Integrand size = 27, antiderivative size = 124 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {2 (37+47 x)}{5 (3+2 x) \left (2+5 x+3 x^2\right )^{3/2}}+\frac {4 (401+462 x)}{5 (3+2 x) \sqrt {2+5 x+3 x^2}}+\frac {4416 \sqrt {2+5 x+3 x^2}}{25 (3+2 x)}+\frac {408 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{25 \sqrt {5}} \]
-2/5*(37+47*x)/(3+2*x)/(3*x^2+5*x+2)^(3/2)+408/125*arctanh(1/10*(7+8*x)*5^ (1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)+4/5*(401+462*x)/(3+2*x)/(3*x^2+5*x+2)^( 1/2)+4416/25*(3*x^2+5*x+2)^(1/2)/(3+2*x)
Time = 0.39 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.73 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2}{125} \left (\frac {5 \sqrt {2+5 x+3 x^2} \left (16667+73215 x+116826 x^2+80100 x^3+19872 x^4\right )}{(1+x)^2 (3+2 x) (2+3 x)^2}+408 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )\right ) \]
(2*((5*Sqrt[2 + 5*x + 3*x^2]*(16667 + 73215*x + 116826*x^2 + 80100*x^3 + 1 9872*x^4))/((1 + x)^2*(3 + 2*x)*(2 + 3*x)^2) + 408*Sqrt[5]*ArcTanh[Sqrt[2/ 5 + x + (3*x^2)/5]/(1 + x)]))/125
Time = 0.30 (sec) , antiderivative size = 130, 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 = {1235, 27, 1235, 27, 1228, 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.
| \(\displaystyle \int \frac {5-x}{(2 x+3)^2 \left (3 x^2+5 x+2\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {2}{15} \int \frac {3 (282 x+343)}{(2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \int \frac {282 x+343}{(2 x+3)^2 \left (3 x^2+5 x+2\right )^{3/2}}dx-\frac {2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1235 |
| \(\displaystyle -\frac {2}{5} \left (-\frac {2}{5} \int \frac {30 (154 x+139)}{(2 x+3)^2 \sqrt {3 x^2+5 x+2}}dx-\frac {2 (462 x+401)}{(2 x+3) \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2}{5} \left (-12 \int \frac {154 x+139}{(2 x+3)^2 \sqrt {3 x^2+5 x+2}}dx-\frac {2 (462 x+401)}{(2 x+3) \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle -\frac {2}{5} \left (-12 \left (\frac {17}{5} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx+\frac {184 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\right )-\frac {2 (462 x+401)}{(2 x+3) \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle -\frac {2}{5} \left (-12 \left (\frac {184 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}-\frac {34}{5} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )\right )-\frac {2 (462 x+401)}{(2 x+3) \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {2}{5} \left (-12 \left (\frac {17 \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{5 \sqrt {5}}+\frac {184 \sqrt {3 x^2+5 x+2}}{5 (2 x+3)}\right )-\frac {2 (462 x+401)}{(2 x+3) \sqrt {3 x^2+5 x+2}}\right )-\frac {2 (47 x+37)}{5 (2 x+3) \left (3 x^2+5 x+2\right )^{3/2}}\) |
(-2*(37 + 47*x))/(5*(3 + 2*x)*(2 + 5*x + 3*x^2)^(3/2)) - (2*((-2*(401 + 46 2*x))/((3 + 2*x)*Sqrt[2 + 5*x + 3*x^2]) - 12*((184*Sqrt[2 + 5*x + 3*x^2])/ (5*(3 + 2*x)) + (17*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/ (5*Sqrt[5]))))/5
3.26.22.3.1 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/(((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[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(f*(b*c*d - b^2*e + 2 *a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x)*((a + b*x + c*x^2)^(p + 1)/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^m *(a + b*x + c*x^2)^(p + 1)*Simp[f*(b*c*d*e*(2*p - m + 2) + b^2*e^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d* m + b*e*m) - b*d*(3*c*d - b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p] )
Time = 0.37 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.59
| method | result | size |
| risch | \(\frac {\frac {39744}{25} x^{4}+6408 x^{3}+\frac {233652}{25} x^{2}+\frac {29286}{5} x +\frac {33334}{25}}{\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}-\frac {408 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{125}\) | \(73\) |
| trager | \(\frac {\frac {39744}{25} x^{4}+6408 x^{3}+\frac {233652}{25} x^{2}+\frac {29286}{5} x +\frac {33334}{25}}{\left (3+2 x \right ) \left (3 x^{2}+5 x +2\right )^{\frac {3}{2}}}+\frac {408 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{125}\) | \(92\) |
| default | \(\frac {17}{5 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}-\frac {16 \left (5+6 x \right )}{5 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}+\frac {\frac {1104}{5}+\frac {6624 x}{25}}{\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}+\frac {204}{25 \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}-\frac {408 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{125}-\frac {13}{10 \left (x +\frac {3}{2}\right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}\) | \(127\) |
2/25*(19872*x^4+80100*x^3+116826*x^2+73215*x+16667)/(3*x^2+5*x+2)^(3/2)/(3 +2*x)-408/125*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2-16*x-19 )^(1/2))
Time = 0.40 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.13 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=\frac {2 \, {\left (102 \, \sqrt {5} {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 5 \, {\left (19872 \, x^{4} + 80100 \, x^{3} + 116826 \, x^{2} + 73215 \, x + 16667\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{125 \, {\left (18 \, x^{5} + 87 \, x^{4} + 164 \, x^{3} + 151 \, x^{2} + 68 \, x + 12\right )}} \]
2/125*(102*sqrt(5)*(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)*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)) + 5*(19872*x^4 + 80100*x^3 + 116826*x^2 + 73215*x + 16667)*sqr t(3*x^2 + 5*x + 2))/(18*x^5 + 87*x^4 + 164*x^3 + 151*x^2 + 68*x + 12)
\[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=- \int \frac {x}{36 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 228 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 589 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 794 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 589 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 228 x \sqrt {3 x^{2} + 5 x + 2} + 36 \sqrt {3 x^{2} + 5 x + 2}}\, dx - \int \left (- \frac {5}{36 x^{6} \sqrt {3 x^{2} + 5 x + 2} + 228 x^{5} \sqrt {3 x^{2} + 5 x + 2} + 589 x^{4} \sqrt {3 x^{2} + 5 x + 2} + 794 x^{3} \sqrt {3 x^{2} + 5 x + 2} + 589 x^{2} \sqrt {3 x^{2} + 5 x + 2} + 228 x \sqrt {3 x^{2} + 5 x + 2} + 36 \sqrt {3 x^{2} + 5 x + 2}}\right )\, dx \]
-Integral(x/(36*x**6*sqrt(3*x**2 + 5*x + 2) + 228*x**5*sqrt(3*x**2 + 5*x + 2) + 589*x**4*sqrt(3*x**2 + 5*x + 2) + 794*x**3*sqrt(3*x**2 + 5*x + 2) + 589*x**2*sqrt(3*x**2 + 5*x + 2) + 228*x*sqrt(3*x**2 + 5*x + 2) + 36*sqrt(3 *x**2 + 5*x + 2)), x) - Integral(-5/(36*x**6*sqrt(3*x**2 + 5*x + 2) + 228* x**5*sqrt(3*x**2 + 5*x + 2) + 589*x**4*sqrt(3*x**2 + 5*x + 2) + 794*x**3*s qrt(3*x**2 + 5*x + 2) + 589*x**2*sqrt(3*x**2 + 5*x + 2) + 228*x*sqrt(3*x** 2 + 5*x + 2) + 36*sqrt(3*x**2 + 5*x + 2)), x)
Time = 0.26 (sec) , antiderivative size = 135, normalized size of antiderivative = 1.09 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {408}{125} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) + \frac {6624 \, x}{25 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} + \frac {5724}{25 \, \sqrt {3 \, x^{2} + 5 \, x + 2}} - \frac {96 \, x}{5 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} - \frac {13}{5 \, {\left (2 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x + 3 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}\right )}} - \frac {63}{5 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}}} \]
-408/125*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs( 2*x + 3) - 2) + 6624/25*x/sqrt(3*x^2 + 5*x + 2) + 5724/25/sqrt(3*x^2 + 5*x + 2) - 96/5*x/(3*x^2 + 5*x + 2)^(3/2) - 13/5/(2*(3*x^2 + 5*x + 2)^(3/2)*x + 3*(3*x^2 + 5*x + 2)^(3/2)) - 63/5/(3*x^2 + 5*x + 2)^(3/2)
Leaf count of result is larger than twice the leaf count of optimal. 235 vs. \(2 (102) = 204\).
Time = 0.31 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.90 \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\frac {24}{125} \, \sqrt {5} {\left (92 \, \sqrt {5} \sqrt {3} - 17 \, \log \left (-\sqrt {5} \sqrt {3} + 4\right )\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {408 \, \sqrt {5} \log \left ({\left | \sqrt {5} {\left (\sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {5}}{2 \, x + 3}\right )} - 4 \right |}\right )}{125 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {8 \, {\left (\frac {\frac {\frac {5 \, {\left (\frac {972}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )} + \frac {13}{{\left (2 \, x + 3\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{2 \, x + 3} - \frac {12324}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} + \frac {9783}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}}{2 \, x + 3} - \frac {2484}{\mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}\right )}}{25 \, {\left (\frac {8}{2 \, x + 3} - \frac {5}{{\left (2 \, x + 3\right )}^{2}} - 3\right )} \sqrt {-\frac {8}{2 \, x + 3} + \frac {5}{{\left (2 \, x + 3\right )}^{2}} + 3}} \]
-24/125*sqrt(5)*(92*sqrt(5)*sqrt(3) - 17*log(-sqrt(5)*sqrt(3) + 4))*sgn(1/ (2*x + 3)) - 408/125*sqrt(5)*log(abs(sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3)) - 4))/sgn(1/(2*x + 3)) + 8/25*(((5*(972/sg n(1/(2*x + 3)) + 13/((2*x + 3)*sgn(1/(2*x + 3))))/(2*x + 3) - 12324/sgn(1/ (2*x + 3)))/(2*x + 3) + 9783/sgn(1/(2*x + 3)))/(2*x + 3) - 2484/sgn(1/(2*x + 3)))/((8/(2*x + 3) - 5/(2*x + 3)^2 - 3)*sqrt(-8/(2*x + 3) + 5/(2*x + 3) ^2 + 3))
Timed out. \[ \int \frac {5-x}{(3+2 x)^2 \left (2+5 x+3 x^2\right )^{5/2}} \, dx=-\int \frac {x-5}{{\left (2\,x+3\right )}^2\,{\left (3\,x^2+5\,x+2\right )}^{5/2}} \,d x \]