Integrand size = 27, antiderivative size = 174 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^2} \, dx=-\frac {(1454315-3037062 x) \sqrt {2+5 x+3 x^2}}{110592}-\frac {(6925-151098 x) \left (2+5 x+3 x^2\right )^{3/2}}{13824}+\frac {(283+8310 x) \left (2+5 x+3 x^2\right )^{5/2}}{1440}-\frac {(47+x) \left (2+5 x+3 x^2\right )^{7/2}}{14 (3+2 x)}+\frac {15434623 \text {arctanh}\left (\frac {5+6 x}{2 \sqrt {3} \sqrt {2+5 x+3 x^2}}\right )}{221184 \sqrt {3}}-\frac {9225}{512} \sqrt {5} \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right ) \]
-1/13824*(6925-151098*x)*(3*x^2+5*x+2)^(3/2)+1/1440*(283+8310*x)*(3*x^2+5* x+2)^(5/2)-1/14*(47+x)*(3*x^2+5*x+2)^(7/2)/(3+2*x)+15434623/663552*arctanh (1/6*(5+6*x)*3^(1/2)/(3*x^2+5*x+2)^(1/2))*3^(1/2)-9225/512*arctanh(1/10*(7 +8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)-1/110592*(1454315-3037062*x)*(3 *x^2+5*x+2)^(1/2)
Time = 0.80 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.71 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^2} \, dx=\frac {-\frac {3 \sqrt {2+5 x+3 x^2} \left (259165107+28017108 x-179819084 x^2-275126016 x^3-273531168 x^4-125632512 x^5-13893120 x^6+7464960 x^7\right )}{3+2 x}-418446000 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )+540211805 \sqrt {3} \text {arctanh}\left (\frac {\sqrt {\frac {2}{3}+\frac {5 x}{3}+x^2}}{1+x}\right )}{11612160} \]
((-3*Sqrt[2 + 5*x + 3*x^2]*(259165107 + 28017108*x - 179819084*x^2 - 27512 6016*x^3 - 273531168*x^4 - 125632512*x^5 - 13893120*x^6 + 7464960*x^7))/(3 + 2*x) - 418446000*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)] + 5 40211805*Sqrt[3]*ArcTanh[Sqrt[2/3 + (5*x)/3 + x^2]/(1 + x)])/11612160
Time = 0.43 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1230, 27, 1231, 25, 1231, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^2} \, dx\) |
| \(\Big \downarrow \) 1230 |
| \(\displaystyle -\frac {1}{8} \int -\frac {2 (277 x+231) \left (3 x^2+5 x+2\right )^{5/2}}{2 x+3}dx-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \int \frac {(277 x+231) \left (3 x^2+5 x+2\right )^{5/2}}{2 x+3}dx-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{360} (8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}-\frac {1}{144} \int -\frac {(50366 x+42339) \left (3 x^2+5 x+2\right )^{3/2}}{2 x+3}dx\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{144} \int \frac {(50366 x+42339) \left (3 x^2+5 x+2\right )^{3/2}}{2 x+3}dx+\frac {1}{360} (8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{144} \left (-\frac {1}{96} \int -\frac {6 (1012354 x+854331) \sqrt {3 x^2+5 x+2}}{2 x+3}dx-\frac {1}{24} (6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{360} (8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{144} \left (\frac {1}{16} \int \frac {(1012354 x+854331) \sqrt {3 x^2+5 x+2}}{2 x+3}dx-\frac {1}{24} (6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{360} (8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
| \(\Big \downarrow \) 1231 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{144} \left (\frac {1}{16} \left (-\frac {1}{48} \int -\frac {2 (30869246 x+26377869)}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} \sqrt {3 x^2+5 x+2} (1454315-3037062 x)\right )-\frac {1}{24} (6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{360} (8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{144} \left (\frac {1}{16} \left (\frac {1}{24} \int \frac {30869246 x+26377869}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx-\frac {1}{12} (1454315-3037062 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{24} (6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{360} (8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
| \(\Big \downarrow \) 1269 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{144} \left (\frac {1}{16} \left (\frac {1}{24} \left (15434623 \int \frac {1}{\sqrt {3 x^2+5 x+2}}dx-19926000 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{12} (1454315-3037062 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{24} (6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{360} (8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
| \(\Big \downarrow \) 1092 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{144} \left (\frac {1}{16} \left (\frac {1}{24} \left (30869246 \int \frac {1}{12-\frac {(6 x+5)^2}{3 x^2+5 x+2}}d\frac {6 x+5}{\sqrt {3 x^2+5 x+2}}-19926000 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{12} (1454315-3037062 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{24} (6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{360} (8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{144} \left (\frac {1}{16} \left (\frac {1}{24} \left (\frac {15434623 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}-19926000 \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {1}{12} (1454315-3037062 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{24} (6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{360} (8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{144} \left (\frac {1}{16} \left (\frac {1}{24} \left (39852000 \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {15434623 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}\right )-\frac {1}{12} (1454315-3037062 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{24} (6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{360} (8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{4} \left (\frac {1}{144} \left (\frac {1}{16} \left (\frac {1}{24} \left (\frac {15434623 \text {arctanh}\left (\frac {6 x+5}{2 \sqrt {3} \sqrt {3 x^2+5 x+2}}\right )}{\sqrt {3}}-3985200 \sqrt {5} \text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )\right )-\frac {1}{12} (1454315-3037062 x) \sqrt {3 x^2+5 x+2}\right )-\frac {1}{24} (6925-151098 x) \left (3 x^2+5 x+2\right )^{3/2}\right )+\frac {1}{360} (8310 x+283) \left (3 x^2+5 x+2\right )^{5/2}\right )-\frac {(x+47) \left (3 x^2+5 x+2\right )^{7/2}}{14 (2 x+3)}\) |
-1/14*((47 + x)*(2 + 5*x + 3*x^2)^(7/2))/(3 + 2*x) + (((283 + 8310*x)*(2 + 5*x + 3*x^2)^(5/2))/360 + (-1/24*((6925 - 151098*x)*(2 + 5*x + 3*x^2)^(3/ 2)) + (-1/12*((1454315 - 3037062*x)*Sqrt[2 + 5*x + 3*x^2]) + ((15434623*Ar cTanh[(5 + 6*x)/(2*Sqrt[3]*Sqrt[2 + 5*x + 3*x^2])])/Sqrt[3] - 3985200*Sqrt [5]*ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])])/24)/16)/144)/4
3.25.53.3.1 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 = 0.34 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.73
| method | result | size |
| risch | \(-\frac {22394880 x^{9}-4354560 x^{8}-431433216 x^{7}-1476542304 x^{6}-2444298912 x^{5}-2462149668 x^{4}-1365296128 x^{3}+557942693 x^{2}+1351859751 x +518330214}{3870720 \left (3+2 x \right ) \sqrt {3 x^{2}+5 x +2}}+\frac {15434623 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}}{663552}+\frac {9225 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{512}\) | \(127\) |
| trager | \(-\frac {\left (7464960 x^{7}-13893120 x^{6}-125632512 x^{5}-273531168 x^{4}-275126016 x^{3}-179819084 x^{2}+28017108 x +259165107\right ) \sqrt {3 x^{2}+5 x +2}}{3870720 \left (3+2 x \right )}+\frac {15434623 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) \ln \left (6 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right ) x +5 \operatorname {RootOf}\left (\textit {\_Z}^{2}-3\right )+6 \sqrt {3 x^{2}+5 x +2}\right )}{663552}-\frac {9225 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{512}\) | \(148\) |
| default | \(-\frac {369 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{140}+\frac {277 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{288}+\frac {25183 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{13824}+\frac {506177 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{110592}+\frac {15434623 \ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}\right ) \sqrt {3}}{663552}-\frac {369 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{80}-\frac {615 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{64}-\frac {9225 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{512}+\frac {9225 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{512}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{10 \left (x +\frac {3}{2}\right )}+\frac {13 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{20}\) | \(232\) |
-1/3870720*(22394880*x^9-4354560*x^8-431433216*x^7-1476542304*x^6-24442989 12*x^5-2462149668*x^4-1365296128*x^3+557942693*x^2+1351859751*x+518330214) /(3+2*x)/(3*x^2+5*x+2)^(1/2)+15434623/663552*ln(1/3*(5/2+3*x)*3^(1/2)+(3*x ^2+5*x+2)^(1/2))*3^(1/2)+9225/512*5^(1/2)*arctanh(2/5*(-7/2-4*x)*5^(1/2)/( 12*(x+3/2)^2-16*x-19)^(1/2))
Time = 0.29 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.91 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^2} \, dx=\frac {540211805 \, \sqrt {3} {\left (2 \, x + 3\right )} \log \left (4 \, \sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (6 \, x + 5\right )} + 72 \, x^{2} + 120 \, x + 49\right ) + 418446000 \, \sqrt {5} {\left (2 \, x + 3\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) - 12 \, {\left (7464960 \, x^{7} - 13893120 \, x^{6} - 125632512 \, x^{5} - 273531168 \, x^{4} - 275126016 \, x^{3} - 179819084 \, x^{2} + 28017108 \, x + 259165107\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{46448640 \, {\left (2 \, x + 3\right )}} \]
1/46448640*(540211805*sqrt(3)*(2*x + 3)*log(4*sqrt(3)*sqrt(3*x^2 + 5*x + 2 )*(6*x + 5) + 72*x^2 + 120*x + 49) + 418446000*sqrt(5)*(2*x + 3)*log(-(4*s qrt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) - 124*x^2 - 212*x - 89)/(4*x^2 + 12 *x + 9)) - 12*(7464960*x^7 - 13893120*x^6 - 125632512*x^5 - 273531168*x^4 - 275126016*x^3 - 179819084*x^2 + 28017108*x + 259165107)*sqrt(3*x^2 + 5*x + 2))/(2*x + 3)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^2} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{4 x^{2} + 12 x + 9}\, dx \]
-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-2 92*x*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-870*x**2*s qrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-1339*x**3*sqrt(3 *x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(-396*x**5*sqrt(3*x**2 + 5*x + 2)/(4*x**2 + 12*x + 9), x) - Integral(27*x**7*sqrt(3*x**2 + 5*x + 2)/(4 *x**2 + 12*x + 9), x)
Time = 0.28 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.10 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^2} \, dx=-\frac {1}{28} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}} + \frac {277}{48} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} x + \frac {283}{1440} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{4 \, {\left (2 \, x + 3\right )}} + \frac {25183}{2304} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {6925}{13824} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} + \frac {506177}{18432} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {15434623}{663552} \, \sqrt {3} \log \left (\sqrt {3} \sqrt {3 \, x^{2} + 5 \, x + 2} + 3 \, x + \frac {5}{2}\right ) + \frac {9225}{512} \, \sqrt {5} \log \left (\frac {\sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2}}{{\left | 2 \, x + 3 \right |}} + \frac {5}{2 \, {\left | 2 \, x + 3 \right |}} - 2\right ) - \frac {1454315}{110592} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
-1/28*(3*x^2 + 5*x + 2)^(7/2) + 277/48*(3*x^2 + 5*x + 2)^(5/2)*x + 283/144 0*(3*x^2 + 5*x + 2)^(5/2) - 13/4*(3*x^2 + 5*x + 2)^(7/2)/(2*x + 3) + 25183 /2304*(3*x^2 + 5*x + 2)^(3/2)*x - 6925/13824*(3*x^2 + 5*x + 2)^(3/2) + 506 177/18432*sqrt(3*x^2 + 5*x + 2)*x + 15434623/663552*sqrt(3)*log(sqrt(3)*sq rt(3*x^2 + 5*x + 2) + 3*x + 5/2) + 9225/512*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3) + 5/2/abs(2*x + 3) - 2) - 1454315/110592*sqrt(3*x ^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 861 vs. \(2 (138) = 276\).
Time = 0.65 (sec) , antiderivative size = 861, normalized size of antiderivative = 4.95 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^2} \, dx=\text {Too large to display} \]
-15434623/663552*sqrt(3)*log(abs(-2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3))/abs(2*sqrt(3) + 2*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + 2*sqrt(5)/(2*x + 3)))*sgn(1/(2*x + 3)) + 9225/512*sqr t(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)) - 1625/512*sqrt(-8/(2*x + 3) + 5/(2*x + 3)^ 2 + 3)*sgn(1/(2*x + 3)) + 1/3870720*(1702084195*(sqrt(-8/(2*x + 3) + 5/(2* x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^13*sgn(1/(2*x + 3)) - 3595838400*sqrt(5 )*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^12*sgn(1/(2 *x + 3)) - 462583100*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2* x + 3))^11*sgn(1/(2*x + 3)) + 12803555520*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/( 2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^10*sgn(1/(2*x + 3)) + 91554292599*(sq rt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^9*sgn(1/(2*x + 3 )) - 132950643840*sqrt(5)*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5 )/(2*x + 3))^8*sgn(1/(2*x + 3)) - 221215739904*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^7*sgn(1/(2*x + 3)) + 432202780800*sqrt(5 )*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^6*sgn(1/(2* x + 3)) + 252015304401*(sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/( 2*x + 3))^5*sgn(1/(2*x + 3)) - 680038027200*sqrt(5)*(sqrt(-8/(2*x + 3) + 5 /(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^4*sgn(1/(2*x + 3)) - 506502404100*( sqrt(-8/(2*x + 3) + 5/(2*x + 3)^2 + 3) + sqrt(5)/(2*x + 3))^3*sgn(1/(2*...
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^2} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^2} \,d x \]