Integrand size = 24, antiderivative size = 117 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=-\frac {7}{16} (775-243 x) \sqrt {2+3 x^2}-\frac {1}{24} (310-153 x) \left (2+3 x^2\right )^{3/2}-\frac {(34+x) \left (2+3 x^2\right )^{5/2}}{10 (3+2 x)}+\frac {18543}{32} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )+\frac {5425}{32} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right ) \]
-1/24*(310-153*x)*(3*x^2+2)^(3/2)-1/10*(34+x)*(3*x^2+2)^(5/2)/(3+2*x)+1854 3/32*arcsinh(1/2*x*6^(1/2))*3^(1/2)+5425/32*arctanh(1/35*(4-9*x)*35^(1/2)/ (3*x^2+2)^(1/2))*35^(1/2)-7/16*(775-243*x)*(3*x^2+2)^(1/2)
Time = 0.55 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.03 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\frac {1}{480} \left (-\frac {2 \sqrt {2+3 x^2} \left (265989+89521 x-19458 x^2+5118 x^3-1836 x^4+216 x^5\right )}{3+2 x}-162750 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )-278145 \sqrt {3} \log \left (-\sqrt {3} x+\sqrt {2+3 x^2}\right )\right ) \]
((-2*Sqrt[2 + 3*x^2]*(265989 + 89521*x - 19458*x^2 + 5118*x^3 - 1836*x^4 + 216*x^5))/(3 + 2*x) - 162750*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]] - 278145*Sqrt[3]*Log[-(Sqrt[3]*x) + Sqrt[2 + 3*x^2]])/480
Time = 0.27 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {681, 27, 682, 27, 682, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 681 |
| \(\displaystyle -\frac {1}{8} \int \frac {8 (1-51 x) \left (3 x^2+2\right )^{3/2}}{2 x+3}dx-\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\int \frac {(1-51 x) \left (3 x^2+2\right )^{3/2}}{2 x+3}dx-\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle -\frac {1}{48} \int \frac {84 (23-243 x) \sqrt {3 x^2+2}}{2 x+3}dx-\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {7}{4} \int \frac {(23-243 x) \sqrt {3 x^2+2}}{2 x+3}dx-\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 682 |
| \(\displaystyle -\frac {7}{4} \left (\frac {1}{24} \int \frac {6 (1642-7947 x)}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {1}{4} \sqrt {3 x^2+2} (775-243 x)\right )-\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {7}{4} \left (\frac {1}{4} \int \frac {1642-7947 x}{(2 x+3) \sqrt {3 x^2+2}}dx+\frac {1}{4} \sqrt {3 x^2+2} (775-243 x)\right )-\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 719 |
| \(\displaystyle -\frac {7}{4} \left (\frac {1}{4} \left (\frac {27125}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {7947}{2} \int \frac {1}{\sqrt {3 x^2+2}}dx\right )+\frac {1}{4} \sqrt {3 x^2+2} (775-243 x)\right )-\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle -\frac {7}{4} \left (\frac {1}{4} \left (\frac {27125}{2} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {2649}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (775-243 x)\right )-\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle -\frac {7}{4} \left (\frac {1}{4} \left (-\frac {27125}{2} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {2649}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (775-243 x)\right )-\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {7}{4} \left (\frac {1}{4} \left (-\frac {2649}{2} \sqrt {3} \text {arcsinh}\left (\sqrt {\frac {3}{2}} x\right )-\frac {775}{2} \sqrt {35} \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )\right )+\frac {1}{4} \sqrt {3 x^2+2} (775-243 x)\right )-\frac {(x+34) \left (3 x^2+2\right )^{5/2}}{10 (2 x+3)}-\frac {1}{24} (310-153 x) \left (3 x^2+2\right )^{3/2}\) |
-1/24*((310 - 153*x)*(2 + 3*x^2)^(3/2)) - ((34 + x)*(2 + 3*x^2)^(5/2))/(10 *(3 + 2*x)) - (7*(((775 - 243*x)*Sqrt[2 + 3*x^2])/4 + ((-2649*Sqrt[3]*ArcS inh[Sqrt[3/2]*x])/2 - (775*Sqrt[35]*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3 *x^2])])/2)/4))/4
3.14.87.3.1 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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ (e^2*(m + 1)*(m + 2*p + 2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] , x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || EqQ[p, 1] || (IntegerQ[p] && !RationalQ[m])) && NeQ[m, -1] && !ILtQ[m + 2 *p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p + 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2))) Int[(d + e*x) ^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x ], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || ! RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Intege rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
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 = 0.33 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.83
| method | result | size |
| risch | \(-\frac {648 x^{7}-5508 x^{6}+15786 x^{5}-62046 x^{4}+278799 x^{3}+759051 x^{2}+179042 x +531978}{240 \left (3+2 x \right ) \sqrt {3 x^{2}+2}}+\frac {18543 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{32}+\frac {5425 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{32}\) | \(97\) |
| trager | \(-\frac {\left (216 x^{5}-1836 x^{4}+5118 x^{3}-19458 x^{2}+89521 x +265989\right ) \sqrt {3 x^{2}+2}}{240 \left (3+2 x \right )}+\frac {5425 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{32}-\frac {21 \operatorname {RootOf}\left (\textit {\_Z}^{2}-2339067\right ) \ln \left (-\operatorname {RootOf}\left (\textit {\_Z}^{2}-2339067\right ) \sqrt {3 x^{2}+2}+2649 x \right )}{32}\) | \(121\) |
| default | \(-\frac {31 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{35}+\frac {51 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{8}+\frac {1701 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{16}+\frac {18543 \,\operatorname {arcsinh}\left (\frac {x \sqrt {6}}{2}\right ) \sqrt {3}}{32}-\frac {155 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{12}-\frac {5425 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{32}+\frac {5425 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{32}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{70 \left (x +\frac {3}{2}\right )}+\frac {39 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{70}\) | \(164\) |
-1/240*(648*x^7-5508*x^6+15786*x^5-62046*x^4+278799*x^3+759051*x^2+179042* x+531978)/(3+2*x)/(3*x^2+2)^(1/2)+18543/32*arcsinh(1/2*x*6^(1/2))*3^(1/2)+ 5425/32*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2 ))
Time = 0.33 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.12 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\frac {278145 \, \sqrt {3} {\left (2 \, x + 3\right )} \log \left (-\sqrt {3} \sqrt {3 \, x^{2} + 2} x - 3 \, x^{2} - 1\right ) + 81375 \, \sqrt {35} {\left (2 \, x + 3\right )} \log \left (\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} - 93 \, x^{2} + 36 \, x - 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 4 \, {\left (216 \, x^{5} - 1836 \, x^{4} + 5118 \, x^{3} - 19458 \, x^{2} + 89521 \, x + 265989\right )} \sqrt {3 \, x^{2} + 2}}{960 \, {\left (2 \, x + 3\right )}} \]
1/960*(278145*sqrt(3)*(2*x + 3)*log(-sqrt(3)*sqrt(3*x^2 + 2)*x - 3*x^2 - 1 ) + 81375*sqrt(35)*(2*x + 3)*log((sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) - 93* x^2 + 36*x - 43)/(4*x^2 + 12*x + 9)) - 4*(216*x^5 - 1836*x^4 + 5118*x^3 - 19458*x^2 + 89521*x + 265989)*sqrt(3*x^2 + 2))/(2*x + 3)
Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.04 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=-\frac {1}{20} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} + \frac {51}{8} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x - \frac {155}{12} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{4 \, {\left (2 \, x + 3\right )}} + \frac {1701}{16} \, \sqrt {3 \, x^{2} + 2} x + \frac {18543}{32} \, \sqrt {3} \operatorname {arsinh}\left (\frac {1}{2} \, \sqrt {6} x\right ) - \frac {5425}{32} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) - \frac {5425}{16} \, \sqrt {3 \, x^{2} + 2} \]
-1/20*(3*x^2 + 2)^(5/2) + 51/8*(3*x^2 + 2)^(3/2)*x - 155/12*(3*x^2 + 2)^(3 /2) - 13/4*(3*x^2 + 2)^(5/2)/(2*x + 3) + 1701/16*sqrt(3*x^2 + 2)*x + 18543 /32*sqrt(3)*arcsinh(1/2*sqrt(6)*x) - 5425/32*sqrt(35)*arcsinh(3/2*sqrt(6)* x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) - 5425/16*sqrt(3*x^2 + 2)
Leaf count of result is larger than twice the leaf count of optimal. 665 vs. \(2 (91) = 182\).
Time = 0.50 (sec) , antiderivative size = 665, normalized size of antiderivative = 5.68 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\frac {5425}{32} \, \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 {18543}{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 {15925}{128} \, \sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + \frac {9 \, {\left (238455 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{9} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 149045 \, \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 )}^{8} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 697600 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{7} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 719040 \, \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 )}^{6} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 4150566 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{5} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 2707250 \, \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 )}^{4} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 6756120 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{3} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 4557000 \, \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 )}^{2} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) + 3563595 \, {\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right ) - 2833425 \, \sqrt {35} \mathrm {sgn}\left (\frac {1}{2 \, x + 3}\right )\right )}}{320 \, {\left ({\left (\sqrt {-\frac {18}{2 \, x + 3} + \frac {35}{{\left (2 \, x + 3\right )}^{2}} + 3} + \frac {\sqrt {35}}{2 \, x + 3}\right )}^{2} - 3\right )}^{5}} \]
5425/32*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)) - 18543/32*sqrt(3)*log(1/2*abs(- 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)) - 15925/128*sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3 )*sgn(1/(2*x + 3)) + 9/320*(238455*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^9*sgn(1/(2*x + 3)) - 149045*sqrt(35)*(sqrt(-18/(2 *x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^8*sgn(1/(2*x + 3)) - 6 97600*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^7*sg n(1/(2*x + 3)) + 719040*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^6*sgn(1/(2*x + 3)) + 4150566*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^5*sgn(1/(2*x + 3)) - 2707250*sqr t(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^4*sg n(1/(2*x + 3)) - 6756120*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt( 35)/(2*x + 3))^3*sgn(1/(2*x + 3)) + 4557000*sqrt(35)*(sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^2*sgn(1/(2*x + 3)) + 3563595*(s qrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))*sgn(1/(2*x + 3)) - 2833425*sqrt(35)*sgn(1/(2*x + 3)))/((sqrt(-18/(2*x + 3) + 35/(2*x + 3)^2 + 3) + sqrt(35)/(2*x + 3))^2 - 3)^5
Time = 0.14 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.18 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^2} \, dx=\frac {18543\,\sqrt {3}\,\mathrm {asinh}\left (\frac {\sqrt {2}\,\sqrt {3}\,x}{2}\right )}{32}-\frac {275027\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{960}-\frac {5425\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{32}+\frac {5425\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{32}-\frac {1393\,\sqrt {3}\,x^2\,\sqrt {x^2+\frac {2}{3}}}{80}+\frac {9\,\sqrt {3}\,x^3\,\sqrt {x^2+\frac {2}{3}}}{2}-\frac {9\,\sqrt {3}\,x^4\,\sqrt {x^2+\frac {2}{3}}}{20}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{128\,\left (x+\frac {3}{2}\right )}+\frac {2133\,\sqrt {3}\,x\,\sqrt {x^2+\frac {2}{3}}}{32} \]
(18543*3^(1/2)*asinh((2^(1/2)*3^(1/2)*x)/2))/32 - (275027*3^(1/2)*(x^2 + 2 /3)^(1/2))/960 - (5425*35^(1/2)*log(x + 3/2))/32 + (5425*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/32 - (1393*3^(1/2)*x^2*(x^2 + 2/3)^(1/2))/80 + (9*3^(1/2)*x^3*(x^2 + 2/3)^(1/2))/2 - (9*3^(1/2)*x^4*( x^2 + 2/3)^(1/2))/20 - (15925*3^(1/2)*(x^2 + 2/3)^(1/2))/(128*(x + 3/2)) + (2133*3^(1/2)*x*(x^2 + 2/3)^(1/2))/32