Integrand size = 27, antiderivative size = 80 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x) (4+3 x)^{3/2}} \, dx=-\frac {8}{1863 \sqrt {4+3 x}}-\frac {203}{108} \sqrt {4+3 x}-\frac {1}{18} (4+3 x)^{3/2}-\frac {1}{405} (4+3 x)^{5/2}+\frac {3969 \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right )}{92 \sqrt {46}} \] Output:
-8/1863/(4+3*x)^(1/2)-203/108*(4+3*x)^(1/2)-1/18*(4+3*x)^(3/2)-1/405*(4+3* x)^(5/2)+3969/4232*46^(1/2)*arctanh(1/23*46^(1/2)*(4+3*x)^(1/2))
Time = 0.16 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.70 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x) (4+3 x)^{3/2}} \, dx=\frac {-106436-91011 x-9522 x^2-828 x^3}{12420 \sqrt {4+3 x}}+\frac {3969 \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {4+3 x}\right )}{92 \sqrt {46}} \] Input:
Integrate[(2 + 3*x + x^2)^2/((5 - 2*x)*(4 + 3*x)^(3/2)),x]
Output:
(-106436 - 91011*x - 9522*x^2 - 828*x^3)/(12420*Sqrt[4 + 3*x]) + (3969*Arc Tanh[Sqrt[2/23]*Sqrt[4 + 3*x]])/(92*Sqrt[46])
Time = 0.37 (sec) , antiderivative size = 84, normalized size of antiderivative = 1.05, 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.
\(\displaystyle \int \frac {\left (x^2+3 x+2\right )^2}{(5-2 x) (3 x+4)^{3/2}} \, dx\) |
\(\Big \downarrow \) 1192 |
\(\displaystyle \frac {2}{81} \int \frac {\left (-(3 x+4)^2-3 x-2\right )^2}{(3 x+4) (23-2 (3 x+4))}d\sqrt {3 x+4}\) |
\(\Big \downarrow \) 1584 |
\(\displaystyle \frac {2}{81} \int \left (-\frac {1}{2} (3 x+4)^2-\frac {27}{4} (3 x+4)-\frac {321489}{184 (2 (3 x+4)-23)}-\frac {609}{8}+\frac {4}{23 (3 x+4)}\right )d\sqrt {3 x+4}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {2}{81} \left (\frac {321489 \text {arctanh}\left (\sqrt {\frac {2}{23}} \sqrt {3 x+4}\right )}{184 \sqrt {46}}-\frac {1}{10} (3 x+4)^{5/2}-\frac {9}{4} (3 x+4)^{3/2}-\frac {609}{8} \sqrt {3 x+4}-\frac {4}{23 \sqrt {3 x+4}}\right )\) |
Input:
Int[(2 + 3*x + x^2)^2/((5 - 2*x)*(4 + 3*x)^(3/2)),x]
Output:
(2*(-4/(23*Sqrt[4 + 3*x]) - (609*Sqrt[4 + 3*x])/8 - (9*(4 + 3*x)^(3/2))/4 - (4 + 3*x)^(5/2)/10 + (321489*ArcTanh[Sqrt[2/23]*Sqrt[4 + 3*x]])/(184*Sqr t[46])))/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 = 1.71 (sec) , antiderivative size = 44, normalized size of antiderivative = 0.55
method | result | size |
risch | \(-\frac {828 x^{3}+9522 x^{2}+91011 x +106436}{12420 \sqrt {3 x +4}}+\frac {3969 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{4232}\) | \(44\) |
pseudoelliptic | \(\frac {535815 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right ) \sqrt {3 x +4}-38088 x^{3}-438012 x^{2}-4186506 x -4896056}{571320 \sqrt {3 x +4}}\) | \(50\) |
derivativedivides | \(-\frac {8}{1863 \sqrt {3 x +4}}-\frac {203 \sqrt {3 x +4}}{108}-\frac {\left (3 x +4\right )^{\frac {3}{2}}}{18}-\frac {\left (3 x +4\right )^{\frac {5}{2}}}{405}+\frac {3969 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{4232}\) | \(56\) |
default | \(-\frac {8}{1863 \sqrt {3 x +4}}-\frac {203 \sqrt {3 x +4}}{108}-\frac {\left (3 x +4\right )^{\frac {3}{2}}}{18}-\frac {\left (3 x +4\right )^{\frac {5}{2}}}{405}+\frac {3969 \sqrt {46}\, \operatorname {arctanh}\left (\frac {\sqrt {46}\, \sqrt {3 x +4}}{23}\right )}{4232}\) | \(56\) |
trager | \(-\frac {828 x^{3}+9522 x^{2}+91011 x +106436}{12420 \sqrt {3 x +4}}-\frac {3969 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right ) \ln \left (-\frac {-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right ) x +92 \sqrt {3 x +4}-31 \operatorname {RootOf}\left (\textit {\_Z}^{2}-46\right )}{-5+2 x}\right )}{8464}\) | \(71\) |
Input:
int((x^2+3*x+2)^2/(5-2*x)/(3*x+4)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/12420*(828*x^3+9522*x^2+91011*x+106436)/(3*x+4)^(1/2)+3969/4232*46^(1/2 )*arctanh(1/23*46^(1/2)*(3*x+4)^(1/2))
Time = 0.08 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.88 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x) (4+3 x)^{3/2}} \, dx=\frac {535815 \, \sqrt {46} {\left (3 \, x + 4\right )} \log \left (\frac {2 \, \sqrt {46} \sqrt {3 \, x + 4} + 6 \, x + 31}{2 \, x - 5}\right ) - 92 \, {\left (828 \, x^{3} + 9522 \, x^{2} + 91011 \, x + 106436\right )} \sqrt {3 \, x + 4}}{1142640 \, {\left (3 \, x + 4\right )}} \] Input:
integrate((x^2+3*x+2)^2/(5-2*x)/(4+3*x)^(3/2),x, algorithm="fricas")
Output:
1/1142640*(535815*sqrt(46)*(3*x + 4)*log((2*sqrt(46)*sqrt(3*x + 4) + 6*x + 31)/(2*x - 5)) - 92*(828*x^3 + 9522*x^2 + 91011*x + 106436)*sqrt(3*x + 4) )/(3*x + 4)
Time = 2.59 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.09 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x) (4+3 x)^{3/2}} \, dx=- \frac {\left (3 x + 4\right )^{\frac {5}{2}}}{405} - \frac {\left (3 x + 4\right )^{\frac {3}{2}}}{18} - \frac {203 \sqrt {3 x + 4}}{108} - \frac {3969 \sqrt {46} \left (\log {\left (\sqrt {3 x + 4} - \frac {\sqrt {46}}{2} \right )} - \log {\left (\sqrt {3 x + 4} + \frac {\sqrt {46}}{2} \right )}\right )}{8464} - \frac {8}{1863 \sqrt {3 x + 4}} \] Input:
integrate((x**2+3*x+2)**2/(5-2*x)/(4+3*x)**(3/2),x)
Output:
-(3*x + 4)**(5/2)/405 - (3*x + 4)**(3/2)/18 - 203*sqrt(3*x + 4)/108 - 3969 *sqrt(46)*(log(sqrt(3*x + 4) - sqrt(46)/2) - log(sqrt(3*x + 4) + sqrt(46)/ 2))/8464 - 8/(1863*sqrt(3*x + 4))
Time = 0.12 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.91 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x) (4+3 x)^{3/2}} \, dx=-\frac {1}{405} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} - \frac {1}{18} \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - \frac {3969}{8464} \, \sqrt {46} \log \left (-\frac {\sqrt {46} - 2 \, \sqrt {3 \, x + 4}}{\sqrt {46} + 2 \, \sqrt {3 \, x + 4}}\right ) - \frac {203}{108} \, \sqrt {3 \, x + 4} - \frac {8}{1863 \, \sqrt {3 \, x + 4}} \] Input:
integrate((x^2+3*x+2)^2/(5-2*x)/(4+3*x)^(3/2),x, algorithm="maxima")
Output:
-1/405*(3*x + 4)^(5/2) - 1/18*(3*x + 4)^(3/2) - 3969/8464*sqrt(46)*log(-(s qrt(46) - 2*sqrt(3*x + 4))/(sqrt(46) + 2*sqrt(3*x + 4))) - 203/108*sqrt(3* x + 4) - 8/1863/sqrt(3*x + 4)
Time = 0.26 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.95 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x) (4+3 x)^{3/2}} \, dx=-\frac {1}{405} \, {\left (3 \, x + 4\right )}^{\frac {5}{2}} - \frac {1}{18} \, {\left (3 \, x + 4\right )}^{\frac {3}{2}} - \frac {3969}{8464} \, \sqrt {46} \log \left (\frac {{\left | -2 \, \sqrt {46} + 4 \, \sqrt {3 \, x + 4} \right |}}{2 \, {\left (\sqrt {46} + 2 \, \sqrt {3 \, x + 4}\right )}}\right ) - \frac {203}{108} \, \sqrt {3 \, x + 4} - \frac {8}{1863 \, \sqrt {3 \, x + 4}} \] Input:
integrate((x^2+3*x+2)^2/(5-2*x)/(4+3*x)^(3/2),x, algorithm="giac")
Output:
-1/405*(3*x + 4)^(5/2) - 1/18*(3*x + 4)^(3/2) - 3969/8464*sqrt(46)*log(1/2 *abs(-2*sqrt(46) + 4*sqrt(3*x + 4))/(sqrt(46) + 2*sqrt(3*x + 4))) - 203/10 8*sqrt(3*x + 4) - 8/1863/sqrt(3*x + 4)
Time = 12.47 (sec) , antiderivative size = 57, normalized size of antiderivative = 0.71 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x) (4+3 x)^{3/2}} \, dx=-\frac {8}{1863\,\sqrt {3\,x+4}}-\frac {203\,\sqrt {3\,x+4}}{108}-\frac {{\left (3\,x+4\right )}^{3/2}}{18}-\frac {{\left (3\,x+4\right )}^{5/2}}{405}-\frac {\sqrt {46}\,\mathrm {atan}\left (\frac {\sqrt {46}\,\sqrt {3\,x+4}\,1{}\mathrm {i}}{23}\right )\,3969{}\mathrm {i}}{4232} \] Input:
int(-(3*x + x^2 + 2)^2/((2*x - 5)*(3*x + 4)^(3/2)),x)
Output:
- (46^(1/2)*atan((46^(1/2)*(3*x + 4)^(1/2)*1i)/23)*3969i)/4232 - 8/(1863*( 3*x + 4)^(1/2)) - (203*(3*x + 4)^(1/2))/108 - (3*x + 4)^(3/2)/18 - (3*x + 4)^(5/2)/405
Time = 0.17 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.89 \[ \int \frac {\left (2+3 x+x^2\right )^2}{(5-2 x) (4+3 x)^{3/2}} \, dx=\frac {-535815 \sqrt {3 x +4}\, \sqrt {46}\, \mathrm {log}\left (2 \sqrt {3 x +4}-\sqrt {46}\right )+535815 \sqrt {3 x +4}\, \sqrt {46}\, \mathrm {log}\left (2 \sqrt {3 x +4}+\sqrt {46}\right )-76176 x^{3}-876024 x^{2}-8373012 x -9792112}{1142640 \sqrt {3 x +4}} \] Input:
int((x^2+3*x+2)^2/(5-2*x)/(4+3*x)^(3/2),x)
Output:
( - 535815*sqrt(3*x + 4)*sqrt(46)*log(2*sqrt(3*x + 4) - sqrt(46)) + 535815 *sqrt(3*x + 4)*sqrt(46)*log(2*sqrt(3*x + 4) + sqrt(46)) - 76176*x**3 - 876 024*x**2 - 8373012*x - 9792112)/(1142640*sqrt(3*x + 4))