Integrand size = 27, antiderivative size = 190 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^4} \, dx=-\frac {7 (37375-78054 x) \sqrt {2+5 x+3 x^2}}{6144}-\frac {7 (5713+1652 x) \left (2+5 x+3 x^2\right )^{3/2}}{768 (3+2 x)}+\frac {7 (1171+414 x) \left (2+5 x+3 x^2\right )^{5/2}}{960 (3+2 x)^2}-\frac {(37+3 x) \left (2+5 x+3 x^2\right )^{7/2}}{30 (3+2 x)^3}+\frac {2776697 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{12288 \sqrt {3}}-\frac {59745 \sqrt {5} \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{1024} \]
-7/768*(5713+1652*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)+7/960*(1171+414*x)*(3*x^2 +5*x+2)^(5/2)/(3+2*x)^2-1/30*(37+3*x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)^3+277669 7/36864*arctanh(1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-59745/102 4*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)-7/6144*(37375- 78054*x)*(3*x^2+5*x+2)^(1/2)
Time = 0.71 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.65 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^4} \, dx=\frac {-\frac {3 \sqrt {2+5 x+3 x^2} \left (61268351+98927312 x+44770416 x^2+746240 x^3-3277520 x^4-1266816 x^5-231552 x^6+82944 x^7\right )}{(3+2 x)^3}-10754100 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )+13883485 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{92160} \]
((-3*Sqrt[2 + 5*x + 3*x^2]*(61268351 + 98927312*x + 44770416*x^2 + 746240* x^3 - 3277520*x^4 - 1266816*x^5 - 231552*x^6 + 82944*x^7))/(3 + 2*x)^3 - 1 0754100*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] + 13883485*Sqrt [3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/92160
Time = 0.44 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.08, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1230, 27, 1230, 27, 1230, 27, 1231, 27, 1269, 1092, 219, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^4} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {7}{120} \int -\frac {2 (207 x+173) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^3}dx-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{60} \int \frac {(207 x+173) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^3}dx-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {7}{60} \left (\frac {(414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}-\frac {5}{64} \int \frac {2 (4956 x+4199) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^2}dx\right )-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{60} \left (\frac {(414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}-\frac {5}{32} \int \frac {(4956 x+4199) \left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^2}dx\right )-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle \frac {7}{60} \left (\frac {(414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}-\frac {5}{32} \left (\frac {(1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{2 (2 x+3)}-\frac {1}{8} \int \frac {6 (26018 x+21957) \sqrt {3 x^2+5 x+2}}{2 x+3}dx\right )\right )-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{60} \left (\frac {(414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}-\frac {5}{32} \left (\frac {(1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{2 (2 x+3)}-\frac {3}{4} \int \frac {(26018 x+21957) \sqrt {3 x^2+5 x+2}}{2 x+3}dx\right )\right )-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {7}{60} \left (\frac {(414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}-\frac {5}{32} \left (\frac {(1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{2 (2 x+3)}-\frac {3}{4} \left (-\frac {1}{48} \int -\frac {2 (793342 x+677913)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} \sqrt {3 x^2+5 x+2} (37375-78054 x)\right )\right )\right )-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7}{60} \left (\frac {(414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}-\frac {5}{32} \left (\frac {(1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{2 (2 x+3)}-\frac {3}{4} \left (\frac {1}{24} \int \frac {793342 x+677913}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (37375-78054 x) \sqrt {3 x^2+5 x+2}\right )\right )\right )-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {7}{60} \left (\frac {(414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}-\frac {5}{32} \left (\frac {(1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{2 (2 x+3)}-\frac {3}{4} \left (\frac {1}{24} \left (396671 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-512100 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{12} (37375-78054 x) \sqrt {3 x^2+5 x+2}\right )\right )\right )-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {7}{60} \left (\frac {(414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}-\frac {5}{32} \left (\frac {(1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{2 (2 x+3)}-\frac {3}{4} \left (\frac {1}{24} \left (793342 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-512100 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{12} (37375-78054 x) \sqrt {3 x^2+5 x+2}\right )\right )\right )-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7}{60} \left (\frac {(414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}-\frac {5}{32} \left (\frac {(1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{2 (2 x+3)}-\frac {3}{4} \left (\frac {1}{24} \left (\frac {396671 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}-512100 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{12} (37375-78054 x) \sqrt {3 x^2+5 x+2}\right )\right )\right )-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {7}{60} \left (\frac {(414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}-\frac {5}{32} \left (\frac {(1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{2 (2 x+3)}-\frac {3}{4} \left (\frac {1}{24} \left (1024200 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {396671 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )-\frac {1}{12} (37375-78054 x) \sqrt {3 x^2+5 x+2}\right )\right )\right )-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {7}{60} \left (\frac {(414 x+1171) \left (3 x^2+5 x+2\right )^{5/2}}{16 (2 x+3)^2}-\frac {5}{32} \left (\frac {(1652 x+5713) \left (3 x^2+5 x+2\right )^{3/2}}{2 (2 x+3)}-\frac {3}{4} \left (\frac {1}{24} \left (\frac {396671 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}-102420 \sqrt {5} \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {1}{12} (37375-78054 x) \sqrt {3 x^2+5 x+2}\right )\right )\right )-\frac {(3 x+37) \left (3 x^2+5 x+2\right )^{7/2}}{30 (2 x+3)^3}\) |
-1/30*((37 + 3*x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x)^3 + (7*(((1171 + 414* x)*(2 + 5*x + 3*x^2)^(5/2))/(16*(3 + 2*x)^2) - (5*(((5713 + 1652*x)*(2 + 5 *x + 3*x^2)^(3/2))/(2*(3 + 2*x)) - (3*(-1/12*((37375 - 78054*x)*Sqrt[2 + 5 *x + 3*x^2]) + ((396671*ArcTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2] )])/Sqrt[3] - 102420*Sqrt[5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3 *x^2])])/24))/4))/32))/60
3.25.55.3.1 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_) + (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[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)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) + e*g*(m + 1)*x)*((a + b*x + 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 + b*x + c*x^2)^(p - 1)*Simp[g*(b*d + 2*a*e + 2*a*e*m + 2*b*d*p) - f*b*e*(m + 2*p + 2) + (g*(2*c*d + b*e + b*e*m + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x], x], x] /; FreeQ[{a, b, 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_.) + (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]
Time = 0.37 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.67
| method | result | size |
| risch | \(-\frac {248832 x^{9}-279936 x^{8}-4792320 x^{7}-16629744 x^{6}-16682512 x^{5}+131487408 x^{4}+522126496 x^{3}+767982445 x^{2}+504196379 x +122536702}{30720 \left (3+2 x \right )^{3} \sqrt {3 x^{2}+5 x +2}}+\frac {2776697 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{36864}+\frac {59745 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1024}\) | \(127\) |
| trager | \(-\frac {\left (82944 x^{7}-231552 x^{6}-1266816 x^{5}-3277520 x^{4}+746240 x^{3}+44770416 x^{2}+98927312 x +61268351\right ) \sqrt {3 x^{2}+5 x +2}}{30720 \left (3+2 x \right )^{3}}+\frac {105 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1618805\right ) \ln \left (\frac {-8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1618805\right ) x +5690 \sqrt {3 x^{2}+5 x +2}-7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-1618805\right )}{3+2 x}\right )}{1024}-\frac {2776697 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (-6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +6 \sqrt {3 x^{2}+5 x +2}-5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )\right )}{36864}\) | \(148\) |
| default | \(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{120 \left (x +\frac {3}{2}\right )^{3}}+\frac {57 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{200 \left (x +\frac {3}{2}\right )^{2}}+\frac {48 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{25}-\frac {96 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{25 \left (x +\frac {3}{2}\right )}+\frac {1253 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{400}+\frac {4529 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{768}+\frac {91063 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{6144}+\frac {2776697 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{36864}+\frac {59745 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{1024}-\frac {11949 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{800}-\frac {3983 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{128}-\frac {1707 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{200}-\frac {59745 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{1024}\) | \(274\) |
-1/30720*(248832*x^9-279936*x^8-4792320*x^7-16629744*x^6-16682512*x^5+1314 87408*x^4+522126496*x^3+767982445*x^2+504196379*x+122536702)/(3+2*x)^3/(3* x^2+5*x+2)^(1/2)+2776697/36864*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x^2+5*x+2)^(1/2 ))*3^(1/2)+59745/1024*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/(12*(x+3/2)^2 -16*x-19)^(1/2))
Time = 0.30 (sec) , antiderivative size = 189, normalized size of antiderivative = 0.99 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^4} \, dx=\frac {13883485 \, \sqrt {3} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 10754100 \, \sqrt {5} {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 12 \, {\left (82944 \, x^{7} - 231552 \, x^{6} - 1266816 \, x^{5} - 3277520 \, x^{4} + 746240 \, x^{3} + 44770416 \, x^{2} + 98927312 \, x + 61268351\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{368640 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} \]
1/368640*(13883485*sqrt(3)*(8*x^3 + 36*x^2 + 54*x + 27)*log(4*sqrt(3)*sqrt (3*x^2 + 5*x + 2)*(6*x + 5) + 72*x^2 + 120*x + 49) + 10754100*sqrt(5)*(8*x ^3 + 36*x^2 + 54*x + 27)*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)) - 12*(82944*x^7 - 231552*x^6 - 1266816*x^5 - 3277520*x^4 + 746240*x^3 + 44770416*x^2 + 98927312*x + 61268 351)*sqrt(3*x^2 + 5*x + 2))/(8*x^3 + 36*x^2 + 54*x + 27)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^4} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{16 x^{4} + 96 x^{3} + 216 x^{2} + 216 x + 81}\, dx \]
-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 2 16*x**2 + 216*x + 81), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(16* x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-1339*x**3*sqrt(3*x **2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral( -1090*x**4*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 216*x**2 + 216*x + 81), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(16*x**4 + 96*x**3 + 2 16*x**2 + 216*x + 81), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2)/(16*x* *4 + 96*x**3 + 216*x**2 + 216*x + 81), x)
Time = 0.28 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.31 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^4} \, dx=-\frac {171}{200} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{15 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} + \frac {57 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {9}{2}}}{50 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {3759}{200} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {581}{800} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {48 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{5 \, {\left (2 \, x + 3\right )}} + \frac {4529}{128} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {1253}{768} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {91063}{1024} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {2776697}{36864} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {59745}{1024} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {261625}{6144} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-171/200*(3*x^2 + 5*x + 2)^(7/2) - 13/15*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27) + 57/50*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) + 3 759/200*(3*x^2 + 5*x + 2)^(5/2)*x + 581/800*(3*x^2 + 5*x + 2)^(5/2) - 48/5 *(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 4529/128*(3*x^2 + 5*x + 2)^(3/2)*x - 1253/768*(3*x^2 + 5*x + 2)^(3/2) + 91063/1024*sqrt(3*x^2 + 5*x + 2)*x + 27 76697/36864*sqrt(3)*log(sqrt(3)*sqrt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 59745 /1024*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 261625/6144*sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 325 vs. \(2 (154) = 308\).
Time = 0.35 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.71 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^4} \, dx=-\frac {1}{30720} \, {\left (2 \, {\left (12 \, {\left (18 \, {\left (24 \, x - 175\right )} x + 4661\right )} x - 218885\right )} x + 1563313\right )} \sqrt {3 \, x^{2} + 5 \, x + 2} - \frac {59745}{1024} \, \sqrt {5} \log \left (\frac {{\left | -4 \, \sqrt {3} x - 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}{{\left | -4 \, \sqrt {3} x + 2 \, \sqrt {5} - 6 \, \sqrt {3} + 4 \, \sqrt {3 \, x^{2} + 5 \, x + 2} \right |}}\right ) - \frac {2776697}{36864} \, \sqrt {3} \log \left ({\left | -2 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} - 5 \right |}\right ) - \frac {5 \, {\left (424596 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 2828550 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 21565510 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 26086815 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 45375675 \, \sqrt {3} x + 10164786 \, \sqrt {3} - 45375675 \, \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}}{1536 \, {\left (2 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 6 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )} + 11\right )}^{3}} \]
-1/30720*(2*(12*(18*(24*x - 175)*x + 4661)*x - 218885)*x + 1563313)*sqrt(3 *x^2 + 5*x + 2) - 59745/1024*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6* sqrt(3) + 4*sqrt(3*x^2 + 5*x + 2))/abs(-4*sqrt(3)*x + 2*sqrt(5) - 6*sqrt(3 ) + 4*sqrt(3*x^2 + 5*x + 2))) - 2776697/36864*sqrt(3)*log(abs(-2*sqrt(3)*( sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) - 5)) - 5/1536*(424596*(sqrt(3)*x - sqr t(3*x^2 + 5*x + 2))^5 + 2828550*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2) )^4 + 21565510*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 26086815*sqrt(3)*(s qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 45375675*sqrt(3)*x + 10164786*sqrt(3 ) - 45375675*sqrt(3*x^2 + 5*x + 2))/(2*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) ^2 + 6*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) + 11)^3
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^4} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^4} \,d x \]