Integrand size = 27, antiderivative size = 110 \[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x) (4+3 x)^3} \, dx=-\frac {\sqrt {2+3 x+x^2}}{46 (4+3 x)^2}+\frac {7 \sqrt {2+3 x+x^2}}{4232 (4+3 x)}+\frac {1667 \arctan \left (\frac {\sqrt {2} (1+x)}{\sqrt {2+3 x+x^2}}\right )}{97336 \sqrt {2}}+\frac {12 \sqrt {7} \text {arctanh}\left (\frac {3 (1+x)}{\sqrt {7} \sqrt {2+3 x+x^2}}\right )}{12167} \] Output:
-1/46*(x^2+3*x+2)^(1/2)/(4+3*x)^2+7*(x^2+3*x+2)^(1/2)/(16928+12696*x)+1667 /194672*2^(1/2)*arctan(2^(1/2)*(1+x)/(x^2+3*x+2)^(1/2))+12/12167*7^(1/2)*a rctanh(3/7*(1+x)*7^(1/2)/(x^2+3*x+2)^(1/2))
Time = 0.47 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.87 \[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x) (4+3 x)^3} \, dx=\frac {(-64+21 x) \sqrt {2+3 x+x^2}}{4232 (4+3 x)^2}-\frac {1667 \arctan \left (\frac {\sqrt {2+3 x+x^2}}{\sqrt {2} (1+x)}\right )}{97336 \sqrt {2}}+\frac {12 \sqrt {7} \text {arctanh}\left (\frac {3 \sqrt {2+3 x+x^2}}{\sqrt {7} (2+x)}\right )}{12167} \] Input:
Integrate[Sqrt[2 + 3*x + x^2]/((5 - 2*x)*(4 + 3*x)^3),x]
Output:
((-64 + 21*x)*Sqrt[2 + 3*x + x^2])/(4232*(4 + 3*x)^2) - (1667*ArcTan[Sqrt[ 2 + 3*x + x^2]/(Sqrt[2]*(1 + x))])/(97336*Sqrt[2]) + (12*Sqrt[7]*ArcTanh[( 3*Sqrt[2 + 3*x + x^2])/(Sqrt[7]*(2 + x))])/12167
Time = 0.64 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.35, 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.
| \(\displaystyle \int \frac {\sqrt {x^2+3 x+2}}{(5-2 x) (3 x+4)^3} \, dx\) |
| \(\Big \downarrow \) 1289 |
| \(\displaystyle \int \left (-\frac {8 \sqrt {x^2+3 x+2}}{12167 (2 x-5)}+\frac {12 \sqrt {x^2+3 x+2}}{12167 (3 x+4)}+\frac {6 \sqrt {x^2+3 x+2}}{529 (3 x+4)^2}+\frac {3 \sqrt {x^2+3 x+2}}{23 (3 x+4)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {4 \sqrt {2} \arctan \left (\frac {x}{2 \sqrt {2} \sqrt {x^2+3 x+2}}\right )}{36501}+\frac {223 \arctan \left (\frac {x}{2 \sqrt {2} \sqrt {x^2+3 x+2}}\right )}{25392 \sqrt {2}}+\frac {6 \sqrt {7} \text {arctanh}\left (\frac {16 x+23}{6 \sqrt {7} \sqrt {x^2+3 x+2}}\right )}{12167}+\frac {3 \sqrt {x^2+3 x+2} x}{184 (3 x+4)^2}-\frac {2 \sqrt {x^2+3 x+2}}{529 (3 x+4)}\) |
Input:
Int[Sqrt[2 + 3*x + x^2]/((5 - 2*x)*(4 + 3*x)^3),x]
Output:
(3*x*Sqrt[2 + 3*x + x^2])/(184*(4 + 3*x)^2) - (2*Sqrt[2 + 3*x + x^2])/(529 *(4 + 3*x)) + (223*ArcTan[x/(2*Sqrt[2]*Sqrt[2 + 3*x + x^2])])/(25392*Sqrt[ 2]) - (4*Sqrt[2]*ArcTan[x/(2*Sqrt[2]*Sqrt[2 + 3*x + x^2])])/36501 + (6*Sqr t[7]*ArcTanh[(23 + 16*x)/(6*Sqrt[7]*Sqrt[2 + 3*x + x^2])])/12167
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 = 2.17 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.84
| method | result | size |
| risch | \(\frac {21 x^{3}-x^{2}-150 x -128}{4232 \left (3 x +4\right )^{2} \sqrt {x^{2}+3 x +2}}+\frac {6 \sqrt {7}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {23}{2}+8 x \right ) \sqrt {7}}{21 \sqrt {4 \left (x -\frac {5}{2}\right )^{2}+32 x -17}}\right )}{12167}+\frac {1667 \sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}\right )}{389344}\) | \(92\) |
| trager | \(\frac {\left (21 x -64\right ) \sqrt {x^{2}+3 x +2}}{4232 \left (3 x +4\right )^{2}}+\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) \ln \left (-\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) x +42 \sqrt {x^{2}+3 x +2}+23 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right )}{-5+2 x}\right )}{12167}-\frac {1667 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {x^{2}+3 x +2}-x}{3 x +4}\right )}{389344}\) | \(114\) |
| default | \(-\frac {2 \sqrt {4 \left (x -\frac {5}{2}\right )^{2}+32 x -17}}{12167}-\frac {16 \ln \left (\frac {3}{2}+x +\sqrt {\left (x -\frac {5}{2}\right )^{2}+8 x -\frac {17}{4}}\right )}{12167}+\frac {6 \sqrt {7}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {23}{2}+8 x \right ) \sqrt {7}}{21 \sqrt {4 \left (x -\frac {5}{2}\right )^{2}+32 x -17}}\right )}{12167}+\frac {\left (\left (x +\frac {4}{3}\right )^{2}+\frac {x}{3}+\frac {2}{9}\right )^{\frac {3}{2}}}{92 \left (x +\frac {4}{3}\right )^{2}}+\frac {117 \left (\left (x +\frac {4}{3}\right )^{2}+\frac {x}{3}+\frac {2}{9}\right )^{\frac {3}{2}}}{8464 \left (x +\frac {4}{3}\right )}-\frac {1667 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}{389344}+\frac {1667 \sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}\right )}{389344}-\frac {117 \left (2 x +3\right ) \sqrt {\left (x +\frac {4}{3}\right )^{2}+\frac {x}{3}+\frac {2}{9}}}{16928}+\frac {16 \ln \left (\frac {3}{2}+x +\sqrt {\left (x +\frac {4}{3}\right )^{2}+\frac {x}{3}+\frac {2}{9}}\right )}{12167}\) | \(183\) |
Input:
int((x^2+3*x+2)^(1/2)/(5-2*x)/(3*x+4)^3,x,method=_RETURNVERBOSE)
Output:
1/4232*(21*x^3-x^2-150*x-128)/(3*x+4)^2/(x^2+3*x+2)^(1/2)+6/12167*7^(1/2)* arctanh(2/21*(23/2+8*x)*7^(1/2)/(4*(x-5/2)^2+32*x-17)^(1/2))+1667/389344*2 ^(1/2)*arctan(3/4*x*2^(1/2)/(9*(x+4/3)^2+3*x+2)^(1/2))
Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.28 \[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x) (4+3 x)^3} \, dx=\frac {5001 \, \sqrt {2} {\left (9 \, x^{2} + 24 \, x + 16\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 4\right )} + \frac {3}{2} \, \sqrt {2} \sqrt {x^{2} + 3 \, x + 2}\right ) + 288 \, \sqrt {7} {\left (9 \, x^{2} + 24 \, x + 16\right )} \log \left (\frac {3 \, \sqrt {7} {\left (16 \, x + 23\right )} + 6 \, \sqrt {x^{2} + 3 \, x + 2} {\left (8 \, \sqrt {7} + 21\right )} + 128 \, x + 184}{2 \, x - 5}\right ) + 2898 \, x^{2} + 138 \, \sqrt {x^{2} + 3 \, x + 2} {\left (21 \, x - 64\right )} + 7728 \, x + 5152}{584016 \, {\left (9 \, x^{2} + 24 \, x + 16\right )}} \] Input:
integrate((x^2+3*x+2)^(1/2)/(5-2*x)/(4+3*x)^3,x, algorithm="fricas")
Output:
1/584016*(5001*sqrt(2)*(9*x^2 + 24*x + 16)*arctan(-1/2*sqrt(2)*(3*x + 4) + 3/2*sqrt(2)*sqrt(x^2 + 3*x + 2)) + 288*sqrt(7)*(9*x^2 + 24*x + 16)*log((3 *sqrt(7)*(16*x + 23) + 6*sqrt(x^2 + 3*x + 2)*(8*sqrt(7) + 21) + 128*x + 18 4)/(2*x - 5)) + 2898*x^2 + 138*sqrt(x^2 + 3*x + 2)*(21*x - 64) + 7728*x + 5152)/(9*x^2 + 24*x + 16)
\[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x) (4+3 x)^3} \, dx=- \int \frac {\sqrt {x^{2} + 3 x + 2}}{54 x^{4} + 81 x^{3} - 252 x^{2} - 592 x - 320}\, dx \] Input:
integrate((x**2+3*x+2)**(1/2)/(5-2*x)/(4+3*x)**3,x)
Output:
-Integral(sqrt(x**2 + 3*x + 2)/(54*x**4 + 81*x**3 - 252*x**2 - 592*x - 320 ), x)
\[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x) (4+3 x)^3} \, dx=\int { -\frac {\sqrt {x^{2} + 3 \, x + 2}}{{\left (3 \, x + 4\right )}^{3} {\left (2 \, x - 5\right )}} \,d x } \] Input:
integrate((x^2+3*x+2)^(1/2)/(5-2*x)/(4+3*x)^3,x, algorithm="maxima")
Output:
-integrate(sqrt(x^2 + 3*x + 2)/((3*x + 4)^3*(2*x - 5)), x)
Leaf count of result is larger than twice the leaf count of optimal. 176 vs. \(2 (86) = 172\).
Time = 0.25 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.60 \[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x) (4+3 x)^3} \, dx=\frac {1667}{194672} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 3 \, \sqrt {x^{2} + 3 \, x + 2} + 4\right )}\right ) - \frac {6}{12167} \, \sqrt {7} \log \left (\frac {{\left | -4 \, x - 6 \, \sqrt {7} + 4 \, \sqrt {x^{2} + 3 \, x + 2} + 10 \right |}}{{\left | -4 \, x + 6 \, \sqrt {7} + 4 \, \sqrt {x^{2} + 3 \, x + 2} + 10 \right |}}\right ) + \frac {531 \, {\left (x - \sqrt {x^{2} + 3 \, x + 2}\right )}^{3} + 2428 \, {\left (x - \sqrt {x^{2} + 3 \, x + 2}\right )}^{2} + 3546 \, x - 3546 \, \sqrt {x^{2} + 3 \, x + 2} + 1656}{12696 \, {\left (3 \, {\left (x - \sqrt {x^{2} + 3 \, x + 2}\right )}^{2} + 8 \, x - 8 \, \sqrt {x^{2} + 3 \, x + 2} + 6\right )}^{2}} \] Input:
integrate((x^2+3*x+2)^(1/2)/(5-2*x)/(4+3*x)^3,x, algorithm="giac")
Output:
1667/194672*sqrt(2)*arctan(-1/2*sqrt(2)*(3*x - 3*sqrt(x^2 + 3*x + 2) + 4)) - 6/12167*sqrt(7)*log(abs(-4*x - 6*sqrt(7) + 4*sqrt(x^2 + 3*x + 2) + 10)/ abs(-4*x + 6*sqrt(7) + 4*sqrt(x^2 + 3*x + 2) + 10)) + 1/12696*(531*(x - sq rt(x^2 + 3*x + 2))^3 + 2428*(x - sqrt(x^2 + 3*x + 2))^2 + 3546*x - 3546*sq rt(x^2 + 3*x + 2) + 1656)/(3*(x - sqrt(x^2 + 3*x + 2))^2 + 8*x - 8*sqrt(x^ 2 + 3*x + 2) + 6)^2
Timed out. \[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x) (4+3 x)^3} \, dx=-\int \frac {\sqrt {x^2+3\,x+2}}{\left (2\,x-5\right )\,{\left (3\,x+4\right )}^3} \,d x \] Input:
int(-(3*x + x^2 + 2)^(1/2)/((2*x - 5)*(3*x + 4)^3),x)
Output:
-int((3*x + x^2 + 2)^(1/2)/((2*x - 5)*(3*x + 4)^3), x)
Time = 0.17 (sec) , antiderivative size = 286, normalized size of antiderivative = 2.60 \[ \int \frac {\sqrt {2+3 x+x^2}}{(5-2 x) (4+3 x)^3} \, dx=\frac {45009 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x^{2}+120024 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x +80016 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right )+2898 \sqrt {x^{2}+3 x +2}\, x -8832 \sqrt {x^{2}+3 x +2}-2592 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right ) x^{2}-6912 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right ) x -4608 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right )+2592 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right ) x^{2}+6912 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right ) x +4608 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right )-39537 x^{2}-105432 x -70288}{5256144 x^{2}+14016384 x +9344256} \] Input:
int((x^2+3*x+2)^(1/2)/(5-2*x)/(4+3*x)^3,x)
Output:
(45009*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2))*x**2 + 120 024*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2))*x + 80016*sqr t(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2)) + 2898*sqrt(x**2 + 3 *x + 2)*x - 8832*sqrt(x**2 + 3*x + 2) - 2592*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) - 3*sqrt(7) + 2*x - 5)*x**2 - 6912*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2 ) - 3*sqrt(7) + 2*x - 5)*x - 4608*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) - 3*s qrt(7) + 2*x - 5) + 2592*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5)*x**2 + 6912*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5)*x + 4608*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5) - 39537*x**2 - 105432*x - 70288)/(584016*(9*x**2 + 24*x + 16))