Integrand size = 25, antiderivative size = 184 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {1}{8 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^2}+\frac {1}{1+2 x+\sqrt {-3-2 x+4 x^2}}-\frac {1-2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )}{4 \left (3+\left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )}+\frac {5}{16} \sqrt {3} \arctan \left (\frac {2 x+\sqrt {-3-2 x+4 x^2}}{\sqrt {3}}\right )-\frac {9}{16} \log \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )+\frac {9}{32} \log \left (x-4 x^2-2 x \sqrt {-3-2 x+4 x^2}\right ) \] Output:
1/8/(1+2*x+(4*x^2-2*x-3)^(1/2))^2+1/(1+2*x+(4*x^2-2*x-3)^(1/2))-(1-4*x-2*( 4*x^2-2*x-3)^(1/2))/(12+4*(2*x+(4*x^2-2*x-3)^(1/2))^2)+5/16*3^(1/2)*arctan (1/3*(2*x+(4*x^2-2*x-3)^(1/2))*3^(1/2))-9/16*ln(1+2*x+(4*x^2-2*x-3)^(1/2)) +9/32*ln(x-4*x^2-2*x*(4*x^2-2*x-3)^(1/2))
Time = 0.51 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.74 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {1}{288} \left (-\frac {18 (17+26 x) \sqrt {-3-2 x+4 x^2}}{(2+3 x)^2}+\frac {2 \left (72+143 x+51 x^2\right )}{x (2+3 x)^2}+90 \sqrt {3} \arctan \left (\frac {-2 x+\sqrt {-3-2 x+4 x^2}}{\sqrt {3}}\right )+81 \log \left (x \left (-1+4 x-2 \sqrt {-3-2 x+4 x^2}\right )\right )-162 \log \left (-5-6 x+3 \sqrt {-3-2 x+4 x^2}\right )\right ) \] Input:
Integrate[1/(x^2*(1 + 2*x + Sqrt[-3 - 2*x + 4*x^2])^3),x]
Output:
((-18*(17 + 26*x)*Sqrt[-3 - 2*x + 4*x^2])/(2 + 3*x)^2 + (2*(72 + 143*x + 5 1*x^2))/(x*(2 + 3*x)^2) + 90*Sqrt[3]*ArcTan[(-2*x + Sqrt[-3 - 2*x + 4*x^2] )/Sqrt[3]] + 81*Log[x*(-1 + 4*x - 2*Sqrt[-3 - 2*x + 4*x^2])] - 162*Log[-5 - 6*x + 3*Sqrt[-3 - 2*x + 4*x^2]])/288
Time = 0.54 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.86, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {7293, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (\sqrt {4 x^2-2 x-3}+2 x+1\right )^3} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {5 \sqrt {4 x^2-2 x-3}}{32 x}+\frac {15 \sqrt {4 x^2-2 x-3}}{32 (3 x+2)}+\frac {15 \sqrt {4 x^2-2 x-3}}{16 (3 x+2)^2}-\frac {\sqrt {4 x^2-2 x-3}}{8 (3 x+2)^3}-\frac {1}{8 x^2}+\frac {9}{32 x}-\frac {27}{32 (3 x+2)}+\frac {37}{48 (3 x+2)^2}-\frac {1}{24 (3 x+2)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {5}{32} \sqrt {3} \arctan \left (\frac {x+3}{\sqrt {3} \sqrt {4 x^2-2 x-3}}\right )-\frac {9}{32} \text {arctanh}\left (\frac {11 x+7}{\sqrt {4 x^2-2 x-3}}\right )-\frac {\sqrt {4 x^2-2 x-3} (11 x+7)}{16 (3 x+2)^2}-\frac {5 \sqrt {4 x^2-2 x-3}}{16 (3 x+2)}+\frac {1}{8 x}-\frac {37}{144 (3 x+2)}+\frac {1}{144 (3 x+2)^2}+\frac {9 \log (x)}{32}-\frac {9}{32} \log (3 x+2)\) |
Input:
Int[1/(x^2*(1 + 2*x + Sqrt[-3 - 2*x + 4*x^2])^3),x]
Output:
1/(8*x) + 1/(144*(2 + 3*x)^2) - 37/(144*(2 + 3*x)) - (5*Sqrt[-3 - 2*x + 4* x^2])/(16*(2 + 3*x)) - ((7 + 11*x)*Sqrt[-3 - 2*x + 4*x^2])/(16*(2 + 3*x)^2 ) - (5*Sqrt[3]*ArcTan[(3 + x)/(Sqrt[3]*Sqrt[-3 - 2*x + 4*x^2])])/32 - (9*A rcTanh[(7 + 11*x)/Sqrt[-3 - 2*x + 4*x^2]])/32 + (9*Log[x])/32 - (9*Log[2 + 3*x])/32
Time = 0.19 (sec) , antiderivative size = 242, normalized size of antiderivative = 1.32
| method | result | size |
| default | \(\frac {1}{144 \left (2+3 x \right )^{2}}+\frac {1}{8 x}+\frac {9 \ln \left (x \right )}{32}-\frac {37}{144 \left (2+3 x \right )}-\frac {9 \ln \left (2+3 x \right )}{32}+\frac {\left (4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}\right )^{\frac {3}{2}}}{48 \left (x +\frac {2}{3}\right )^{2}}-\frac {\left (4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}\right )^{\frac {3}{2}}}{4 \left (x +\frac {2}{3}\right )}-\frac {9 \sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}{32}+\frac {9 \,\operatorname {arctanh}\left (\frac {-21-33 x}{\sqrt {36 \left (x +\frac {2}{3}\right )^{2}-66 x -43}}\right )}{32}+\frac {\left (8 x -2\right ) \sqrt {4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}}}{8}-\frac {5 \sqrt {4 x^{2}-2 x -3}}{32}+\frac {5 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 x^{2}-2 x -3}\right ) \sqrt {4}}{128}+\frac {5 \sqrt {3}\, \arctan \left (\frac {\left (-6-2 x \right ) \sqrt {3}}{6 \sqrt {4 x^{2}-2 x -3}}\right )}{32}-\frac {5 \ln \left (\frac {\left (4 x -1\right ) \sqrt {4}}{4}+\sqrt {4 \left (x +\frac {2}{3}\right )^{2}-\frac {22 x}{3}-\frac {43}{9}}\right ) \sqrt {4}}{128}\) | \(242\) |
| trager | \(-\frac {\left (x -1\right ) \left (266 x^{2}+479 x +200\right )}{400 x \left (2+3 x \right )^{2}}-\frac {\left (17+26 x \right ) \sqrt {4 x^{2}-2 x -3}}{16 \left (2+3 x \right )^{2}}+\frac {9 \ln \left (-\frac {1875 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right )^{2} x -1875 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right )^{2}-4625 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right ) \sqrt {4 x^{2}-2 x -3}-725 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right ) x -1650 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right )+340 \sqrt {4 x^{2}-2 x -3}-682 x -363}{2+3 x}\right )}{16}-\frac {75 \ln \left (-\frac {1875 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right )^{2} x -1875 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right )^{2}-4625 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right ) \sqrt {4 x^{2}-2 x -3}-725 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right ) x -1650 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right )+340 \sqrt {4 x^{2}-2 x -3}-682 x -363}{2+3 x}\right ) \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right )}{16}+\frac {75 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right ) \ln \left (-\frac {1875 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right )^{2} x -1875 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right )^{2}+4625 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right ) \sqrt {4 x^{2}-2 x -3}+275 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right ) x +2100 \operatorname {RootOf}\left (1875 \textit {\_Z}^{2}-225 \textit {\_Z} +13\right )-215 \sqrt {4 x^{2}-2 x -3}-742 x -588}{2+3 x}\right )}{16}\) | \(417\) |
Input:
int(1/x^2/(1+2*x+(4*x^2-2*x-3)^(1/2))^3,x,method=_RETURNVERBOSE)
Output:
1/144/(2+3*x)^2+1/8/x+9/32*ln(x)-37/144/(2+3*x)-9/32*ln(2+3*x)+1/48/(x+2/3 )^2*(4*(x+2/3)^2-22/3*x-43/9)^(3/2)-1/4/(x+2/3)*(4*(x+2/3)^2-22/3*x-43/9)^ (3/2)-9/32*(36*(x+2/3)^2-66*x-43)^(1/2)+9/32*arctanh(9/2*(-14/3-22/3*x)/(3 6*(x+2/3)^2-66*x-43)^(1/2))+1/8*(8*x-2)*(4*(x+2/3)^2-22/3*x-43/9)^(1/2)-5/ 32*(4*x^2-2*x-3)^(1/2)+5/128*ln(1/4*(4*x-1)*4^(1/2)+(4*x^2-2*x-3)^(1/2))*4 ^(1/2)+5/32*3^(1/2)*arctan(1/6*(-6-2*x)*3^(1/2)/(4*x^2-2*x-3)^(1/2))-5/128 *ln(1/4*(4*x-1)*4^(1/2)+(4*(x+2/3)^2-22/3*x-43/9)^(1/2))*4^(1/2)
Time = 0.10 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.14 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=-\frac {936 \, x^{3} - 90 \, \sqrt {3} {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \arctan \left (-\frac {2}{3} \, \sqrt {3} x + \frac {1}{3} \, \sqrt {3} \sqrt {4 \, x^{2} - 2 \, x - 3}\right ) + 1146 \, x^{2} + 81 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \log \left (3 \, x + 2\right ) - 81 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \log \left (x\right ) - 81 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \log \left (-2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1\right ) + 81 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )} \log \left (-6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5\right ) + 18 \, {\left (26 \, x^{2} + 17 \, x\right )} \sqrt {4 \, x^{2} - 2 \, x - 3} + 130 \, x - 144}{288 \, {\left (9 \, x^{3} + 12 \, x^{2} + 4 \, x\right )}} \] Input:
integrate(1/x^2/(1+2*x+(4*x^2-2*x-3)^(1/2))^3,x, algorithm="fricas")
Output:
-1/288*(936*x^3 - 90*sqrt(3)*(9*x^3 + 12*x^2 + 4*x)*arctan(-2/3*sqrt(3)*x + 1/3*sqrt(3)*sqrt(4*x^2 - 2*x - 3)) + 1146*x^2 + 81*(9*x^3 + 12*x^2 + 4*x )*log(3*x + 2) - 81*(9*x^3 + 12*x^2 + 4*x)*log(x) - 81*(9*x^3 + 12*x^2 + 4 *x)*log(-2*x + sqrt(4*x^2 - 2*x - 3) - 1) + 81*(9*x^3 + 12*x^2 + 4*x)*log( -6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5) + 18*(26*x^2 + 17*x)*sqrt(4*x^2 - 2*x - 3) + 130*x - 144)/(9*x^3 + 12*x^2 + 4*x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\int \frac {1}{x^{2} \left (2 x + \sqrt {4 x^{2} - 2 x - 3} + 1\right )^{3}}\, dx \] Input:
integrate(1/x**2/(1+2*x+(4*x**2-2*x-3)**(1/2))**3,x)
Output:
Integral(1/(x**2*(2*x + sqrt(4*x**2 - 2*x - 3) + 1)**3), x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right )}^{3} x^{2}} \,d x } \] Input:
integrate(1/x^2/(1+2*x+(4*x^2-2*x-3)^(1/2))^3,x, algorithm="maxima")
Output:
integrate(1/((2*x + sqrt(4*x^2 - 2*x - 3) + 1)^3*x^2), x)
Time = 0.17 (sec) , antiderivative size = 218, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {5}{16} \, \sqrt {3} \arctan \left (-\frac {1}{3} \, \sqrt {3} {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}\right ) - \frac {435 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 1681 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 4312 \, x - 2156 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 919}{36 \, {\left (3 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 16 \, x - 8 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 5\right )}^{2}} + \frac {51 \, x^{2} + 143 \, x + 72}{144 \, {\left (3 \, x + 2\right )}^{2} x} - \frac {9}{32} \, \log \left ({\left | 3 \, x + 2 \right |}\right ) + \frac {9}{32} \, \log \left ({\left | x \right |}\right ) + \frac {9}{32} \, \log \left ({\left | -2 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} - 1 \right |}\right ) - \frac {9}{32} \, \log \left ({\left | -6 \, x + 3 \, \sqrt {4 \, x^{2} - 2 \, x - 3} - 5 \right |}\right ) \] Input:
integrate(1/x^2/(1+2*x+(4*x^2-2*x-3)^(1/2))^3,x, algorithm="giac")
Output:
5/16*sqrt(3)*arctan(-1/3*sqrt(3)*(2*x - sqrt(4*x^2 - 2*x - 3))) - 1/36*(43 5*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 1681*(2*x - sqrt(4*x^2 - 2*x - 3))^2 + 4312*x - 2156*sqrt(4*x^2 - 2*x - 3) + 919)/(3*(2*x - sqrt(4*x^2 - 2*x - 3 ))^2 + 16*x - 8*sqrt(4*x^2 - 2*x - 3) + 5)^2 + 1/144*(51*x^2 + 143*x + 72) /((3*x + 2)^2*x) - 9/32*log(abs(3*x + 2)) + 9/32*log(abs(x)) + 9/32*log(ab s(-2*x + sqrt(4*x^2 - 2*x - 3) - 1)) - 9/32*log(abs(-6*x + 3*sqrt(4*x^2 - 2*x - 3) - 5))
Timed out. \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\int \frac {1}{x^2\,{\left (2\,x+\sqrt {4\,x^2-2\,x-3}+1\right )}^3} \,d x \] Input:
int(1/(x^2*(2*x + (4*x^2 - 2*x - 3)^(1/2) + 1)^3),x)
Output:
int(1/(x^2*(2*x + (4*x^2 - 2*x - 3)^(1/2) + 1)^3), x)
Time = 0.29 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.98 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {-3-2 x+4 x^2}\right )^3} \, dx=\frac {33078060 \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {4 x^{2}-2 x -3}+2 x}{\sqrt {3}}\right ) x^{3}+44104080 \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {4 x^{2}-2 x -3}+2 x}{\sqrt {3}}\right ) x^{2}+14701360 \sqrt {3}\, \mathit {atan} \left (\frac {\sqrt {4 x^{2}-2 x -3}+2 x}{\sqrt {3}}\right ) x -19111768 \sqrt {4 x^{2}-2 x -3}\, x^{2}-12496156 \sqrt {4 x^{2}-2 x -3}\, x -29770254 \,\mathrm {log}\left (3 x +2\right ) x^{3}-39693672 \,\mathrm {log}\left (3 x +2\right ) x^{2}-13231224 \,\mathrm {log}\left (3 x +2\right ) x -29770254 \,\mathrm {log}\left (\frac {26 \sqrt {4 x^{2}-2 x -3}+52 x +26}{\sqrt {13}}\right ) x^{3}-39693672 \,\mathrm {log}\left (\frac {26 \sqrt {4 x^{2}-2 x -3}+52 x +26}{\sqrt {13}}\right ) x^{2}-13231224 \,\mathrm {log}\left (\frac {26 \sqrt {4 x^{2}-2 x -3}+52 x +26}{\sqrt {13}}\right ) x +29770254 \,\mathrm {log}\left (\frac {6 \sqrt {4 x^{2}-2 x -3}+12 x +10}{\sqrt {13}}\right ) x^{3}+39693672 \,\mathrm {log}\left (\frac {6 \sqrt {4 x^{2}-2 x -3}+12 x +10}{\sqrt {13}}\right ) x^{2}+13231224 \,\mathrm {log}\left (\frac {6 \sqrt {4 x^{2}-2 x -3}+12 x +10}{\sqrt {13}}\right ) x +29770254 \,\mathrm {log}\left (x \right ) x^{3}+39693672 \,\mathrm {log}\left (x \right ) x^{2}+13231224 \,\mathrm {log}\left (x \right ) x +27414617 x^{3}+40718208 x^{2}+23863688 x +5880544}{11761088 x \left (9 x^{2}+12 x +4\right )} \] Input:
int(1/x^2/(1+2*x+(4*x^2-2*x-3)^(1/2))^3,x)
Output:
(33078060*sqrt(3)*atan((sqrt(4*x**2 - 2*x - 3) + 2*x)/sqrt(3))*x**3 + 4410 4080*sqrt(3)*atan((sqrt(4*x**2 - 2*x - 3) + 2*x)/sqrt(3))*x**2 + 14701360* sqrt(3)*atan((sqrt(4*x**2 - 2*x - 3) + 2*x)/sqrt(3))*x - 19111768*sqrt(4*x **2 - 2*x - 3)*x**2 - 12496156*sqrt(4*x**2 - 2*x - 3)*x - 29770254*log(3*x + 2)*x**3 - 39693672*log(3*x + 2)*x**2 - 13231224*log(3*x + 2)*x - 297702 54*log((26*sqrt(4*x**2 - 2*x - 3) + 52*x + 26)/sqrt(13))*x**3 - 39693672*l og((26*sqrt(4*x**2 - 2*x - 3) + 52*x + 26)/sqrt(13))*x**2 - 13231224*log(( 26*sqrt(4*x**2 - 2*x - 3) + 52*x + 26)/sqrt(13))*x + 29770254*log((6*sqrt( 4*x**2 - 2*x - 3) + 12*x + 10)/sqrt(13))*x**3 + 39693672*log((6*sqrt(4*x** 2 - 2*x - 3) + 12*x + 10)/sqrt(13))*x**2 + 13231224*log((6*sqrt(4*x**2 - 2 *x - 3) + 12*x + 10)/sqrt(13))*x + 29770254*log(x)*x**3 + 39693672*log(x)* x**2 + 13231224*log(x)*x + 27414617*x**3 + 40718208*x**2 + 23863688*x + 58 80544)/(11761088*x*(9*x**2 + 12*x + 4))