Integrand size = 27, antiderivative size = 108 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {91287}{32} \sqrt {4+3 x}-\frac {1323}{16} (4+3 x)^{3/2}-\frac {2795}{648} (4+3 x)^{5/2}-\frac {29}{108} (4+3 x)^{7/2}-\frac {1}{54} (4+3 x)^{9/2}-\frac {1}{891} (4+3 x)^{11/2}+\frac {91287}{32} \sqrt {\frac {23}{2}} \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right ) \] Output:
-91287/32*(4+3*x)^(1/2)-1323/16*(4+3*x)^(3/2)-2795/648*(4+3*x)^(5/2)-29/10 8*(4+3*x)^(7/2)-1/54*(4+3*x)^(9/2)-1/891*(4+3*x)^(11/2)+91287/64*46^(1/2)* arctanh(1/23*46^(1/2)*(4+3*x)^(1/2))
Time = 0.13 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.63 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {\sqrt {4+3 x} \left (31130887+3885042 x+858060 x^2+191016 x^3+31536 x^4+2592 x^5\right )}{9504}+\frac {91287}{32} \sqrt {\frac {23}{2}} \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right ) \] Input:
Integrate[((4 + 3*x)^(3/2)*(2 + 3*x + x^2)^2)/(5 - 2*x),x]
Output:
-1/9504*(Sqrt[4 + 3*x]*(31130887 + 3885042*x + 858060*x^2 + 191016*x^3 + 3 1536*x^4 + 2592*x^5)) + (91287*Sqrt[23/2]*ArcTanh[Sqrt[2/23]*Sqrt[4 + 3*x] ])/32
Time = 0.38 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {1192, 1584, 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.
\(\displaystyle \int \frac {(3 x+4)^{3/2} \left (x^2+3 x+2\right )^2}{5-2 x} \, dx\) |
\(\Big \downarrow \) 1192 |
\(\displaystyle \frac {2}{81} \int \frac {(3 x+4)^2 \left (-(3 x+4)^2-3 x-2\right )^2}{23-2 (3 x+4)}d\sqrt {3 x+4}\) |
\(\Big \downarrow \) 1584 |
\(\displaystyle \frac {2}{81} \int \left (-\frac {1}{2} (3 x+4)^5-\frac {27}{4} (3 x+4)^4-\frac {609}{8} (3 x+4)^3-\frac {13975}{16} (3 x+4)^2-\frac {321489}{32} (3 x+4)+\frac {170067681}{64 (23-2 (3 x+4))}-\frac {7394247}{64}\right )d\sqrt {3 x+4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{81} \left (\frac {7394247}{64} \sqrt {\frac {23}{2}} \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )-\frac {1}{22} (3 x+4)^{11/2}-\frac {3}{4} (3 x+4)^{9/2}-\frac {87}{8} (3 x+4)^{7/2}-\frac {2795}{16} (3 x+4)^{5/2}-\frac {107163}{32} (3 x+4)^{3/2}-\frac {7394247}{64} \sqrt {3 x+4}\right )\) |
Input:
Int[((4 + 3*x)^(3/2)*(2 + 3*x + x^2)^2)/(5 - 2*x),x]
Output:
(2*((-7394247*Sqrt[4 + 3*x])/64 - (107163*(4 + 3*x)^(3/2))/32 - (2795*(4 + 3*x)^(5/2))/16 - (87*(4 + 3*x)^(7/2))/8 - (3*(4 + 3*x)^(9/2))/4 - (4 + 3* x)^(11/2)/22 + (7394247*Sqrt[23/2]*ArcTanh[Sqrt[2/23]*Sqrt[4 + 3*x]])/64)) /81
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^( 2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + ( c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(f*x)^m*(d + e*x^2)^q* (a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, q}, x] && NeQ[ b^2 - 4*a*c, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Time = 1.69 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.50
method | result | size |
risch | \(-\frac {\left (2592 x^{5}+31536 x^{4}+191016 x^{3}+858060 x^{2}+3885042 x +31130887\right ) \sqrt {3 x +4}}{9504}+\frac {91287 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{64}\) | \(54\) |
pseudoelliptic | \(-\frac {\left (2592 x^{5}+31536 x^{4}+191016 x^{3}+858060 x^{2}+3885042 x +31130887\right ) \sqrt {3 x +4}}{9504}+\frac {91287 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{64}\) | \(54\) |
derivativedivides | \(-\frac {91287 \sqrt {3 x +4}}{32}-\frac {1323 \left (3 x +4\right )^{\frac {3}{2}}}{16}-\frac {2795 \left (3 x +4\right )^{\frac {5}{2}}}{648}-\frac {29 \left (3 x +4\right )^{\frac {7}{2}}}{108}-\frac {\left (3 x +4\right )^{\frac {9}{2}}}{54}-\frac {\left (3 x +4\right )^{\frac {11}{2}}}{891}+\frac {91287 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{64}\) | \(74\) |
default | \(-\frac {91287 \sqrt {3 x +4}}{32}-\frac {1323 \left (3 x +4\right )^{\frac {3}{2}}}{16}-\frac {2795 \left (3 x +4\right )^{\frac {5}{2}}}{648}-\frac {29 \left (3 x +4\right )^{\frac {7}{2}}}{108}-\frac {\left (3 x +4\right )^{\frac {9}{2}}}{54}-\frac {\left (3 x +4\right )^{\frac {11}{2}}}{891}+\frac {91287 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{64}\) | \(74\) |
trager | \(\left (-\frac {3}{11} x^{5}-\frac {73}{22} x^{4}-\frac {2653}{132} x^{3}-\frac {7945}{88} x^{2}-\frac {647507}{1584} x -\frac {31130887}{9504}\right ) \sqrt {3 x +4}+\frac {91287 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right ) \ln \left (-\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right ) x +92 \sqrt {3 x +4}+31 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right )}{-5+2 x}\right )}{128}\) | \(80\) |
Input:
int((3*x+4)^(3/2)*(x^2+3*x+2)^2/(5-2*x),x,method=_RETURNVERBOSE)
Output:
-1/9504*(2592*x^5+31536*x^4+191016*x^3+858060*x^2+3885042*x+31130887)*(3*x +4)^(1/2)+91287/64*46^(1/2)*arctanh(1/23*46^(1/2)*(3*x+4)^(1/2))
Time = 0.07 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.61 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {1}{9504} \, {\left (2592 \, x^{5} + 31536 \, x^{4} + 191016 \, x^{3} + 858060 \, x^{2} + 3885042 \, x + 31130887\right )} \sqrt {3 \, x + 4} + \frac {91287}{64} \, \sqrt {\frac {23}{2}} \log \left (\frac {4 \, \sqrt {\frac {23}{2}} \sqrt {3 \, x + 4} + 6 \, x + 31}{2 \, x - 5}\right ) \] Input:
integrate((4+3*x)^(3/2)*(x^2+3*x+2)^2/(5-2*x),x, algorithm="fricas")
Output:
-1/9504*(2592*x^5 + 31536*x^4 + 191016*x^3 + 858060*x^2 + 3885042*x + 3113 0887)*sqrt(3*x + 4) + 91287/64*sqrt(23/2)*log((4*sqrt(23/2)*sqrt(3*x + 4) + 6*x + 31)/(2*x - 5))
Time = 2.23 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.02 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=- \frac {\left (3 x + 4\right )^{\frac {11}{2}}}{891} - \frac {\left (3 x + 4\right )^{\frac {9}{2}}}{54} - \frac {29 \left (3 x + 4\right )^{\frac {7}{2}}}{108} - \frac {2795 \left (3 x + 4\right )^{\frac {5}{2}}}{648} - \frac {1323 \left (3 x + 4\right )^{\frac {3}{2}}}{16} - \frac {91287 \sqrt {3 x + 4}}{32} - \frac {91287 \sqrt {46} \left (\log {\left (\sqrt {3 x + 4} - \frac {\sqrt {46}}{2} \right )} - \log {\left (\sqrt {3 x + 4} + \frac {\sqrt {46}}{2} \right )}\right )}{128} \] Input:
integrate((4+3*x)**(3/2)*(x**2+3*x+2)**2/(5-2*x),x)
Output:
-(3*x + 4)**(11/2)/891 - (3*x + 4)**(9/2)/54 - 29*(3*x + 4)**(7/2)/108 - 2 795*(3*x + 4)**(5/2)/648 - 1323*(3*x + 4)**(3/2)/16 - 91287*sqrt(3*x + 4)/ 32 - 91287*sqrt(46)*(log(sqrt(3*x + 4) - sqrt(46)/2) - log(sqrt(3*x + 4) + sqrt(46)/2))/128
Time = 0.11 (sec) , antiderivative size = 91, normalized size of antiderivative = 0.84 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {1}{891} \, {\left (3 \, x + 4\right )}^{\frac {11}{2}} - \frac {1}{54} \, {\left (3 \, x + 4\right )}^{\frac {9}{2}} - \frac {29}{108} \, {\left (3 \, x + 4\right )}^{\frac {7}{2}} - \frac {2795}{648} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} - \frac {1323}{16} \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - \frac {91287}{128} \, \sqrt {46} \log \left (-\frac {\sqrt {46} - 2 \, \sqrt {3 \, x + 4}}{\sqrt {46} + 2 \, \sqrt {3 \, x + 4}}\right ) - \frac {91287}{32} \, \sqrt {3 \, x + 4} \] Input:
integrate((4+3*x)^(3/2)*(x^2+3*x+2)^2/(5-2*x),x, algorithm="maxima")
Output:
-1/891*(3*x + 4)^(11/2) - 1/54*(3*x + 4)^(9/2) - 29/108*(3*x + 4)^(7/2) - 2795/648*(3*x + 4)^(5/2) - 1323/16*(3*x + 4)^(3/2) - 91287/128*sqrt(46)*lo g(-(sqrt(46) - 2*sqrt(3*x + 4))/(sqrt(46) + 2*sqrt(3*x + 4))) - 91287/32*s qrt(3*x + 4)
Time = 0.25 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.87 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {1}{891} \, {\left (3 \, x + 4\right )}^{\frac {11}{2}} - \frac {1}{54} \, {\left (3 \, x + 4\right )}^{\frac {9}{2}} - \frac {29}{108} \, {\left (3 \, x + 4\right )}^{\frac {7}{2}} - \frac {2795}{648} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} - \frac {1323}{16} \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - \frac {91287}{128} \, \sqrt {46} \log \left (\frac {{\left | -2 \, \sqrt {46} + 4 \, \sqrt {3 \, x + 4} \right |}}{2 \, {\left (\sqrt {46} + 2 \, \sqrt {3 \, x + 4}\right )}}\right ) - \frac {91287}{32} \, \sqrt {3 \, x + 4} \] Input:
integrate((4+3*x)^(3/2)*(x^2+3*x+2)^2/(5-2*x),x, algorithm="giac")
Output:
-1/891*(3*x + 4)^(11/2) - 1/54*(3*x + 4)^(9/2) - 29/108*(3*x + 4)^(7/2) - 2795/648*(3*x + 4)^(5/2) - 1323/16*(3*x + 4)^(3/2) - 91287/128*sqrt(46)*lo g(1/2*abs(-2*sqrt(46) + 4*sqrt(3*x + 4))/(sqrt(46) + 2*sqrt(3*x + 4))) - 9 1287/32*sqrt(3*x + 4)
Time = 0.07 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.69 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {91287\,\sqrt {3\,x+4}}{32}-\frac {1323\,{\left (3\,x+4\right )}^{3/2}}{16}-\frac {2795\,{\left (3\,x+4\right )}^{5/2}}{648}-\frac {29\,{\left (3\,x+4\right )}^{7/2}}{108}-\frac {{\left (3\,x+4\right )}^{9/2}}{54}-\frac {{\left (3\,x+4\right )}^{11/2}}{891}-\frac {\sqrt {46}\,\mathrm {atan}\left (\frac {\sqrt {46}\,\sqrt {3\,x+4}\,1{}\mathrm {i}}{23}\right )\,91287{}\mathrm {i}}{64} \] Input:
int(-((3*x + 4)^(3/2)*(3*x + x^2 + 2)^2)/(2*x - 5),x)
Output:
- (46^(1/2)*atan((46^(1/2)*(3*x + 4)^(1/2)*1i)/23)*91287i)/64 - (91287*(3* x + 4)^(1/2))/32 - (1323*(3*x + 4)^(3/2))/16 - (2795*(3*x + 4)^(5/2))/648 - (29*(3*x + 4)^(7/2))/108 - (3*x + 4)^(9/2)/54 - (3*x + 4)^(11/2)/891
Time = 0.18 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.89 \[ \int \frac {(4+3 x)^{3/2} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {3 \sqrt {3 x +4}\, x^{5}}{11}-\frac {73 \sqrt {3 x +4}\, x^{4}}{22}-\frac {2653 \sqrt {3 x +4}\, x^{3}}{132}-\frac {7945 \sqrt {3 x +4}\, x^{2}}{88}-\frac {647507 \sqrt {3 x +4}\, x}{1584}-\frac {31130887 \sqrt {3 x +4}}{9504}-\frac {91287 \sqrt {46}\, \mathrm {log}\left (2 \sqrt {3 x +4}-\sqrt {46}\right )}{128}+\frac {91287 \sqrt {46}\, \mathrm {log}\left (2 \sqrt {3 x +4}+\sqrt {46}\right )}{128} \] Input:
int((4+3*x)^(3/2)*(x^2+3*x+2)^2/(5-2*x),x)
Output:
( - 10368*sqrt(3*x + 4)*x**5 - 126144*sqrt(3*x + 4)*x**4 - 764064*sqrt(3*x + 4)*x**3 - 3432240*sqrt(3*x + 4)*x**2 - 15540168*sqrt(3*x + 4)*x - 12452 3548*sqrt(3*x + 4) - 27112239*sqrt(46)*log(2*sqrt(3*x + 4) - sqrt(46)) + 2 7112239*sqrt(46)*log(2*sqrt(3*x + 4) + sqrt(46)))/38016