Integrand size = 40, antiderivative size = 172 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=-\frac {(85448933-14243732 x) \sqrt {3-x+2 x^2}}{32768}-\frac {(909513-226052 x) \left (3-x+2 x^2\right )^{3/2}}{18432}-\frac {839}{960} \left (3-x+2 x^2\right )^{5/2}-\frac {3667 \left (3-x+2 x^2\right )^{5/2}}{576 (5+2 x)}+\frac {5}{96} (5+2 x) \left (3-x+2 x^2\right )^{5/2}-\frac {982669459 \text {arcsinh}\left (\frac {1-4 x}{\sqrt {23}}\right )}{65536 \sqrt {2}}+\frac {959625 \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {3-x+2 x^2}}\right )}{64 \sqrt {2}} \]
-1/18432*(909513-226052*x)*(2*x^2-x+3)^(3/2)-839/960*(2*x^2-x+3)^(5/2)-366 7/576*(2*x^2-x+3)^(5/2)/(5+2*x)+5/96*(5+2*x)*(2*x^2-x+3)^(5/2)-982669459/1 31072*arcsinh(1/23*(1-4*x)*23^(1/2))*2^(1/2)+959625/128*arctanh(1/24*(17-2 2*x)*2^(1/2)/(2*x^2-x+3)^(1/2))*2^(1/2)-1/32768*(85448933-14243732*x)*(2*x ^2-x+3)^(1/2)
Time = 0.72 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.70 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=\frac {\frac {4 \sqrt {3-x+2 x^2} \left (-6814208295-1404323114 x+182033816 x^2-35369408 x^3+8283904 x^4-1798144 x^5+409600 x^6\right )}{5+2 x}-29479680000 \sqrt {2} \text {arctanh}\left (\frac {1}{6} \left (5+2 x-\sqrt {6-2 x+4 x^2}\right )\right )-14740041885 \sqrt {2} \log \left (1-4 x+2 \sqrt {6-2 x+4 x^2}\right )}{1966080} \]
((4*Sqrt[3 - x + 2*x^2]*(-6814208295 - 1404323114*x + 182033816*x^2 - 3536 9408*x^3 + 8283904*x^4 - 1798144*x^5 + 409600*x^6))/(5 + 2*x) - 2947968000 0*Sqrt[2]*ArcTanh[(5 + 2*x - Sqrt[6 - 2*x + 4*x^2])/6] - 14740041885*Sqrt[ 2]*Log[1 - 4*x + 2*Sqrt[6 - 2*x + 4*x^2]])/1966080
Time = 0.61 (sec) , antiderivative size = 184, normalized size of antiderivative = 1.07, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2181, 27, 2184, 27, 2184, 27, 1231, 27, 1231, 27, 1269, 1090, 222, 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.
| \(\displaystyle \int \frac {\left (2 x^2-x+3\right )^{3/2} \left (5 x^4-x^3+3 x^2+x+2\right )}{(2 x+5)^2} \, dx\) |
| \(\Big \downarrow \) 2181 |
| \(\displaystyle -\frac {1}{72} \int \frac {\left (2 x^2-x+3\right )^{3/2} \left (-2880 x^3+7776 x^2-79840 x+26675\right )}{16 (2 x+5)}dx-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\int \frac {\left (2 x^2-x+3\right )^{3/2} \left (-2880 x^3+7776 x^2-79840 x+26675\right )}{2 x+5}dx}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {1}{96} \int \frac {96 \left (2 x^2-x+3\right )^{3/2} \left (20136 x^2-67720 x+24725\right )}{2 x+5}dx}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\int \frac {\left (2 x^2-x+3\right )^{3/2} \left (20136 x^2-67720 x+24725\right )}{2 x+5}dx}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 2184 |
| \(\displaystyle \frac {-\frac {1}{40} \int \frac {80 (18655-56513 x) \left (2 x^2-x+3\right )^{3/2}}{2 x+5}dx+60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {5034}{5} \left (2 x^2-x+3\right )^{5/2}}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-2 \int \frac {(18655-56513 x) \left (2 x^2-x+3\right )^{3/2}}{2 x+5}dx+60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {5034}{5} \left (2 x^2-x+3\right )^{5/2}}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {-2 \left (\frac {1}{32} (909513-226052 x) \left (2 x^2-x+3\right )^{3/2}-\frac {1}{64} \int -\frac {9 (2667335-7121866 x) \sqrt {2 x^2-x+3}}{2 x+5}dx\right )+60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {5034}{5} \left (2 x^2-x+3\right )^{5/2}}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-2 \left (\frac {9}{64} \int \frac {(2667335-7121866 x) \sqrt {2 x^2-x+3}}{2 x+5}dx+\frac {1}{32} (909513-226052 x) \left (2 x^2-x+3\right )^{3/2}\right )+60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {5034}{5} \left (2 x^2-x+3\right )^{5/2}}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {-2 \left (\frac {9}{64} \left (\frac {1}{8} (85448933-14243732 x) \sqrt {2 x^2-x+3}-\frac {1}{32} \int -\frac {2 (982588705-1965338918 x)}{(2 x+5) \sqrt {2 x^2-x+3}}dx\right )+\frac {1}{32} (909513-226052 x) \left (2 x^2-x+3\right )^{3/2}\right )+60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {5034}{5} \left (2 x^2-x+3\right )^{5/2}}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {-2 \left (\frac {9}{64} \left (\frac {1}{16} \int \frac {982588705-1965338918 x}{(2 x+5) \sqrt {2 x^2-x+3}}dx+\frac {1}{8} \sqrt {2 x^2-x+3} (85448933-14243732 x)\right )+\frac {1}{32} (909513-226052 x) \left (2 x^2-x+3\right )^{3/2}\right )+60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {5034}{5} \left (2 x^2-x+3\right )^{5/2}}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {-2 \left (\frac {9}{64} \left (\frac {1}{16} \left (5895936000 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-982669459 \int \frac {1}{\sqrt {2 x^2-x+3}}dx\right )+\frac {1}{8} \sqrt {2 x^2-x+3} (85448933-14243732 x)\right )+\frac {1}{32} (909513-226052 x) \left (2 x^2-x+3\right )^{3/2}\right )+60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {5034}{5} \left (2 x^2-x+3\right )^{5/2}}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 1090 |
| \(\displaystyle \frac {-2 \left (\frac {9}{64} \left (\frac {1}{16} \left (5895936000 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {982669459 \int \frac {1}{\sqrt {\frac {1}{23} (4 x-1)^2+1}}d(4 x-1)}{\sqrt {46}}\right )+\frac {1}{8} \sqrt {2 x^2-x+3} (85448933-14243732 x)\right )+\frac {1}{32} (909513-226052 x) \left (2 x^2-x+3\right )^{3/2}\right )+60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {5034}{5} \left (2 x^2-x+3\right )^{5/2}}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 222 |
| \(\displaystyle \frac {-2 \left (\frac {9}{64} \left (\frac {1}{16} \left (5895936000 \int \frac {1}{(2 x+5) \sqrt {2 x^2-x+3}}dx-\frac {982669459 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {2}}\right )+\frac {1}{8} \sqrt {2 x^2-x+3} (85448933-14243732 x)\right )+\frac {1}{32} (909513-226052 x) \left (2 x^2-x+3\right )^{3/2}\right )+60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {5034}{5} \left (2 x^2-x+3\right )^{5/2}}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {-2 \left (\frac {9}{64} \left (\frac {1}{16} \left (-11791872000 \int \frac {1}{288-\frac {(17-22 x)^2}{2 x^2-x+3}}d\frac {17-22 x}{\sqrt {2 x^2-x+3}}-\frac {982669459 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {2}}\right )+\frac {1}{8} \sqrt {2 x^2-x+3} (85448933-14243732 x)\right )+\frac {1}{32} (909513-226052 x) \left (2 x^2-x+3\right )^{3/2}\right )+60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {5034}{5} \left (2 x^2-x+3\right )^{5/2}}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {-2 \left (\frac {9}{64} \left (\frac {1}{16} \left (-\frac {982669459 \text {arcsinh}\left (\frac {4 x-1}{\sqrt {23}}\right )}{\sqrt {2}}-491328000 \sqrt {2} \text {arctanh}\left (\frac {17-22 x}{12 \sqrt {2} \sqrt {2 x^2-x+3}}\right )\right )+\frac {1}{8} \sqrt {2 x^2-x+3} (85448933-14243732 x)\right )+\frac {1}{32} (909513-226052 x) \left (2 x^2-x+3\right )^{3/2}\right )+60 (2 x+5) \left (2 x^2-x+3\right )^{5/2}-\frac {5034}{5} \left (2 x^2-x+3\right )^{5/2}}{1152}-\frac {3667 \left (2 x^2-x+3\right )^{5/2}}{576 (2 x+5)}\) |
(-3667*(3 - x + 2*x^2)^(5/2))/(576*(5 + 2*x)) + ((-5034*(3 - x + 2*x^2)^(5 /2))/5 + 60*(5 + 2*x)*(3 - x + 2*x^2)^(5/2) - 2*(((909513 - 226052*x)*(3 - x + 2*x^2)^(3/2))/32 + (9*(((85448933 - 14243732*x)*Sqrt[3 - x + 2*x^2])/ 8 + ((-982669459*ArcSinh[(-1 + 4*x)/Sqrt[23]])/Sqrt[2] - 491328000*Sqrt[2] *ArcTanh[(17 - 22*x)/(12*Sqrt[2]*Sqrt[3 - x + 2*x^2])])/16))/64))/1152
3.4.37.3.1 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[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/(2*c*(-4* (c/(b^2 - 4*a*c)))^p) Subst[Int[Simp[1 - x^2/(b^2 - 4*a*c), x]^p, x], x, b + 2*c*x], x] /; FreeQ[{a, b, c, p}, x] && GtQ[4*a - b^2/c, 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[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*(c*d + 2*c*d*p - b*e*p) + g*c*e*(m + 2*p + 1)*x)*((a + b*x + c*x^2)^p/ (c*e^2*(m + 2*p + 1)*(m + 2*p + 2))), x] - Simp[p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)) Int[(d + e*x)^m*(a + b*x + c*x^2)^(p - 1)*Simp[c*e*f*(b*d - 2* a*e)*(m + 2*p + 2) + g*(a*e*(b*e - 2*c*d*m + b*e*m) + b*d*(b*e*p - c*d - 2* c*d*p)) + (c*e*f*(2*c*d - b*e)*(m + 2*p + 2) + g*(b^2*e^2*(p + m + 1) - 2*c ^2*d^2*(1 + 2*p) - c*e*(b*d*(m - 2*p) + 2*a*e*(m + 2*p + 1))))*x, x], x], x ] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] || !R ationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) && !ILtQ[m + 2*p, 0] && (Integer Q[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[g/e Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Simp[(e*f - d*g)/e Int[(d + e*x)^m*(a + b*x + c*x^2)^ p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x] && !IGtQ[m, 0]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_ ), x_Symbol] :> With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = Polynomi alRemainder[Pq, d + e*x, x]}, Simp[(e*R*(d + e*x)^(m + 1)*(a + b*x + c*x^2) ^(p + 1))/((m + 1)*(c*d^2 - b*d*e + a*e^2)), x] + Simp[1/((m + 1)*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*ExpandToSum[(m + 1)*(c*d^2 - b*d*e + a*e^2)*Qx + c*d*R*(m + 1) - b*e*R*(m + p + 2) - c*e*R *(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, e, p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1]
Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p _), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1)) Int[(d + e*x)^m*(a + b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(b*d*e*(p + 1) + a*e^2*(m + q - 1) - c *d^2*(m + q + 2*p + 1) - e*(2*c*d - b*e)*(m + q + p)*x), x], x], x] /; GtQ[ q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, p}, x] && Pol yQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && !(IGt Q[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))
Timed out.
hanged
Time = 0.27 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.89 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=\frac {14740041885 \, \sqrt {2} {\left (2 \, x + 5\right )} \log \left (-4 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (4 \, x - 1\right )} - 32 \, x^{2} + 16 \, x - 25\right ) + 14739840000 \, \sqrt {2} {\left (2 \, x + 5\right )} \log \left (\frac {24 \, \sqrt {2} \sqrt {2 \, x^{2} - x + 3} {\left (22 \, x - 17\right )} - 1060 \, x^{2} + 1036 \, x - 1153}{4 \, x^{2} + 20 \, x + 25}\right ) + 8 \, {\left (409600 \, x^{6} - 1798144 \, x^{5} + 8283904 \, x^{4} - 35369408 \, x^{3} + 182033816 \, x^{2} - 1404323114 \, x - 6814208295\right )} \sqrt {2 \, x^{2} - x + 3}}{3932160 \, {\left (2 \, x + 5\right )}} \]
1/3932160*(14740041885*sqrt(2)*(2*x + 5)*log(-4*sqrt(2)*sqrt(2*x^2 - x + 3 )*(4*x - 1) - 32*x^2 + 16*x - 25) + 14739840000*sqrt(2)*(2*x + 5)*log((24* sqrt(2)*sqrt(2*x^2 - x + 3)*(22*x - 17) - 1060*x^2 + 1036*x - 1153)/(4*x^2 + 20*x + 25)) + 8*(409600*x^6 - 1798144*x^5 + 8283904*x^4 - 35369408*x^3 + 182033816*x^2 - 1404323114*x - 6814208295)*sqrt(2*x^2 - x + 3))/(2*x + 5 )
\[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=\int \frac {\left (2 x^{2} - x + 3\right )^{\frac {3}{2}} \cdot \left (5 x^{4} - x^{3} + 3 x^{2} + x + 2\right )}{\left (2 x + 5\right )^{2}}\, dx \]
Time = 0.28 (sec) , antiderivative size = 161, normalized size of antiderivative = 0.94 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=\frac {5}{48} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} x - \frac {589}{960} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {5}{2}} + \frac {9059}{1536} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} x - \frac {185827}{6144} \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}} + \frac {3560933}{8192} \, \sqrt {2 \, x^{2} - x + 3} x + \frac {982669459}{131072} \, \sqrt {2} \operatorname {arsinh}\left (\frac {4}{23} \, \sqrt {23} x - \frac {1}{23} \, \sqrt {23}\right ) - \frac {959625}{128} \, \sqrt {2} \operatorname {arsinh}\left (\frac {22 \, \sqrt {23} x}{23 \, {\left | 2 \, x + 5 \right |}} - \frac {17 \, \sqrt {23}}{23 \, {\left | 2 \, x + 5 \right |}}\right ) - \frac {85448933}{32768} \, \sqrt {2 \, x^{2} - x + 3} - \frac {3667 \, {\left (2 \, x^{2} - x + 3\right )}^{\frac {3}{2}}}{32 \, {\left (2 \, x + 5\right )}} \]
5/48*(2*x^2 - x + 3)^(5/2)*x - 589/960*(2*x^2 - x + 3)^(5/2) + 9059/1536*( 2*x^2 - x + 3)^(3/2)*x - 185827/6144*(2*x^2 - x + 3)^(3/2) + 3560933/8192* sqrt(2*x^2 - x + 3)*x + 982669459/131072*sqrt(2)*arcsinh(4/23*sqrt(23)*x - 1/23*sqrt(23)) - 959625/128*sqrt(2)*arcsinh(22/23*sqrt(23)*x/abs(2*x + 5) - 17/23*sqrt(23)/abs(2*x + 5)) - 85448933/32768*sqrt(2*x^2 - x + 3) - 366 7/32*(2*x^2 - x + 3)^(3/2)/(2*x + 5)
Leaf count of result is larger than twice the leaf count of optimal. 707 vs. \(2 (137) = 274\).
Time = 0.36 (sec) , antiderivative size = 707, normalized size of antiderivative = 4.11 \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=\text {Too large to display} \]
1/1966080*sqrt(2)*(14739840000*log(12*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 72/(2*x + 5) - 11)*sgn(1/(2*x + 5)) + 14740041885*log(abs(sqrt(-11/ (2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5) + 1))*sgn(1/(2*x + 5)) - 147 40041885*log(abs(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5) - 1))*sgn(1/(2*x + 5)) - 2027704320*sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) *sgn(1/(2*x + 5)) + 2*(45496763235*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^11*sgn(1/(2*x + 5)) - 126553743360*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^10*sgn(1/(2*x + 5)) + 44062768335*(sqr t(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^9*sgn(1/(2*x + 5)) + 33178982400*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^8*sgn (1/(2*x + 5)) + 294206421582*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6 /(2*x + 5))^7*sgn(1/(2*x + 5)) - 463672074240*(sqrt(-11/(2*x + 5) + 36/(2* x + 5)^2 + 1) + 6/(2*x + 5))^6*sgn(1/(2*x + 5)) + 35099942478*(sqrt(-11/(2 *x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^5*sgn(1/(2*x + 5)) + 17132461 0560*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^4*sgn(1/(2*x + 5)) + 60059281615*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^3*sgn(1/(2*x + 5)) - 105051009024*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^2*sgn(1/(2*x + 5)) - 5210329245*(sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))*sgn(1/(2*x + 5)) + 17058392064*sgn(1/( 2*x + 5)))/((sqrt(-11/(2*x + 5) + 36/(2*x + 5)^2 + 1) + 6/(2*x + 5))^2 ...
Timed out. \[ \int \frac {\left (3-x+2 x^2\right )^{3/2} \left (2+x+3 x^2-x^3+5 x^4\right )}{(5+2 x)^2} \, dx=\int \frac {{\left (2\,x^2-x+3\right )}^{3/2}\,\left (5\,x^4-x^3+3\,x^2+x+2\right )}{{\left (2\,x+5\right )}^2} \,d x \]