:
integrate(1/(-3*x^2+4*x+2)^3,x, algorithm="fricas")
Output:
-1/16000*(1620*x^3 - 27*sqrt(10)*(9*x^4 - 24*x^3 + 4*x^2 + 16*x + 4)*log(( 9*x^2 + 2*sqrt(10)*(3*x - 2) - 12*x + 14)/(3*x^2 - 4*x - 2)) - 3240*x^2 - 840*x + 1520)/(9*x^4 - 24*x^3 + 4*x^2 + 16*x + 4)
Time = 0.08 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\left (2+4 x-3 x^2\right )^3} \, dx=- \frac {81 x^{3} - 162 x^{2} - 42 x + 76}{7200 x^{4} - 19200 x^{3} + 3200 x^{2} + 12800 x + 3200} + \frac {27 \sqrt {10} \log {\left (x - \frac {2}{3} + \frac {\sqrt {10}}{3} \right )}}{16000} - \frac {27 \sqrt {10} \log {\left (x - \frac {\sqrt {10}}{3} - \frac {2}{3} \right )}}{16000} \] Input:
integrate(1/(-3*x**2+4*x+2)**3,x)
Output:
-(81*x**3 - 162*x**2 - 42*x + 76)/(7200*x**4 - 19200*x**3 + 3200*x**2 + 12 800*x + 3200) + 27*sqrt(10)*log(x - 2/3 + sqrt(10)/3)/16000 - 27*sqrt(10)* log(x - sqrt(10)/3 - 2/3)/16000
Time = 0.11 (sec) , antiderivative size = 67, normalized size of antiderivative = 1.05 \[ \int \frac {1}{\left (2+4 x-3 x^2\right )^3} \, dx=-\frac {27}{16000} \, \sqrt {10} \log \left (\frac {3 \, x - \sqrt {10} - 2}{3 \, x + \sqrt {10} - 2}\right ) - \frac {81 \, x^{3} - 162 \, x^{2} - 42 \, x + 76}{800 \, {\left (9 \, x^{4} - 24 \, x^{3} + 4 \, x^{2} + 16 \, x + 4\right )}} \] Input:
integrate(1/(-3*x^2+4*x+2)^3,x, algorithm="maxima")
Output:
-27/16000*sqrt(10)*log((3*x - sqrt(10) - 2)/(3*x + sqrt(10) - 2)) - 1/800* (81*x^3 - 162*x^2 - 42*x + 76)/(9*x^4 - 24*x^3 + 4*x^2 + 16*x + 4)
Time = 0.18 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.95 \[ \int \frac {1}{\left (2+4 x-3 x^2\right )^3} \, dx=-\frac {27}{16000} \, \sqrt {10} \log \left (\frac {{\left | 6 \, x - 2 \, \sqrt {10} - 4 \right |}}{{\left | 6 \, x + 2 \, \sqrt {10} - 4 \right |}}\right ) - \frac {81 \, x^{3} - 162 \, x^{2} - 42 \, x + 76}{800 \, {\left (3 \, x^{2} - 4 \, x - 2\right )}^{2}} \] Input:
integrate(1/(-3*x^2+4*x+2)^3,x, algorithm="giac")
Output:
-27/16000*sqrt(10)*log(abs(6*x - 2*sqrt(10) - 4)/abs(6*x + 2*sqrt(10) - 4) ) - 1/800*(81*x^3 - 162*x^2 - 42*x + 76)/(3*x^2 - 4*x - 2)^2
Time = 0.09 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.81 \[ \int \frac {1}{\left (2+4 x-3 x^2\right )^3} \, dx=\frac {27\,\sqrt {10}\,\mathrm {atanh}\left (\sqrt {10}\,\left (\frac {3\,x}{10}-\frac {1}{5}\right )\right )}{8000}+\frac {-\frac {9\,x^3}{800}+\frac {9\,x^2}{400}+\frac {7\,x}{1200}-\frac {19}{1800}}{x^4-\frac {8\,x^3}{3}+\frac {4\,x^2}{9}+\frac {16\,x}{9}+\frac {4}{9}} \] Input:
int(1/(4*x - 3*x^2 + 2)^3,x)
Output:
(27*10^(1/2)*atanh(10^(1/2)*((3*x)/10 - 1/5)))/8000 + ((7*x)/1200 + (9*x^2 )/400 - (9*x^3)/800 - 19/1800)/((16*x)/9 + (4*x^2)/9 - (8*x^3)/3 + x^4 + 4 /9)
Time = 0.21 (sec) , antiderivative size = 188, normalized size of antiderivative = 2.94 \[ \int \frac {1}{\left (2+4 x-3 x^2\right )^3} \, dx=\frac {-486 \sqrt {10}\, \mathrm {log}\left (-\sqrt {10}+3 x -2\right ) x^{4}+1296 \sqrt {10}\, \mathrm {log}\left (-\sqrt {10}+3 x -2\right ) x^{3}-216 \sqrt {10}\, \mathrm {log}\left (-\sqrt {10}+3 x -2\right ) x^{2}-864 \sqrt {10}\, \mathrm {log}\left (-\sqrt {10}+3 x -2\right ) x -216 \sqrt {10}\, \mathrm {log}\left (-\sqrt {10}+3 x -2\right )+486 \sqrt {10}\, \mathrm {log}\left (\sqrt {10}+3 x -2\right ) x^{4}-1296 \sqrt {10}\, \mathrm {log}\left (\sqrt {10}+3 x -2\right ) x^{3}+216 \sqrt {10}\, \mathrm {log}\left (\sqrt {10}+3 x -2\right ) x^{2}+864 \sqrt {10}\, \mathrm {log}\left (\sqrt {10}+3 x -2\right ) x +216 \sqrt {10}\, \mathrm {log}\left (\sqrt {10}+3 x -2\right )-1215 x^{4}+5940 x^{2}-480 x -3580}{288000 x^{4}-768000 x^{3}+128000 x^{2}+512000 x +128000} \] Input:
int(1/(-3*x^2+4*x+2)^3,x)
Output:
( - 486*sqrt(10)*log( - sqrt(10) + 3*x - 2)*x**4 + 1296*sqrt(10)*log( - sq rt(10) + 3*x - 2)*x**3 - 216*sqrt(10)*log( - sqrt(10) + 3*x - 2)*x**2 - 86 4*sqrt(10)*log( - sqrt(10) + 3*x - 2)*x - 216*sqrt(10)*log( - sqrt(10) + 3 *x - 2) + 486*sqrt(10)*log(sqrt(10) + 3*x - 2)*x**4 - 1296*sqrt(10)*log(sq rt(10) + 3*x - 2)*x**3 + 216*sqrt(10)*log(sqrt(10) + 3*x - 2)*x**2 + 864*s qrt(10)*log(sqrt(10) + 3*x - 2)*x + 216*sqrt(10)*log(sqrt(10) + 3*x - 2) - 1215*x**4 + 5940*x**2 - 480*x - 3580)/(32000*(9*x**4 - 24*x**3 + 4*x**2 + 16*x + 4))