Integrand size = 21, antiderivative size = 104 \[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=-\frac {9+73 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )}{10 \left (1+2 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )^2\right )}-\frac {13}{4} \arctan \left (\frac {1}{2} \left (1+5 \left (2 x+\sqrt {-3-2 x+4 x^2}\right )\right )\right ) \] Output:
-1/10*(9+146*x+73*(4*x^2-2*x-3)^(1/2))/(1+4*x+2*(4*x^2-2*x-3)^(1/2)+5*(2*x +(4*x^2-2*x-3)^(1/2))^2)-13/4*arctan(1/2+5*x+5/2*(4*x^2-2*x-3)^(1/2))
Time = 0.35 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.75 \[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {44+13 x+5 (8+7 x) \sqrt {-3-2 x+4 x^2}+65 \left (4+8 x+5 x^2\right ) \arctan \left (\frac {3}{2}+x-\frac {1}{2} \sqrt {-3-2 x+4 x^2}\right )}{20 \left (4+8 x+5 x^2\right )} \] Input:
Integrate[(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2])^(-2),x]
Output:
(44 + 13*x + 5*(8 + 7*x)*Sqrt[-3 - 2*x + 4*x^2] + 65*(4 + 8*x + 5*x^2)*Arc Tan[3/2 + x - Sqrt[-3 - 2*x + 4*x^2]/2])/(20*(4 + 8*x + 5*x^2))
Time = 0.45 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.23, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, 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.
\(\displaystyle \int \frac {1}{\left (\sqrt {4 x^2-2 x-3}+3 x+1\right )^2} \, dx\) |
\(\Big \downarrow \) 7293 |
\(\displaystyle \int \left (-\frac {6 \sqrt {4 x^2-2 x-3} x}{\left (5 x^2+8 x+4\right )^2}+\frac {13}{5 \left (5 x^2+8 x+4\right )}-\frac {2 (42 x+31)}{5 \left (5 x^2+8 x+4\right )^2}-\frac {2 \sqrt {4 x^2-2 x-3}}{\left (5 x^2+8 x+4\right )^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {13}{8} \arctan \left (\frac {7 x+8}{2 \sqrt {4 x^2-2 x-3}}\right )+\frac {13}{8} \arctan \left (\frac {5 x}{2}+2\right )+\frac {3 \sqrt {4 x^2-2 x-3} (x+1)}{5 x^2+8 x+4}+\frac {13 x+44}{20 \left (5 x^2+8 x+4\right )}-\frac {(5 x+4) \sqrt {4 x^2-2 x-3}}{4 \left (5 x^2+8 x+4\right )}\) |
Input:
Int[(1 + 3*x + Sqrt[-3 - 2*x + 4*x^2])^(-2),x]
Output:
(44 + 13*x)/(20*(4 + 8*x + 5*x^2)) + (3*(1 + x)*Sqrt[-3 - 2*x + 4*x^2])/(4 + 8*x + 5*x^2) - ((4 + 5*x)*Sqrt[-3 - 2*x + 4*x^2])/(4*(4 + 8*x + 5*x^2)) + (13*ArcTan[2 + (5*x)/2])/8 + (13*ArcTan[(8 + 7*x)/(2*Sqrt[-3 - 2*x + 4* x^2])])/8
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.21 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.07
method | result | size |
trager | \(-\frac {\left (15+11 x \right ) x}{4 \left (5 x^{2}+8 x +4\right )}+\frac {\left (8+7 x \right ) \sqrt {4 x^{2}-2 x -3}}{20 x^{2}+32 x +16}+\frac {13 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) \ln \left (-\frac {7 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +8 \operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right )-2 \sqrt {4 x^{2}-2 x -3}}{\operatorname {RootOf}\left (\textit {\_Z}^{2}+1\right ) x +2 x +2}\right )}{8}\) | \(111\) |
default | \(\text {Expression too large to display}\) | \(3159\) |
Input:
int(1/(1+3*x+(4*x^2-2*x-3)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
-1/4*(15+11*x)*x/(5*x^2+8*x+4)+1/4*(8+7*x)/(5*x^2+8*x+4)*(4*x^2-2*x-3)^(1/ 2)+13/8*RootOf(_Z^2+1)*ln(-(7*RootOf(_Z^2+1)*x+8*RootOf(_Z^2+1)-2*(4*x^2-2 *x-3)^(1/2))/(RootOf(_Z^2+1)*x+2*x+2))
Time = 0.07 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {28 \, x^{2} + 13 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \arctan \left (\frac {5}{2} \, x + 2\right ) - 13 \, {\left (5 \, x^{2} + 8 \, x + 4\right )} \arctan \left (-\frac {70 \, x^{2} - 5 \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (7 \, x + 8\right )} + 112 \, x + 56}{2 \, {\left (42 \, x + 31\right )}}\right ) + 2 \, \sqrt {4 \, x^{2} - 2 \, x - 3} {\left (7 \, x + 8\right )} + 50 \, x + 40}{8 \, {\left (5 \, x^{2} + 8 \, x + 4\right )}} \] Input:
integrate(1/(1+3*x+(4*x^2-2*x-3)^(1/2))^2,x, algorithm="fricas")
Output:
1/8*(28*x^2 + 13*(5*x^2 + 8*x + 4)*arctan(5/2*x + 2) - 13*(5*x^2 + 8*x + 4 )*arctan(-1/2*(70*x^2 - 5*sqrt(4*x^2 - 2*x - 3)*(7*x + 8) + 112*x + 56)/(4 2*x + 31)) + 2*sqrt(4*x^2 - 2*x - 3)*(7*x + 8) + 50*x + 40)/(5*x^2 + 8*x + 4)
\[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\int \frac {1}{\left (3 x + \sqrt {4 x^{2} - 2 x - 3} + 1\right )^{2}}\, dx \] Input:
integrate(1/(1+3*x+(4*x**2-2*x-3)**(1/2))**2,x)
Output:
Integral((3*x + sqrt(4*x**2 - 2*x - 3) + 1)**(-2), x)
\[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\int { \frac {1}{{\left (3 \, x + \sqrt {4 \, x^{2} - 2 \, x - 3} + 1\right )}^{2}} \,d x } \] Input:
integrate(1/(1+3*x+(4*x^2-2*x-3)^(1/2))^2,x, algorithm="maxima")
Output:
integrate((3*x + sqrt(4*x^2 - 2*x - 3) + 1)^(-2), x)
Leaf count of result is larger than twice the leaf count of optimal. 224 vs. \(2 (85) = 170\).
Time = 0.12 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.15 \[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {99 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 786 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 1458 \, x - 729 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 166}{10 \, {\left (5 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{4} + 32 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{3} + 78 \, {\left (2 \, x - \sqrt {4 \, x^{2} - 2 \, x - 3}\right )}^{2} + 64 \, x - 32 \, \sqrt {4 \, x^{2} - 2 \, x - 3} + 13\right )}} + \frac {13 \, x + 44}{20 \, {\left (5 \, x^{2} + 8 \, x + 4\right )}} + \frac {13}{8} \, \arctan \left (\frac {5}{2} \, x + 2\right ) - \frac {13}{8} \, \arctan \left (-x + \frac {1}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {3}{2}\right ) - \frac {13}{8} \, \arctan \left (-5 \, x + \frac {5}{2} \, \sqrt {4 \, x^{2} - 2 \, x - 3} - \frac {1}{2}\right ) \] Input:
integrate(1/(1+3*x+(4*x^2-2*x-3)^(1/2))^2,x, algorithm="giac")
Output:
1/10*(99*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 786*(2*x - sqrt(4*x^2 - 2*x - 3 ))^2 + 1458*x - 729*sqrt(4*x^2 - 2*x - 3) + 166)/(5*(2*x - sqrt(4*x^2 - 2* x - 3))^4 + 32*(2*x - sqrt(4*x^2 - 2*x - 3))^3 + 78*(2*x - sqrt(4*x^2 - 2* x - 3))^2 + 64*x - 32*sqrt(4*x^2 - 2*x - 3) + 13) + 1/20*(13*x + 44)/(5*x^ 2 + 8*x + 4) + 13/8*arctan(5/2*x + 2) - 13/8*arctan(-x + 1/2*sqrt(4*x^2 - 2*x - 3) - 3/2) - 13/8*arctan(-5*x + 5/2*sqrt(4*x^2 - 2*x - 3) - 1/2)
Timed out. \[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\int \frac {1}{{\left (3\,x+\sqrt {4\,x^2-2\,x-3}+1\right )}^2} \,d x \] Input:
int(1/(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1)^2,x)
Output:
int(1/(3*x + (4*x^2 - 2*x - 3)^(1/2) + 1)^2, x)
Time = 0.35 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.98 \[ \int \frac {1}{\left (1+3 x+\sqrt {-3-2 x+4 x^2}\right )^2} \, dx=\frac {-3818100 \mathit {atan} \left (\frac {\sqrt {4 x^{2}-2 x -3}}{2}+x +\frac {3}{2}\right ) x^{2}-6108960 \mathit {atan} \left (\frac {\sqrt {4 x^{2}-2 x -3}}{2}+x +\frac {3}{2}\right ) x -3054480 \mathit {atan} \left (\frac {\sqrt {4 x^{2}-2 x -3}}{2}+x +\frac {3}{2}\right )-3818100 \mathit {atan} \left (\frac {5 \sqrt {4 x^{2}-2 x -3}}{2}+5 x +\frac {1}{2}\right ) x^{2}-6108960 \mathit {atan} \left (\frac {5 \sqrt {4 x^{2}-2 x -3}}{2}+5 x +\frac {1}{2}\right ) x -3054480 \mathit {atan} \left (\frac {5 \sqrt {4 x^{2}-2 x -3}}{2}+5 x +\frac {1}{2}\right )+3818100 \mathit {atan} \left (\frac {5 x}{2}+2\right ) x^{2}+6108960 \mathit {atan} \left (\frac {5 x}{2}+2\right ) x +3054480 \mathit {atan} \left (\frac {5 x}{2}+2\right )+822360 \sqrt {4 x^{2}-2 x -3}\, x +939840 \sqrt {4 x^{2}-2 x -3}-412945 x^{2}-355264 x +703468}{2349600 x^{2}+3759360 x +1879680} \] Input:
int(1/(1+3*x+(4*x^2-2*x-3)^(1/2))^2,x)
Output:
( - 3818100*atan((sqrt(4*x**2 - 2*x - 3) + 2*x + 3)/2)*x**2 - 6108960*atan ((sqrt(4*x**2 - 2*x - 3) + 2*x + 3)/2)*x - 3054480*atan((sqrt(4*x**2 - 2*x - 3) + 2*x + 3)/2) - 3818100*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2 )*x**2 - 6108960*atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2)*x - 3054480 *atan((5*sqrt(4*x**2 - 2*x - 3) + 10*x + 1)/2) + 3818100*atan((5*x + 4)/2) *x**2 + 6108960*atan((5*x + 4)/2)*x + 3054480*atan((5*x + 4)/2) + 822360*s qrt(4*x**2 - 2*x - 3)*x + 939840*sqrt(4*x**2 - 2*x - 3) - 412945*x**2 - 35 5264*x + 703468)/(469920*(5*x**2 + 8*x + 4))