Integrand size = 27, antiderivative size = 135 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=\frac {3 (93+43 x) \sqrt {2+5 x+3 x^2}}{16 (3+2 x)}-\frac {(8+x) \left (2+5 x+3 x^2\right )^{3/2}}{4 (3+2 x)^2}-\frac {343}{64} \sqrt {3} \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )+\frac {1329 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{64 \sqrt {5}} \]
-1/4*(8+x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^2-343/64*arctanh(1/6*(5+6*x)*3^(1/2 )/(3*x^2+5*x+2)^(1/2))*3^(1/2)+1329/320*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^ 2+5*x+2)^(1/2))*5^(1/2)+3/16*(93+43*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)
Time = 0.42 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.76 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=\frac {1}{160} \left (-\frac {10 \sqrt {2+5 x+3 x^2} \left (-773-777 x-142 x^2+12 x^3\right )}{(3+2 x)^2}+1329 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )-1715 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \]
((-10*Sqrt[2 + 5*x + 3*x^2]*(-773 - 777*x - 142*x^2 + 12*x^3))/(3 + 2*x)^2 + 1329*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] - 1715*Sqrt[3]* ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/160
Time = 0.33 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.07, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1230, 27, 1230, 27, 1269, 1092, 219, 1154, 219}
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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^3} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {3}{32} \int -\frac {4 (43 x+36) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx-\frac {(x+8) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{8} \int \frac {(43 x+36) \sqrt {3 x^2+5 x+2}}{(2 x+3)^2}dx-\frac {(x+8) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {3}{8} \left (\frac {(43 x+93) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}-\frac {1}{8} \int \frac {2 (343 x+293)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{8} \left (\frac {(43 x+93) \sqrt {3 x^2+5 x+2}}{2 (2 x+3)}-\frac {1}{4} \int \frac {343 x+293}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {3}{8} \left (\frac {1}{4} \left (\frac {443}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {343}{2} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )+\frac {\sqrt {3 x^2+5 x+2} (43 x+93)}{2 (2 x+3)}\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {3}{8} \left (\frac {1}{4} \left (\frac {443}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-343 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (43 x+93)}{2 (2 x+3)}\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{8} \left (\frac {1}{4} \left (\frac {443}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {343 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (43 x+93)}{2 (2 x+3)}\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {3}{8} \left (\frac {1}{4} \left (-443 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )-\frac {343 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (43 x+93)}{2 (2 x+3)}\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{8} \left (\frac {1}{4} \left (\frac {443 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{2 \sqrt {5}}-\frac {343 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+5 x+2} (43 x+93)}{2 (2 x+3)}\right )-\frac {(x+8) \left (3 x^2+5 x+2\right )^{3/2}}{4 (2 x+3)^2}\) |
-1/4*((8 + x)*(2 + 5*x + 3*x^2)^(3/2))/(3 + 2*x)^2 + (3*(((93 + 43*x)*Sqrt [2 + 5*x + 3*x^2])/(2*(3 + 2*x)) + ((-343*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqr t[2 + 5*x + 3*x^2])])/(2*Sqrt[3]) + (443*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt [2 + 5*x + 3*x^2])])/(2*Sqrt[5]))/4))/8
3.25.25.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Simp[2 Subst[I nt[1/(4*c - x^2), x], x, (b + 2*c*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a , b, c}, x]
Int[1/(((d_.) + (e_.)*(x_))*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x_Sym bol] :> Simp[-2 Subst[Int[1/(4*c*d^2 - 4*b*d*e + 4*a*e^2 - x^2), x], x, ( 2*a*e - b*d - (2*c*d - b*e)*x)/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c , d, e}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/(e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, - 1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ [m + 2*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 0.34 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.79
| method | result | size |
| risch | \(-\frac {36 x^{5}-366 x^{4}-3017 x^{3}-6488 x^{2}-5419 x -1546}{16 \left (3+2 x \right )^{2} \sqrt {3 x^{2}+5 x +2}}-\frac {343 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{64}-\frac {1329 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{320}\) | \(107\) |
| trager | \(-\frac {\left (12 x^{3}-142 x^{2}-777 x -773\right ) \sqrt {3 x^{2}+5 x +2}}{16 \left (3+2 x \right )^{2}}-\frac {343 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{64}+\frac {1329 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{320}\) | \(128\) |
| default | \(\frac {31 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{50 \left (x +\frac {3}{2}\right )}+\frac {443 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{200}-\frac {171 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{160}-\frac {343 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{64}+\frac {1329 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{320}-\frac {1329 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{320}-\frac {31 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{100}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{40 \left (x +\frac {3}{2}\right )^{2}}\) | \(179\) |
-1/16*(36*x^5-366*x^4-3017*x^3-6488*x^2-5419*x-1546)/(3+2*x)^2/(3*x^2+5*x+ 2)^(1/2)-343/64*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-1329 /320*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19)^(1/2))
Time = 0.29 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.13 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=\frac {1715 \, \sqrt {3} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 1329 \, \sqrt {5} {\left (4 \, x^{2} + 12 \, x + 9\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 40 \, {\left (12 \, x^{3} - 142 \, x^{2} - 777 \, x - 773\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{640 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} \]
1/640*(1715*sqrt(3)*(4*x^2 + 12*x + 9)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2 )*(6*x + 5) + 72*x^2 + 120*x + 49) + 1329*sqrt(5)*(4*x^2 + 12*x + 9)*log(( 4*sqrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12*x + 9)) - 40*(12*x^3 - 142*x^2 - 777*x - 773)*sqrt(3*x^2 + 5*x + 2))/( 4*x^2 + 12*x + 9)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=- \int \left (- \frac {10 \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {23 x \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \left (- \frac {10 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\right )\, dx - \int \frac {3 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{8 x^{3} + 36 x^{2} + 54 x + 27}\, dx \]
-Integral(-10*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-23*x*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(-10*x**2*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27), x) - Integral(3*x**3*sqrt(3*x**2 + 5*x + 2)/(8*x**3 + 36*x**2 + 54*x + 27) , x)
Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.19 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=\frac {39}{40} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{10 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {513}{80} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x - \frac {343}{64} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) - \frac {1329}{320} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {237}{80} \, \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {31 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}{20 \, {\left (2 \, x + 3\right )}} \]
39/40*(3*x^2 + 5*x + 2)^(3/2) - 13/10*(3*x^2 + 5*x + 2)^(5/2)/(4*x^2 + 12* x + 9) - 513/80*sqrt(3*x^2 + 5*x + 2)*x - 343/64*sqrt(3)*log(sqrt(3)*sqrt( 3*x^2 + 5*x + 2) + 3*x + 5/2) - 1329/320*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) + 237/80*sqrt(3*x^2 + 5*x + 2) + 31/20*(3*x^2 + 5*x + 2)^(3/2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 259 vs. \(2 (107) = 214\).
Time = 0.33 (sec) , antiderivative size = 259, normalized size of antiderivative = 1.92 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=-\frac {1}{32} \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x - 89\right )} + \frac {1329}{320} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) + \frac {343}{64} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {5 \, {\left (510 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 1869 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6259 \, \sqrt {3} x + 2209 \, \sqrt {3} - 6259 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{32 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{2}} \]
-1/32*sqrt(3*x^2 + 5*x + 2)*(6*x - 89) + 1329/320*sqrt(5)*log(abs(-4*sqrt( 3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))) + 343/64*sqrt(3)*log(ab s(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) + 5/32*(510*(sqrt(3 )*x - sqrt(3*x^2 + 5*x + 2))^3 + 1869*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5* x + 2))^2 + 6259*sqrt(3)*x + 2209*sqrt(3) - 6259*sqrt(3*x^2 + 5*x + 2))/(2 *(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^2
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{3/2}}{(3+2 x)^3} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{3/2}}{{\left (2\,x+3\right )}^3} \,d x \]