Integrand size = 27, antiderivative size = 87 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (3+2 x)^2 (121+139 x)}{9 \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (2481+2834 x)}{9 \sqrt {2+5 x+3 x^2}}-\frac {16 \text {arctanh}\left (\frac {\sqrt {3} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{9 \sqrt {3}} \] Output:
-2/9*(3+2*x)^2*(121+139*x)/(3*x^2+5*x+2)^(3/2)+4/9*(2481+2834*x)/(3*x^2+5* x+2)^(1/2)-16/27*arctanh(3^(1/2)*(1+x)/(3*x^2+5*x+2)^(1/2))*3^(1/2)
Time = 0.41 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2}{27} \left (-\frac {3 \sqrt {2+5 x+3 x^2} \left (8835+33443 x+41074 x^2+16448 x^3\right )}{(1+x)^2 (2+3 x)^2}+8 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \] Input:
Integrate[((5 - x)*(3 + 2*x)^3)/(2 + 5*x + 3*x^2)^(5/2),x]
Output:
(-2*((-3*Sqrt[2 + 5*x + 3*x^2]*(8835 + 33443*x + 41074*x^2 + 16448*x^3))/( (1 + x)^2*(2 + 3*x)^2) + 8*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)]))/27
Time = 0.37 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1233, 25, 1224, 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.
| \(\displaystyle \int \frac {(5-x) (2 x+3)^3}{\left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1233 |
| \(\displaystyle \frac {2}{9} \int -\frac {(2 x+3) (6 x+359)}{\left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (2 x+3)^2 (139 x+121)}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2}{9} \int \frac {(2 x+3) (6 x+359)}{\left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (139 x+121) (2 x+3)^2}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1224 |
| \(\displaystyle -\frac {2}{9} \left (4 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {2 (2834 x+2481)}{\sqrt {3 x^2+5 x+2}}\right )-\frac {2 (139 x+121) (2 x+3)^2}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle -\frac {2}{9} \left (8 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {2 (2834 x+2481)}{\sqrt {3 x^2+5 x+2}}\right )-\frac {2 (139 x+121) (2 x+3)^2}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2}{9} \left (\frac {4 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}-\frac {2 (2834 x+2481)}{\sqrt {3 x^2+5 x+2}}\right )-\frac {2 (139 x+121) (2 x+3)^2}{9 \left (3 x^2+5 x+2\right )^{3/2}}\) |
Input:
Int[((5 - x)*(3 + 2*x)^3)/(2 + 5*x + 3*x^2)^(5/2),x]
Output:
(-2*(3 + 2*x)^2*(121 + 139*x))/(9*(2 + 5*x + 3*x^2)^(3/2)) - (2*((-2*(2481 + 2834*x))/Sqrt[2 + 5*x + 3*x^2] + (4*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[3]))/9
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_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*( x_)^2)^(p_), x_Symbol] :> Simp[(-(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - ( b^2*e*g - b*c*(e*f + d*g) + 2*c*(c*d*f - a*e*g))*x))*((a + b*x + c*x^2)^(p + 1)/(c*(p + 1)*(b^2 - 4*a*c))), x] - Simp[(b^2*e*g*(p + 2) - 2*a*c*e*g + c *(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(c*(p + 1)*(b^2 - 4*a*c)) Int[(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && LtQ[p, - 1] && !(IntegerQ[p] && NeQ[a, 0] && NiceSqrtQ[b^2 - 4*a*c])
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 = 1.62 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.69
| method | result | size |
| risch | \(\frac {\frac {32896}{9} x^{3}+\frac {82148}{9} x^{2}+\frac {66886}{9} x +\frac {5890}{3}}{\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {8 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{27}\) | \(60\) |
| trager | \(\frac {\frac {32896}{9} x^{3}+\frac {82148}{9} x^{2}+\frac {66886}{9} x +\frac {5890}{3}}{\left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{27}\) | \(72\) |
| default | \(-\frac {4033 \left (6 x +5\right )}{486 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {\frac {32864 x}{27}+\frac {82160}{81}}{\sqrt {3 x^{2}+5 x +2}}-\frac {10855}{486 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {607 x}{27 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {32 x^{2}}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {8 x^{3}}{9 \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {8 x}{9 \sqrt {3 x^{2}+5 x +2}}-\frac {20}{27 \sqrt {3 x^{2}+5 x +2}}-\frac {8 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{27}\) | \(161\) |
Input:
int((5-x)*(2*x+3)^3/(3*x^2+5*x+2)^(5/2),x,method=_RETURNVERBOSE)
Output:
2/9*(16448*x^3+41074*x^2+33443*x+8835)/(3*x^2+5*x+2)^(3/2)-8/27*ln(1/3*(5/ 2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)
Time = 0.12 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.29 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (2 \, \sqrt {3} {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 3 \, {\left (16448 \, x^{3} + 41074 \, x^{2} + 33443 \, x + 8835\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{27 \, {\left (9 \, x^{4} + 30 \, x^{3} + 37 \, x^{2} + 20 \, x + 4\right )}} \] Input:
integrate((5-x)*(3+2*x)^3/(3*x^2+5*x+2)^(5/2),x, algorithm="fricas")
Output:
2/27*(2*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*(16448*x^3 + 41074*x^ 2 + 33443*x + 8835)*sqrt(3*x^2 + 5*x + 2))/(9*x^4 + 30*x^3 + 37*x^2 + 20*x + 4)
\[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \left (- \frac {243 x}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {126 x^{2}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \left (- \frac {4 x^{3}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {8 x^{4}}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {135}{9 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 30 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 37 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 20 x \sqrt {3 x^{2} + 5 x + 2} + 4 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \] Input:
integrate((5-x)*(3+2*x)**3/(3*x**2+5*x+2)**(5/2),x)
Output:
-Integral(-243*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(-126*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(-4* 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)), x) - Integral(8*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(-135/(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. 197 vs. \(2 (71) = 142\).
Time = 0.12 (sec) , antiderivative size = 197, normalized size of antiderivative = 2.26 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {8}{27} \, x {\left (\frac {1410 \, x}{\sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {9 \, x^{2}}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} + \frac {1175}{\sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {55 \, x}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {46}{{\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}}\right )} - \frac {8}{27} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {3760}{27} \, \sqrt {3 \, x^{2} + 5 \, x + 2} + \frac {42272 \, x}{27 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {4 \, x^{2}}{3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} + \frac {11680}{9 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {2318 \, x}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {2030}{27 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \] Input:
integrate((5-x)*(3+2*x)^3/(3*x^2+5*x+2)^(5/2),x, algorithm="maxima")
Output:
8/27*x*(1410*x/sqrt(3*x^2 + 5*x + 2) + 9*x^2/(3*x^2 + 5*x + 2)^(3/2) + 117 5/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)) - 8/27*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 3760/27*sqrt(3*x^2 + 5*x + 2) + 42272/27*x/sqrt(3*x^2 + 5*x + 2) - 4/3*x^ 2/(3*x^2 + 5*x + 2)^(3/2) + 11680/9/sqrt(3*x^2 + 5*x + 2) - 2318/27*x/(3*x ^2 + 5*x + 2)^(3/2) - 2030/27/(3*x^2 + 5*x + 2)^(3/2)
Time = 0.39 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.72 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {8}{27} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) + \frac {2 \, {\left ({\left (2 \, {\left (8224 \, x + 20537\right )} x + 33443\right )} x + 8835\right )}}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \] Input:
integrate((5-x)*(3+2*x)^3/(3*x^2+5*x+2)^(5/2),x, algorithm="giac")
Output:
8/27*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) + 2/9*((2*(8224*x + 20537)*x + 33443)*x + 8835)/(3*x^2 + 5*x + 2)^(3/2)
Timed out. \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {{\left (2\,x+3\right )}^3\,\left (x-5\right )}{{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \] Input:
int(-((2*x + 3)^3*(x - 5))/(5*x + 3*x^2 + 2)^(5/2),x)
Output:
-int(((2*x + 3)^3*(x - 5))/(5*x + 3*x^2 + 2)^(5/2), x)
Time = 0.19 (sec) , antiderivative size = 249, normalized size of antiderivative = 2.86 \[ \int \frac {(5-x) (3+2 x)^3}{\left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {296064 \sqrt {3 x^{2}+5 x +2}\, x^{3}+739332 \sqrt {3 x^{2}+5 x +2}\, x^{2}+601974 \sqrt {3 x^{2}+5 x +2}\, x +159030 \sqrt {3 x^{2}+5 x +2}-216 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right ) x^{4}-720 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right ) x^{3}-888 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right ) x^{2}-480 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right ) x -96 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right )-349344 \sqrt {3}\, x^{4}-1164480 \sqrt {3}\, x^{3}-1436192 \sqrt {3}\, x^{2}-776320 \sqrt {3}\, x -155264 \sqrt {3}}{729 x^{4}+2430 x^{3}+2997 x^{2}+1620 x +324} \] Input:
int((5-x)*(3+2*x)^3/(3*x^2+5*x+2)^(5/2),x)
Output:
(2*(148032*sqrt(3*x**2 + 5*x + 2)*x**3 + 369666*sqrt(3*x**2 + 5*x + 2)*x** 2 + 300987*sqrt(3*x**2 + 5*x + 2)*x + 79515*sqrt(3*x**2 + 5*x + 2) - 108*s qrt(3)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5)*x**4 - 360*sqrt(3)* log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5)*x**3 - 444*sqrt(3)*log(2*s qrt(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5)*x**2 - 240*sqrt(3)*log(2*sqrt(3*x **2 + 5*x + 2)*sqrt(3) + 6*x + 5)*x - 48*sqrt(3)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5) - 174672*sqrt(3)*x**4 - 582240*sqrt(3)*x**3 - 71809 6*sqrt(3)*x**2 - 388160*sqrt(3)*x - 77632*sqrt(3)))/(81*(9*x**4 + 30*x**3 + 37*x**2 + 20*x + 4))