Integrand size = 24, antiderivative size = 202 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{11}} \, dx=-\frac {1977291 (4-9 x) \sqrt {2+3 x^2}}{514714375000 (3+2 x)^2}-\frac {219699 (4-9 x) \left (2+3 x^2\right )^{3/2}}{14706125000 (3+2 x)^4}-\frac {73233 (4-9 x) \left (2+3 x^2\right )^{5/2}}{1050437500 (3+2 x)^6}-\frac {13 \left (2+3 x^2\right )^{7/2}}{350 (3+2 x)^{10}}-\frac {1171 \left (2+3 x^2\right )^{7/2}}{110250 (3+2 x)^9}-\frac {4393 \left (2+3 x^2\right )^{7/2}}{1715000 (3+2 x)^8}-\frac {739619 \left (2+3 x^2\right )^{7/2}}{1260525000 (3+2 x)^7}-\frac {5931873 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {2+3 x^2}}\right )}{257357187500 \sqrt {35}} \]
-219699/14706125000*(4-9*x)*(3*x^2+2)^(3/2)/(3+2*x)^4-73233/1050437500*(4- 9*x)*(3*x^2+2)^(5/2)/(3+2*x)^6-13/350*(3*x^2+2)^(7/2)/(3+2*x)^10-1171/1102 50*(3*x^2+2)^(7/2)/(3+2*x)^9-4393/1715000*(3*x^2+2)^(7/2)/(3+2*x)^8-739619 /1260525000*(3*x^2+2)^(7/2)/(3+2*x)^7-5931873/9007501562500*arctanh(1/35*( 4-9*x)*35^(1/2)/(3*x^2+2)^(1/2))*35^(1/2)-1977291/514714375000*(4-9*x)*(3* x^2+2)^(1/2)/(3+2*x)^2
Time = 2.50 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.56 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{11}} \, dx=\frac {-\frac {35 \sqrt {2+3 x^2} \left (5288003538036+5421307926571 x+18888919063956 x^2+4704132871221 x^3+11369945485836 x^4-3078520541586 x^5+1541962687104 x^6+544524933294 x^7+101311348104 x^8+7968937464 x^9\right )}{(3+2 x)^{10}}+213547428 \sqrt {35} \text {arctanh}\left (\frac {3 \sqrt {3}+2 \sqrt {3} x-2 \sqrt {2+3 x^2}}{\sqrt {35}}\right )}{162135028125000} \]
((-35*Sqrt[2 + 3*x^2]*(5288003538036 + 5421307926571*x + 18888919063956*x^ 2 + 4704132871221*x^3 + 11369945485836*x^4 - 3078520541586*x^5 + 154196268 7104*x^6 + 544524933294*x^7 + 101311348104*x^8 + 7968937464*x^9))/(3 + 2*x )^10 + 213547428*Sqrt[35]*ArcTanh[(3*Sqrt[3] + 2*Sqrt[3]*x - 2*Sqrt[2 + 3* x^2])/Sqrt[35]])/162135028125000
Time = 0.37 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {688, 25, 688, 27, 688, 25, 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.
| \(\displaystyle \int \frac {(5-x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^{11}} \, dx\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle -\frac {1}{350} \int -\frac {(410-117 x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^{10}}dx-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{350} \int \frac {(410-117 x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^{10}}dx-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {1}{350} \left (-\frac {1}{315} \int -\frac {6 (4833-1171 x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^9}dx-\frac {1171 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{350} \left (\frac {2}{105} \int \frac {(4833-1171 x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^9}dx-\frac {1171 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 688 |
| \(\displaystyle \frac {1}{350} \left (\frac {2}{105} \left (-\frac {1}{280} \int -\frac {(310504-39537 x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^8}dx-\frac {13179 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\right )-\frac {1171 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{350} \left (\frac {2}{105} \left (\frac {1}{280} \int \frac {(310504-39537 x) \left (3 x^2+2\right )^{5/2}}{(2 x+3)^8}dx-\frac {13179 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\right )-\frac {1171 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 679 |
| \(\displaystyle \frac {1}{350} \left (\frac {2}{105} \left (\frac {1}{280} \left (\frac {2636388}{35} \int \frac {\left (3 x^2+2\right )^{5/2}}{(2 x+3)^7}dx-\frac {739619 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {13179 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\right )-\frac {1171 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 486 |
| \(\displaystyle \frac {1}{350} \left (\frac {2}{105} \left (\frac {1}{280} \left (\frac {2636388}{35} \left (\frac {1}{7} \int \frac {\left (3 x^2+2\right )^{3/2}}{(2 x+3)^5}dx-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {739619 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {13179 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\right )-\frac {1171 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 486 |
| \(\displaystyle \frac {1}{350} \left (\frac {2}{105} \left (\frac {1}{280} \left (\frac {2636388}{35} \left (\frac {1}{7} \left (\frac {9}{70} \int \frac {\sqrt {3 x^2+2}}{(2 x+3)^3}dx-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {739619 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {13179 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\right )-\frac {1171 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 486 |
| \(\displaystyle \frac {1}{350} \left (\frac {2}{105} \left (\frac {1}{280} \left (\frac {2636388}{35} \left (\frac {1}{7} \left (\frac {9}{70} \left (\frac {3}{35} \int \frac {1}{(2 x+3) \sqrt {3 x^2+2}}dx-\frac {(4-9 x) \sqrt {3 x^2+2}}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {739619 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {13179 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\right )-\frac {1171 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 488 |
| \(\displaystyle \frac {1}{350} \left (\frac {2}{105} \left (\frac {1}{280} \left (\frac {2636388}{35} \left (\frac {1}{7} \left (\frac {9}{70} \left (-\frac {3}{35} \int \frac {1}{35-\frac {(4-9 x)^2}{3 x^2+2}}d\frac {4-9 x}{\sqrt {3 x^2+2}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {739619 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {13179 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\right )-\frac {1171 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {1}{350} \left (\frac {2}{105} \left (\frac {1}{280} \left (\frac {2636388}{35} \left (\frac {1}{7} \left (\frac {9}{70} \left (-\frac {3 \text {arctanh}\left (\frac {4-9 x}{\sqrt {35} \sqrt {3 x^2+2}}\right )}{35 \sqrt {35}}-\frac {\sqrt {3 x^2+2} (4-9 x)}{70 (2 x+3)^2}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{3/2}}{140 (2 x+3)^4}\right )-\frac {(4-9 x) \left (3 x^2+2\right )^{5/2}}{210 (2 x+3)^6}\right )-\frac {739619 \left (3 x^2+2\right )^{7/2}}{245 (2 x+3)^7}\right )-\frac {13179 \left (3 x^2+2\right )^{7/2}}{280 (2 x+3)^8}\right )-\frac {1171 \left (3 x^2+2\right )^{7/2}}{315 (2 x+3)^9}\right )-\frac {13 \left (3 x^2+2\right )^{7/2}}{350 (2 x+3)^{10}}\) |
(-13*(2 + 3*x^2)^(7/2))/(350*(3 + 2*x)^10) + ((-1171*(2 + 3*x^2)^(7/2))/(3 15*(3 + 2*x)^9) + (2*((-13179*(2 + 3*x^2)^(7/2))/(280*(3 + 2*x)^8) + ((-73 9619*(2 + 3*x^2)^(7/2))/(245*(3 + 2*x)^7) + (2636388*(-1/210*((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)/280))/105 )/350
3.14.96.3.1 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 = 0.51 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.52
| method | result | size |
| risch | \(-\frac {23906812392 x^{11}+303934044312 x^{10}+1649512674810 x^{9}+4828510757520 x^{8}-8146511758170 x^{7}+37193761831716 x^{6}+7955357530491 x^{5}+79406648163540 x^{4}+25672189522155 x^{3}+53641848742020 x^{2}+10842615853142 x +10576007076072}{4632429375000 \left (3+2 x \right )^{10} \sqrt {3 x^{2}+2}}-\frac {5931873 \sqrt {35}\, \operatorname {arctanh}\left (\frac {2 \left (4-9 x \right ) \sqrt {35}}{35 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}\right )}{9007501562500}\) | \(105\) |
| trager | \(-\frac {\left (7968937464 x^{9}+101311348104 x^{8}+544524933294 x^{7}+1541962687104 x^{6}-3078520541586 x^{5}+11369945485836 x^{4}+4704132871221 x^{3}+18888919063956 x^{2}+5421307926571 x +5288003538036\right ) \sqrt {3 x^{2}+2}}{4632429375000 \left (3+2 x \right )^{10}}-\frac {5931873 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) \ln \left (\frac {-9 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right ) x +35 \sqrt {3 x^{2}+2}+4 \operatorname {RootOf}\left (\textit {\_Z}^{2}-35\right )}{3+2 x}\right )}{9007501562500}\) | \(111\) |
| default | \(\frac {3954582 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{78815638671875}-\frac {13 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{358400 \left (x +\frac {3}{2}\right )^{10}}-\frac {1171 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{56448000 \left (x +\frac {3}{2}\right )^{9}}-\frac {4393 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{439040000 \left (x +\frac {3}{2}\right )^{8}}-\frac {739619 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{161347200000 \left (x +\frac {3}{2}\right )^{7}}-\frac {73233 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{33614000000 \left (x +\frac {3}{2}\right )^{6}}-\frac {659097 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{588245000000 \left (x +\frac {3}{2}\right )^{5}}-\frac {6371271 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{10294287500000 \left (x +\frac {3}{2}\right )^{4}}-\frac {65250603 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{180150031250000 \left (x +\frac {3}{2}\right )^{3}}-\frac {709847469 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{3152625546875000 \left (x +\frac {3}{2}\right )^{2}}+\frac {24718114791 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{55170947070312500}-\frac {8239371597 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {7}{2}}}{55170947070312500 \left (x +\frac {3}{2}\right )}+\frac {694029141 x \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {3}{2}}}{630525109375000}+\frac {53386857 x \sqrt {3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}}}{18015003125000}-\frac {5931873 \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 )}{9007501562500}+\frac {5931873 \sqrt {12 \left (x +\frac {3}{2}\right )^{2}-36 x -19}}{9007501562500}+\frac {47454984 \left (3 \left (x +\frac {3}{2}\right )^{2}-9 x -\frac {19}{4}\right )^{\frac {5}{2}}}{13792736767578125}\) | \(341\) |
-1/4632429375000*(23906812392*x^11+303934044312*x^10+1649512674810*x^9+482 8510757520*x^8-8146511758170*x^7+37193761831716*x^6+7955357530491*x^5+7940 6648163540*x^4+25672189522155*x^3+53641848742020*x^2+10842615853142*x+1057 6007076072)/(3+2*x)^10/(3*x^2+2)^(1/2)-5931873/9007501562500*35^(1/2)*arct anh(2/35*(4-9*x)*35^(1/2)/(12*(x+3/2)^2-36*x-19)^(1/2))
Time = 0.38 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.03 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{11}} \, dx=\frac {53386857 \, \sqrt {35} {\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )} \log \left (-\frac {\sqrt {35} \sqrt {3 \, x^{2} + 2} {\left (9 \, x - 4\right )} + 93 \, x^{2} - 36 \, x + 43}{4 \, x^{2} + 12 \, x + 9}\right ) - 35 \, {\left (7968937464 \, x^{9} + 101311348104 \, x^{8} + 544524933294 \, x^{7} + 1541962687104 \, x^{6} - 3078520541586 \, x^{5} + 11369945485836 \, x^{4} + 4704132871221 \, x^{3} + 18888919063956 \, x^{2} + 5421307926571 \, x + 5288003538036\right )} \sqrt {3 \, x^{2} + 2}}{162135028125000 \, {\left (1024 \, x^{10} + 15360 \, x^{9} + 103680 \, x^{8} + 414720 \, x^{7} + 1088640 \, x^{6} + 1959552 \, x^{5} + 2449440 \, x^{4} + 2099520 \, x^{3} + 1180980 \, x^{2} + 393660 \, x + 59049\right )}} \]
1/162135028125000*(53386857*sqrt(35)*(1024*x^10 + 15360*x^9 + 103680*x^8 + 414720*x^7 + 1088640*x^6 + 1959552*x^5 + 2449440*x^4 + 2099520*x^3 + 1180 980*x^2 + 393660*x + 59049)*log(-(sqrt(35)*sqrt(3*x^2 + 2)*(9*x - 4) + 93* x^2 - 36*x + 43)/(4*x^2 + 12*x + 9)) - 35*(7968937464*x^9 + 101311348104*x ^8 + 544524933294*x^7 + 1541962687104*x^6 - 3078520541586*x^5 + 1136994548 5836*x^4 + 4704132871221*x^3 + 18888919063956*x^2 + 5421307926571*x + 5288 003538036)*sqrt(3*x^2 + 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)
Timed out. \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{11}} \, dx=\text {Timed out} \]
Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (167) = 334\).
Time = 0.30 (sec) , antiderivative size = 497, normalized size of antiderivative = 2.46 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{11}} \, dx=\frac {2129542407}{3152625546875000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}} - \frac {13 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{350 \, {\left (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 )}} - \frac {1171 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{110250 \, {\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 {4393 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1715000 \, {\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 {739619 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{1260525000 \, {\left (128 \, x^{7} + 1344 \, x^{6} + 6048 \, x^{5} + 15120 \, x^{4} + 22680 \, x^{3} + 20412 \, x^{2} + 10206 \, x + 2187\right )}} - \frac {73233 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{525218750 \, {\left (64 \, x^{6} + 576 \, x^{5} + 2160 \, x^{4} + 4320 \, x^{3} + 4860 \, x^{2} + 2916 \, x + 729\right )}} - \frac {659097 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{18382656250 \, {\left (32 \, x^{5} + 240 \, x^{4} + 720 \, x^{3} + 1080 \, x^{2} + 810 \, x + 243\right )}} - \frac {6371271 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{643392968750 \, {\left (16 \, x^{4} + 96 \, x^{3} + 216 \, x^{2} + 216 \, x + 81\right )}} - \frac {65250603 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{22518753906250 \, {\left (8 \, x^{3} + 36 \, x^{2} + 54 \, x + 27\right )}} - \frac {709847469 \, {\left (3 \, x^{2} + 2\right )}^{\frac {7}{2}}}{788156386718750 \, {\left (4 \, x^{2} + 12 \, x + 9\right )}} + \frac {694029141}{630525109375000} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} x + \frac {3954582}{78815638671875} \, {\left (3 \, x^{2} + 2\right )}^{\frac {3}{2}} - \frac {8239371597 \, {\left (3 \, x^{2} + 2\right )}^{\frac {5}{2}}}{3152625546875000 \, {\left (2 \, x + 3\right )}} + \frac {53386857}{18015003125000} \, \sqrt {3 \, x^{2} + 2} x + \frac {5931873}{9007501562500} \, \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 {5931873}{4503750781250} \, \sqrt {3 \, x^{2} + 2} \]
2129542407/3152625546875000*(3*x^2 + 2)^(5/2) - 13/350*(3*x^2 + 2)^(7/2)/( 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) - 1171/110 250*(3*x^2 + 2)^(7/2)/(512*x^9 + 6912*x^8 + 41472*x^7 + 145152*x^6 + 32659 2*x^5 + 489888*x^4 + 489888*x^3 + 314928*x^2 + 118098*x + 19683) - 4393/17 15000*(3*x^2 + 2)^(7/2)/(256*x^8 + 3072*x^7 + 16128*x^6 + 48384*x^5 + 9072 0*x^4 + 108864*x^3 + 81648*x^2 + 34992*x + 6561) - 739619/1260525000*(3*x^ 2 + 2)^(7/2)/(128*x^7 + 1344*x^6 + 6048*x^5 + 15120*x^4 + 22680*x^3 + 2041 2*x^2 + 10206*x + 2187) - 73233/525218750*(3*x^2 + 2)^(7/2)/(64*x^6 + 576* x^5 + 2160*x^4 + 4320*x^3 + 4860*x^2 + 2916*x + 729) - 659097/18382656250* (3*x^2 + 2)^(7/2)/(32*x^5 + 240*x^4 + 720*x^3 + 1080*x^2 + 810*x + 243) - 6371271/643392968750*(3*x^2 + 2)^(7/2)/(16*x^4 + 96*x^3 + 216*x^2 + 216*x + 81) - 65250603/22518753906250*(3*x^2 + 2)^(7/2)/(8*x^3 + 36*x^2 + 54*x + 27) - 709847469/788156386718750*(3*x^2 + 2)^(7/2)/(4*x^2 + 12*x + 9) + 69 4029141/630525109375000*(3*x^2 + 2)^(3/2)*x + 3954582/78815638671875*(3*x^ 2 + 2)^(3/2) - 8239371597/3152625546875000*(3*x^2 + 2)^(5/2)/(2*x + 3) + 5 3386857/18015003125000*sqrt(3*x^2 + 2)*x + 5931873/9007501562500*sqrt(35)* arcsinh(3/2*sqrt(6)*x/abs(2*x + 3) - 2/3*sqrt(6)/abs(2*x + 3)) + 5931873/4 503750781250*sqrt(3*x^2 + 2)
Leaf count of result is larger than twice the leaf count of optimal. 547 vs. \(2 (167) = 334\).
Time = 0.34 (sec) , antiderivative size = 547, normalized size of antiderivative = 2.71 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{11}} \, dx=\frac {5931873}{9007501562500} \, \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 (56242944 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{19} + 4808771712 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{18} + 60161202432 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{17} + 2449600006086 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{16} + 650003734476 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{15} + 11324343251586 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{14} - 43249498138224 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{13} - 114750161469717 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{12} - 263561308381422 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{11} - 64560900263031 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{10} - 173527579922724 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{9} + 409007369125548 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{8} - 812515292998272 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{7} + 775661489485344 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{6} - 309262645005696 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{5} + 53888888658816 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{4} - 21200045958144 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{3} + 6293205518848 \, {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )}^{2} - 348990277632 \, \sqrt {3} {\left (\sqrt {3} x - \sqrt {3 \, x^{2} + 2}\right )} + 25185777664\right )}}{65883440000000 \, {\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 )}^{10}} \]
5931873/9007501562500*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))) - 9/65883440000000*sqrt(3)*(56242944*sqrt(3)*(sqrt(3)*x - sqrt(3*x^ 2 + 2))^19 + 4808771712*(sqrt(3)*x - sqrt(3*x^2 + 2))^18 + 60161202432*sqr t(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^17 + 2449600006086*(sqrt(3)*x - sqrt(3* x^2 + 2))^16 + 650003734476*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^15 + 113 24343251586*(sqrt(3)*x - sqrt(3*x^2 + 2))^14 - 43249498138224*sqrt(3)*(sqr t(3)*x - sqrt(3*x^2 + 2))^13 - 114750161469717*(sqrt(3)*x - sqrt(3*x^2 + 2 ))^12 - 263561308381422*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^11 - 6456090 0263031*(sqrt(3)*x - sqrt(3*x^2 + 2))^10 - 173527579922724*sqrt(3)*(sqrt(3 )*x - sqrt(3*x^2 + 2))^9 + 409007369125548*(sqrt(3)*x - sqrt(3*x^2 + 2))^8 - 812515292998272*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^7 + 7756614894853 44*(sqrt(3)*x - sqrt(3*x^2 + 2))^6 - 309262645005696*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^5 + 53888888658816*(sqrt(3)*x - sqrt(3*x^2 + 2))^4 - 2120 0045958144*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2))^3 + 6293205518848*(sqrt(3 )*x - sqrt(3*x^2 + 2))^2 - 348990277632*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) + 25185777664)/((sqrt(3)*x - sqrt(3*x^2 + 2))^2 + 3*sqrt(3)*(sqrt(3)*x - sqrt(3*x^2 + 2)) - 2)^10
Time = 10.92 (sec) , antiderivative size = 449, normalized size of antiderivative = 2.22 \[ \int \frac {(5-x) \left (2+3 x^2\right )^{5/2}}{(3+2 x)^{11}} \, dx=\frac {5931873\,\sqrt {35}\,\ln \left (x+\frac {3}{2}\right )}{9007501562500}-\frac {5931873\,\sqrt {35}\,\ln \left (x-\frac {\sqrt {3}\,\sqrt {35}\,\sqrt {x^2+\frac {2}{3}}}{9}-\frac {4}{9}\right )}{9007501562500}-\frac {43213\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{655360\,\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 {4728159\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{430259200000\,\left (x^4+6\,x^3+\frac {27\,x^2}{2}+\frac {27\,x}{2}+\frac {81}{16}\right )}+\frac {36029\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{589824\,\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 {27428781\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{24586240000\,\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 {3185\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{131072\,\left (x^{10}+15\,x^9+\frac {405\,x^8}{4}+405\,x^7+\frac {8505\,x^6}{8}+\frac {15309\,x^5}{8}+\frac {76545\,x^4}{32}+\frac {32805\,x^3}{16}+\frac {295245\,x^2}{256}+\frac {98415\,x}{256}+\frac {59049}{1024}\right )}-\frac {110679687\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{65883440000000\,\left (x+\frac {3}{2}\right )}-\frac {2988711\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{280985600\,\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 {4975641\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{3764768000000\,\left (x^2+3\,x+\frac {9}{4}\right )}+\frac {1785563\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{48168960\,\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 {5833857\,\sqrt {3}\,\sqrt {x^2+\frac {2}{3}}}{1075648000000\,\left (x^3+\frac {9\,x^2}{2}+\frac {27\,x}{4}+\frac {27}{8}\right )} \]
(5931873*35^(1/2)*log(x + 3/2))/9007501562500 - (5931873*35^(1/2)*log(x - (3^(1/2)*35^(1/2)*(x^2 + 2/3)^(1/2))/9 - 4/9))/9007501562500 - (43213*3^(1 /2)*(x^2 + 2/3)^(1/2))/(655360*((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)) + (4728159* 3^(1/2)*(x^2 + 2/3)^(1/2))/(430259200000*((27*x)/2 + (27*x^2)/2 + 6*x^3 + x^4 + 81/16)) + (36029*3^(1/2)*(x^2 + 2/3)^(1/2))/(589824*((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)) + (27428781*3^(1/2)*(x^2 + 2/3)^(1/2))/(24586240000*((405*x)/16 + (135*x^2)/4 + (45*x^3)/2 + (15*x^4) /2 + x^5 + 243/32)) - (3185*3^(1/2)*(x^2 + 2/3)^(1/2))/(131072*((98415*x)/ 256 + (295245*x^2)/256 + (32805*x^3)/16 + (76545*x^4)/32 + (15309*x^5)/8 + (8505*x^6)/8 + 405*x^7 + (405*x^8)/4 + 15*x^9 + x^10 + 59049/1024)) - (11 0679687*3^(1/2)*(x^2 + 2/3)^(1/2))/(65883440000000*(x + 3/2)) - (2988711*3 ^(1/2)*(x^2 + 2/3)^(1/2))/(280985600*((729*x)/16 + (1215*x^2)/16 + (135*x^ 3)/2 + (135*x^4)/4 + 9*x^5 + x^6 + 729/64)) + (4975641*3^(1/2)*(x^2 + 2/3) ^(1/2))/(3764768000000*(3*x + x^2 + 9/4)) + (1785563*3^(1/2)*(x^2 + 2/3)^( 1/2))/(48168960*((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)) + (5833857*3^(1/2)*(x^2 + 2 /3)^(1/2))/(1075648000000*((27*x)/4 + (9*x^2)/2 + x^3 + 27/8))