Integrand size = 25, antiderivative size = 271 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=\frac {\sqrt {2}-\sqrt {2+3 x+5 x^2}}{2 \left (2-\sqrt {2}\right ) x}-\frac {31 \left (1-\sqrt {2}\right ) x}{8 \left (4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}\right )}+\frac {2 \arctan \left (\frac {3-4 \sqrt {2}-\frac {2 \left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{\sqrt {3}}\right )}{\sqrt {3}}-\frac {1}{4} \left (12-7 \sqrt {2}\right ) \log \left (\frac {4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}}{x}\right )+3 \log \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right ) \] Output:
1/2*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/(2-2^(1/2))/x-31*(1-2^(1/2))*x/(32+24*x- 16*2^(1/2)*(5*x^2+3*x+2)^(1/2))+2/3*arctan(1/3*(3-4*2^(1/2)-2*(1-2^(1/2))* (2^(1/2)-(5*x^2+3*x+2)^(1/2))/x)*3^(1/2))*3^(1/2)-1/4*(12-7*2^(1/2))*ln((4 +3*x-2*2^(1/2)*(5*x^2+3*x+2)^(1/2))/x)+3*ln(1-5*2^(1/2)-(3-4*2^(1/2))*(2^( 1/2)-(5*x^2+3*x+2)^(1/2))/x+(1-2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))^2/x^ 2)
Time = 2.68 (sec) , antiderivative size = 293, normalized size of antiderivative = 1.08 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=\frac {1}{x}-\frac {\sqrt {2+3 x+5 x^2}}{x}+\frac {2 \arctan \left (\frac {\sqrt {29-12 \sqrt {5}}+2 \left (-5+2 \sqrt {5}\right ) x+2 \left (-2+\sqrt {5}\right ) \sqrt {2+3 x+5 x^2}}{\sqrt {3}}\right )}{\sqrt {3}}+\frac {7 \text {arctanh}\left (\frac {\sqrt {5} x-\sqrt {2+3 x+5 x^2}}{\sqrt {2}}\right )}{\sqrt {2}}+6 \text {arctanh}\left (\frac {-20974964966+9380288779 \sqrt {5}-51657237985 x+23101819444 \sqrt {5} x-74388271700 x^2+33267446160 \sqrt {5} x^2+5 \sqrt {5} (1173630259+2975530868 x) \sqrt {2+3 x+5 x^2}-4 (3280396399+8316861540 x) \sqrt {2+3 x+5 x^2}}{-20974964966+9380288779 \sqrt {5}-51657237979 x+23101819444 \sqrt {5} x-74388271680 x^2+33267446160 \sqrt {5} x^2-4 (3280396399+8316861540 x) \sqrt {2+3 x+5 x^2}+\sqrt {5} (5868151295+14877654336 x) \sqrt {2+3 x+5 x^2}}\right ) \] Input:
Integrate[1/(x^2*(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])),x]
Output:
x^(-1) - Sqrt[2 + 3*x + 5*x^2]/x + (2*ArcTan[(Sqrt[29 - 12*Sqrt[5]] + 2*(- 5 + 2*Sqrt[5])*x + 2*(-2 + Sqrt[5])*Sqrt[2 + 3*x + 5*x^2])/Sqrt[3]])/Sqrt[ 3] + (7*ArcTanh[(Sqrt[5]*x - Sqrt[2 + 3*x + 5*x^2])/Sqrt[2]])/Sqrt[2] + 6* ArcTanh[(-20974964966 + 9380288779*Sqrt[5] - 51657237985*x + 23101819444*S qrt[5]*x - 74388271700*x^2 + 33267446160*Sqrt[5]*x^2 + 5*Sqrt[5]*(11736302 59 + 2975530868*x)*Sqrt[2 + 3*x + 5*x^2] - 4*(3280396399 + 8316861540*x)*S qrt[2 + 3*x + 5*x^2])/(-20974964966 + 9380288779*Sqrt[5] - 51657237979*x + 23101819444*Sqrt[5]*x - 74388271680*x^2 + 33267446160*Sqrt[5]*x^2 - 4*(32 80396399 + 8316861540*x)*Sqrt[2 + 3*x + 5*x^2] + Sqrt[5]*(5868151295 + 148 77654336*x)*Sqrt[2 + 3*x + 5*x^2])]
Time = 0.60 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.69, 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.
| \(\displaystyle \int \frac {1}{x^2 \left (\sqrt {5 x^2+3 x+2}+2 x+1\right )} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {\sqrt {5 x^2+3 x+2} x}{x^2-x+1}+\frac {3 x-2}{x^2-x+1}+\frac {\sqrt {5 x^2+3 x+2}}{x}+\frac {\sqrt {5 x^2+3 x+2}}{x^2}-\frac {1}{x^2}-\frac {3}{x}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )}{\sqrt {3}}+\frac {\arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{\sqrt {3}}+3 \text {arctanh}\left (\frac {2 x+1}{\sqrt {5 x^2+3 x+2}}\right )-\sqrt {2} \text {arctanh}\left (\frac {3 x+4}{2 \sqrt {2} \sqrt {5 x^2+3 x+2}}\right )-\frac {3 \text {arctanh}\left (\frac {3 x+4}{2 \sqrt {2} \sqrt {5 x^2+3 x+2}}\right )}{2 \sqrt {2}}-\frac {\sqrt {5 x^2+3 x+2}}{x}+\frac {3}{2} \log \left (x^2-x+1\right )+\frac {1}{x}-3 \log (x)\) |
Input:
Int[1/(x^2*(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])),x]
Output:
x^(-1) - Sqrt[2 + 3*x + 5*x^2]/x + ArcTan[(1 - 2*x)/Sqrt[3]]/Sqrt[3] - Arc Tan[(5 - 4*x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])]/Sqrt[3] + 3*ArcTanh[(1 + 2* x)/Sqrt[2 + 3*x + 5*x^2]] - (3*ArcTanh[(4 + 3*x)/(2*Sqrt[2]*Sqrt[2 + 3*x + 5*x^2])])/(2*Sqrt[2]) - Sqrt[2]*ArcTanh[(4 + 3*x)/(2*Sqrt[2]*Sqrt[2 + 3*x + 5*x^2])] - 3*Log[x] + (3*Log[1 - x + x^2])/2
Leaf count of result is larger than twice the leaf count of optimal. \(580\) vs. \(2(222)=444\).
Time = 0.62 (sec) , antiderivative size = 581, normalized size of antiderivative = 2.14
| method | result | size |
| default | \(-\frac {\left (5 x^{2}+3 x +2\right )^{\frac {3}{2}}}{2 x}+\frac {3 \sqrt {5 x^{2}+3 x +2}}{4}-\frac {7 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\left (4+3 x \right ) \sqrt {2}}{4 \sqrt {5 x^{2}+3 x +2}}\right )}{4}+\frac {\left (10 x +3\right ) \sqrt {5 x^{2}+3 x +2}}{4}-\frac {5 \sqrt {7}\, \sqrt {16}\, \sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}\, \left (2 \sqrt {3}\, \arctan \left (\frac {4 \sqrt {3}\, \sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}\, \left (-\frac {5}{4}+x \right )}{3 \left (\frac {4 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+31\right ) \left (-\frac {1}{2}-x \right )}\right )-3 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}}{14}\right )\right )}{294 \sqrt {\frac {\frac {4 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+31}{\left (\frac {-\frac {5}{4}+x}{-\frac {1}{2}-x}+1\right )^{2}}}\, \left (\frac {-\frac {5}{4}+x}{-\frac {1}{2}-x}+1\right )}+\frac {\sqrt {7}\, \sqrt {16}\, \sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}\, \left (\sqrt {3}\, \arctan \left (\frac {4 \sqrt {3}\, \sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}\, \left (-\frac {5}{4}+x \right )}{3 \left (\frac {4 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+31\right ) \left (-\frac {1}{2}-x \right )}\right )+9 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}}{14}\right )\right )}{196 \sqrt {\frac {\frac {4 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+31}{\left (\frac {-\frac {5}{4}+x}{-\frac {1}{2}-x}+1\right )^{2}}}\, \left (\frac {-\frac {5}{4}+x}{-\frac {1}{2}-x}+1\right )}+\frac {\sqrt {7}\, \sqrt {16}\, \sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}\, \left (5 \sqrt {3}\, \arctan \left (\frac {4 \sqrt {3}\, \sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}\, \left (-\frac {5}{4}+x \right )}{3 \left (\frac {4 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+31\right ) \left (-\frac {1}{2}-x \right )}\right )+3 \,\operatorname {arctanh}\left (\frac {\sqrt {\frac {28 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+217}}{14}\right )\right )}{294 \sqrt {\frac {\frac {4 \left (-\frac {5}{4}+x \right )^{2}}{\left (-\frac {1}{2}-x \right )^{2}}+31}{\left (\frac {-\frac {5}{4}+x}{-\frac {1}{2}-x}+1\right )^{2}}}\, \left (\frac {-\frac {5}{4}+x}{-\frac {1}{2}-x}+1\right )}+\frac {3 \ln \left (x^{2}-x +1\right )}{2}-\frac {\sqrt {3}\, \arctan \left (\frac {\left (2 x -1\right ) \sqrt {3}}{3}\right )}{3}+\frac {1}{x}-3 \ln \left (x \right )\) | \(581\) |
| trager | \(\text {Expression too large to display}\) | \(1636\) |
Input:
int(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2)),x,method=_RETURNVERBOSE)
Output:
-1/2/x*(5*x^2+3*x+2)^(3/2)+3/4*(5*x^2+3*x+2)^(1/2)-7/4*2^(1/2)*arctanh(1/4 *(4+3*x)*2^(1/2)/(5*x^2+3*x+2)^(1/2))+1/4*(10*x+3)*(5*x^2+3*x+2)^(1/2)-5/2 94*7^(1/2)*16^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(1/2)*(2*3^(1/2)*arctan (4/3*3^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(1/2)/(4*(-5/4+x)^2/(-1/2-x)^2 +31)*(-5/4+x)/(-1/2-x))-3*arctanh(1/14*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(1/2 )))/((4*(-5/4+x)^2/(-1/2-x)^2+31)/((-5/4+x)/(-1/2-x)+1)^2)^(1/2)/((-5/4+x) /(-1/2-x)+1)+1/196*7^(1/2)*16^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(1/2)*( 3^(1/2)*arctan(4/3*3^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+217)^(1/2)/(4*(-5/4+x )^2/(-1/2-x)^2+31)*(-5/4+x)/(-1/2-x))+9*arctanh(1/14*(28*(-5/4+x)^2/(-1/2- x)^2+217)^(1/2)))/((4*(-5/4+x)^2/(-1/2-x)^2+31)/((-5/4+x)/(-1/2-x)+1)^2)^( 1/2)/((-5/4+x)/(-1/2-x)+1)+1/294*7^(1/2)*16^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^ 2+217)^(1/2)*(5*3^(1/2)*arctan(4/3*3^(1/2)*(28*(-5/4+x)^2/(-1/2-x)^2+217)^ (1/2)/(4*(-5/4+x)^2/(-1/2-x)^2+31)*(-5/4+x)/(-1/2-x))+3*arctanh(1/14*(28*( -5/4+x)^2/(-1/2-x)^2+217)^(1/2)))/((4*(-5/4+x)^2/(-1/2-x)^2+31)/((-5/4+x)/ (-1/2-x)+1)^2)^(1/2)/((-5/4+x)/(-1/2-x)+1)+3/2*ln(x^2-x+1)-1/3*3^(1/2)*arc tan(1/3*(2*x-1)*3^(1/2))+1/x-3*ln(x)
Time = 0.08 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.05 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=-\frac {8 \, \sqrt {3} x \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 4 \, \sqrt {3} x \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} + 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + 4 \, \sqrt {3} x \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} - 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) - 21 \, \sqrt {2} x \log \left (\frac {4 \, \sqrt {2} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (3 \, x + 4\right )} - 49 \, x^{2} - 48 \, x - 32}{x^{2}}\right ) - 36 \, x \log \left (x^{2} - x + 1\right ) + 72 \, x \log \left (x\right ) - 18 \, x \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) + 18 \, x \log \left (\frac {9 \, x^{2} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) + 24 \, \sqrt {5 \, x^{2} + 3 \, x + 2} - 24}{24 \, x} \] Input:
integrate(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2)),x, algorithm="fricas")
Output:
-1/24*(8*sqrt(3)*x*arctan(1/3*sqrt(3)*(2*x - 1)) + 4*sqrt(3)*x*arctan(1/3* (4*sqrt(3)*sqrt(5*x^2 + 3*x + 2)*(4*x - 5) + 31*sqrt(3)*(x^2 - 2*x))/(11*x ^2 - 12*x - 8)) + 4*sqrt(3)*x*arctan(1/3*(4*sqrt(3)*sqrt(5*x^2 + 3*x + 2)* (4*x - 5) - 31*sqrt(3)*(x^2 - 2*x))/(11*x^2 - 12*x - 8)) - 21*sqrt(2)*x*lo g((4*sqrt(2)*sqrt(5*x^2 + 3*x + 2)*(3*x + 4) - 49*x^2 - 48*x - 32)/x^2) - 36*x*log(x^2 - x + 1) + 72*x*log(x) - 18*x*log((9*x^2 + 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7*x + 3)/x^2) + 18*x*log((9*x^2 - 2*sqrt(5*x^2 + 3*x + 2 )*(2*x + 1) + 7*x + 3)/x^2) + 24*sqrt(5*x^2 + 3*x + 2) - 24)/x
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=\int \frac {1}{x^{2} \cdot \left (2 x + \sqrt {5 x^{2} + 3 x + 2} + 1\right )}\, dx \] Input:
integrate(1/x**2/(1+2*x+(5*x**2+3*x+2)**(1/2)),x)
Output:
Integral(1/(x**2*(2*x + sqrt(5*x**2 + 3*x + 2) + 1)), x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=\int { \frac {1}{{\left (2 \, x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 1\right )} x^{2}} \,d x } \] Input:
integrate(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2)),x, algorithm="maxima")
Output:
integrate(1/((2*x + sqrt(5*x^2 + 3*x + 2) + 1)*x^2), x)
Time = 0.18 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.42 \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=-\frac {1}{3} \, \sqrt {3} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + \frac {7}{4} \, \sqrt {2} \log \left (-\frac {{\left | -2 \, \sqrt {5} x - 2 \, \sqrt {2} + 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} \right |}}{2 \, {\left (\sqrt {5} x - \sqrt {2} - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}}\right ) + \frac {{\left (\sqrt {5} + 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} - 4}{\sqrt {15} + 2 \, \sqrt {3}}\right )}{\sqrt {15} + 2 \, \sqrt {3}} - \frac {{\left (\sqrt {5} - 2\right )} \arctan \left (-\frac {2 \, \sqrt {5} x - \sqrt {5} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} + 4}{\sqrt {15} - 2 \, \sqrt {3}}\right )}{\sqrt {15} - 2 \, \sqrt {3}} + \frac {3 \, \sqrt {5} x + 4 \, \sqrt {5} - 3 \, \sqrt {5 \, x^{2} + 3 \, x + 2}}{{\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - 2} + \frac {1}{x} + \frac {3}{2} \, \log \left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} {\left (\sqrt {5} + 4\right )} + 5 \, \sqrt {5} + 12\right ) - \frac {3}{2} \, \log \left ({\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )}^{2} - {\left (\sqrt {5} x - \sqrt {5 \, x^{2} + 3 \, x + 2}\right )} {\left (\sqrt {5} - 4\right )} - 5 \, \sqrt {5} + 12\right ) + \frac {3}{2} \, \log \left (x^{2} - x + 1\right ) - 3 \, \log \left ({\left | x \right |}\right ) \] Input:
integrate(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2)),x, algorithm="giac")
Output:
-1/3*sqrt(3)*arctan(1/3*sqrt(3)*(2*x - 1)) + 7/4*sqrt(2)*log(-1/2*abs(-2*s qrt(5)*x - 2*sqrt(2) + 2*sqrt(5*x^2 + 3*x + 2))/(sqrt(5)*x - sqrt(2) - sqr t(5*x^2 + 3*x + 2))) + (sqrt(5) + 2)*arctan(-(2*sqrt(5)*x - sqrt(5) - 2*sq rt(5*x^2 + 3*x + 2) - 4)/(sqrt(15) + 2*sqrt(3)))/(sqrt(15) + 2*sqrt(3)) - (sqrt(5) - 2)*arctan(-(2*sqrt(5)*x - sqrt(5) - 2*sqrt(5*x^2 + 3*x + 2) + 4 )/(sqrt(15) - 2*sqrt(3)))/(sqrt(15) - 2*sqrt(3)) + (3*sqrt(5)*x + 4*sqrt(5 ) - 3*sqrt(5*x^2 + 3*x + 2))/((sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 - 2) + 1/x + 3/2*log((sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))^2 - (sqrt(5)*x - sqrt(5 *x^2 + 3*x + 2))*(sqrt(5) + 4) + 5*sqrt(5) + 12) - 3/2*log((sqrt(5)*x - sq rt(5*x^2 + 3*x + 2))^2 - (sqrt(5)*x - sqrt(5*x^2 + 3*x + 2))*(sqrt(5) - 4) - 5*sqrt(5) + 12) + 3/2*log(x^2 - x + 1) - 3*log(abs(x))
Timed out. \[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=\int \frac {1}{x^2\,\left (2\,x+\sqrt {5\,x^2+3\,x+2}+1\right )} \,d x \] Input:
int(1/(x^2*(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)),x)
Output:
int(1/(x^2*(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)), x)
\[ \int \frac {1}{x^2 \left (1+2 x+\sqrt {2+3 x+5 x^2}\right )} \, dx=\int \frac {1}{\sqrt {5 x^{2}+3 x +2}\, x^{2}+2 x^{3}+x^{2}}d x \] Input:
int(1/x^2/(1+2*x+(5*x^2+3*x+2)^(1/2)),x)
Output:
int(1/(sqrt(5*x**2 + 3*x + 2)*x**2 + 2*x**3 + x**2),x)