Integrand size = 27, antiderivative size = 133 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {4 \sqrt {2+3 x+x^2}}{33327 (5-2 x)^2}+\frac {1744 \sqrt {2+3 x+x^2}}{16096941 (5-2 x)}+\frac {81 \sqrt {2+3 x+x^2}}{24334 (4+3 x)}+\frac {1269 \arctan \left (\frac {\sqrt {2} (1+x)}{\sqrt {2+3 x+x^2}}\right )}{559682 \sqrt {2}}+\frac {562228 \text {arctanh}\left (\frac {3 (1+x)}{\sqrt {7} \sqrt {2+3 x+x^2}}\right )}{1110688929 \sqrt {7}} \] Output:
4/33327*(x^2+3*x+2)^(1/2)/(5-2*x)^2+1744*(x^2+3*x+2)^(1/2)/(80484705-32193 882*x)+81*(x^2+3*x+2)^(1/2)/(97336+73002*x)+1269/1119364*2^(1/2)*arctan(2^ (1/2)*(1+x)/(x^2+3*x+2)^(1/2))+562228/7774822503*7^(1/2)*arctanh(3/7*(1+x) *7^(1/2)/(x^2+3*x+2)^(1/2))
Time = 0.67 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.81 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {\sqrt {2+3 x+x^2} \left (2764291-2107252 x+407724 x^2\right )}{32193882 (-5+2 x)^2 (4+3 x)}-\frac {1269 \arctan \left (\frac {\sqrt {2+3 x+x^2}}{\sqrt {2} (1+x)}\right )}{559682 \sqrt {2}}+\frac {562228 \text {arctanh}\left (\frac {3 \sqrt {2+3 x+x^2}}{\sqrt {7} (2+x)}\right )}{1110688929 \sqrt {7}} \] Input:
Integrate[1/((5 - 2*x)^3*(4 + 3*x)^2*Sqrt[2 + 3*x + x^2]),x]
Output:
(Sqrt[2 + 3*x + x^2]*(2764291 - 2107252*x + 407724*x^2))/(32193882*(-5 + 2 *x)^2*(4 + 3*x)) - (1269*ArcTan[Sqrt[2 + 3*x + x^2]/(Sqrt[2]*(1 + x))])/(5 59682*Sqrt[2]) + (562228*ArcTanh[(3*Sqrt[2 + 3*x + x^2])/(Sqrt[7]*(2 + x)) ])/(1110688929*Sqrt[7])
Time = 0.61 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.28, 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.
| \(\displaystyle \int \frac {1}{(5-2 x)^3 (3 x+4)^2 \sqrt {x^2+3 x+2}} \, dx\) |
| \(\Big \downarrow \) 1289 |
| \(\displaystyle \int \left (-\frac {108}{279841 (2 x-5) \sqrt {x^2+3 x+2}}+\frac {162}{279841 (3 x+4) \sqrt {x^2+3 x+2}}+\frac {24}{12167 (2 x-5)^2 \sqrt {x^2+3 x+2}}+\frac {27}{12167 (3 x+4)^2 \sqrt {x^2+3 x+2}}-\frac {4}{529 (2 x-5)^3 \sqrt {x^2+3 x+2}}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {81 \sqrt {2} \arctan \left (\frac {x}{2 \sqrt {2} \sqrt {x^2+3 x+2}}\right )}{279841}+\frac {27 \arctan \left (\frac {x}{2 \sqrt {2} \sqrt {x^2+3 x+2}}\right )}{48668 \sqrt {2}}+\frac {281114 \text {arctanh}\left (\frac {16 x+23}{6 \sqrt {7} \sqrt {x^2+3 x+2}}\right )}{1110688929 \sqrt {7}}+\frac {1744 \sqrt {x^2+3 x+2}}{16096941 (5-2 x)}+\frac {81 \sqrt {x^2+3 x+2}}{24334 (3 x+4)}+\frac {4 \sqrt {x^2+3 x+2}}{33327 (5-2 x)^2}\) |
Input:
Int[1/((5 - 2*x)^3*(4 + 3*x)^2*Sqrt[2 + 3*x + x^2]),x]
Output:
(4*Sqrt[2 + 3*x + x^2])/(33327*(5 - 2*x)^2) + (1744*Sqrt[2 + 3*x + x^2])/( 16096941*(5 - 2*x)) + (81*Sqrt[2 + 3*x + x^2])/(24334*(4 + 3*x)) + (27*Arc Tan[x/(2*Sqrt[2]*Sqrt[2 + 3*x + x^2])])/(48668*Sqrt[2]) + (81*Sqrt[2]*ArcT an[x/(2*Sqrt[2]*Sqrt[2 + 3*x + x^2])])/279841 + (281114*ArcTanh[(23 + 16*x )/(6*Sqrt[7]*Sqrt[2 + 3*x + x^2])])/(1110688929*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 = 1.51 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.78
| method | result | size |
| risch | \(\frac {407724 x^{4}-884080 x^{3}-2742017 x^{2}+4078369 x +5528582}{32193882 \left (-5+2 x \right )^{2} \sqrt {x^{2}+3 x +2}\, \left (3 x +4\right )}+\frac {281114 \sqrt {7}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {23}{2}+8 x \right ) \sqrt {7}}{21 \sqrt {4 \left (x -\frac {5}{2}\right )^{2}+32 x -17}}\right )}{7774822503}+\frac {1269 \sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}\right )}{2238728}\) | \(104\) |
| default | \(\frac {\sqrt {\left (x -\frac {5}{2}\right )^{2}+8 x -\frac {17}{4}}}{33327 \left (x -\frac {5}{2}\right )^{2}}-\frac {872 \sqrt {\left (x -\frac {5}{2}\right )^{2}+8 x -\frac {17}{4}}}{16096941 \left (x -\frac {5}{2}\right )}+\frac {281114 \sqrt {7}\, \operatorname {arctanh}\left (\frac {2 \left (\frac {23}{2}+8 x \right ) \sqrt {7}}{21 \sqrt {4 \left (x -\frac {5}{2}\right )^{2}+32 x -17}}\right )}{7774822503}+\frac {27 \sqrt {\left (x +\frac {4}{3}\right )^{2}+\frac {x}{3}+\frac {2}{9}}}{24334 \left (x +\frac {4}{3}\right )}+\frac {1269 \sqrt {2}\, \arctan \left (\frac {3 x \sqrt {2}}{4 \sqrt {9 \left (x +\frac {4}{3}\right )^{2}+3 x +2}}\right )}{2238728}\) | \(115\) |
| trager | \(\frac {\left (407724 x^{2}-2107252 x +2764291\right ) \sqrt {x^{2}+3 x +2}}{32193882 \left (-5+2 x \right )^{2} \left (3 x +4\right )}-\frac {281114 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) \ln \left (\frac {16 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right ) x +23 \operatorname {RootOf}\left (\textit {\_Z}^{2}-7\right )-42 \sqrt {x^{2}+3 x +2}}{-5+2 x}\right )}{7774822503}-\frac {1269 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \ln \left (-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}+2\right ) \sqrt {x^{2}+3 x +2}-x}{3 x +4}\right )}{2238728}\) | \(125\) |
Input:
int(1/(5-2*x)^3/(3*x+4)^2/(x^2+3*x+2)^(1/2),x,method=_RETURNVERBOSE)
Output:
1/32193882*(407724*x^4-884080*x^3-2742017*x^2+4078369*x+5528582)/(-5+2*x)^ 2/(x^2+3*x+2)^(1/2)/(3*x+4)+281114/7774822503*7^(1/2)*arctanh(2/21*(23/2+8 *x)*7^(1/2)/(4*(x-5/2)^2+32*x-17)^(1/2))+1269/2238728*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 = 166, normalized size of antiderivative = 1.25 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {393861384 \, x^{3} + 35256627 \, \sqrt {2} {\left (12 \, x^{3} - 44 \, x^{2} - 5 \, x + 100\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (3 \, x + 4\right )} + \frac {3}{2} \, \sqrt {2} \sqrt {x^{2} + 3 \, x + 2}\right ) + 1124456 \, \sqrt {7} {\left (12 \, x^{3} - 44 \, x^{2} - 5 \, x + 100\right )} \log \left (\frac {3 \, \sqrt {7} {\left (16 \, x + 23\right )} + 6 \, \sqrt {x^{2} + 3 \, x + 2} {\left (8 \, \sqrt {7} + 21\right )} + 128 \, x + 184}{2 \, x - 5}\right ) - 1444158408 \, x^{2} + 966 \, {\left (407724 \, x^{2} - 2107252 \, x + 2764291\right )} \sqrt {x^{2} + 3 \, x + 2} - 164108910 \, x + 3282178200}{31099290012 \, {\left (12 \, x^{3} - 44 \, x^{2} - 5 \, x + 100\right )}} \] Input:
integrate(1/(5-2*x)^3/(4+3*x)^2/(x^2+3*x+2)^(1/2),x, algorithm="fricas")
Output:
1/31099290012*(393861384*x^3 + 35256627*sqrt(2)*(12*x^3 - 44*x^2 - 5*x + 1 00)*arctan(-1/2*sqrt(2)*(3*x + 4) + 3/2*sqrt(2)*sqrt(x^2 + 3*x + 2)) + 112 4456*sqrt(7)*(12*x^3 - 44*x^2 - 5*x + 100)*log((3*sqrt(7)*(16*x + 23) + 6* sqrt(x^2 + 3*x + 2)*(8*sqrt(7) + 21) + 128*x + 184)/(2*x - 5)) - 144415840 8*x^2 + 966*(407724*x^2 - 2107252*x + 2764291)*sqrt(x^2 + 3*x + 2) - 16410 8910*x + 3282178200)/(12*x^3 - 44*x^2 - 5*x + 100)
\[ \int \frac {1}{(5-2 x)^3 (4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=- \int \frac {1}{72 x^{5} \sqrt {x^{2} + 3 x + 2} - 348 x^{4} \sqrt {x^{2} + 3 x + 2} + 38 x^{3} \sqrt {x^{2} + 3 x + 2} + 1515 x^{2} \sqrt {x^{2} + 3 x + 2} - 600 x \sqrt {x^{2} + 3 x + 2} - 2000 \sqrt {x^{2} + 3 x + 2}}\, dx \] Input:
integrate(1/(5-2*x)**3/(4+3*x)**2/(x**2+3*x+2)**(1/2),x)
Output:
-Integral(1/(72*x**5*sqrt(x**2 + 3*x + 2) - 348*x**4*sqrt(x**2 + 3*x + 2) + 38*x**3*sqrt(x**2 + 3*x + 2) + 1515*x**2*sqrt(x**2 + 3*x + 2) - 600*x*sq rt(x**2 + 3*x + 2) - 2000*sqrt(x**2 + 3*x + 2)), x)
\[ \int \frac {1}{(5-2 x)^3 (4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\int { -\frac {1}{\sqrt {x^{2} + 3 \, x + 2} {\left (3 \, x + 4\right )}^{2} {\left (2 \, x - 5\right )}^{3}} \,d x } \] Input:
integrate(1/(5-2*x)^3/(4+3*x)^2/(x^2+3*x+2)^(1/2),x, algorithm="maxima")
Output:
-integrate(1/(sqrt(x^2 + 3*x + 2)*(3*x + 4)^2*(2*x - 5)^3), x)
Leaf count of result is larger than twice the leaf count of optimal. 549 vs. \(2 (105) = 210\).
Time = 0.40 (sec) , antiderivative size = 549, normalized size of antiderivative = 4.13 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx =\text {Too large to display} \] Input:
integrate(1/(5-2*x)^3/(4+3*x)^2/(x^2+3*x+2)^(1/2),x, algorithm="giac")
Output:
-140557/7774822503*sqrt(14)*sqrt(2)*log(abs(-12*sqrt(14) - 10*(2*sqrt(2)*s qrt(1/(3*x + 4) - 2/(3*x + 4)^2 + 1) - 3)/(4/(3*x + 4) - 1) + 46)/abs(12*s qrt(14) - 10*(2*sqrt(2)*sqrt(1/(3*x + 4) - 2/(3*x + 4)^2 + 1) - 3)/(4/(3*x + 4) - 1) + 46))/sgn(1/(3*x + 4)) + 1/111068892900*(1510998300*sqrt(14)*s qrt(2)*arcsin(1/3) - 2140580925*sqrt(14)*arcsin(1/3) - 674673600*sqrt(2)*l og((3*sqrt(14) - 5*sqrt(2) - 4)/(3*sqrt(14) + 5*sqrt(2) + 4)) - 1922192202 *sqrt(14)*sqrt(2) + 2713337232*sqrt(14) + 955787600*log((3*sqrt(14) - 5*sq rt(2) - 4)/(3*sqrt(14) + 5*sqrt(2) + 4)))*sgn(1/(3*x + 4))/(17*sqrt(14)*sq rt(2) - 24*sqrt(14)) + 1269/2238728*sqrt(2)*arcsin(-4/3/(3*x + 4) + 1/3)/s gn(1/(3*x + 4)) + 27/24334*sqrt(1/(3*x + 4) - 2/(3*x + 4)^2 + 1)/sgn(1/(3* x + 4)) + 8/402423525*(66575*sqrt(2) - 325565*sqrt(2)*(2*sqrt(2)*sqrt(1/(3 *x + 4) - 2/(3*x + 4)^2 + 1) - 3)^3/(4/(3*x + 4) - 1)^3 + 2884029*sqrt(2)* (2*sqrt(2)*sqrt(1/(3*x + 4) - 2/(3*x + 4)^2 + 1) - 3)^2/(4/(3*x + 4) - 1)^ 2 - 899415*sqrt(2)*(2*sqrt(2)*sqrt(1/(3*x + 4) - 2/(3*x + 4)^2 + 1) - 3)/( 4/(3*x + 4) - 1))/((5*(2*sqrt(2)*sqrt(1/(3*x + 4) - 2/(3*x + 4)^2 + 1) - 3 )^2/(4/(3*x + 4) - 1)^2 - 46*(2*sqrt(2)*sqrt(1/(3*x + 4) - 2/(3*x + 4)^2 + 1) - 3)/(4/(3*x + 4) - 1) + 5)^2*sgn(1/(3*x + 4)))
Timed out. \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=-\int \frac {1}{{\left (2\,x-5\right )}^3\,{\left (3\,x+4\right )}^2\,\sqrt {x^2+3\,x+2}} \,d x \] Input:
int(-1/((2*x - 5)^3*(3*x + 4)^2*(3*x + x^2 + 2)^(1/2)),x)
Output:
-int(1/((2*x - 5)^3*(3*x + 4)^2*(3*x + x^2 + 2)^(1/2)), x)
Time = 0.19 (sec) , antiderivative size = 395, normalized size of antiderivative = 2.97 \[ \int \frac {1}{(5-2 x)^3 (4+3 x)^2 \sqrt {2+3 x+x^2}} \, dx=\frac {2961556668 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x^{3}-10859041116 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x^{2}-1233981945 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right ) x +24679638900 \sqrt {2}\, \mathit {atan} \left (\frac {3 \sqrt {x^{2}+3 x +2}+3 x +4}{\sqrt {2}}\right )+2757029688 \sqrt {x^{2}+3 x +2}\, x^{2}-14249238024 \sqrt {x^{2}+3 x +2}\, x +18692135742 \sqrt {x^{2}+3 x +2}-94454304 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right ) x^{3}+346332448 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right ) x^{2}+39355960 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right ) x -787119200 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}-3 \sqrt {7}+2 x -5\right )+94454304 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right ) x^{3}-346332448 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right ) x^{2}-39355960 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right ) x +787119200 \sqrt {7}\, \mathrm {log}\left (2 \sqrt {x^{2}+3 x +2}+3 \sqrt {7}+2 x -5\right )-2757591072 x^{3}+10111167264 x^{2}+1148996280 x -22979925600}{2612340361008 x^{3}-9578581323696 x^{2}-1088475150420 x +21769503008400} \] Input:
int(1/(5-2*x)^3/(4+3*x)^2/(x^2+3*x+2)^(1/2),x)
Output:
(2961556668*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2))*x**3 - 10859041116*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2))*x** 2 - 1233981945*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2))*x + 24679638900*sqrt(2)*atan((3*sqrt(x**2 + 3*x + 2) + 3*x + 4)/sqrt(2)) + 2 757029688*sqrt(x**2 + 3*x + 2)*x**2 - 14249238024*sqrt(x**2 + 3*x + 2)*x + 18692135742*sqrt(x**2 + 3*x + 2) - 94454304*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) - 3*sqrt(7) + 2*x - 5)*x**3 + 346332448*sqrt(7)*log(2*sqrt(x**2 + 3* x + 2) - 3*sqrt(7) + 2*x - 5)*x**2 + 39355960*sqrt(7)*log(2*sqrt(x**2 + 3* x + 2) - 3*sqrt(7) + 2*x - 5)*x - 787119200*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) - 3*sqrt(7) + 2*x - 5) + 94454304*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5)*x**3 - 346332448*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5)*x**2 - 39355960*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5)*x + 787119200*sqrt(7)*log(2*sqrt(x**2 + 3*x + 2) + 3*sqrt(7) + 2*x - 5) - 2757591072*x**3 + 10111167264*x**2 + 1148996280*x - 22979925600)/(217695030084*(12*x**3 - 44*x**2 - 5*x + 100))