Integrand size = 28, antiderivative size = 280 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=-\frac {54281308 \sqrt {1-2 x} \sqrt {3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac {1876198516 \sqrt {1-2 x} \sqrt {3+5 x}}{750891141 (2+3 x)^{3/2}}+\frac {129922578224 \sqrt {1-2 x} \sqrt {3+5 x}}{5256237987 \sqrt {2+3 x}}-\frac {2622980 \sqrt {1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac {60080 \sqrt {1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}-\frac {129922578224 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{477839817 \sqrt {33}}-\frac {3894280616 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{477839817 \sqrt {33}} \]
-2/39*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2)+370/1287*(1-2*x)^(3/2)*(3 +5*x)^(5/2)/(2+3*x)^(11/2)-129922578224/15768713961*EllipticE(1/7*21^(1/2) *(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-3894280616/15768713961*EllipticF( 1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2622980/1702701*(3+5* x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(7/2)+60080/34749*(3+5*x)^(5/2)*(1-2*x)^(1/ 2)/(2+3*x)^(9/2)-54281308/35756721*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/ 2)+1876198516/750891141*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+12992257 8224/5256237987*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.94 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.41 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (4382625184685+39086872650957 x+145238558453649 x^2+287874442427697 x^3+321056742490902 x^4+191022825888450 x^5+47356779762648 x^6\right )}{4 (2+3 x)^{13/2}}+i \sqrt {33} \left (16240322278 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-16727107355 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{15768713961} \]
(8*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(4382625184685 + 39086872650957*x + 145 238558453649*x^2 + 287874442427697*x^3 + 321056742490902*x^4 + 19102282588 8450*x^5 + 47356779762648*x^6))/(4*(2 + 3*x)^(13/2)) + I*Sqrt[33]*(1624032 2278*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 16727107355*EllipticF[I *ArcSinh[Sqrt[9 + 15*x]], -2/33])))/15768713961
Time = 0.34 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {108, 27, 167, 167, 27, 167, 27, 167, 27, 169, 169, 27, 176, 123, 129}
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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^{15/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{39} \int -\frac {5 (1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{2 (3 x+2)^{13/2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{39} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{(3 x+2)^{13/2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \int \frac {(778-335 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^{11/2}}dx-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \left (\frac {6008 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}-\frac {2}{27} \int -\frac {(75089-84090 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} (3 x+2)^{9/2}}dx\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \left (\frac {1}{27} \int \frac {(75089-84090 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^{9/2}}dx+\frac {6008 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \left (\frac {1}{27} \left (\frac {2}{147} \int \frac {3 (2568189-2932880 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^{7/2}}dx-\frac {262298 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {6008 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \left (\frac {1}{27} \left (\frac {1}{49} \int \frac {(2568189-2932880 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^{7/2}}dx-\frac {262298 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {6008 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \left (\frac {1}{27} \left (\frac {1}{49} \left (\frac {2}{105} \int \frac {85587083-106144190 x}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {27140654 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {262298 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {6008 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \left (\frac {1}{27} \left (\frac {1}{49} \left (\frac {1}{105} \int \frac {85587083-106144190 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {27140654 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {262298 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {6008 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \left (\frac {1}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {2}{21} \int \frac {3849941996-2345248145 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {938099258 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {27140654 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {262298 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {6008 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \left (\frac {1}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {2}{21} \left (\frac {2}{7} \int \frac {5 (32480644556 x+20559313903)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {32480644556 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {938099258 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {27140654 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {262298 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {6008 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \left (\frac {1}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {2}{21} \left (\frac {5}{7} \int \frac {32480644556 x+20559313903}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {32480644556 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {938099258 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {27140654 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {262298 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {6008 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \left (\frac {1}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {5354635847}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {32480644556}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {32480644556 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {938099258 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {27140654 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {262298 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {6008 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \left (\frac {1}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {5354635847}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {32480644556}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {32480644556 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {938099258 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {27140654 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {262298 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {6008 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle -\frac {5}{39} \left (-\frac {2}{33} \left (\frac {1}{27} \left (\frac {1}{49} \left (\frac {1}{105} \left (\frac {2}{21} \left (\frac {5}{7} \left (-\frac {973570154}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {32480644556}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {32480644556 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {938099258 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {27140654 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}\right )-\frac {262298 \sqrt {1-2 x} (5 x+3)^{3/2}}{49 (3 x+2)^{7/2}}\right )+\frac {6008 \sqrt {1-2 x} (5 x+3)^{5/2}}{27 (3 x+2)^{9/2}}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
(-2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(39*(2 + 3*x)^(13/2)) - (5*((-74*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) - (2*((6008*Sqrt[1 - 2* x]*(3 + 5*x)^(5/2))/(27*(2 + 3*x)^(9/2)) + ((-262298*Sqrt[1 - 2*x]*(3 + 5* x)^(3/2))/(49*(2 + 3*x)^(7/2)) + ((-27140654*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/ (105*(2 + 3*x)^(5/2)) + ((938099258*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(3/2)) + (2*((32480644556*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3* x]) + (5*((-32480644556*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x ]], 35/33])/5 - (973570154*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5))/7))/21)/105)/49)/27))/33))/39
3.28.84.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 0] && (IntegersQ[2*m, 2 *n, 2*p] || IntegersQ[m, n + p] || IntegersQ[p, m + n])
Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ )]), x_] :> Simp[(2/b)*Rt[-(b*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x] /Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] && !L tQ[-(b*c - a*d)/d, 0] && !(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d ), 0] && GtQ[d/(d*e - c*f), 0] && !LtQ[(b*c - a*d)/b, 0])
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x _)]), x_] :> Simp[2*(Rt[-b/d, 2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[ Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ [(b*e - a*f)/b, 0] && PosQ[-b/d] && !(SimplerQ[c + d*x, a + b*x] && GtQ[(d *e - c*f)/d, 0] && GtQ[-d/b, 0]) && !(SimplerQ[c + d*x, a + b*x] && GtQ[(( -b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) && !(SimplerQ[e + f*x, a + b*x] && GtQ [((-d)*e + c*f)/f, 0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f /b]))
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Simp[1/(b*(b*e - a*f)*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b* c*(f*g - e*h)*(m + 1) + (b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h )*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[(b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n *(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[ 2*m, 2*n, 2*p]
Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]* Sqrt[(e_) + (f_.)*(x_)]), x_] :> Simp[h/f Int[Sqrt[e + f*x]/(Sqrt[a + b*x ]*Sqrt[c + d*x]), x], x] + Simp[(f*g - e*h)/f Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && Sim plerQ[a + b*x, e + f*x] && SimplerQ[c + d*x, e + f*x]
Time = 1.31 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.25
| method | result | size |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6908733 \left (\frac {2}{3}+x \right )^{7}}+\frac {518 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{938223 \left (\frac {2}{3}+x \right )^{6}}-\frac {478462 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{75996063 \left (\frac {2}{3}+x \right )^{5}}+\frac {17427370 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1241269029 \left (\frac {2}{3}+x \right )^{4}}+\frac {13028276 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2896294401 \left (\frac {2}{3}+x \right )^{3}}+\frac {1876198516 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6758020269 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {1299225782240}{5256237987} x^{2}-\frac {129922578224}{5256237987} x +\frac {129922578224}{1752079329}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {164474511224 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{110380997727 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {259845156448 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{110380997727 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(350\) |
| default | \(\frac {2 \left (47356779762648 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-45989031044484 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+189427119050592 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-183956124177936 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+315711865084320 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-306593540296560 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+280632768963840 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-272527591374720 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+140316384481920 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-136263795687360 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+1420703392879440 x^{8}+37417702528512 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-36337012183296 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+5872755115941444 x^{7}+4157522503168 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4037445798144 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+9778559734528578 x^{6}+7880198067307566 x^{5}+2331269398474443 x^{4}-982548126959616 x^{3}-1058407652589420 x^{2}-338633978304558 x -39443626662165\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{15768713961 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {13}{2}}}\) | \(694\) |
-1/(10*x^2+x-3)/(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-(-1+2*x)*(3+5* x)*(2+3*x))^(1/2)*(-98/6908733*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^7+518/ 938223*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^6-478462/75996063*(-30*x^3-23* x^2+7*x+6)^(1/2)/(2/3+x)^5+17427370/1241269029*(-30*x^3-23*x^2+7*x+6)^(1/2 )/(2/3+x)^4+13028276/2896294401*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+187 6198516/6758020269*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+129922578224/157 68713961*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+164474511224/1103 80997727*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7 *x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))+259845156448/11038099 7727*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6 )^(1/2)*(-7/6*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+1/2*EllipticF((10+1 5*x)^(1/2),1/35*70^(1/2))))
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 188, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\frac {2 \, {\left (135 \, {\left (47356779762648 \, x^{6} + 191022825888450 \, x^{5} + 321056742490902 \, x^{4} + 287874442427697 \, x^{3} + 145238558453649 \, x^{2} + 39086872650957 \, x + 4382625184685\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1103283426482 \, \sqrt {-30} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 2923258010040 \, \sqrt {-30} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{709592128245 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
2/709592128245*(135*(47356779762648*x^6 + 191022825888450*x^5 + 3210567424 90902*x^4 + 287874442427697*x^3 + 145238558453649*x^2 + 39086872650957*x + 4382625184685)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 1103283426482 *sqrt(-30)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 604 8*x^2 + 1344*x + 128)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90 ) + 2923258010040*sqrt(-30)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)*weierstrassZeta(1159/675, 38998/911 25, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(2187*x^7 + 10 206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {15}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {15}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{15/2}} \,d x \]