Integrand size = 27, antiderivative size = 110 \[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x)^3 (4+3 x)} \, dx=\frac {\sqrt {2+3 x+x^2}}{46 (5-2 x)^2}+\frac {97 \sqrt {2+3 x+x^2}}{33327 (5-2 x)}-\frac {6 \sqrt {2} \arctan \left (\frac {\sqrt {2} (1+x)}{\sqrt {2+3 x+x^2}}\right )}{12167}+\frac {1361 \text {arctanh}\left (\frac {3 (1+x)}{\sqrt {7} \sqrt {2+3 x+x^2}}\right )}{4599126 \sqrt {7}} \] Output:
1/46*(x^2+3*x+2)^(1/2)/(5-2*x)^2+97*(x^2+3*x+2)^(1/2)/(166635-66654*x)-6/1 2167*2^(1/2)*arctan(2^(1/2)*(1+x)/(x^2+3*x+2)^(1/2))+1361/32193882*7^(1/2) *arctanh(3/7*(1+x)*7^(1/2)/(x^2+3*x+2)^(1/2))
Time = 0.53 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x)^3 (4+3 x)} \, dx=\frac {(2419-388 x) \sqrt {2+3 x+x^2}}{66654 (-5+2 x)^2}+\frac {6 \sqrt {2} \arctan \left (\frac {\sqrt {2+3 x+x^2}}{\sqrt {2} (1+x)}\right )}{12167}+\frac {1361 \text {arctanh}\left (\frac {3 \sqrt {2+3 x+x^2}}{\sqrt {7} (2+x)}\right )}{4599126 \sqrt {7}} \] Input:
Integrate[Sqrt[2 + 3*x + x^2]/((5 - 2*x)^3*(4 + 3*x)),x]
Output:
((2419 - 388*x)*Sqrt[2 + 3*x + x^2])/(66654*(-5 + 2*x)^2) + (6*Sqrt[2]*Arc Tan[Sqrt[2 + 3*x + x^2]/(Sqrt[2]*(1 + x))])/12167 + (1361*ArcTanh[(3*Sqrt[ 2 + 3*x + x^2])/(Sqrt[7]*(2 + x))])/(4599126*Sqrt[7])
Time = 0.65 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.42, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {1289, 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.
| \(\displaystyle \int \frac {\sqrt {x^2+3 x+2}}{(5-2 x)^3 (3 x+4)} \, dx\) |
| \(\Big \downarrow \) 1289 |
| \(\displaystyle \int \left (-\frac {18 \sqrt {x^2+3 x+2}}{12167 (2 x-5)}+\frac {27 \sqrt {x^2+3 x+2}}{12167 (3 x+4)}+\frac {6 \sqrt {x^2+3 x+2}}{529 (2 x-5)^2}-\frac {2 \sqrt {x^2+3 x+2}}{23 (2 x-5)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {3 \sqrt {2} \arctan \left (\frac {x}{2 \sqrt {2} \sqrt {x^2+3 x+2}}\right )}{12167}+\frac {27 \sqrt {7} \text {arctanh}\left (\frac {16 x+23}{6 \sqrt {7} \sqrt {x^2+3 x+2}}\right )}{24334}-\frac {3047 \text {arctanh}\left (\frac {16 x+23}{6 \sqrt {7} \sqrt {x^2+3 x+2}}\right )}{399924 \sqrt {7}}+\frac {\sqrt {x^2+3 x+2} (16 x+23)}{2898 (5-2 x)^2}+\frac {3 \sqrt {x^2+3 x+2}}{529 (5-2 x)}\) |
Input:
Int[Sqrt[2 + 3*x + x^2]/((5 - 2*x)^3*(4 + 3*x)),x]
Output:
(3*Sqrt[2 + 3*x + x^2])/(529*(5 - 2*x)) + ((23 + 16*x)*Sqrt[2 + 3*x + x^2] )/(2898*(5 - 2*x)^2) - (3*Sqrt[2]*ArcTan[x/(2*Sqrt[2]*Sqrt[2 + 3*x + x^2]) ])/12167 - (3047*ArcTanh[(23 + 16*x)/(6*Sqrt[7]*Sqrt[2 + 3*x + x^2])])/(39 9924*Sqrt[7]) + (27*Sqrt[7]*ArcTanh[(23 + 16*x)/(6*Sqrt[7]*Sqrt[2 + 3*x + x^2])])/24334
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_.) + (b_.)*(x _) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && ( IntegerQ[p] || (ILtQ[m, 0] && ILtQ[n, 0]))
Time = 2.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(-\frac {388 x^{3}-1255 x^{2}-6481 x -4838}{66654 \left (-5+2 x \right )^{2} \sqrt {x^{2}+3 x +2}}+\frac {1361 \sqrt {7}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {23}{2}+8 x \right ) \sqrt {7}}{21 \sqrt {4 \left (x -\frac {5}{2}\right )^{2}+32 x -17}}\right )}{64387764}-\frac {3 \sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}\right )}{12167}\) | \(92\) |
| trager | \(-\frac {\left (388 x -2419\right ) \sqrt {x^{2}+3 x +2}}{66654 \left (-5+2 x \right )^{2}}-\frac {1361 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) \ln \left (-\frac {-16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) x +42 \sqrt {x^{2}+3 x +2}-23 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right )}{-5+2 x}\right )}{64387764}-\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {x^{2}+3 x +2}+x}{3 x +4}\right )}{12167}\) | \(111\) |
| default | \(\frac {\left (\left (x -\frac {5}{2}\right )^{2}+8 x -\frac {17}{4}\right )^{\frac {3}{2}}}{2898 \left (x -\frac {5}{2}\right )^{2}}-\frac {562 \left (\left (x -\frac {5}{2}\right )^{2}+8 x -\frac {17}{4}\right )^{\frac {3}{2}}}{2099601 \left (x -\frac {5}{2}\right )}-\frac {1361 \sqrt {4 \left (x -\frac {5}{2}\right )^{2}+32 x -17}}{193163292}+\frac {1361 \sqrt {7}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {23}{2}+8 x \right ) \sqrt {7}}{21 \sqrt {4 \left (x -\frac {5}{2}\right )^{2}+32 x -17}}\right )}{64387764}+\frac {281 \left (2 x +3\right ) \sqrt {\left (x -\frac {5}{2}\right )^{2}+8 x -\frac {17}{4}}}{2099601}-\frac {3 \ln \left (\frac {3}{2}+x +\sqrt {\left (x -\frac {5}{2}\right )^{2}+8 x -\frac {17}{4}}\right )}{24334}+\frac {3 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}{12167}+\frac {3 \ln \left (\frac {3}{2}+x +\sqrt {\left (x +\frac {4}{3}\right )^{2}+\frac {x}{3}+\frac {2}{9}}\right )}{24334}-\frac {3 \sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}\right )}{12167}\) | \(183\) |
Input:
int((x^2+3*x+2)^(1/2)/(5-2*x)^3/(3*x+4),x,method=_RETURNVERBOSE)
Output:
-1/66654*(388*x^3-1255*x^2-6481*x-4838)/(-5+2*x)^2/(x^2+3*x+2)^(1/2)+1361/ 64387764*7^(1/2)*arctanh(2/21*(23/2+8*x)*7^(1/2)/(4*(x-5/2)^2+32*x-17)^(1/ 2))-3/12167*2^(1/2)*arctan(3/4*x*2^(1/2)/(9*(x+4/3)^2+3*x+2)^(1/2))
Time = 0.09 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x)^3 (4+3 x)} \, dx=-\frac {31752 \, \sqrt {2} {\left (4 \, x^{2} - 20 \, x + 25\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 4\right )} + \frac {3}{2} \, \sqrt {2} \sqrt {x^{2} + 3 \, x + 2}\right ) - 1361 \, \sqrt {7} {\left (4 \, x^{2} - 20 \, x + 25\right )} \log \left (\frac {3 \, \sqrt {7} {\left (16 \, x + 23\right )} + 6 \, \sqrt {x^{2} + 3 \, x + 2} {\left (8 \, \sqrt {7} + 21\right )} + 128 \, x + 184}{2 \, x - 5}\right ) + 374808 \, x^{2} + 966 \, \sqrt {x^{2} + 3 \, x + 2} {\left (388 \, x - 2419\right )} - 1874040 \, x + 2342550}{64387764 \, {\left (4 \, x^{2} - 20 \, x + 25\right )}} \] Input:
integrate((x^2+3*x+2)^(1/2)/(5-2*x)^3/(4+3*x),x, algorithm="fricas")
Output:
-1/64387764*(31752*sqrt(2)*(4*x^2 - 20*x + 25)*arctan(-1/2*sqrt(2)*(3*x + 4) + 3/2*sqrt(2)*sqrt(x^2 + 3*x + 2)) - 1361*sqrt(7)*(4*x^2 - 20*x + 25)*l og((3*sqrt(7)*(16*x + 23) + 6*sqrt(x^2 + 3*x + 2)*(8*sqrt(7) + 21) + 128*x + 184)/(2*x - 5)) + 374808*x^2 + 966*sqrt(x^2 + 3*x + 2)*(388*x - 2419) - 1874040*x + 2342550)/(4*x^2 - 20*x + 25)
\[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x)^3 (4+3 x)} \, dx=- \int \frac {\sqrt {x^{2} + 3 x + 2}}{24 x^{4} - 148 x^{3} + 210 x^{2} + 225 x - 500}\, dx \] Input:
integrate((x**2+3*x+2)**(1/2)/(5-2*x)**3/(4+3*x),x)
Output:
-Integral(sqrt(x**2 + 3*x + 2)/(24*x**4 - 148*x**3 + 210*x**2 + 225*x - 50 0), x)
\[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x)^3 (4+3 x)} \, dx=\int { -\frac {\sqrt {x^{2} + 3 \, x + 2}}{{\left (3 \, x + 4\right )} {\left (2 \, x - 5\right )}^{3}} \,d x } \] Input:
integrate((x^2+3*x+2)^(1/2)/(5-2*x)^3/(4+3*x),x, algorithm="maxima")
Output:
-integrate(sqrt(x^2 + 3*x + 2)/((3*x + 4)*(2*x - 5)^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (86) = 172\).
Time = 0.29 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.60 \[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x)^3 (4+3 x)} \, dx=-\frac {6}{12167} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 3 \, \sqrt {x^{2} + 3 \, x + 2} + 4\right )}\right ) - \frac {1361}{64387764} \, \sqrt {7} \log \left (\frac {{\left | -4 \, x - 6 \, \sqrt {7} + 4 \, \sqrt {x^{2} + 3 \, x + 2} + 10 \right |}}{{\left | -4 \, x + 6 \, \sqrt {7} + 4 \, \sqrt {x^{2} + 3 \, x + 2} + 10 \right |}}\right ) + \frac {206 \, {\left (x - \sqrt {x^{2} + 3 \, x + 2}\right )}^{3} - 24099 \, {\left (x - \sqrt {x^{2} + 3 \, x + 2}\right )}^{2} - 70635 \, x + 70635 \, \sqrt {x^{2} + 3 \, x + 2} - 51083}{66654 \, {\left (2 \, {\left (x - \sqrt {x^{2} + 3 \, x + 2}\right )}^{2} - 10 \, x + 10 \, \sqrt {x^{2} + 3 \, x + 2} - 19\right )}^{2}} \] Input:
integrate((x^2+3*x+2)^(1/2)/(5-2*x)^3/(4+3*x),x, algorithm="giac")
Output:
-6/12167*sqrt(2)*arctan(-1/2*sqrt(2)*(3*x - 3*sqrt(x^2 + 3*x + 2) + 4)) - 1361/64387764*sqrt(7)*log(abs(-4*x - 6*sqrt(7) + 4*sqrt(x^2 + 3*x + 2) + 1 0)/abs(-4*x + 6*sqrt(7) + 4*sqrt(x^2 + 3*x + 2) + 10)) + 1/66654*(206*(x - sqrt(x^2 + 3*x + 2))^3 - 24099*(x - sqrt(x^2 + 3*x + 2))^2 - 70635*x + 70 635*sqrt(x^2 + 3*x + 2) - 51083)/(2*(x - sqrt(x^2 + 3*x + 2))^2 - 10*x + 1 0*sqrt(x^2 + 3*x + 2) - 19)^2
Timed out. \[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x)^3 (4+3 x)} \, dx=\int -\frac {\sqrt {x^2+3\,x+2}}{{\left (2\,x-5\right )}^3\,\left (3\,x+4\right )} \,d x \] Input:
int(-(3*x + x^2 + 2)^(1/2)/((2*x - 5)^3*(3*x + 4)),x)
Output:
int(-(3*x + x^2 + 2)^(1/2)/((2*x - 5)^3*(3*x + 4)), x)
Time = 0.20 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.60 \[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x)^3 (4+3 x)} \, dx=\frac {-2032128 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x^{2}+10160640 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x -12700800 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right )-5996928 \sqrt {x^{2}+3 x +2}\, x +37388064 \sqrt {x^{2}+3 x +2}-87104 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right ) x^{2}+435520 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right ) x -544400 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right )+87104 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right ) x^{2}-435520 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right ) x +544400 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right )+5797932 x^{2}-28989660 x +36237075}{4120816896 x^{2}-20604084480 x +25755105600} \] Input:
int((x^2+3*x+2)^(1/2)/(5-2*x)^3/(4+3*x),x)
Output:
( - 2032128*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2))*x**2 + 10160640*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2))*x - 12 700800*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2)) - 5996928* sqrt(x**2 + 3*x + 2)*x + 37388064*sqrt(x**2 + 3*x + 2) - 87104*sqrt(7)*log (2*sqrt(x**2 + 3*x + 2) - 3*sqrt(7) + 2*x - 5)*x**2 + 435520*sqrt(7)*log(2 *sqrt(x**2 + 3*x + 2) - 3*sqrt(7) + 2*x - 5)*x - 544400*sqrt(7)*log(2*sqrt (x**2 + 3*x + 2) - 3*sqrt(7) + 2*x - 5) + 87104*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5)*x**2 - 435520*sqrt(7)*log(2*sqrt(x**2 + 3* x + 2) + 3*sqrt(7) + 2*x - 5)*x + 544400*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2 ) + 3*sqrt(7) + 2*x - 5) + 5797932*x**2 - 28989660*x + 36237075)/(10302042 24*(4*x**2 - 20*x + 25))