Integrand size = 23, antiderivative size = 352 \[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=-\frac {2 \left (53+16 \sqrt {2}-\frac {2 \left (10+3 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}\right )}{3 \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right )}-\frac {158 \arctan \left (\frac {3-4 \sqrt {2}-\frac {2 \left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}}{\sqrt {3}}\right )}{3 \sqrt {3}}+8 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {2}-\sqrt {2+3 x+5 x^2}}{\sqrt {5} x}\right )-9 \log \left (\frac {4+3 x-2 \sqrt {2} \sqrt {2+3 x+5 x^2}}{x^2}\right )+9 \log \left (1-5 \sqrt {2}-\frac {\left (3-4 \sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )}{x}+\frac {\left (1-\sqrt {2}\right ) \left (\sqrt {2}-\sqrt {2+3 x+5 x^2}\right )^2}{x^2}\right ) \] Output:
(-106-32*2^(1/2)+4*(10+3*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x)/(3-15*2 ^(1/2)-3*(3-4*2^(1/2))*(2^(1/2)-(5*x^2+3*x+2)^(1/2))/x+3*(1-2^(1/2))*(2^(1 /2)-(5*x^2+3*x+2)^(1/2))^2/x^2)-158/9*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)+8*arctanh(1/5*(2^(1/2) -(5*x^2+3*x+2)^(1/2))*5^(1/2)/x)*5^(1/2)-9*ln((4+3*x-2*2^(1/2)*(5*x^2+3*x+ 2)^(1/2))/x^2)+9*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)
Result contains complex when optimal does not.
Time = 5.71 (sec) , antiderivative size = 1086, normalized size of antiderivative = 3.09 \[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx =\text {Too large to display} \] Input:
Integrate[x/(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^2,x]
Output:
-((4 + 22*x)/(3 - 3*x + 3*x^2)) + (2*(4 + x)*Sqrt[2 + 3*x + 5*x^2])/(3*(1 - x + x^2)) - 4*Sqrt[5]*ArcSinh[(3 + 10*x)/Sqrt[31]] + (79*ArcTan[(-1 + 2* x)/Sqrt[3]])/(3*Sqrt[3]) + (7*(11 - (19*I)*Sqrt[3])*ArcTan[(3*(9*(1321 + ( 800*I)*Sqrt[3]) + 6*(6043 - (1240*I)*Sqrt[3])*x + (-1637 + (2192*I)*Sqrt[3 ])*x^2 + (-21450 - (28272*I)*Sqrt[3])*x^3 + (2189 + (28880*I)*Sqrt[3])*x^4 ))/(33120*I + 4143*Sqrt[3] - 3*(16720*I + 10667*Sqrt[3])*x^4 + 3612*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2] + x^2*(23568*I + 51389*Sqrt[3] - 8428*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + 2*x*(11784*I + 9497 *Sqrt[3] + 4214*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + 2*x^3*(- 47688*I + 26297*Sqrt[3] + 6020*Sqrt[3 - (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x ^2]))])/(6*Sqrt[3 - (12*I)*Sqrt[3]]) - (((7*I)/6)*(-11*I + 19*Sqrt[3])*Arc Tan[(3*(-11889 + (7200*I)*Sqrt[3] + (-36258 - (7440*I)*Sqrt[3])*x + (1637 + (2192*I)*Sqrt[3])*x^2 + 6*(3575 - (4712*I)*Sqrt[3])*x^3 + (-2189 + (2888 0*I)*Sqrt[3])*x^4))/(-33120*I + 4143*Sqrt[3] + (50160*I - 32001*Sqrt[3])*x ^4 + 3612*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2] + x^2*(-23568*I + 51389*Sqrt[3] - 8428*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + 2* x*(-11784*I + 9497*Sqrt[3] + 4214*Sqrt[3 + (12*I)*Sqrt[3]]*Sqrt[2 + 3*x + 5*x^2]) + 2*x^3*(47688*I + 26297*Sqrt[3] + 6020*Sqrt[3 + (12*I)*Sqrt[3]]*S qrt[2 + 3*x + 5*x^2]))])/Sqrt[3 + (12*I)*Sqrt[3]] + (9*Log[1 - x + x^2])/2 - (7*(-11*I + 19*Sqrt[3])*Log[16*(1 - x + x^2)^2])/(12*Sqrt[3 + (12*I)...
Time = 0.62 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.55, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, 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.
| \(\displaystyle \int \frac {x}{\left (\sqrt {5 x^2+3 x+2}+2 x+1\right )^2} \, dx\) |
| \(\Big \downarrow \) 7293 |
| \(\displaystyle \int \left (-\frac {6 \sqrt {5 x^2+3 x+2} x}{\left (x^2-x+1\right )^2}-\frac {4 \sqrt {5 x^2+3 x+2}}{x^2-x+1}+\frac {4 \sqrt {5 x^2+3 x+2}}{\left (x^2-x+1\right )^2}+\frac {9 x+16}{x^2-x+1}+\frac {2 (5 x-8)}{\left (x^2-x+1\right )^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -4 \sqrt {5} \text {arcsinh}\left (\frac {10 x+3}{\sqrt {31}}\right )+\frac {79 \arctan \left (\frac {5-4 x}{\sqrt {3} \sqrt {5 x^2+3 x+2}}\right )}{3 \sqrt {3}}-\frac {79 \arctan \left (\frac {1-2 x}{\sqrt {3}}\right )}{3 \sqrt {3}}+9 \text {arctanh}\left (\frac {2 x+1}{\sqrt {5 x^2+3 x+2}}\right )-\frac {4 \sqrt {5 x^2+3 x+2} (1-2 x)}{3 \left (x^2-x+1\right )}+\frac {2 (2-x) \sqrt {5 x^2+3 x+2}}{x^2-x+1}-\frac {2 (11 x+2)}{3 \left (x^2-x+1\right )}+\frac {9}{2} \log \left (x^2-x+1\right )\) |
Input:
Int[x/(1 + 2*x + Sqrt[2 + 3*x + 5*x^2])^2,x]
Output:
(-2*(2 + 11*x))/(3*(1 - x + x^2)) - (4*(1 - 2*x)*Sqrt[2 + 3*x + 5*x^2])/(3 *(1 - x + x^2)) + (2*(2 - x)*Sqrt[2 + 3*x + 5*x^2])/(1 - x + x^2) - 4*Sqrt [5]*ArcSinh[(3 + 10*x)/Sqrt[31]] - (79*ArcTan[(1 - 2*x)/Sqrt[3]])/(3*Sqrt[ 3]) + (79*ArcTan[(5 - 4*x)/(Sqrt[3]*Sqrt[2 + 3*x + 5*x^2])])/(3*Sqrt[3]) + 9*ArcTanh[(1 + 2*x)/Sqrt[2 + 3*x + 5*x^2]] + (9*Log[1 - x + x^2])/2
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.65 (sec) , antiderivative size = 1195, normalized size of antiderivative = 3.39
| method | result | size |
| trager | \(\text {Expression too large to display}\) | \(1195\) |
| default | \(\text {Expression too large to display}\) | \(2886\) |
Input:
int(x/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x,method=_RETURNVERBOSE)
Output:
2/3*(2*x-13)*x/(x^2-x+1)+2/3*(4+x)/(x^2-x+1)*(5*x^2+3*x+2)^(1/2)+2/3*RootO f(4*_Z^2+108*_Z+9)*ln((324*RootOf(3*_Z^2-81*_Z+2107)^2*RootOf(4*_Z^2+108*_ Z+9)^2*x+27108*RootOf(3*_Z^2-81*_Z+2107)^2*RootOf(4*_Z^2+108*_Z+9)*x+61404 *RootOf(3*_Z^2-81*_Z+2107)*RootOf(4*_Z^2+108*_Z+9)^2*x+242109*RootOf(3*_Z^ 2-81*_Z+2107)^2*x+346968*RootOf(3*_Z^2-81*_Z+2107)*RootOf(4*_Z^2+108*_Z+9) *(5*x^2+3*x+2)^(1/2)-146052*RootOf(3*_Z^2-81*_Z+2107)*RootOf(4*_Z^2+108*_Z +9)*x-9357040*RootOf(4*_Z^2+108*_Z+9)^2*x+12533508*(5*x^2+3*x+2)^(1/2)*Roo tOf(3*_Z^2-81*_Z+2107)+972648*RootOf(3*_Z^2-81*_Z+2107)*RootOf(4*_Z^2+108* _Z+9)-34449381*RootOf(3*_Z^2-81*_Z+2107)*x+80468136*RootOf(4*_Z^2+108*_Z+9 )*(5*x^2+3*x+2)^(1/2)-419049960*RootOf(4*_Z^2+108*_Z+9)*x+9888588*RootOf(3 *_Z^2-81*_Z+2107)+2162148156*(5*x^2+3*x+2)^(1/2)-97084680*RootOf(4*_Z^2+10 8*_Z+9)-4686181920*x-2248708140)/(3*RootOf(3*_Z^2-81*_Z+2107)*x-80*x+79))- 2/3*ln((324*RootOf(3*_Z^2-81*_Z+2107)^2*RootOf(4*_Z^2+108*_Z+9)^2*x-9612*R ootOf(3*_Z^2-81*_Z+2107)^2*RootOf(4*_Z^2+108*_Z+9)*x+61404*RootOf(3*_Z^2-8 1*_Z+2107)*RootOf(4*_Z^2+108*_Z+9)^2*x-253611*RootOf(3*_Z^2-81*_Z+2107)^2* x-346968*RootOf(3*_Z^2-81*_Z+2107)*RootOf(4*_Z^2+108*_Z+9)*(5*x^2+3*x+2)^( 1/2)+3461868*RootOf(3*_Z^2-81*_Z+2107)*RootOf(4*_Z^2+108*_Z+9)*x-9357040*R ootOf(4*_Z^2+108*_Z+9)^2*x+3165372*(5*x^2+3*x+2)^(1/2)*RootOf(3*_Z^2-81*_Z +2107)-972648*RootOf(3*_Z^2-81*_Z+2107)*RootOf(4*_Z^2+108*_Z+9)+14257539*R ootOf(3*_Z^2-81*_Z+2107)*x-80468136*RootOf(4*_Z^2+108*_Z+9)*(5*x^2+3*x+...
Time = 0.10 (sec) , antiderivative size = 337, normalized size of antiderivative = 0.96 \[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\frac {316 \, \sqrt {3} {\left (x^{2} - x + 1\right )} \arctan \left (\frac {1}{3} \, \sqrt {3} {\left (2 \, x - 1\right )}\right ) + 158 \, \sqrt {3} {\left (x^{2} - x + 1\right )} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} + 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + 158 \, \sqrt {3} {\left (x^{2} - x + 1\right )} \arctan \left (\frac {4 \, \sqrt {3} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (4 \, x - 5\right )} - 31 \, \sqrt {3} {\left (x^{2} - 2 \, x\right )}}{3 \, {\left (11 \, x^{2} - 12 \, x - 8\right )}}\right ) + 72 \, \sqrt {5} {\left (x^{2} - x + 1\right )} \log \left (4 \, \sqrt {5} \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (10 \, x + 3\right )} - 200 \, x^{2} - 120 \, x - 49\right ) + 162 \, {\left (x^{2} - x + 1\right )} \log \left (x^{2} - x + 1\right ) + 81 \, {\left (x^{2} - x + 1\right )} \log \left (\frac {9 \, x^{2} + 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) - 81 \, {\left (x^{2} - x + 1\right )} \log \left (\frac {9 \, x^{2} - 2 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (2 \, x + 1\right )} + 7 \, x + 3}{x^{2}}\right ) + 24 \, \sqrt {5 \, x^{2} + 3 \, x + 2} {\left (x + 4\right )} - 264 \, x - 48}{36 \, {\left (x^{2} - x + 1\right )}} \] Input:
integrate(x/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x, algorithm="fricas")
Output:
1/36*(316*sqrt(3)*(x^2 - x + 1)*arctan(1/3*sqrt(3)*(2*x - 1)) + 158*sqrt(3 )*(x^2 - x + 1)*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)) + 158*sqrt(3)*(x^2 - x + 1)*arc tan(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)) + 72*sqrt(5)*(x^2 - x + 1)*log(4*sqrt(5)*sqrt(5*x^ 2 + 3*x + 2)*(10*x + 3) - 200*x^2 - 120*x - 49) + 162*(x^2 - x + 1)*log(x^ 2 - x + 1) + 81*(x^2 - x + 1)*log((9*x^2 + 2*sqrt(5*x^2 + 3*x + 2)*(2*x + 1) + 7*x + 3)/x^2) - 81*(x^2 - x + 1)*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)*(x + 4) - 264*x - 48 )/(x^2 - x + 1)
\[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\int \frac {x}{\left (2 x + \sqrt {5 x^{2} + 3 x + 2} + 1\right )^{2}}\, dx \] Input:
integrate(x/(1+2*x+(5*x**2+3*x+2)**(1/2))**2,x)
Output:
Integral(x/(2*x + sqrt(5*x**2 + 3*x + 2) + 1)**2, x)
\[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\int { \frac {x}{{\left (2 \, x + \sqrt {5 \, x^{2} + 3 \, x + 2} + 1\right )}^{2}} \,d x } \] Input:
integrate(x/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x, algorithm="maxima")
Output:
integrate(x/(2*x + sqrt(5*x^2 + 3*x + 2) + 1)^2, x)
Exception generated. \[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(x/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{923521,[8]%%%}+%%%{%%{[3694084,0]:[1,0,-5]%%},[7]%%%}+%%%{ 42481966,
Timed out. \[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\int \frac {x}{{\left (2\,x+\sqrt {5\,x^2+3\,x+2}+1\right )}^2} \,d x \] Input:
int(x/(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)^2,x)
Output:
int(x/(2*x + (3*x + 5*x^2 + 2)^(1/2) + 1)^2, x)
\[ \int \frac {x}{\left (1+2 x+\sqrt {2+3 x+5 x^2}\right )^2} \, dx=\int \frac {x}{4 \sqrt {5 x^{2}+3 x +2}\, x +2 \sqrt {5 x^{2}+3 x +2}+9 x^{2}+7 x +3}d x \] Input:
int(x/(1+2*x+(5*x^2+3*x+2)^(1/2))^2,x)
Output:
int(x/(4*sqrt(5*x**2 + 3*x + 2)*x + 2*sqrt(5*x**2 + 3*x + 2) + 9*x**2 + 7* x + 3),x)