Integrand size = 27, antiderivative size = 259 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{13}} \, dx=-\frac {175119 (7+8 x) \sqrt {2+5 x+3 x^2}}{20480000000 (3+2 x)^2}+\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{512000000 (3+2 x)^4}-\frac {58373 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{32000000 (3+2 x)^6}+\frac {25017 (7+8 x) \left (2+5 x+3 x^2\right )^{7/2}}{800000 (3+2 x)^8}-\frac {13 \left (2+5 x+3 x^2\right )^{9/2}}{60 (3+2 x)^{12}}-\frac {12 \left (2+5 x+3 x^2\right )^{9/2}}{55 (3+2 x)^{11}}-\frac {2067 \left (2+5 x+3 x^2\right )^{9/2}}{11000 (3+2 x)^{10}}-\frac {6379 \left (2+5 x+3 x^2\right )^{9/2}}{41250 (3+2 x)^9}+\frac {175119 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{40960000000 \sqrt {5}} \]
58373/512000000*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4-58373/32000000*(7+8* x)*(3*x^2+5*x+2)^(5/2)/(3+2*x)^6+25017/800000*(7+8*x)*(3*x^2+5*x+2)^(7/2)/ (3+2*x)^8-13/60*(3*x^2+5*x+2)^(9/2)/(3+2*x)^12-12/55*(3*x^2+5*x+2)^(9/2)/( 3+2*x)^11-2067/11000*(3*x^2+5*x+2)^(9/2)/(3+2*x)^10-6379/41250*(3*x^2+5*x+ 2)^(9/2)/(3+2*x)^9+175119/204800000000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2 +5*x+2)^(1/2))*5^(1/2)-175119/20480000000*(7+8*x)*(3*x^2+5*x+2)^(1/2)/(3+2 *x)^2
Time = 0.87 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.44 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{13}} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (2531527640959+25843081681156 x+111795175925940 x^2+271870111600160 x^3+412855931529440 x^4+410468875350912 x^5+273282692080768 x^6+123629135656960 x^7+38544695427840 x^8+8182662620160 x^9+1044584776704 x^{10}+60734693376 x^{11}\right )}{(3+2 x)^{12}}+5778927 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{3379200000000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(2531527640959 + 25843081681156*x + 111795175925 940*x^2 + 271870111600160*x^3 + 412855931529440*x^4 + 410468875350912*x^5 + 273282692080768*x^6 + 123629135656960*x^7 + 38544695427840*x^8 + 8182662 620160*x^9 + 1044584776704*x^10 + 60734693376*x^11))/(3 + 2*x)^12 + 577892 7*Sqrt[5]*ArcTanh[Sqrt[2/5 + x + (3*x^2)/5]/(1 + x)])/3379200000000
Time = 0.52 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.14, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.481, Rules used = {1237, 27, 1237, 27, 1237, 25, 1228, 1152, 1152, 1152, 1152, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{13}} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {1}{60} \int -\frac {9 (41-26 x) \left (3 x^2+5 x+2\right )^{7/2}}{2 (2 x+3)^{12}}dx-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{40} \int \frac {(41-26 x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{12}}dx-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {3}{40} \left (-\frac {1}{55} \int -\frac {5 (401-192 x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{11}}dx-\frac {32 \left (3 x^2+5 x+2\right )^{9/2}}{11 (2 x+3)^{11}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{40} \left (\frac {1}{11} \int \frac {(401-192 x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{11}}dx-\frac {32 \left (3 x^2+5 x+2\right )^{9/2}}{11 (2 x+3)^{11}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {3}{40} \left (\frac {1}{11} \left (-\frac {1}{50} \int -\frac {(19315-4134 x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{10}}dx-\frac {689 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {32 \left (3 x^2+5 x+2\right )^{9/2}}{11 (2 x+3)^{11}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {3}{40} \left (\frac {1}{11} \left (\frac {1}{50} \int \frac {(19315-4134 x) \left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^{10}}dx-\frac {689 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {32 \left (3 x^2+5 x+2\right )^{9/2}}{11 (2 x+3)^{11}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {3}{40} \left (\frac {1}{11} \left (\frac {1}{50} \left (\frac {91729}{5} \int \frac {\left (3 x^2+5 x+2\right )^{7/2}}{(2 x+3)^9}dx-\frac {51032 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {689 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {32 \left (3 x^2+5 x+2\right )^{9/2}}{11 (2 x+3)^{11}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{40} \left (\frac {1}{11} \left (\frac {1}{50} \left (\frac {91729}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \int \frac {\left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7}dx\right )-\frac {51032 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {689 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {32 \left (3 x^2+5 x+2\right )^{9/2}}{11 (2 x+3)^{11}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{40} \left (\frac {1}{11} \left (\frac {1}{50} \left (\frac {91729}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}-\frac {1}{24} \int \frac {\left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx\right )\right )-\frac {51032 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {689 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {32 \left (3 x^2+5 x+2\right )^{9/2}}{11 (2 x+3)^{11}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{40} \left (\frac {1}{11} \left (\frac {1}{50} \left (\frac {91729}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \int \frac {\sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {51032 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {689 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {32 \left (3 x^2+5 x+2\right )^{9/2}}{11 (2 x+3)^{11}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {3}{40} \left (\frac {1}{11} \left (\frac {1}{50} \left (\frac {91729}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {51032 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {689 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {32 \left (3 x^2+5 x+2\right )^{9/2}}{11 (2 x+3)^{11}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {3}{40} \left (\frac {1}{11} \left (\frac {1}{50} \left (\frac {91729}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{20} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (8 x+7)}{20 (2 x+3)^2}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {51032 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {689 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {32 \left (3 x^2+5 x+2\right )^{9/2}}{11 (2 x+3)^{11}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {3}{40} \left (\frac {1}{11} \left (\frac {1}{50} \left (\frac {91729}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{7/2}}{80 (2 x+3)^8}-\frac {7}{160} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {\text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )\right )-\frac {51032 \left (3 x^2+5 x+2\right )^{9/2}}{45 (2 x+3)^9}\right )-\frac {689 \left (3 x^2+5 x+2\right )^{9/2}}{25 (2 x+3)^{10}}\right )-\frac {32 \left (3 x^2+5 x+2\right )^{9/2}}{11 (2 x+3)^{11}}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{9/2}}{60 (2 x+3)^{12}}\) |
(-13*(2 + 5*x + 3*x^2)^(9/2))/(60*(3 + 2*x)^12) + (3*((-32*(2 + 5*x + 3*x^ 2)^(9/2))/(11*(3 + 2*x)^11) + ((-689*(2 + 5*x + 3*x^2)^(9/2))/(25*(3 + 2*x )^10) + ((-51032*(2 + 5*x + 3*x^2)^(9/2))/(45*(3 + 2*x)^9) + (91729*(((7 + 8*x)*(2 + 5*x + 3*x^2)^(7/2))/(80*(3 + 2*x)^8) - (7*(((7 + 8*x)*(2 + 5*x + 3*x^2)^(5/2))/(60*(3 + 2*x)^6) + (-1/40*((7 + 8*x)*(2 + 5*x + 3*x^2)^(3/ 2))/(3 + 2*x)^4 + (3*(((7 + 8*x)*Sqrt[2 + 5*x + 3*x^2])/(20*(3 + 2*x)^2) - ArcTanh[(7 + 8*x)/(2*Sqrt[5]*Sqrt[2 + 5*x + 3*x^2])]/(40*Sqrt[5])))/80)/2 4))/160))/5)/50)/11))/40
3.25.64.3.1 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[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S ymbol] :> Simp[(-(d + e*x)^(m + 1))*(d*b - 2*a*e + (2*c*d - b*e)*x)*((a + b *x + c*x^2)^p/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Simp[p*((b^2 - 4*a *c)/(2*(m + 1)*(c*d^2 - b*d*e + a*e^2))) Int[(d + e*x)^(m + 2)*(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[m + 2*p + 2, 0] && GtQ[p, 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[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + b*x + c*x^2)^(p + 1)/(2*(p + 1)*(c*d^2 - b*d*e + a*e^2))), x] - Simp[(b*(e *f + d*g) - 2*(c*d*f + a*e*g))/(2*(c*d^2 - b*d*e + a*e^2)) Int[(d + e*x)^ (m + 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, m, p}, x ] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c _.)*(x_)^2)^(p_.), x_Symbol] :> Simp[(e*f - d*g)*(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*Simp[ (c*d*f - f*b*e + a*e*g)*(m + 1) + b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && LtQ[m, -1 ] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 0.60 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.46
| method | result | size |
| risch | \(\frac {182204080128 x^{13}+3437427796992 x^{12}+29892381130752 x^{11}+158636568937728 x^{10}+579976209350400 x^{9}+1515083145382784 x^{8}+2845078357770496 x^{7}+3837477555504416 x^{6}+3700827743149504 x^{5}+2520447948837500 x^{4}+1180245347873488 x^{3}+360400343180537 x^{2}+64343801567107 x +5063055281918}{675840000000 \left (3+2 x \right )^{12} \sqrt {3 x^{2}+5 x +2}}-\frac {175119 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{204800000000}\) | \(118\) |
| trager | \(\frac {\left (60734693376 x^{11}+1044584776704 x^{10}+8182662620160 x^{9}+38544695427840 x^{8}+123629135656960 x^{7}+273282692080768 x^{6}+410468875350912 x^{5}+412855931529440 x^{4}+271870111600160 x^{3}+111795175925940 x^{2}+25843081681156 x +2531527640959\right ) \sqrt {3 x^{2}+5 x +2}}{675840000000 \left (3+2 x \right )^{12}}+\frac {175119 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{204800000000}\) | \(127\) |
| default | \(-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{245760 \left (x +\frac {3}{2}\right )^{12}}-\frac {158441 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{128000000 \left (x +\frac {3}{2}\right )^{6}}-\frac {775527 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{400000000 \left (x +\frac {3}{2}\right )^{5}}-\frac {48057657 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{16000000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {46022941 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{10000000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {1395223107 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{200000000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {261602769 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{50000000000}-\frac {261602769 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{25000000000 \left (x +\frac {3}{2}\right )}-\frac {101744139 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{200000000000}+\frac {1692817 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{32000000000}-\frac {175119 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{25600000000}-\frac {175119 \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 )}{204800000000}-\frac {2067 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{11264000 \left (x +\frac {3}{2}\right )^{10}}-\frac {3 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{28160 \left (x +\frac {3}{2}\right )^{11}}+\frac {175119 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{800000000000}+\frac {58373 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{128000000000}+\frac {25017 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{200000000000}-\frac {25017 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{32000000 \left (x +\frac {3}{2}\right )^{7}}+\frac {175119 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{204800000000}-\frac {25017 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{51200000 \left (x +\frac {3}{2}\right )^{8}}-\frac {6379 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {9}{2}}}{21120000 \left (x +\frac {3}{2}\right )^{9}}\) | \(432\) |
1/675840000000*(182204080128*x^13+3437427796992*x^12+29892381130752*x^11+1 58636568937728*x^10+579976209350400*x^9+1515083145382784*x^8+2845078357770 496*x^7+3837477555504416*x^6+3700827743149504*x^5+2520447948837500*x^4+118 0245347873488*x^3+360400343180537*x^2+64343801567107*x+5063055281918)/(3+2 *x)^12/(3*x^2+5*x+2)^(1/2)-175119/204800000000*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 = 245, normalized size of antiderivative = 0.95 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{13}} \, dx=\frac {5778927 \, \sqrt {5} {\left (4096 \, x^{12} + 73728 \, x^{11} + 608256 \, x^{10} + 3041280 \, x^{9} + 10264320 \, x^{8} + 24634368 \, x^{7} + 43110144 \, x^{6} + 55427328 \, x^{5} + 51963120 \, x^{4} + 34642080 \, x^{3} + 15588936 \, x^{2} + 4251528 \, x + 531441\right )} \log \left (\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} + 124 \, x^{2} + 212 \, x + 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (60734693376 \, x^{11} + 1044584776704 \, x^{10} + 8182662620160 \, x^{9} + 38544695427840 \, x^{8} + 123629135656960 \, x^{7} + 273282692080768 \, x^{6} + 410468875350912 \, x^{5} + 412855931529440 \, x^{4} + 271870111600160 \, x^{3} + 111795175925940 \, x^{2} + 25843081681156 \, x + 2531527640959\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{13516800000000 \, {\left (4096 \, x^{12} + 73728 \, x^{11} + 608256 \, x^{10} + 3041280 \, x^{9} + 10264320 \, x^{8} + 24634368 \, x^{7} + 43110144 \, x^{6} + 55427328 \, x^{5} + 51963120 \, x^{4} + 34642080 \, x^{3} + 15588936 \, x^{2} + 4251528 \, x + 531441\right )}} \]
1/13516800000000*(5778927*sqrt(5)*(4096*x^12 + 73728*x^11 + 608256*x^10 + 3041280*x^9 + 10264320*x^8 + 24634368*x^7 + 43110144*x^6 + 55427328*x^5 + 51963120*x^4 + 34642080*x^3 + 15588936*x^2 + 4251528*x + 531441)*log((4*sq rt(5)*sqrt(3*x^2 + 5*x + 2)*(8*x + 7) + 124*x^2 + 212*x + 89)/(4*x^2 + 12* x + 9)) + 20*(60734693376*x^11 + 1044584776704*x^10 + 8182662620160*x^9 + 38544695427840*x^8 + 123629135656960*x^7 + 273282692080768*x^6 + 410468875 350912*x^5 + 412855931529440*x^4 + 271870111600160*x^3 + 111795175925940*x ^2 + 25843081681156*x + 2531527640959)*sqrt(3*x^2 + 5*x + 2))/(4096*x^12 + 73728*x^11 + 608256*x^10 + 3041280*x^9 + 10264320*x^8 + 24634368*x^7 + 43 110144*x^6 + 55427328*x^5 + 51963120*x^4 + 34642080*x^3 + 15588936*x^2 + 4 251528*x + 531441)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{13}} \, dx=- \int \left (- \frac {40 \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {292 x \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {870 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {1339 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {1090 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \left (- \frac {396 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\right )\, dx - \int \frac {27 x^{7} \sqrt {3 x^{2} + 5 x + 2}}{8192 x^{13} + 159744 x^{12} + 1437696 x^{11} + 7907328 x^{10} + 29652480 x^{9} + 80061696 x^{8} + 160123392 x^{7} + 240185088 x^{6} + 270208224 x^{5} + 225173520 x^{4} + 135104112 x^{3} + 55269864 x^{2} + 13817466 x + 1594323}\, dx \]
-Integral(-40*sqrt(3*x**2 + 5*x + 2)/(8192*x**13 + 159744*x**12 + 1437696* x**11 + 7907328*x**10 + 29652480*x**9 + 80061696*x**8 + 160123392*x**7 + 2 40185088*x**6 + 270208224*x**5 + 225173520*x**4 + 135104112*x**3 + 5526986 4*x**2 + 13817466*x + 1594323), x) - Integral(-292*x*sqrt(3*x**2 + 5*x + 2 )/(8192*x**13 + 159744*x**12 + 1437696*x**11 + 7907328*x**10 + 29652480*x* *9 + 80061696*x**8 + 160123392*x**7 + 240185088*x**6 + 270208224*x**5 + 22 5173520*x**4 + 135104112*x**3 + 55269864*x**2 + 13817466*x + 1594323), x) - Integral(-870*x**2*sqrt(3*x**2 + 5*x + 2)/(8192*x**13 + 159744*x**12 + 1 437696*x**11 + 7907328*x**10 + 29652480*x**9 + 80061696*x**8 + 160123392*x **7 + 240185088*x**6 + 270208224*x**5 + 225173520*x**4 + 135104112*x**3 + 55269864*x**2 + 13817466*x + 1594323), x) - Integral(-1339*x**3*sqrt(3*x** 2 + 5*x + 2)/(8192*x**13 + 159744*x**12 + 1437696*x**11 + 7907328*x**10 + 29652480*x**9 + 80061696*x**8 + 160123392*x**7 + 240185088*x**6 + 27020822 4*x**5 + 225173520*x**4 + 135104112*x**3 + 55269864*x**2 + 13817466*x + 15 94323), x) - Integral(-1090*x**4*sqrt(3*x**2 + 5*x + 2)/(8192*x**13 + 1597 44*x**12 + 1437696*x**11 + 7907328*x**10 + 29652480*x**9 + 80061696*x**8 + 160123392*x**7 + 240185088*x**6 + 270208224*x**5 + 225173520*x**4 + 13510 4112*x**3 + 55269864*x**2 + 13817466*x + 1594323), x) - Integral(-396*x**5 *sqrt(3*x**2 + 5*x + 2)/(8192*x**13 + 159744*x**12 + 1437696*x**11 + 79073 28*x**10 + 29652480*x**9 + 80061696*x**8 + 160123392*x**7 + 240185088*x...
Leaf count of result is larger than twice the leaf count of optimal. 726 vs. \(2 (217) = 434\).
Time = 0.31 (sec) , antiderivative size = 726, normalized size of antiderivative = 2.80 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{13}} \, dx=\text {Too large to display} \]
4185669321/200000000000*(3*x^2 + 5*x + 2)^(7/2) - 13/60*(3*x^2 + 5*x + 2)^ (9/2)/(4096*x^12 + 73728*x^11 + 608256*x^10 + 3041280*x^9 + 10264320*x^8 + 24634368*x^7 + 43110144*x^6 + 55427328*x^5 + 51963120*x^4 + 34642080*x^3 + 15588936*x^2 + 4251528*x + 531441) - 12/55*(3*x^2 + 5*x + 2)^(9/2)/(2048 *x^11 + 33792*x^10 + 253440*x^9 + 1140480*x^8 + 3421440*x^7 + 7185024*x^6 + 10777536*x^5 + 11547360*x^4 + 8660520*x^3 + 4330260*x^2 + 1299078*x + 17 7147) - 2067/11000*(3*x^2 + 5*x + 2)^(9/2)/(1024*x^10 + 15360*x^9 + 103680 *x^8 + 414720*x^7 + 1088640*x^6 + 1959552*x^5 + 2449440*x^4 + 2099520*x^3 + 1180980*x^2 + 393660*x + 59049) - 6379/41250*(3*x^2 + 5*x + 2)^(9/2)/(51 2*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 4898 88*x^3 + 314928*x^2 + 118098*x + 19683) - 25017/200000*(3*x^2 + 5*x + 2)^( 9/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561) - 25017/250000*(3*x^2 + 5*x + 2)^(9/2)/(128* x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 158441/2000000*(3*x^2 + 5*x + 2)^(9/2)/(64*x^6 + 576*x^5 + 2160*x^ 4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 775527/12500000*(3*x^2 + 5*x + 2 )^(9/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 48057657/1 000000000*(3*x^2 + 5*x + 2)^(9/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 46022941/1250000000*(3*x^2 + 5*x + 2)^(9/2)/(8*x^3 + 36*x^2 + 54*x + 27 ) - 1395223107/50000000000*(3*x^2 + 5*x + 2)^(9/2)/(4*x^2 + 12*x + 9) -...
Leaf count of result is larger than twice the leaf count of optimal. 716 vs. \(2 (217) = 434\).
Time = 0.35 (sec) , antiderivative size = 716, normalized size of antiderivative = 2.76 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{13}} \, dx=\frac {175119}{204800000000} \, \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 {11835242496 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{23} + 408315866112 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{22} + 20039038086144 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{21} + 535243596890112 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{20} + 13859706456921600 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{19} + 31535346744025344 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{18} - 789031961976842496 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} - 7977976824329385984 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} - 113078650509677476096 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 358779889050339715200 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} - 2538162771649151164032 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} - 4660243350382625915904 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} - 20499122524155108829248 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} - 24347916060701730772704 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} - 70788415443572756925600 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} - 56076083911431114398208 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} - 108598043564223524909928 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} - 56663550021725424101412 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} - 70668287639831997261828 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} - 22876037084903247115200 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} - 16680770211437743348146 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} - 2864949797863813201587 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} - 930278306769206446269 \, \sqrt {3} x - 47729262032858665512 \, \sqrt {3} + 930278306769206446269 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{675840000000 \, {\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 )}^{12}} \]
175119/204800000000*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*sqr t(3*x^2 + 5*x + 2))) - 1/675840000000*(11835242496*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^23 + 408315866112*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2)) ^22 + 20039038086144*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^21 + 535243596890 112*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^20 + 13859706456921600*(sq rt(3)*x - sqrt(3*x^2 + 5*x + 2))^19 + 31535346744025344*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^18 - 789031961976842496*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^17 - 7977976824329385984*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 - 113078650509677476096*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^15 - 358779889050339715200*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^14 - 25 38162771649151164032*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 - 466024335038 2625915904*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 - 20499122524155 108829248*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 - 24347916060701730772704 *sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 - 70788415443572756925600* (sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 - 56076083911431114398208*sqrt(3)*(s qrt(3)*x - sqrt(3*x^2 + 5*x + 2))^8 - 108598043564223524909928*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^7 - 56663550021725424101412*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^6 - 70668287639831997261828*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 - 22876037084903247115200*sqrt(3)*(sqrt(3)*x - sqrt(3*x^...
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{7/2}}{(3+2 x)^{13}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{7/2}}{{\left (2\,x+3\right )}^{13}} \,d x \]