Integrand size = 27, antiderivative size = 115 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (3+2 x)^3 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (3+2 x) (6809+7976 x)}{27 \sqrt {2+5 x+3 x^2}}-\frac {6848}{9} \sqrt {2+5 x+3 x^2}+\frac {152 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{27 \sqrt {3}} \]
-2/9*(3+2*x)^3*(121+139*x)/(3*x^2+5*x+2)^(3/2)+152/81*arctanh(1/6*(5+6*x)* 3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)+4/27*(3+2*x)*(6809+7976*x)/(3*x^2+5*x +2)^(1/2)-6848/9*(3*x^2+5*x+2)^(1/2)
Time = 0.39 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.72 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2}{81} \left (\frac {3 \sqrt {2+5 x+3 x^2} \left (-30819-118153 x-146180 x^2-58720 x^3+72 x^4\right )}{(1+x)^2 (2+3 x)^2}-152 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \]
(-2*((3*Sqrt[2 + 5*x + 3*x^2]*(-30819 - 118153*x - 146180*x^2 - 58720*x^3 + 72*x^4))/((1 + x)^2*(2 + 3*x)^2) - 152*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/ 3 + x^2]/(1 + x)]))/81
Time = 0.27 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {1233, 25, 1233, 27, 1160, 1092, 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.
| \(\displaystyle \int \frac {(5-x) (2 x+3)^4}{\left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2}{9} \int -\frac {(117-272 x) (2 x+3)^2}{\left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (2 x+3)^3 (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2}{9} \int \frac {(117-272 x) (2 x+3)^2}{\left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle -\frac {2}{9} \left (\frac {2}{3} \int \frac {2 (7704 x+6401)}{\sqrt {3 x^2+5 x+2}}dx-\frac {2 (2 x+3) (7976 x+6809)}{3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{9} \left (\frac {4}{3} \int \frac {7704 x+6401}{\sqrt {3 x^2+5 x+2}}dx-\frac {2 (2 x+3) (7976 x+6809)}{3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1160 |
| \(\displaystyle -\frac {2}{9} \left (\frac {4}{3} \left (2568 \sqrt {3 x^2+5 x+2}-19 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx\right )-\frac {2 (2 x+3) (7976 x+6809)}{3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {2}{9} \left (\frac {4}{3} \left (2568 \sqrt {3 x^2+5 x+2}-38 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}\right )-\frac {2 (2 x+3) (7976 x+6809)}{3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2}{9} \left (\frac {4}{3} \left (2568 \sqrt {3 x^2+5 x+2}-\frac {19 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )-\frac {2 (2 x+3) (7976 x+6809)}{3 \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (139 x+121) (2 x+3)^3}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
(-2*(3 + 2*x)^3*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (2*((-2*(3 + 2*x)*(6809 + 7976*x))/(3*Sqrt[2 + 5*x + 3*x^2]) + (4*(2568*Sqrt[2 + 5*x + 3*x^2] - (19*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[3] ))/3))/9
3.26.16.3.1 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[((d_.) + (e_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol ] :> Simp[e*((a + b*x + c*x^2)^(p + 1)/(2*c*(p + 1))), x] + Simp[(2*c*d - b *e)/(2*c) Int[(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, p}, x] && NeQ[p, -1]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(-(d + e*x)^(m - 1))*(a + b*x + c*x^2) ^(p + 1)*((2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g - c *(b*e*f + b*d*g + 2*a*e*g))*x)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[1/(c*( p + 1)*(b^2 - 4*a*c)) Int[(d + e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Sim p[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a*e*(e*f *(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*( m + p + 1) + 2*c^2*d*f*(m + 2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2* p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] && RationalQ[a, b, c, d, e, f, g]) | | !ILtQ[m + 2*p + 3, 0])
Time = 0.36 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.57
| method | result | size |
| risch | \(-\frac {2 \left (72 x^{4}-58720 x^{3}-146180 x^{2}-118153 x -30819\right )}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {152 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{81}\) | \(65\) |
| trager | \(-\frac {2 \left (72 x^{4}-58720 x^{3}-146180 x^{2}-118153 x -30819\right )}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {152 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{81}\) | \(77\) |
| default | \(-\frac {16181 \left (5+6 x \right )}{1458 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {\frac {295120}{243}+\frac {118048 x}{81}}{\sqrt {3 x^{2}+5 x +2}}-\frac {145763}{1458 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {16 x^{4}}{3 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {152 x^{3}}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {2380 x^{2}}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {14639 x}{81 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {152 x}{27 \sqrt {3 x^{2}+5 x +2}}+\frac {380}{81 \sqrt {3 x^{2}+5 x +2}}+\frac {152 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{81}\) | \(178\) |
-2/27*(72*x^4-58720*x^3-146180*x^2-118153*x-30819)/(3*x^2+5*x+2)^(3/2)+152 /81*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)
Time = 0.28 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.02 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (38 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 3 \, {\left (72 \, x^{4} - 58720 \, x^{3} - 146180 \, x^{2} - 118153 \, x - 30819\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{81 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \]
2/81*(38*sqrt(3)*(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)*log(4*sqrt(3)*sqrt(3 *x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) - 3*(72*x^4 - 58720*x^3 - 146180*x^2 - 118153*x - 30819)*sqrt(3*x^2 + 5*x + 2))/(9*x^4 + 30*x^3 + 3 7*x^2 + 20*x + 4)
\[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {999 x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {864 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {264 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {16 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \frac {16 x^{5}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {405}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
-Integral(-999*x/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5* x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4* sqrt(3*x**2 + 5*x + 2)), x) - Integral(-864*x**2/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-26 4*x**3/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 3 7*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x** 2 + 5*x + 2)), x) - Integral(16*x**4/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x **3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3* x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(16*x**5/(9*x**4 *sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3* x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x) - Integral(-405/(9*x**4*sqrt(3*x**2 + 5*x + 2) + 30*x**3*sqrt(3*x**2 + 5*x + 2) + 37*x**2*sqrt(3*x**2 + 5*x + 2) + 20*x*sqrt(3*x**2 + 5*x + 2) + 4*sqrt(3*x**2 + 5*x + 2)), x)
Leaf count of result is larger than twice the leaf count of optimal. 214 vs. \(2 (93) = 186\).
Time = 0.26 (sec) , antiderivative size = 214, normalized size of antiderivative = 1.86 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {16 \, x^{4}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {152}{81} \, x {\left (\frac {1410 \, x}{\sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {9 \, x^{2}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} + \frac {1175}{\sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {55 \, x}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {46}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}\right )} + \frac {152}{81} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) + \frac {71440}{81} \, \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {60704 \, x}{81 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {920 \, x^{2}}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {15680}{27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {13066 \, x}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {6766}{81 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
-16/3*x^4/(3*x^2 + 5*x + 2)^(3/2) - 152/81*x*(1410*x/sqrt(3*x^2 + 5*x + 2) + 9*x^2/(3*x^2 + 5*x + 2)^(3/2) + 1175/sqrt(3*x^2 + 5*x + 2) - 55*x/(3*x^ 2 + 5*x + 2)^(3/2) - 46/(3*x^2 + 5*x + 2)^(3/2)) + 152/81*sqrt(3)*log(2*sq rt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) + 71440/81*sqrt(3*x^2 + 5*x + 2) - 60704/81*x/sqrt(3*x^2 + 5*x + 2) - 920/9*x^2/(3*x^2 + 5*x + 2)^(3/2) - 156 80/27/sqrt(3*x^2 + 5*x + 2) - 13066/81*x/(3*x^2 + 5*x + 2)^(3/2) - 6766/81 /(3*x^2 + 5*x + 2)^(3/2)
Time = 0.28 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.59 \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {152}{81} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {2 \, {\left ({\left (4 \, {\left (2 \, {\left (9 \, x - 7340\right )} x - 36545\right )} x - 118153\right )} x - 30819\right )}}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
-152/81*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5 )) - 2/27*((4*(2*(9*x - 7340)*x - 36545)*x - 118153)*x - 30819)/(3*x^2 + 5 *x + 2)^(3/2)
Timed out. \[ \int \frac {(5-x) (3+2 x)^4}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {{\left (2\,x+3\right )}^4\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]