Integrand size = 27, antiderivative size = 95 \[ \int \frac {\sqrt {4+3 x} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {3969}{16} \sqrt {4+3 x}-\frac {13975 (4+3 x)^{3/2}}{1944}-\frac {203}{540} (4+3 x)^{5/2}-\frac {1}{42} (4+3 x)^{7/2}-\frac {1}{729} (4+3 x)^{9/2}+\frac {3969}{16} \sqrt {\frac {23}{2}} \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right ) \] Output:
-3969/16*(4+3*x)^(1/2)-13975/1944*(4+3*x)^(3/2)-203/540*(4+3*x)^(5/2)-1/42 *(4+3*x)^(7/2)-1/729*(4+3*x)^(9/2)+3969/32*46^(1/2)*arctanh(1/23*46^(1/2)* (4+3*x)^(1/2))
Time = 0.12 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.66 \[ \int \frac {\sqrt {4+3 x} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {\sqrt {4+3 x} \left (116228963+14317242 x+2914812 x^2+504360 x^3+45360 x^4\right )}{408240}+\frac {3969}{16} \sqrt {\frac {23}{2}} \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right ) \] Input:
Integrate[(Sqrt[4 + 3*x]*(2 + 3*x + x^2)^2)/(5 - 2*x),x]
Output:
-1/408240*(Sqrt[4 + 3*x]*(116228963 + 14317242*x + 2914812*x^2 + 504360*x^ 3 + 45360*x^4)) + (3969*Sqrt[23/2]*ArcTanh[Sqrt[2/23]*Sqrt[4 + 3*x]])/16
Time = 0.36 (sec) , antiderivative size = 99, 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.
\(\displaystyle \int \frac {\sqrt {3 x+4} \left (x^2+3 x+2\right )^2}{5-2 x} \, dx\) |
\(\Big \downarrow \) 1192 |
\(\displaystyle \frac {2}{81} \int \frac {(3 x+4) \left (-(3 x+4)^2-3 x-2\right )^2}{23-2 (3 x+4)}d\sqrt {3 x+4}\) |
\(\Big \downarrow \) 1584 |
\(\displaystyle \frac {2}{81} \int \left (-\frac {1}{2} (3 x+4)^4-\frac {27}{4} (3 x+4)^3-\frac {609}{8} (3 x+4)^2-\frac {13975}{16} (3 x+4)+\frac {7394247}{32 (23-2 (3 x+4))}-\frac {321489}{32}\right )d\sqrt {3 x+4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{81} \left (\frac {321489}{32} \sqrt {\frac {23}{2}} \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )-\frac {1}{18} (3 x+4)^{9/2}-\frac {27}{28} (3 x+4)^{7/2}-\frac {609}{40} (3 x+4)^{5/2}-\frac {13975}{48} (3 x+4)^{3/2}-\frac {321489}{32} \sqrt {3 x+4}\right )\) |
Input:
Int[(Sqrt[4 + 3*x]*(2 + 3*x + x^2)^2)/(5 - 2*x),x]
Output:
(2*((-321489*Sqrt[4 + 3*x])/32 - (13975*(4 + 3*x)^(3/2))/48 - (609*(4 + 3* x)^(5/2))/40 - (27*(4 + 3*x)^(7/2))/28 - (4 + 3*x)^(9/2)/18 + (321489*Sqrt [23/2]*ArcTanh[Sqrt[2/23]*Sqrt[4 + 3*x]])/32))/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 = 1.72 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.52
method | result | size |
risch | \(-\frac {\left (45360 x^{4}+504360 x^{3}+2914812 x^{2}+14317242 x +116228963\right ) \sqrt {3 x +4}}{408240}+\frac {3969 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{32}\) | \(49\) |
pseudoelliptic | \(-\frac {\left (45360 x^{4}+504360 x^{3}+2914812 x^{2}+14317242 x +116228963\right ) \sqrt {3 x +4}}{408240}+\frac {3969 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{32}\) | \(49\) |
derivativedivides | \(-\frac {3969 \sqrt {3 x +4}}{16}-\frac {13975 \left (3 x +4\right )^{\frac {3}{2}}}{1944}-\frac {203 \left (3 x +4\right )^{\frac {5}{2}}}{540}-\frac {\left (3 x +4\right )^{\frac {7}{2}}}{42}-\frac {\left (3 x +4\right )^{\frac {9}{2}}}{729}+\frac {3969 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{32}\) | \(65\) |
default | \(-\frac {3969 \sqrt {3 x +4}}{16}-\frac {13975 \left (3 x +4\right )^{\frac {3}{2}}}{1944}-\frac {203 \left (3 x +4\right )^{\frac {5}{2}}}{540}-\frac {\left (3 x +4\right )^{\frac {7}{2}}}{42}-\frac {\left (3 x +4\right )^{\frac {9}{2}}}{729}+\frac {3969 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{32}\) | \(65\) |
trager | \(\left (-\frac {1}{9} x^{4}-\frac {467}{378} x^{3}-\frac {26989}{3780} x^{2}-\frac {2386207}{68040} x -\frac {116228963}{408240}\right ) \sqrt {3 x +4}+\frac {3969 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right ) \ln \left (-\frac {6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right ) x +92 \sqrt {3 x +4}+31 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right )}{-5+2 x}\right )}{64}\) | \(75\) |
Input:
int((3*x+4)^(1/2)*(x^2+3*x+2)^2/(5-2*x),x,method=_RETURNVERBOSE)
Output:
-1/408240*(45360*x^4+504360*x^3+2914812*x^2+14317242*x+116228963)*(3*x+4)^ (1/2)+3969/32*46^(1/2)*arctanh(1/23*46^(1/2)*(3*x+4)^(1/2))
Time = 0.08 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.64 \[ \int \frac {\sqrt {4+3 x} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {1}{408240} \, {\left (45360 \, x^{4} + 504360 \, x^{3} + 2914812 \, x^{2} + 14317242 \, x + 116228963\right )} \sqrt {3 \, x + 4} + \frac {3969}{32} \, \sqrt {\frac {23}{2}} \log \left (\frac {4 \, \sqrt {\frac {23}{2}} \sqrt {3 \, x + 4} + 6 \, x + 31}{2 \, x - 5}\right ) \] Input:
integrate((4+3*x)^(1/2)*(x^2+3*x+2)^2/(5-2*x),x, algorithm="fricas")
Output:
-1/408240*(45360*x^4 + 504360*x^3 + 2914812*x^2 + 14317242*x + 116228963)* sqrt(3*x + 4) + 3969/32*sqrt(23/2)*log((4*sqrt(23/2)*sqrt(3*x + 4) + 6*x + 31)/(2*x - 5))
Time = 1.87 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.04 \[ \int \frac {\sqrt {4+3 x} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=- \frac {\left (3 x + 4\right )^{\frac {9}{2}}}{729} - \frac {\left (3 x + 4\right )^{\frac {7}{2}}}{42} - \frac {203 \left (3 x + 4\right )^{\frac {5}{2}}}{540} - \frac {13975 \left (3 x + 4\right )^{\frac {3}{2}}}{1944} - \frac {3969 \sqrt {3 x + 4}}{16} - \frac {3969 \sqrt {46} \left (\log {\left (\sqrt {3 x + 4} - \frac {\sqrt {46}}{2} \right )} - \log {\left (\sqrt {3 x + 4} + \frac {\sqrt {46}}{2} \right )}\right )}{64} \] Input:
integrate((4+3*x)**(1/2)*(x**2+3*x+2)**2/(5-2*x),x)
Output:
-(3*x + 4)**(9/2)/729 - (3*x + 4)**(7/2)/42 - 203*(3*x + 4)**(5/2)/540 - 1 3975*(3*x + 4)**(3/2)/1944 - 3969*sqrt(3*x + 4)/16 - 3969*sqrt(46)*(log(sq rt(3*x + 4) - sqrt(46)/2) - log(sqrt(3*x + 4) + sqrt(46)/2))/64
Time = 0.11 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {4+3 x} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {1}{729} \, {\left (3 \, x + 4\right )}^{\frac {9}{2}} - \frac {1}{42} \, {\left (3 \, x + 4\right )}^{\frac {7}{2}} - \frac {203}{540} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} - \frac {13975}{1944} \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - \frac {3969}{64} \, \sqrt {46} \log \left (-\frac {\sqrt {46} - 2 \, \sqrt {3 \, x + 4}}{\sqrt {46} + 2 \, \sqrt {3 \, x + 4}}\right ) - \frac {3969}{16} \, \sqrt {3 \, x + 4} \] Input:
integrate((4+3*x)^(1/2)*(x^2+3*x+2)^2/(5-2*x),x, algorithm="maxima")
Output:
-1/729*(3*x + 4)^(9/2) - 1/42*(3*x + 4)^(7/2) - 203/540*(3*x + 4)^(5/2) - 13975/1944*(3*x + 4)^(3/2) - 3969/64*sqrt(46)*log(-(sqrt(46) - 2*sqrt(3*x + 4))/(sqrt(46) + 2*sqrt(3*x + 4))) - 3969/16*sqrt(3*x + 4)
Time = 0.25 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {4+3 x} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {1}{729} \, {\left (3 \, x + 4\right )}^{\frac {9}{2}} - \frac {1}{42} \, {\left (3 \, x + 4\right )}^{\frac {7}{2}} - \frac {203}{540} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} - \frac {13975}{1944} \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - \frac {3969}{64} \, \sqrt {46} \log \left (\frac {{\left | -2 \, \sqrt {46} + 4 \, \sqrt {3 \, x + 4} \right |}}{2 \, {\left (\sqrt {46} + 2 \, \sqrt {3 \, x + 4}\right )}}\right ) - \frac {3969}{16} \, \sqrt {3 \, x + 4} \] Input:
integrate((4+3*x)^(1/2)*(x^2+3*x+2)^2/(5-2*x),x, algorithm="giac")
Output:
-1/729*(3*x + 4)^(9/2) - 1/42*(3*x + 4)^(7/2) - 203/540*(3*x + 4)^(5/2) - 13975/1944*(3*x + 4)^(3/2) - 3969/64*sqrt(46)*log(1/2*abs(-2*sqrt(46) + 4* sqrt(3*x + 4))/(sqrt(46) + 2*sqrt(3*x + 4))) - 3969/16*sqrt(3*x + 4)
Time = 12.63 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.69 \[ \int \frac {\sqrt {4+3 x} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {3969\,\sqrt {3\,x+4}}{16}-\frac {13975\,{\left (3\,x+4\right )}^{3/2}}{1944}-\frac {203\,{\left (3\,x+4\right )}^{5/2}}{540}-\frac {{\left (3\,x+4\right )}^{7/2}}{42}-\frac {{\left (3\,x+4\right )}^{9/2}}{729}-\frac {\sqrt {46}\,\mathrm {atan}\left (\frac {\sqrt {46}\,\sqrt {3\,x+4}\,1{}\mathrm {i}}{23}\right )\,3969{}\mathrm {i}}{32} \] Input:
int(-((3*x + 4)^(1/2)*(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)*3969i)/32 - (3969*(3*x + 4)^(1/2))/16 - (13975*(3*x + 4)^(3/2))/1944 - (203*(3*x + 4)^(5/2))/540 - (3*x + 4)^(7/2)/42 - (3*x + 4)^(9/2)/729
Time = 0.17 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.89 \[ \int \frac {\sqrt {4+3 x} \left (2+3 x+x^2\right )^2}{5-2 x} \, dx=-\frac {\sqrt {3 x +4}\, x^{4}}{9}-\frac {467 \sqrt {3 x +4}\, x^{3}}{378}-\frac {26989 \sqrt {3 x +4}\, x^{2}}{3780}-\frac {2386207 \sqrt {3 x +4}\, x}{68040}-\frac {116228963 \sqrt {3 x +4}}{408240}-\frac {3969 \sqrt {46}\, \mathrm {log}\left (2 \sqrt {3 x +4}-\sqrt {46}\right )}{64}+\frac {3969 \sqrt {46}\, \mathrm {log}\left (2 \sqrt {3 x +4}+\sqrt {46}\right )}{64} \] Input:
int((4+3*x)^(1/2)*(x^2+3*x+2)^2/(5-2*x),x)
Output:
( - 181440*sqrt(3*x + 4)*x**4 - 2017440*sqrt(3*x + 4)*x**3 - 11659248*sqrt (3*x + 4)*x**2 - 57268968*sqrt(3*x + 4)*x - 464915852*sqrt(3*x + 4) - 1012 69035*sqrt(46)*log(2*sqrt(3*x + 4) - sqrt(46)) + 101269035*sqrt(46)*log(2* sqrt(3*x + 4) + sqrt(46)))/1632960