Integrand size = 27, antiderivative size = 131 \[ \int \frac {1}{(5-2 x)^3 (4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=\frac {-2573-1115 x}{250047 \sqrt {2+3 x+x^2}}+\frac {8 \sqrt {2+3 x+x^2}}{91287 (5-2 x)^2}+\frac {272 \sqrt {2+3 x+x^2}}{2699487 (5-2 x)}-\frac {243 \arctan \left (\frac {\sqrt {2} (1+x)}{\sqrt {2+3 x+x^2}}\right )}{12167 \sqrt {2}}+\frac {1834552 \text {arctanh}\left (\frac {3 (1+x)}{\sqrt {7} \sqrt {2+3 x+x^2}}\right )}{3042321849 \sqrt {7}} \] Output:
1/250047*(-2573-1115*x)/(x^2+3*x+2)^(1/2)+8/91287*(x^2+3*x+2)^(1/2)/(5-2*x )^2+272*(x^2+3*x+2)^(1/2)/(13497435-5398974*x)-243/24334*2^(1/2)*arctan(2^ (1/2)*(1+x)/(x^2+3*x+2)^(1/2))+1834552/21296252943*7^(1/2)*arctanh(3/7*(1+ x)*7^(1/2)/(x^2+3*x+2)^(1/2))
Time = 0.64 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.89 \[ \int \frac {1}{(5-2 x)^3 (4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=\frac {\sqrt {2+3 x+x^2} \left (-11290487+4219283 x+2116832 x^2-795332 x^3\right )}{44091621 (1+x) (2+x) (-5+2 x)^2}+\frac {243 \arctan \left (\frac {\sqrt {2+3 x+x^2}}{\sqrt {2} (1+x)}\right )}{12167 \sqrt {2}}+\frac {1834552 \text {arctanh}\left (\frac {3 \sqrt {2+3 x+x^2}}{\sqrt {7} (2+x)}\right )}{3042321849 \sqrt {7}} \] Input:
Integrate[1/((5 - 2*x)^3*(4 + 3*x)*(2 + 3*x + x^2)^(3/2)),x]
Output:
(Sqrt[2 + 3*x + x^2]*(-11290487 + 4219283*x + 2116832*x^2 - 795332*x^3))/( 44091621*(1 + x)*(2 + x)*(-5 + 2*x)^2) + (243*ArcTan[Sqrt[2 + 3*x + x^2]/( Sqrt[2]*(1 + x))])/(12167*Sqrt[2]) + (1834552*ArcTanh[(3*Sqrt[2 + 3*x + x^ 2])/(Sqrt[7]*(2 + x))])/(3042321849*Sqrt[7])
Time = 0.66 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.61, 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.
| \(\displaystyle \int \frac {1}{(5-2 x)^3 (3 x+4) \left (x^2+3 x+2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 1289 |
| \(\displaystyle \int \left (\frac {27}{12167 (3 x+4) \left (x^2+3 x+2\right )^{3/2}}-\frac {18}{12167 \left (x^2+3 x+2\right )^{3/2} (2 x-5)}+\frac {6}{529 \left (x^2+3 x+2\right )^{3/2} (2 x-5)^2}-\frac {2}{23 \left (x^2+3 x+2\right )^{3/2} (2 x-5)^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {243 \arctan \left (\frac {x}{2 \sqrt {2} \sqrt {x^2+3 x+2}}\right )}{24334 \sqrt {2}}+\frac {917276 \text {arctanh}\left (\frac {16 x+23}{6 \sqrt {7} \sqrt {x^2+3 x+2}}\right )}{3042321849 \sqrt {7}}-\frac {27 (x+3)}{12167 \sqrt {x^2+3 x+2}}+\frac {135712 \sqrt {x^2+3 x+2}}{44091621 (5-2 x)}+\frac {1048 \sqrt {x^2+3 x+2}}{91287 (5-2 x)^2}-\frac {4 (16 x+25)}{11109 (5-2 x) \sqrt {x^2+3 x+2}}-\frac {4 (16 x+25)}{1449 (5-2 x)^2 \sqrt {x^2+3 x+2}}-\frac {4 (16 x+25)}{85169 \sqrt {x^2+3 x+2}}\) |
Input:
Int[1/((5 - 2*x)^3*(4 + 3*x)*(2 + 3*x + x^2)^(3/2)),x]
Output:
(-27*(3 + x))/(12167*Sqrt[2 + 3*x + x^2]) - (4*(25 + 16*x))/(85169*Sqrt[2 + 3*x + x^2]) - (4*(25 + 16*x))/(1449*(5 - 2*x)^2*Sqrt[2 + 3*x + x^2]) - ( 4*(25 + 16*x))/(11109*(5 - 2*x)*Sqrt[2 + 3*x + x^2]) + (1048*Sqrt[2 + 3*x + x^2])/(91287*(5 - 2*x)^2) + (135712*Sqrt[2 + 3*x + x^2])/(44091621*(5 - 2*x)) - (243*ArcTan[x/(2*Sqrt[2]*Sqrt[2 + 3*x + x^2])])/(24334*Sqrt[2]) + (917276*ArcTanh[(23 + 16*x)/(6*Sqrt[7]*Sqrt[2 + 3*x + x^2])])/(3042321849* Sqrt[7])
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 = 1.42 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.70
| method | result | size |
| risch | \(-\frac {795332 x^{3}-2116832 x^{2}-4219283 x +11290487}{44091621 \left (-5+2 x \right )^{2} \sqrt {x^{2}+3 x +2}}+\frac {917276 \sqrt {7}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {23}{2}+8 x \right ) \sqrt {7}}{21 \sqrt {4 \left (x -\frac {5}{2}\right )^{2}+32 x -17}}\right )}{21296252943}-\frac {243 \sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}\right )}{48668}\) | \(92\) |
| trager | \(-\frac {795332 x^{3}-2116832 x^{2}-4219283 x +11290487}{44091621 \left (-5+2 x \right )^{2} \sqrt {x^{2}+3 x +2}}+\frac {917276 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) \ln \left (-\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) x +42 \sqrt {x^{2}+3 x +2}+23 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right )}{-5+2 x}\right )}{21296252943}-\frac {243 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {x^{2}+3 x +2}+x}{3 x +4}\right )}{48668}\) | \(121\) |
| default | \(\frac {1}{2898 \left (x -\frac {5}{2}\right )^{2} \sqrt {\left (x -\frac {5}{2}\right )^{2}+8 x -\frac {17}{4}}}-\frac {1298}{2099601 \left (x -\frac {5}{2}\right ) \sqrt {\left (x -\frac {5}{2}\right )^{2}+8 x -\frac {17}{4}}}-\frac {458638}{1014107283 \sqrt {\left (x -\frac {5}{2}\right )^{2}+8 x -\frac {17}{4}}}-\frac {1161368 \left (2 x +3\right )}{1014107283 \sqrt {\left (x -\frac {5}{2}\right )^{2}+8 x -\frac {17}{4}}}+\frac {917276 \sqrt {7}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {23}{2}+8 x \right ) \sqrt {7}}{21 \sqrt {4 \left (x -\frac {5}{2}\right )^{2}+32 x -17}}\right )}{21296252943}-\frac {81}{24334 \sqrt {\left (x +\frac {4}{3}\right )^{2}+\frac {x}{3}+\frac {2}{9}}}-\frac {27 \left (2 x +3\right )}{24334 \sqrt {\left (x +\frac {4}{3}\right )^{2}+\frac {x}{3}+\frac {2}{9}}}-\frac {243 \sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}\right )}{48668}\) | \(162\) |
Input:
int(1/(5-2*x)^3/(3*x+4)/(x^2+3*x+2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/44091621*(795332*x^3-2116832*x^2-4219283*x+11290487)/(-5+2*x)^2/(x^2+3* x+2)^(1/2)+917276/21296252943*7^(1/2)*arctanh(2/21*(23/2+8*x)*7^(1/2)/(4*( x-5/2)^2+32*x-17)^(1/2))-243/48668*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 = 191, normalized size of antiderivative = 1.46 \[ \int \frac {1}{(5-2 x)^3 (4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=-\frac {768290712 \, x^{4} - 1536581424 \, x^{3} + 425329947 \, \sqrt {2} {\left (4 \, x^{4} - 8 \, x^{3} - 27 \, x^{2} + 35 \, x + 50\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 4\right )} + \frac {3}{2} \, \sqrt {2} \sqrt {x^{2} + 3 \, x + 2}\right ) - 1834552 \, \sqrt {7} {\left (4 \, x^{4} - 8 \, x^{3} - 27 \, x^{2} + 35 \, x + 50\right )} \log \left (\frac {3 \, \sqrt {7} {\left (16 \, x + 23\right )} + 6 \, \sqrt {x^{2} + 3 \, x + 2} {\left (8 \, \sqrt {7} + 21\right )} + 128 \, x + 184}{2 \, x - 5}\right ) - 5185962306 \, x^{2} + 966 \, {\left (795332 \, x^{3} - 2116832 \, x^{2} - 4219283 \, x + 11290487\right )} \sqrt {x^{2} + 3 \, x + 2} + 6722543730 \, x + 9603633900}{42592505886 \, {\left (4 \, x^{4} - 8 \, x^{3} - 27 \, x^{2} + 35 \, x + 50\right )}} \] Input:
integrate(1/(5-2*x)^3/(4+3*x)/(x^2+3*x+2)^(3/2),x, algorithm="fricas")
Output:
-1/42592505886*(768290712*x^4 - 1536581424*x^3 + 425329947*sqrt(2)*(4*x^4 - 8*x^3 - 27*x^2 + 35*x + 50)*arctan(-1/2*sqrt(2)*(3*x + 4) + 3/2*sqrt(2)* sqrt(x^2 + 3*x + 2)) - 1834552*sqrt(7)*(4*x^4 - 8*x^3 - 27*x^2 + 35*x + 50 )*log((3*sqrt(7)*(16*x + 23) + 6*sqrt(x^2 + 3*x + 2)*(8*sqrt(7) + 21) + 12 8*x + 184)/(2*x - 5)) - 5185962306*x^2 + 966*(795332*x^3 - 2116832*x^2 - 4 219283*x + 11290487)*sqrt(x^2 + 3*x + 2) + 6722543730*x + 9603633900)/(4*x ^4 - 8*x^3 - 27*x^2 + 35*x + 50)
\[ \int \frac {1}{(5-2 x)^3 (4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=- \int \frac {1}{24 x^{6} \sqrt {x^{2} + 3 x + 2} - 76 x^{5} \sqrt {x^{2} + 3 x + 2} - 186 x^{4} \sqrt {x^{2} + 3 x + 2} + 559 x^{3} \sqrt {x^{2} + 3 x + 2} + 595 x^{2} \sqrt {x^{2} + 3 x + 2} - 1050 x \sqrt {x^{2} + 3 x + 2} - 1000 \sqrt {x^{2} + 3 x + 2}}\, dx \] Input:
integrate(1/(5-2*x)**3/(4+3*x)/(x**2+3*x+2)**(3/2),x)
Output:
-Integral(1/(24*x**6*sqrt(x**2 + 3*x + 2) - 76*x**5*sqrt(x**2 + 3*x + 2) - 186*x**4*sqrt(x**2 + 3*x + 2) + 559*x**3*sqrt(x**2 + 3*x + 2) + 595*x**2* sqrt(x**2 + 3*x + 2) - 1050*x*sqrt(x**2 + 3*x + 2) - 1000*sqrt(x**2 + 3*x + 2)), x)
\[ \int \frac {1}{(5-2 x)^3 (4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=\int { -\frac {1}{{\left (x^{2} + 3 \, x + 2\right )}^{\frac {3}{2}} {\left (3 \, x + 4\right )} {\left (2 \, x - 5\right )}^{3}} \,d x } \] Input:
integrate(1/(5-2*x)^3/(4+3*x)/(x^2+3*x+2)^(3/2),x, algorithm="maxima")
Output:
-integrate(1/((x^2 + 3*x + 2)^(3/2)*(3*x + 4)*(2*x - 5)^3), x)
Time = 0.27 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.47 \[ \int \frac {1}{(5-2 x)^3 (4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=-\frac {243}{24334} \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x - 3 \, \sqrt {x^{2} + 3 \, x + 2} + 4\right )}\right ) - \frac {917276}{21296252943} \, \sqrt {7} \log \left (\frac {{\left | -4 \, x - 6 \, \sqrt {7} + 4 \, \sqrt {x^{2} + 3 \, x + 2} + 10 \right |}}{{\left | -4 \, x + 6 \, \sqrt {7} + 4 \, \sqrt {x^{2} + 3 \, x + 2} + 10 \right |}}\right ) - \frac {1115 \, x + 2573}{250047 \, \sqrt {x^{2} + 3 \, x + 2}} + \frac {8 \, {\left (3394 \, {\left (x - \sqrt {x^{2} + 3 \, x + 2}\right )}^{3} - 15429 \, {\left (x - \sqrt {x^{2} + 3 \, x + 2}\right )}^{2} - 66237 \, x + 66237 \, \sqrt {x^{2} + 3 \, x + 2} - 53245\right )}}{18896409 \, {\left (2 \, {\left (x - \sqrt {x^{2} + 3 \, x + 2}\right )}^{2} - 10 \, x + 10 \, \sqrt {x^{2} + 3 \, x + 2} - 19\right )}^{2}} \] Input:
integrate(1/(5-2*x)^3/(4+3*x)/(x^2+3*x+2)^(3/2),x, algorithm="giac")
Output:
-243/24334*sqrt(2)*arctan(-1/2*sqrt(2)*(3*x - 3*sqrt(x^2 + 3*x + 2) + 4)) - 917276/21296252943*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)) - 1/250047*( 1115*x + 2573)/sqrt(x^2 + 3*x + 2) + 8/18896409*(3394*(x - sqrt(x^2 + 3*x + 2))^3 - 15429*(x - sqrt(x^2 + 3*x + 2))^2 - 66237*x + 66237*sqrt(x^2 + 3 *x + 2) - 53245)/(2*(x - sqrt(x^2 + 3*x + 2))^2 - 10*x + 10*sqrt(x^2 + 3*x + 2) - 19)^2
Timed out. \[ \int \frac {1}{(5-2 x)^3 (4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=\int -\frac {1}{{\left (2\,x-5\right )}^3\,\left (3\,x+4\right )\,{\left (x^2+3\,x+2\right )}^{3/2}} \,d x \] Input:
int(-1/((2*x - 5)^3*(3*x + 4)*(3*x + x^2 + 2)^(3/2)),x)
Output:
int(-1/((2*x - 5)^3*(3*x + 4)*(3*x + x^2 + 2)^(3/2)), x)
Time = 0.29 (sec) , antiderivative size = 504, normalized size of antiderivative = 3.85 \[ \int \frac {1}{(5-2 x)^3 (4+3 x) \left (2+3 x+x^2\right )^{3/2}} \, dx=\frac {-13610558304 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x^{4}+27221116608 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x^{3}+91871268552 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x^{2}-119092385160 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x -170131978800 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right )-6146325696 \sqrt {x^{2}+3 x +2}\, x^{3}+16358877696 \sqrt {x^{2}+3 x +2}\, x^{2}+32606619024 \sqrt {x^{2}+3 x +2}\, x -87252883536 \sqrt {x^{2}+3 x +2}-58705664 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right ) x^{4}+117411328 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right ) x^{3}+396263232 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right ) x^{2}-513674560 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right ) x -733820800 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right )+58705664 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right ) x^{4}-117411328 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right ) x^{3}-396263232 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right ) x^{2}+513674560 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right ) x +733820800 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right )+5502167916 x^{4}-11004335832 x^{3}-37139633433 x^{2}+48143969265 x +68777098950}{1362960188352 x^{4}-2725920376704 x^{3}-9199981271376 x^{2}+11925901648080 x +17037002354400} \] Input:
int(1/(5-2*x)^3/(4+3*x)/(x^2+3*x+2)^(3/2),x)
Output:
( - 13610558304*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2))*x **4 + 27221116608*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2)) *x**3 + 91871268552*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2 ))*x**2 - 119092385160*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqr t(2))*x - 170131978800*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqr t(2)) - 6146325696*sqrt(x**2 + 3*x + 2)*x**3 + 16358877696*sqrt(x**2 + 3*x + 2)*x**2 + 32606619024*sqrt(x**2 + 3*x + 2)*x - 87252883536*sqrt(x**2 + 3*x + 2) - 58705664*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) - 3*sqrt(7) + 2*x - 5)*x**4 + 117411328*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) - 3*sqrt(7) + 2*x - 5)*x**3 + 396263232*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) - 3*sqrt(7) + 2*x - 5)*x**2 - 513674560*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) - 3*sqrt(7) + 2* x - 5)*x - 733820800*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) - 3*sqrt(7) + 2*x - 5) + 58705664*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5)* x**4 - 117411328*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5) *x**3 - 396263232*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5 )*x**2 + 513674560*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5)*x + 733820800*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5) + 5502167916*x**4 - 11004335832*x**3 - 37139633433*x**2 + 48143969265*x + 68777098950)/(340740047088*(4*x**4 - 8*x**3 - 27*x**2 + 35*x + 50))