Integrand size = 24, antiderivative size = 180 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=-\frac {25623 (4-9 x) \sqrt {2+3 x^2}}{1470612500 (3+2 x)^2}-\frac {2847 (4-9 x) \left (2+3 x^2\right )^{3/2}}{42017500 (3+2 x)^4}-\frac {949 (4-9 x) \left (2+3 x^2\right )^{5/2}}{3001250 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{315 (3+2 x)^9}-\frac {27 \left (2+3 x^2\right )^{7/2}}{2450 (3+2 x)^8}-\frac {4741 \left (2+3 x^2\right )^{7/2}}{1800750 (3+2 x)^7}-\frac {76869 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{735306250 \sqrt {35}} \]
-2847/42017500*(4-9*x)*(3*x^2+2)^(3/2)/(3+2*x)^4-949/3001250*(4-9*x)*(3*x^ 2+2)^(5/2)/(3+2*x)^6-13/315*(3*x^2+2)^(7/2)/(3+2*x)^9-27/2450*(3*x^2+2)^(7 /2)/(3+2*x)^8-4741/1800750*(3*x^2+2)^(7/2)/(3+2*x)^7-76869/25735718750*arc tanh(1/35*(4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)-25623/1470612500*(4-9 *x)*(3*x^2+2)^(1/2)/(3+2*x)^2
Time = 2.09 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.60 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {-\frac {35 \sqrt {2+3 x^2} \left (15948113036+11990965797 x+42455611758 x^2-11567526201 x^3+9750269970 x^4-25197346566 x^5-620594352 x^6+30006612 x^7+10968696 x^8\right )}{(3+2 x)^9}+2767284 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{463242937500} \]
((-35*Sqrt[2 + 3*x^2]*(15948113036 + 11990965797*x + 42455611758*x^2 - 115 67526201*x^3 + 9750269970*x^4 - 25197346566*x^5 - 620594352*x^6 + 30006612 *x^7 + 10968696*x^8))/(3 + 2*x)^9 + 2767284*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3*x^2])/Sqrt[35]])/463242937500
Time = 0.34 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.14, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.417, Rules used = {688, 27, 688, 27, 679, 486, 486, 486, 488, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^{10}} \, dx\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {1}{315} \int -\frac {3 (123-26 x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^9}dx-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \int \frac {(123-26 x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^9}dx-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {1}{105} \left (-\frac {1}{280} \int -\frac {4 (2006-243 x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^8}dx-\frac {81 \left (3 x^2+2\right )^{7/2}}{70 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{70} \int \frac {(2006-243 x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^8}dx-\frac {81 \left (3 x^2+2\right )^{7/2}}{70 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{70} \left (\frac {17082}{35} \int \frac {\left (3 x^2+2\right )^{5/2}}{(2 x+3)^7}dx-\frac {4741 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {81 \left (3 x^2+2\right )^{7/2}}{70 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\) |
| \(\Big \downarrow \) 486 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{70} \left (\frac {17082}{35} \left (\frac {1}{7} \int \frac {\left (3 x^2+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {4741 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {81 \left (3 x^2+2\right )^{7/2}}{70 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\) |
| \(\Big \downarrow \) 486 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{70} \left (\frac {17082}{35} \left (\frac {1}{7} \left (\frac {9}{70} \int \frac {\sqrt {3 x^2+2}}{(2 x+3)^3}dx-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {4741 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {81 \left (3 x^2+2\right )^{7/2}}{70 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\) |
| \(\Big \downarrow \) 486 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{70} \left (\frac {17082}{35} \left (\frac {1}{7} \left (\frac {9}{70} \left (\frac {3}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {(4-9 x) \sqrt {3 x^2+2}}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {4741 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {81 \left (3 x^2+2\right )^{7/2}}{70 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{70} \left (\frac {17082}{35} \left (\frac {1}{7} \left (\frac {9}{70} \left (-\frac {3}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {4741 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {81 \left (3 x^2+2\right )^{7/2}}{70 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{105} \left (\frac {1}{70} \left (\frac {17082}{35} \left (\frac {1}{7} \left (\frac {9}{70} \left (-\frac {3 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {4741 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {81 \left (3 x^2+2\right )^{7/2}}{70 (2 x+3)^8}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\) |
(-13*(2 + 3*x^2)^(7/2))/(315*(3 + 2*x)^9) + ((-81*(2 + 3*x^2)^(7/2))/(70*( 3 + 2*x)^8) + ((-4741*(2 + 3*x^2)^(7/2))/(245*(3 + 2*x)^7) + (17082*(-1/21 0*((4 - 9*x)*(2 + 3*x^2)^(5/2))/(3 + 2*x)^6 + (-1/140*((4 - 9*x)*(2 + 3*x^ 2)^(3/2))/(3 + 2*x)^4 + (9*(-1/70*((4 - 9*x)*Sqrt[2 + 3*x^2])/(3 + 2*x)^2 - (3*ArcTanh[(4 - 9*x)/(Sqrt[35]*Sqrt[2 + 3*x^2])])/(35*Sqrt[35])))/70)/7) )/35)/70)/105
3.14.95.3.1 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[((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[ (c + d*x)^(n + 1)*(a*d - b*c*x)*((a + b*x^2)^p/((n + 1)*(b*c^2 + a*d^2))), x] - Simp[2*a*b*(p/((n + 1)*(b*c^2 + a*d^2))) Int[(c + d*x)^(n + 2)*(a + b*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[n + 2*p + 2, 0] && GtQ[p, 0]
Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ [{a, b, c, d}, x]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 )/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( (m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2)) Int[(d + e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Time = 0.48 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.56
| method | result | size |
| risch | \(-\frac {32906088 x^{10}+90019836 x^{9}-1839845664 x^{8}-75532026474 x^{7}+28009621206 x^{6}-85097271735 x^{5}+146867375214 x^{4}+12837844989 x^{3}+132755562624 x^{2}+23981931594 x +31896226072}{13235512500 \left (3+2 x \right )^{9} \sqrt {3 x^{2}+2}}-\frac {76869 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{25735718750}\) | \(100\) |
| trager | \(-\frac {\left (10968696 x^{8}+30006612 x^{7}-620594352 x^{6}-25197346566 x^{5}+9750269970 x^{4}-11567526201 x^{3}+42455611758 x^{2}+11990965797 x +15948113036\right ) \sqrt {3 x^{2}+2}}{13235512500 \left (3+2 x \right )^{9}}-\frac {76869 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{25735718750}\) | \(106\) |
| default | \(\frac {102492 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{450375078125}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{161280 \left (x +\frac {3}{2}\right )^{9}}-\frac {27 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{627200 \left (x +\frac {3}{2}\right )^{8}}-\frac {4741 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{230496000 \left (x +\frac {3}{2}\right )^{7}}-\frac {949 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{96040000 \left (x +\frac {3}{2}\right )^{6}}-\frac {8541 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{1680700000 \left (x +\frac {3}{2}\right )^{5}}-\frac {82563 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{29412250000 \left (x +\frac {3}{2}\right )^{4}}-\frac {845559 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{514714375000 \left (x +\frac {3}{2}\right )^{3}}-\frac {9198657 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{9007501562500 \left (x +\frac {3}{2}\right )^{2}}+\frac {320313123 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{157631277343750}-\frac {106771041 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{157631277343750 \left (x +\frac {3}{2}\right )}+\frac {8993673 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{1801500312500}+\frac {691821 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{51471437500}-\frac {76869 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{25735718750}+\frac {76869 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{25735718750}+\frac {1229904 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{78815638671875}\) | \(320\) |
-1/13235512500*(32906088*x^10+90019836*x^9-1839845664*x^8-75532026474*x^7+ 28009621206*x^6-85097271735*x^5+146867375214*x^4+12837844989*x^3+132755562 624*x^2+23981931594*x+31896226072)/(3+2*x)^9/(3*x^2+2)^(1/2)-76869/2573571 8750*35^(1/2)*arctanh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
Time = 0.65 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.08 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {691821 \, \sqrt {35} {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (10968696 \, x^{8} + 30006612 \, x^{7} - 620594352 \, x^{6} - 25197346566 \, x^{5} + 9750269970 \, x^{4} - 11567526201 \, x^{3} + 42455611758 \, x^{2} + 11990965797 \, x + 15948113036\right )} \sqrt {3 \, x^{2} + 2}}{463242937500 \, {\left (512 \, x^{9} + 6912 \, x^{8} + 41472 \, x^{7} + 145152 \, x^{6} + 326592 \, x^{5} + 489888 \, x^{4} + 489888 \, x^{3} + 314928 \, x^{2} + 118098 \, x + 19683\right )}} \]
1/463242937500*(691821*sqrt(35)*(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) *log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93*x^2 - 36*x + 43)/(4*x^2 + 1 2*x + 9)) - 35*(10968696*x^8 + 30006612*x^7 - 620594352*x^6 - 25197346566* x^5 + 9750269970*x^4 - 11567526201*x^3 + 42455611758*x^2 + 11990965797*x + 15948113036)*sqrt(3*x^2 + 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)
Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (149) = 298\).
Time = 0.30 (sec) , antiderivative size = 434, normalized size of antiderivative = 2.41 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {27595971}{9007501562500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{315 \, {\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 {27 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{2450 \, {\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 {4741 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1800750 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {949 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1500625 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {8541 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{52521875 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {82563 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1838265625 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {845559 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{64339296875 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {9198657 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{2251875390625 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {8993673}{1801500312500} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {102492}{450375078125} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {106771041 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{9007501562500 \, {\left (2 \, x + 3\right )}} + \frac {691821}{51471437500} \, \sqrt {3 \, x^{2} + 2} x + \frac {76869}{25735718750} \, \sqrt {35} \operatorname {arsinh}\left (\frac {3 \, \sqrt {6} x}{2 \, {\left | 2 \, x + 3 \right |}} - \frac {2 \, \sqrt {6}}{3 \, {\left | 2 \, x + 3 \right |}}\right ) + \frac {76869}{12867859375} \, \sqrt {3 \, x^{2} + 2} \]
27595971/9007501562500*(3*x^2 + 2)^(5/2) - 13/315*(3*x^2 + 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) - 27/2450*(3*x^2 + 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) - 4741/1800750*(3*x^2 + 2)^(7/2)/(128*x^7 + 1344*x^6 + 604 8*x^5 + 15120*x^4 + 22680*x^3 + 20412*x^2 + 10206*x + 2187) - 949/1500625* (3*x^2 + 2)^(7/2)/(64*x^6 + 576*x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 291 6*x + 729) - 8541/52521875*(3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 82563/1838265625*(3*x^2 + 2)^(7/2)/(16*x^4 + 96 *x^3 + 216*x^2 + 216*x + 81) - 845559/64339296875*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 9198657/2251875390625*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) + 8993673/1801500312500*(3*x^2 + 2)^(3/2)*x + 102492/45037507812 5*(3*x^2 + 2)^(3/2) - 106771041/9007501562500*(3*x^2 + 2)^(5/2)/(2*x + 3) + 691821/51471437500*sqrt(3*x^2 + 2)*x + 76869/25735718750*sqrt(35)*arcsin h(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 76869/128678593 75*sqrt(3*x^2 + 2)
Leaf count of result is larger than twice the leaf count of optimal. 502 vs. \(2 (149) = 298\).
Time = 0.33 (sec) , antiderivative size = 502, normalized size of antiderivative = 2.79 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {76869}{25735718750} \, \sqrt {35} \log \left (-\frac {{\left | -2 \, \sqrt {3} x - \sqrt {35} - 3 \, \sqrt {3} + 2 \, \sqrt {3 \, x^{2} + 2} \right |}}{2 \, \sqrt {3} x - \sqrt {35} + 3 \, \sqrt {3} - 2 \, \sqrt {3 \, x^{2} + 2}}\right ) - \frac {9 \, \sqrt {3} {\left (364416 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{17} + 27877824 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{16} + 1042205258 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{15} - 956098170 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{14} + 1003625490 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{13} - 85987901496 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{12} - 60468401868 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} - 331045664193 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} - 22913148915 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} - 544736640510 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} + 284856270864 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} - 908850124224 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} + 90616216992 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} - 115517223360 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 52895204480 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} - 565618176 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 140708352 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 17333248\right )}}{94119200000 \, {\left ({\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} + 3 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} - 2\right )}^{9}} \]
76869/25735718750*sqrt(35)*log(-abs(-2*sqrt(3)*x - sqrt(35) - 3*sqrt(3) + 2*sqrt(3*x^2 + 2))/(2*sqrt(3)*x - sqrt(35) + 3*sqrt(3) - 2*sqrt(3*x^2 + 2) )) - 9/94119200000*sqrt(3)*(364416*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^1 7 + 27877824*(sqrt(3)*x - sqrt(3*x^2 + 2))^16 + 1042205258*sqrt(3)*(sqrt(3 )*x - sqrt(3*x^2 + 2))^15 - 956098170*(sqrt(3)*x - sqrt(3*x^2 + 2))^14 + 1 003625490*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^13 - 85987901496*(sqrt(3)* x - sqrt(3*x^2 + 2))^12 - 60468401868*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2) )^11 - 331045664193*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 - 22913148915*sqrt(3) *(sqrt(3)*x - sqrt(3*x^2 + 2))^9 - 544736640510*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 + 284856270864*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 - 90885012422 4*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 + 90616216992*sqrt(3)*(sqrt(3)*x - sqrt( 3*x^2 + 2))^5 - 115517223360*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 52895204480 *sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 - 565618176*(sqrt(3)*x - sqrt(3*x ^2 + 2))^2 + 140708352*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 17333248)/( (sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^9
Time = 0.18 (sec) , antiderivative size = 385, normalized size of antiderivative = 2.14 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{10}} \, dx=\frac {76869\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{25735718750}-\frac {76869\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{25735718750}+\frac {4515\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{32768\,\left (x^8+12\,x^7+63\,x^6+189\,x^5+\frac {2835\,x^4}{8}+\frac {1701\,x^3}{4}+\frac {5103\,x^2}{16}+\frac {2187\,x}{16}+\frac {6561}{256}\right )}+\frac {1838301\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{614656000\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}-\frac {15925\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{294912\,\left (x^9+\frac {27\,x^8}{2}+81\,x^7+\frac {567\,x^6}{2}+\frac {5103\,x^5}{8}+\frac {15309\,x^4}{16}+\frac {15309\,x^3}{16}+\frac {19683\,x^2}{32}+\frac {59049\,x}{256}+\frac {19683}{512}\right )}-\frac {923241\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{35123200\,\left (x^5+\frac {15\,x^4}{2}+\frac {45\,x^3}{2}+\frac {135\,x^2}{4}+\frac {405\,x}{16}+\frac {243}{32}\right )}-\frac {152343\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{94119200000\,\left (x+\frac {3}{2}\right )}+\frac {35213\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{401408\,\left (x^6+9\,x^5+\frac {135\,x^4}{4}+\frac {135\,x^3}{2}+\frac {1215\,x^2}{16}+\frac {729\,x}{16}+\frac {729}{64}\right )}+\frac {80649\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{5378240000\,\left (x^2+3\,x+\frac {9}{4}\right )}-\frac {52201\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{344064\,\left (x^7+\frac {21\,x^6}{2}+\frac {189\,x^5}{4}+\frac {945\,x^4}{8}+\frac {2835\,x^3}{16}+\frac {5103\,x^2}{32}+\frac {5103\,x}{64}+\frac {2187}{128}\right )}+\frac {55473\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1536640000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \]
(76869*35^(1/2)*log(x + 3/2))/25735718750 - (76869*35^(1/2)*log(x - (3^(1/ 2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/25735718750 + (4515*3^(1/2)*(x^2 + 2/3)^(1/2))/(32768*((2187*x)/16 + (5103*x^2)/16 + (1701*x^3)/4 + (2835*x ^4)/8 + 189*x^5 + 63*x^6 + 12*x^7 + x^8 + 6561/256)) + (1838301*3^(1/2)*(x ^2 + 2/3)^(1/2))/(614656000*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) - (15925*3^(1/2)*(x^2 + 2/3)^(1/2))/(294912*((59049*x)/256 + (19683*x^2)/ 32 + (15309*x^3)/16 + (15309*x^4)/16 + (5103*x^5)/8 + (567*x^6)/2 + 81*x^7 + (27*x^8)/2 + x^9 + 19683/512)) - (923241*3^(1/2)*(x^2 + 2/3)^(1/2))/(35 123200*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4)/2 + x^5 + 243/32) ) - (152343*3^(1/2)*(x^2 + 2/3)^(1/2))/(94119200000*(x + 3/2)) + (35213*3^ (1/2)*(x^2 + 2/3)^(1/2))/(401408*((729*x)/16 + (1215*x^2)/16 + (135*x^3)/2 + (135*x^4)/4 + 9*x^5 + x^6 + 729/64)) + (80649*3^(1/2)*(x^2 + 2/3)^(1/2) )/(5378240000*(3*x + x^2 + 9/4)) - (52201*3^(1/2)*(x^2 + 2/3)^(1/2))/(3440 64*((5103*x)/64 + (5103*x^2)/32 + (2835*x^3)/16 + (945*x^4)/8 + (189*x^5)/ 4 + (21*x^6)/2 + x^7 + 2187/128)) + (55473*3^(1/2)*(x^2 + 2/3)^(1/2))/(153 6640000*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))