Integrand size = 27, antiderivative size = 204 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {6167 (7+8 x) \sqrt {2+5 x+3 x^2}}{25600000 (3+2 x)^2}-\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{3/2}}{1920000 (3+2 x)^4}+\frac {6167 (7+8 x) \left (2+5 x+3 x^2\right )^{5/2}}{120000 (3+2 x)^6}-\frac {13 \left (2+5 x+3 x^2\right )^{7/2}}{45 (3+2 x)^9}-\frac {527 \left (2+5 x+3 x^2\right )^{7/2}}{1800 (3+2 x)^8}-\frac {1321 \left (2+5 x+3 x^2\right )^{7/2}}{5250 (3+2 x)^7}-\frac {6167 \text {arctanh}\left (\frac {7+8 x}{2 \sqrt {5} \sqrt {2+5 x+3 x^2}}\right )}{51200000 \sqrt {5}} \]
-6167/1920000*(7+8*x)*(3*x^2+5*x+2)^(3/2)/(3+2*x)^4+6167/120000*(7+8*x)*(3 *x^2+5*x+2)^(5/2)/(3+2*x)^6-13/45*(3*x^2+5*x+2)^(7/2)/(3+2*x)^9-527/1800*( 3*x^2+5*x+2)^(7/2)/(3+2*x)^8-1321/5250*(3*x^2+5*x+2)^(7/2)/(3+2*x)^7-6167/ 256000000*arctanh(1/10*(7+8*x)*5^(1/2)/(3*x^2+5*x+2)^(1/2))*5^(1/2)+6167/2 5600000*(7+8*x)*(3*x^2+5*x+2)^(1/2)/(3+2*x)^2
Time = 0.72 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.48 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {\frac {5 \sqrt {2+5 x+3 x^2} \left (6706847909+43246799138 x+117870367452 x^2+173974546136 x^3+149661252080 x^4+76435267296 x^5+23288995392 x^6+4204480128 x^7+333241344 x^8\right )}{(3+2 x)^9}-388521 \sqrt {5} \text {arctanh}\left (\frac {\sqrt {\frac {2}{5}+x+\frac {3 x^2}{5}}}{1+x}\right )}{8064000000} \]
((5*Sqrt[2 + 5*x + 3*x^2]*(6706847909 + 43246799138*x + 117870367452*x^2 + 173974546136*x^3 + 149661252080*x^4 + 76435267296*x^5 + 23288995392*x^6 + 4204480128*x^7 + 333241344*x^8))/(3 + 2*x)^9 - 388521*Sqrt[5]*ArcTanh[Sqr t[2/5 + x + (3*x^2)/5]/(1 + x)])/8064000000
Time = 0.42 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.12, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.370, Rules used = {1237, 27, 1237, 27, 1228, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^{10}} \, dx\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle -\frac {1}{45} \int -\frac {(293-156 x) \left (3 x^2+5 x+2\right )^{5/2}}{2 (2 x+3)^9}dx-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{90} \int \frac {(293-156 x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^9}dx-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 1237 |
| \(\displaystyle \frac {1}{90} \left (-\frac {1}{40} \int -\frac {3 (3703-1054 x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^8}dx-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{20 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{90} \left (\frac {3}{40} \int \frac {(3703-1054 x) \left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^8}dx-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{20 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 1228 |
| \(\displaystyle \frac {1}{90} \left (\frac {3}{40} \left (\frac {18501}{5} \int \frac {\left (3 x^2+5 x+2\right )^{5/2}}{(2 x+3)^7}dx-\frac {10568 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\right )-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{20 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{90} \left (\frac {3}{40} \left (\frac {18501}{5} \left (\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}-\frac {1}{24} \int \frac {\left (3 x^2+5 x+2\right )^{3/2}}{(2 x+3)^5}dx\right )-\frac {10568 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\right )-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{20 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{90} \left (\frac {3}{40} \left (\frac {18501}{5} \left (\frac {1}{24} \left (\frac {3}{80} \int \frac {\sqrt {3 x^2+5 x+2}}{(2 x+3)^3}dx-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )-\frac {10568 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\right )-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{20 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 1152 |
| \(\displaystyle \frac {1}{90} \left (\frac {3}{40} \left (\frac {18501}{5} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {1}{40} \int \frac {1}{(2 x+3) \sqrt {3 x^2+5 x+2}}dx\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )-\frac {10568 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\right )-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{20 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 1154 |
| \(\displaystyle \frac {1}{90} \left (\frac {3}{40} \left (\frac {18501}{5} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {1}{20} \int \frac {1}{20-\frac {(8 x+7)^2}{3 x^2+5 x+2}}d\left (-\frac {8 x+7}{\sqrt {3 x^2+5 x+2}}\right )+\frac {\sqrt {3 x^2+5 x+2} (8 x+7)}{20 (2 x+3)^2}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )-\frac {10568 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\right )-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{20 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{90} \left (\frac {3}{40} \left (\frac {18501}{5} \left (\frac {1}{24} \left (\frac {3}{80} \left (\frac {(8 x+7) \sqrt {3 x^2+5 x+2}}{20 (2 x+3)^2}-\frac {\text {arctanh}\left (\frac {8 x+7}{2 \sqrt {5} \sqrt {3 x^2+5 x+2}}\right )}{40 \sqrt {5}}\right )-\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{3/2}}{40 (2 x+3)^4}\right )+\frac {(8 x+7) \left (3 x^2+5 x+2\right )^{5/2}}{60 (2 x+3)^6}\right )-\frac {10568 \left (3 x^2+5 x+2\right )^{7/2}}{35 (2 x+3)^7}\right )-\frac {527 \left (3 x^2+5 x+2\right )^{7/2}}{20 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+5 x+2\right )^{7/2}}{45 (2 x+3)^9}\) |
(-13*(2 + 5*x + 3*x^2)^(7/2))/(45*(3 + 2*x)^9) + ((-527*(2 + 5*x + 3*x^2)^ (7/2))/(20*(3 + 2*x)^8) + (3*((-10568*(2 + 5*x + 3*x^2)^(7/2))/(35*(3 + 2* x)^7) + (18501*(((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)/24))/5))/40)/90
3.25.46.3.1 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 = 0.45 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.50
| method | result | size |
| risch | \(\frac {999724032 x^{10}+14279647104 x^{9}+91555869504 x^{8}+354159739104 x^{7}+877738083504 x^{6}+1423100433400 x^{5}+1522806337196 x^{4}+1067041326946 x^{3}+472095274321 x^{2}+120027837821 x +13413695818}{1612800000 \left (3+2 x \right )^{9} \sqrt {3 x^{2}+5 x +2}}+\frac {6167 \sqrt {5}\, \operatorname {arctanh}\left (\frac {2 \left (-\frac {7}{2}-4 x \right ) \sqrt {5}}{5 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}\right )}{256000000}\) | \(103\) |
| trager | \(\frac {\left (333241344 x^{8}+4204480128 x^{7}+23288995392 x^{6}+76435267296 x^{5}+149661252080 x^{4}+173974546136 x^{3}+117870367452 x^{2}+43246799138 x +6706847909\right ) \sqrt {3 x^{2}+5 x +2}}{1612800000 \left (3+2 x \right )^{9}}-\frac {6167 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) \ln \left (\frac {8 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right ) x +10 \sqrt {3 x^{2}+5 x +2}+7 \operatorname {RootOf}\left (\textit {\_Z}^{2}-5\right )}{3+2 x}\right )}{256000000}\) | \(112\) |
| default | \(-\frac {1321 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{672000 \left (x +\frac {3}{2}\right )^{7}}-\frac {6167 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1920000 \left (x +\frac {3}{2}\right )^{6}}-\frac {6167 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1200000 \left (x +\frac {3}{2}\right )^{5}}-\frac {129507 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{16000000 \left (x +\frac {3}{2}\right )^{4}}-\frac {376187 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{30000000 \left (x +\frac {3}{2}\right )^{3}}-\frac {11464453 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{600000000 \left (x +\frac {3}{2}\right )^{2}}+\frac {3583027 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{250000000}-\frac {3583027 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{125000000 \left (x +\frac {3}{2}\right )}-\frac {178843 \left (5+6 x \right ) \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{120000000}+\frac {6167 \left (5+6 x \right ) \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}}}{32000000}+\frac {6167 \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 )}{256000000}-\frac {6167 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {5}{2}}}{1000000000}-\frac {6167 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {3}{2}}}{480000000}-\frac {527 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{460800 \left (x +\frac {3}{2}\right )^{8}}-\frac {6167 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-16 x -19}}{256000000}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-4 x -\frac {19}{4}\right )^{\frac {7}{2}}}{23040 \left (x +\frac {3}{2}\right )^{9}}\) | \(332\) |
1/1612800000*(999724032*x^10+14279647104*x^9+91555869504*x^8+354159739104* x^7+877738083504*x^6+1423100433400*x^5+1522806337196*x^4+1067041326946*x^3 +472095274321*x^2+120027837821*x+13413695818)/(3+2*x)^9/(3*x^2+5*x+2)^(1/2 )+6167/256000000*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 = 201, normalized size of antiderivative = 0.99 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {388521 \, \sqrt {5} {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )} \log \left (-\frac {4 \, \sqrt {5} \sqrt {3 \, x^{2} + 5 \, x + 2} {\left (8 \, x + 7\right )} - 124 \, x^{2} - 212 \, x - 89}{4 \, x^{2} + 12 \, x + 9}\right ) + 20 \, {\left (333241344 \, x^{8} + 4204480128 \, x^{7} + 23288995392 \, x^{6} + 76435267296 \, x^{5} + 149661252080 \, x^{4} + 173974546136 \, x^{3} + 117870367452 \, x^{2} + 43246799138 \, x + 6706847909\right )} \sqrt {3 \, x^{2} + 5 \, x + 2}}{32256000000 \, {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} \]
1/32256000000*(388521*sqrt(5)*(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x + 19683)*l og(-(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)) + 20*(333241344*x^8 + 4204480128*x^7 + 23288995392*x^6 + 76435267296*x^5 + 149661252080*x^4 + 173974546136*x^3 + 117870367452*x^2 + 43246799138*x + 6706847909)*sqrt(3*x^2 + 5*x + 2))/(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x + 19683)
\[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=- \int \left (- \frac {20 \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {96 x \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {165 x^{2} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {113 x^{3} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \left (- \frac {15 x^{4} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\right )\, dx - \int \frac {9 x^{5} \sqrt {3 x^{2} + 5 x + 2}}{1024 x^{10} + 15360 x^{9} + 103680 x^{8} + 414720 x^{7} + 1088640 x^{6} + 1959552 x^{5} + 2449440 x^{4} + 2099520 x^{3} + 1180980 x^{2} + 393660 x + 59049}\, dx \]
-Integral(-20*sqrt(3*x**2 + 5*x + 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), x) - Integral(-96*x*sqrt(3*x**2 + 5* x + 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), x) - Integral(-165*x**2*sqrt(3*x**2 + 5*x + 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), x) - Integral(-113 *x**3*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 4147 20*x**7 + 1088640*x**6 + 1959552*x**5 + 2449440*x**4 + 2099520*x**3 + 1180 980*x**2 + 393660*x + 59049), x) - Integral(-15*x**4*sqrt(3*x**2 + 5*x + 2 )/(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 19 59552*x**5 + 2449440*x**4 + 2099520*x**3 + 1180980*x**2 + 393660*x + 59049 ), x) - Integral(9*x**5*sqrt(3*x**2 + 5*x + 2)/(1024*x**10 + 15360*x**9 + 103680*x**8 + 414720*x**7 + 1088640*x**6 + 1959552*x**5 + 2449440*x**4 + 2 099520*x**3 + 1180980*x**2 + 393660*x + 59049), x)
Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (170) = 340\).
Time = 0.29 (sec) , antiderivative size = 484, normalized size of antiderivative = 2.37 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {11464453}{200000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{45 \, {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} - \frac {527 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{1800 \, {\left (256 \, x^{8} + 3072 \, x^{7} + 16128 \, x^{6} + 48384 \, x^{5} + 90720 \, x^{4} + 108864 \, x^{3} + 81648 \, x^{2} + 34992 \, x + 6561\right )}} - \frac {1321 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{5250 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {6167 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{30000 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {6167 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{37500 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {129507 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{1000000 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {376187 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{3750000 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {11464453 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {7}{2}}}{150000000 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} - \frac {178843}{20000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} x - \frac {3583027}{480000000} \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {3}{2}} - \frac {3583027 \, {\left (3 \, x^{2} + 5 \, x + 2\right )}^{\frac {5}{2}}}{50000000 \, {\left (2 \, x + 3\right )}} + \frac {18501}{16000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} x + \frac {6167}{256000000} \, \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 {117173}{128000000} \, \sqrt {3 \, x^{2} + 5 \, x + 2} \]
11464453/200000000*(3*x^2 + 5*x + 2)^(5/2) - 13/45*(3*x^2 + 5*x + 2)^(7/2) /(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 326592*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x + 19683) - 527/1800*(3*x^2 + 5*x + 2)^( 7/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 90720*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561) - 1321/5250*(3*x^2 + 5*x + 2)^(7/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 218 7) - 6167/30000*(3*x^2 + 5*x + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 432 0*x^3 + 4860*x^2 + 2916*x + 729) - 6167/37500*(3*x^2 + 5*x + 2)^(7/2)/(32* x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 129507/1000000*(3*x^2 + 5*x + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 376187/3750000 *(3*x^2 + 5*x + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 11464453/150000000 *(3*x^2 + 5*x + 2)^(7/2)/(4*x^2 + 12*x + 9) - 178843/20000000*(3*x^2 + 5*x + 2)^(3/2)*x - 3583027/480000000*(3*x^2 + 5*x + 2)^(3/2) - 3583027/500000 00*(3*x^2 + 5*x + 2)^(5/2)/(2*x + 3) + 18501/16000000*sqrt(3*x^2 + 5*x + 2 )*x + 6167/256000000*sqrt(5)*log(sqrt(5)*sqrt(3*x^2 + 5*x + 2)/abs(2*x + 3 ) + 5/2/abs(2*x + 3) - 2) + 117173/128000000*sqrt(3*x^2 + 5*x + 2)
Leaf count of result is larger than twice the leaf count of optimal. 563 vs. \(2 (170) = 340\).
Time = 0.40 (sec) , antiderivative size = 563, normalized size of antiderivative = 2.76 \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=-\frac {6167}{256000000} \, \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 {99461376 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{17} + 2536265088 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{16} - 83954355072 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{15} - 341000936640 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{14} + 17778066768000 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{13} + 177356386111968 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{12} + 2399974462831392 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{11} + 6844601123556624 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{10} + 41172892580130560 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{9} + 60936872688585000 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{8} + 204498063708405624 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{7} + 174436297943297292 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{6} + 339439601929212792 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{5} + 164994557892929730 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{4} + 174936772514694750 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{3} + 42504221165006223 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 5 \, x + 2}\right )}^{2} + 19065836258759367 \, \sqrt {3} x + 1323473153587704 \, \sqrt {3} - 19065836258759367 \, \sqrt {3 \, x^{2} + 5 \, x + 2}}{1612800000 \, {\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 )}^{9}} \]
-6167/256000000*sqrt(5)*log(abs(-4*sqrt(3)*x - 2*sqrt(5) - 6*sqrt(3) + 4*s qrt(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))) + 1/1612800000*(99461376*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^17 + 2536265088*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^16 - 83954 355072*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^15 - 341000936640*sqrt(3)*(sqrt (3)*x - sqrt(3*x^2 + 5*x + 2))^14 + 17778066768000*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^13 + 177356386111968*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^12 + 2399974462831392*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^11 + 6844601 123556624*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^10 + 411728925801305 60*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^9 + 60936872688585000*sqrt(3)*(sqrt (3)*x - sqrt(3*x^2 + 5*x + 2))^8 + 204498063708405624*(sqrt(3)*x - sqrt(3* x^2 + 5*x + 2))^7 + 174436297943297292*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5 *x + 2))^6 + 339439601929212792*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^5 + 16 4994557892929730*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^4 + 174936772 514694750*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^3 + 42504221165006223*sqrt(3 )*(sqrt(3)*x - sqrt(3*x^2 + 5*x + 2))^2 + 19065836258759367*sqrt(3)*x + 13 23473153587704*sqrt(3) - 19065836258759367*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)^9
Timed out. \[ \int \frac {(5-x) \left (2+5 x+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=-\int \frac {\left (x-5\right )\,{\left (3\,x^2+5\,x+2\right )}^{5/2}}{{\left (2\,x+3\right )}^{10}} \,d x \]