Integrand size = 25, antiderivative size = 57 \[ \int \frac {(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {2 (121+139 x)}{3 \sqrt {2+5 x+3 x^2}}-\frac {4 \text {arctanh}\left (\frac {\sqrt {3} (1+x)}{\sqrt {2+5 x+3 x^2}}\right )}{3 \sqrt {3}} \] Output:
1/3*(-242-278*x)/(3*x^2+5*x+2)^(1/2)-4/9*arctanh(3^(1/2)*(1+x)/(3*x^2+5*x+ 2)^(1/2))*3^(1/2)
Time = 0.32 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96 \[ \int \frac {(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {2}{9} \left (\frac {363+417 x}{\sqrt {2+5 x+3 x^2}}+2 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )\right ) \] Input:
Integrate[((5 - x)*(3 + 2*x))/(2 + 5*x + 3*x^2)^(3/2),x]
Output:
(-2*((363 + 417*x)/Sqrt[2 + 5*x + 3*x^2] + 2*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5 *x)/3 + x^2]/(1 + x)]))/9
Time = 0.31 (sec) , antiderivative size = 62, normalized size of antiderivative = 1.09, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {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.
\(\displaystyle \int \frac {(5-x) (2 x+3)}{\left (3 x^2+5 x+2\right )^{3/2}} \, dx\) |
\(\Big \downarrow \) 1224 |
\(\displaystyle -\frac {2}{3} \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-\frac {2 (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}\) |
\(\Big \downarrow \) 1092 |
\(\displaystyle -\frac {4}{3} \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-\frac {2 (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}\) |
\(\Big \downarrow \) 219 |
\(\displaystyle -\frac {2 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{3 \sqrt {3}}-\frac {2 (139 x+121)}{3 \sqrt {3 x^2+5 x+2}}\) |
Input:
Int[((5 - x)*(3 + 2*x))/(2 + 5*x + 3*x^2)^(3/2),x]
Output:
(-2*(121 + 139*x))/(3*Sqrt[2 + 5*x + 3*x^2]) - (2*ArcTanh[(5 + 6*x)/(2*Sqr t[3]*Sqrt[2 + 5*x + 3*x^2])])/(3*Sqrt[3])
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])
Time = 1.45 (sec) , antiderivative size = 50, normalized size of antiderivative = 0.88
method | result | size |
risch | \(-\frac {2 \left (121+139 x \right )}{3 \sqrt {3 x^{2}+5 x +2}}-\frac {2 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{9}\) | \(50\) |
trager | \(-\frac {2 \left (121+139 x \right )}{3 \sqrt {3 x^{2}+5 x +2}}-\frac {2 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{9}\) | \(62\) |
default | \(-\frac {140 \left (6 x +5\right )}{9 \sqrt {3 x^{2}+5 x +2}}-\frac {26}{9 \sqrt {3 x^{2}+5 x +2}}+\frac {2 x}{3 \sqrt {3 x^{2}+5 x +2}}-\frac {2 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{9}\) | \(79\) |
Input:
int((5-x)*(2*x+3)/(3*x^2+5*x+2)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/3*(121+139*x)/(3*x^2+5*x+2)^(1/2)-2/9*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5 *x+2)^(1/2))*3^(1/2)
Time = 0.11 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.42 \[ \int \frac {(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {\sqrt {3} {\left (3 \, x^{2} + 5 \, x + 2\right )} \log \left (-4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) - 6 \, \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (139 \, x + 121\right )}}{9 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}} \] Input:
integrate((5-x)*(3+2*x)/(3*x^2+5*x+2)^(3/2),x, algorithm="fricas")
Output:
1/9*(sqrt(3)*(3*x^2 + 5*x + 2)*log(-4*sqrt(3)*sqrt(3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) - 6*sqrt(3*x^2 + 5*x + 2)*(139*x + 121))/(3*x^2 + 5*x + 2)
\[ \int \frac {(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=- \int \left (- \frac {7 x}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx - \int \frac {2 x^{2}}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {15}{3 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 5 x \sqrt {3 x^{2} + 5 x + 2} + 2 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \] Input:
integrate((5-x)*(3+2*x)/(3*x**2+5*x+2)**(3/2),x)
Output:
-Integral(-7*x/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x) - Integral(2*x**2/(3*x**2*sqrt(3*x**2 + 5 *x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2*sqrt(3*x**2 + 5*x + 2)), x) - Int egral(-15/(3*x**2*sqrt(3*x**2 + 5*x + 2) + 5*x*sqrt(3*x**2 + 5*x + 2) + 2* sqrt(3*x**2 + 5*x + 2)), x)
Time = 0.11 (sec) , antiderivative size = 58, normalized size of antiderivative = 1.02 \[ \int \frac {(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=-\frac {2}{9} \, \sqrt {3} \log \left (2 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 6 \, x + 5\right ) - \frac {278 \, x}{3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {242}{3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} \] Input:
integrate((5-x)*(3+2*x)/(3*x^2+5*x+2)^(3/2),x, algorithm="maxima")
Output:
-2/9*sqrt(3)*log(2*sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 6*x + 5) - 278/3*x/sqrt (3*x^2 + 5*x + 2) - 242/3/sqrt(3*x^2 + 5*x + 2)
Time = 0.32 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.95 \[ \int \frac {(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {2}{9} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {2 \, {\left (139 \, x + 121\right )}}{3 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} \] Input:
integrate((5-x)*(3+2*x)/(3*x^2+5*x+2)^(3/2),x, algorithm="giac")
Output:
2/9*sqrt(3)*log(abs(-2*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 2/3*(139*x + 121)/sqrt(3*x^2 + 5*x + 2)
Time = 13.11 (sec) , antiderivative size = 78, normalized size of antiderivative = 1.37 \[ \int \frac {(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {352\,x}{3\,\sqrt {3\,x^2+5\,x+2}}-\frac {6\,\left (35\,x+29\right )}{\sqrt {3\,x^2+5\,x+2}}-\frac {2\,\sqrt {3}\,\ln \left (\sqrt {3\,x^2+5\,x+2}+\frac {\sqrt {3}\,\left (3\,x+\frac {5}{2}\right )}{3}\right )}{9}+\frac {280}{3\,\sqrt {3\,x^2+5\,x+2}} \] Input:
int(-((2*x + 3)*(x - 5))/(5*x + 3*x^2 + 2)^(3/2),x)
Output:
(352*x)/(3*(5*x + 3*x^2 + 2)^(1/2)) - (6*(35*x + 29))/(5*x + 3*x^2 + 2)^(1 /2) - (2*3^(1/2)*log((5*x + 3*x^2 + 2)^(1/2) + (3^(1/2)*(3*x + 5/2))/3))/9 + 280/(3*(5*x + 3*x^2 + 2)^(1/2))
Time = 0.24 (sec) , antiderivative size = 137, normalized size of antiderivative = 2.40 \[ \int \frac {(5-x) (3+2 x)}{\left (2+5 x+3 x^2\right )^{3/2}} \, dx=\frac {-834 \sqrt {3 x^{2}+5 x +2}\, x -726 \sqrt {3 x^{2}+5 x +2}-6 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right ) x^{2}-10 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right ) x -4 \sqrt {3}\, \mathrm {log}\left (2 \sqrt {3 x^{2}+5 x +2}\, \sqrt {3}+6 x +5\right )-834 \sqrt {3}\, x^{2}-1390 \sqrt {3}\, x -556 \sqrt {3}}{27 x^{2}+45 x +18} \] Input:
int((5-x)*(3+2*x)/(3*x^2+5*x+2)^(3/2),x)
Output:
(2*( - 417*sqrt(3*x**2 + 5*x + 2)*x - 363*sqrt(3*x**2 + 5*x + 2) - 3*sqrt( 3)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5)*x**2 - 5*sqrt(3)*log(2* sqrt(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5)*x - 2*sqrt(3)*log(2*sqrt(3*x**2 + 5*x + 2)*sqrt(3) + 6*x + 5) - 417*sqrt(3)*x**2 - 695*sqrt(3)*x - 278*sqr t(3)))/(9*(3*x**2 + 5*x + 2))