Integrand size = 24, antiderivative size = 106 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=\frac {3 (2943+4097 x) \sqrt {2+3 x^2}}{19600 (3+2 x)^2}+\frac {(54+491 x) \left (2+3 x^2\right )^{3/2}}{840 (3+2 x)^4}-\frac {3}{32} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {39663 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{39200 \sqrt {35}} \]
1/840*(54+491*x)*(3*x^2+2)^(3/2)/(3+2*x)^4-3/32*arcsinh(1/2*x*6^(1/2))*3^( 1/2)-39663/1372000*arctanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2) +3/19600*(2943+4097*x)*(3*x^2+2)^(1/2)/(3+2*x)^2
Time = 1.02 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.07 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=\frac {\sqrt {2+3 x^2} \left (245943+718441 x+559764 x^2+250602 x^3\right )}{58800 (3+2 x)^4}+\frac {39663 \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{19600 \sqrt {35}}+\frac {3}{32} \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right ) \]
(Sqrt[2 + 3*x^2]*(245943 + 718441*x + 559764*x^2 + 250602*x^3))/(58800*(3 + 2*x)^4) + (39663*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/S qrt[35]])/(19600*Sqrt[35]) + (3*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2] ])/32
Time = 0.26 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {680, 27, 680, 27, 719, 222, 488, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+2\right )^{3/2}}{(2 x+3)^5} \, dx\) |
| \(\Big \downarrow \) 680 |
| \(\displaystyle \frac {(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}-\frac {\int -\frac {24 (39-35 x) \sqrt {3 x^2+2}}{(2 x+3)^3}dx}{1120}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{140} \int \frac {(39-35 x) \sqrt {3 x^2+2}}{(2 x+3)^3}dx+\frac {(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}\) |
| \(\Big \downarrow \) 680 |
| \(\displaystyle \frac {3}{140} \left (\frac {(4097 x+2943) \sqrt {3 x^2+2}}{140 (2 x+3)^2}-\frac {1}{560} \int -\frac {12 (366-1225 x)}{(2 x+3) \sqrt {3 x^2+2}}dx\right )+\frac {(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{140} \left (\frac {3}{140} \int \frac {366-1225 x}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {\sqrt {3 x^2+2} (4097 x+2943)}{140 (2 x+3)^2}\right )+\frac {(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle \frac {3}{140} \left (\frac {3}{140} \left (\frac {4407}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {1225}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )+\frac {\sqrt {3 x^2+2} (4097 x+2943)}{140 (2 x+3)^2}\right )+\frac {(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {3}{140} \left (\frac {3}{140} \left (\frac {4407}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {1225 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+2} (4097 x+2943)}{140 (2 x+3)^2}\right )+\frac {(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {3}{140} \left (\frac {3}{140} \left (-\frac {4407}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {1225 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}\right )+\frac {\sqrt {3 x^2+2} (4097 x+2943)}{140 (2 x+3)^2}\right )+\frac {(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{140} \left (\frac {3}{140} \left (-\frac {1225 \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )}{2 \sqrt {3}}-\frac {4407 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{2 \sqrt {35}}\right )+\frac {\sqrt {3 x^2+2} (4097 x+2943)}{140 (2 x+3)^2}\right )+\frac {(491 x+54) \left (3 x^2+2\right )^{3/2}}{840 (2 x+3)^4}\) |
((54 + 491*x)*(2 + 3*x^2)^(3/2))/(840*(3 + 2*x)^4) + (3*(((2943 + 4097*x)* Sqrt[2 + 3*x^2])/(140*(3 + 2*x)^2) + (3*((-1225*ArcSinh[Sqrt[3/2]*x])/(2*S qrt[3]) - (4407*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(2*Sqrt[35] )))/140))/140
3.14.77.3.1 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_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt [a])]/Rt[b, 2], x] /; FreeQ[{a, b}, x] && GtQ[a, 0] && PosQ[b]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)))*((d*g - e*f*(m + 2))*(c*d^2 + a*e^2) - 2*c*d^2*p*(e* f - d*g) - e*(g*(m + 1)*(c*d^2 + a*e^2) + 2*c*d*p*(e*f - d*g))*x), x] - Sim p[p/(e^2*(m + 1)*(m + 2)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 2)*(a + c*x^ 2)^(p - 1)*Simp[2*a*c*e*(e*f - d*g)*(m + 2) - c*(2*c*d*(d*g*(2*p + 1) - e*f *(m + 2*p + 2)) - 2*a*e^2*g*(m + 1))*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && GtQ[p, 0] && LtQ[m, -2] && LtQ[m + 2*p, 0] && !ILtQ[m + 2*p + 3 , 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Time = 0.76 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.82
| method | result | size |
| risch | \(\frac {751806 x^{5}+1679292 x^{4}+2656527 x^{3}+1857357 x^{2}+1436882 x +491886}{58800 \left (3+2 x \right )^{4} \sqrt {3 x^{2}+2}}-\frac {3 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{32}-\frac {39663 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1372000}\) | \(87\) |
| trager | \(\frac {\left (250602 x^{3}+559764 x^{2}+718441 x +245943\right ) \sqrt {3 x^{2}+2}}{58800 \left (3+2 x \right )^{4}}+\frac {39663 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}-4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{1372000}+\frac {3 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \sqrt {3 x^{2}+2}+3 x \right )}{32}\) | \(111\) |
| default | \(-\frac {211 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{117600 \left (x +\frac {3}{2}\right )^{3}}-\frac {999 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{686000 \left (x +\frac {3}{2}\right )^{2}}-\frac {5779 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{12005000 \left (x +\frac {3}{2}\right )}+\frac {13221 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{6002500}-\frac {7227 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{686000}-\frac {3 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{32}+\frac {39663 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{1372000}-\frac {39663 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{1372000}+\frac {17337 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{12005000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{2240 \left (x +\frac {3}{2}\right )^{4}}\) | \(194\) |
1/58800*(751806*x^5+1679292*x^4+2656527*x^3+1857357*x^2+1436882*x+491886)/ (3+2*x)^4/(3*x^2+2)^(1/2)-3/32*arcsinh(1/2*x*6^(1/2))*3^(1/2)-39663/137200 0*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
Time = 0.29 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.57 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=\frac {385875 \, \sqrt {3} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 118989 \, \sqrt {35} {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) + 140 \, {\left (250602 \, x^{3} + 559764 \, x^{2} + 718441 \, x + 245943\right )} \sqrt {3 \, x^{2} + 2}}{8232000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} \]
1/8232000*(385875*sqrt(3)*(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)*log(sqr t(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1) + 118989*sqrt(35)*(16*x^4 + 96*x^3 + 2 16*x^2 + 216*x + 81)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 3 6*x + 43)/(4*x^2 + 12*x + 9)) + 140*(250602*x^3 + 559764*x^2 + 718441*x + 245943)*sqrt(3*x^2 + 2))/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81)
Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 183 vs. \(2 (84) = 168\).
Time = 0.28 (sec) , antiderivative size = 183, normalized size of antiderivative = 1.73 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=\frac {2997}{686000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{140 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {211 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{14700 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {999 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{171500 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {7227}{686000} \, \sqrt {3 \, x^{2} + 2} x - \frac {3}{32} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) + \frac {39663}{1372000} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {39663}{686000} \, \sqrt {3 \, x^{2} + 2} - \frac {5779 \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}}}{686000 \, {\left (2 \, x + 3\right )}} \]
2997/686000*(3*x^2 + 2)^(3/2) - 13/140*(3*x^2 + 2)^(5/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 211/14700*(3*x^2 + 2)^(5/2)/(8*x^3 + 36*x^2 + 54 *x + 27) - 999/171500*(3*x^2 + 2)^(5/2)/(4*x^2 + 12*x + 9) - 7227/686000*s qrt(3*x^2 + 2)*x - 3/32*sqrt(3)*arcsinh(1/2*sqrt(6)*x) + 39663/1372000*sqr t(35)*arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 396 63/686000*sqrt(3*x^2 + 2) - 5779/686000*(3*x^2 + 2)^(3/2)/(2*x + 3)
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (84) = 168\).
Time = 0.41 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.31 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=-\frac {39663}{1372000} \, \sqrt {35} \log \left (\sqrt {35} {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} - 9\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {3}{32} \, \sqrt {3} \log \left (\frac {{\left | -2 \, \sqrt {3} + 2 \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {2 \, \sqrt {35}}{2 \, x + 3} \right |}}{2 \, {\left (\sqrt {3} + \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}}\right ) \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - \frac {1}{470400} \, {\left (\frac {35 \, {\left (\frac {35 \, {\left (\frac {1365 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )}{2 \, x + 3} - 1193 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} + 16227 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{2 \, x + 3} - 125301 \, \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )} \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} \]
-39663/1372000*sqrt(35)*log(sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3)) - 9)*sgn(1/(2*x + 3)) + 3/32*sqrt(3)*log(1/2*ab s(-2*sqrt(3) + 2*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + 2*sqrt(35)/(2* x + 3))/(sqrt(3) + sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2* x + 3)))*sgn(1/(2*x + 3)) - 1/470400*(35*(35*(1365*sgn(1/(2*x + 3))/(2*x + 3) - 1193*sgn(1/(2*x + 3)))/(2*x + 3) + 16227*sgn(1/(2*x + 3)))/(2*x + 3) - 125301*sgn(1/(2*x + 3)))*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3)
Time = 0.12 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.46 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{3/2}}{(3+2 x)^5} \, dx=\frac {39663\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{1372000}-\frac {3\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{32}-\frac {39663\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{1372000}-\frac {455\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1024\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {41767\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{156800\,\left (x+\frac {3}{2}\right )}-\frac {5409\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{8960\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {1193\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1536\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \]
(39663*35^(1/2)*log(x + 3/2))/1372000 - (3*3^(1/2)*asinh((2^(1/2)*3^(1/2)* x)/2))/32 - (39663*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/1372000 - (455*3^(1/2)*(x^2 + 2/3)^(1/2))/(1024*((27*x)/2 + (27*x ^2)/2 + 6*x^3 + x^4 + 81/16)) + (41767*3^(1/2)*(x^2 + 2/3)^(1/2))/(156800* (x + 3/2)) - (5409*3^(1/2)*(x^2 + 2/3)^(1/2))/(8960*(3*x + x^2 + 9/4)) + ( 1193*3^(1/2)*(x^2 + 2/3)^(1/2))/(1536*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))