Integrand size = 28, antiderivative size = 280 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=-\frac {2174468 \sqrt {1-2 x} \sqrt {3+5 x}}{11918907 (2+3 x)^{7/2}}+\frac {73596464 \sqrt {1-2 x} \sqrt {3+5 x}}{417161745 (2+3 x)^{5/2}}+\frac {3523482724 \sqrt {1-2 x} \sqrt {3+5 x}}{2920132215 (2+3 x)^{3/2}}+\frac {245282464136 \sqrt {1-2 x} \sqrt {3+5 x}}{20440925505 \sqrt {2+3 x}}-\frac {20992 \sqrt {1-2 x} (3+5 x)^{3/2}}{81081 (2+3 x)^{9/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac {362 \sqrt {1-2 x} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}-\frac {245282464136 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1858265955 \sqrt {33}}-\frac {7391549624 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1858265955 \sqrt {33}} \]
-2/39*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2)-245282464136/61322776515* EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-7391549624/ 61322776515*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2) -20992/81081*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2)+362/1287*(3+5*x)^(5 /2)*(1-2*x)^(1/2)/(2+3*x)^(11/2)-2174468/11918907*(1-2*x)^(1/2)*(3+5*x)^(1 /2)/(2+3*x)^(7/2)+73596464/417161745*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^( 5/2)+3523482724/2920132215*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+24528 2464136/20440925505*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.21 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.41 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\frac {\frac {96 \sqrt {1-2 x} \sqrt {3+5 x} \left (8272877174903+73802680969881 x+274263621177573 x^2+543590753927373 x^3+606171513555828 x^4+360618554767050 x^5+89405458177572 x^6\right )}{(2+3 x)^{13/2}}+128 i \sqrt {33} \left (30660308017 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-31584251720 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{981164424240} \]
((96*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(8272877174903 + 73802680969881*x + 27426 3621177573*x^2 + 543590753927373*x^3 + 606171513555828*x^4 + 3606185547670 50*x^5 + 89405458177572*x^6))/(2 + 3*x)^(13/2) + (128*I)*Sqrt[33]*(3066030 8017*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 31584251720*EllipticF[I *ArcSinh[Sqrt[9 + 15*x]], -2/33]))/981164424240
Time = 0.35 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.14, number of steps used = 16, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.571, Rules used = {108, 27, 167, 25, 167, 27, 167, 27, 169, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^{5/2}}{(3 x+2)^{15/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{39} \int \frac {(7-80 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{2 (3 x+2)^{13/2}}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{39} \int \frac {(7-80 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{(3 x+2)^{13/2}}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{39} \left (\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac {2}{33} \int -\frac {(2818-3645 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^{11/2}}dx\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \int \frac {(2818-3645 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} (3 x+2)^{11/2}}dx+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \left (\frac {2}{189} \int \frac {3 (100989-120325 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} (3 x+2)^{9/2}}dx-\frac {10496 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{63} \int \frac {(100989-120325 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^{9/2}}dx-\frac {10496 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{63} \left (\frac {2}{147} \int \frac {4600522-2298775 x}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {1087234 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {10496 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{63} \left (\frac {1}{147} \int \frac {4600522-2298775 x}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {1087234 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {10496 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {2}{35} \int \frac {3 (170962787-183991160 x)}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {36798232 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1087234 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {10496 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {3}{35} \int \frac {170962787-183991160 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {36798232 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1087234 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {10496 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \int \frac {7283867069-4404353405 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {1761741362 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {36798232 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1087234 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {10496 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {2}{7} \int \frac {5 (61320616034 x+38825045767)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {61320616034 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {1761741362 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {36798232 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1087234 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {10496 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \int \frac {61320616034 x+38825045767}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {61320616034 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {1761741362 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {36798232 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1087234 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {10496 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {10163380733}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {61320616034}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {61320616034 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {1761741362 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {36798232 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1087234 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {10496 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {10163380733}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {61320616034}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {61320616034 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {1761741362 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {36798232 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1087234 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {10496 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{39} \left (\frac {2}{33} \left (\frac {1}{63} \left (\frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (-\frac {1847887406}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {61320616034}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {61320616034 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {1761741362 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {36798232 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1087234 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )-\frac {10496 \sqrt {1-2 x} (5 x+3)^{3/2}}{63 (3 x+2)^{9/2}}\right )+\frac {362 \sqrt {1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}\) |
(-2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(39*(2 + 3*x)^(13/2)) + ((362*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) + (2*((-10496*Sqrt[1 - 2*x]* (3 + 5*x)^(3/2))/(63*(2 + 3*x)^(9/2)) + ((-1087234*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(147*(2 + 3*x)^(7/2)) + ((36798232*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35* (2 + 3*x)^(5/2)) + (3*((1761741362*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3 *x)^(3/2)) + (2*((61320616034*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x ]) + (5*((-61320616034*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x] ], 35/33])/5 - (1847887406*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5))/7))/21))/35)/147)/63))/33)/39
3.28.28.3.1 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 = 1.29 (sec) , antiderivative size = 350, normalized size of antiderivative = 1.25
| method | result | size |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {1946 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8444007 \left (\frac {2}{3}+x \right )^{6}}-\frac {145118 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{59108049 \left (\frac {2}{3}+x \right )^{5}}+\frac {3126842 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{965431467 \left (\frac {2}{3}+x \right )^{4}}+\frac {73596464 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{11263367115 \left (\frac {2}{3}+x \right )^{3}}+\frac {3523482724 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{26281189935 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {490564928272}{4088185101} x^{2}-\frac {245282464136}{20440925505} x +\frac {245282464136}{6813641835}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {310600366136 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{429259435605 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {490564928272 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{429259435605 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2302911 \left (\frac {2}{3}+x \right )^{7}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(350\) |
| default | \(-\frac {2 \left (86836839271776 \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}-89405458177572 \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}+347347357087104 \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}-357621832710288 \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}+578912261811840 \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}-596036387850480 \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}+514588677166080 \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}-529810122533760 \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}+257294338583040 \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}-264905061266880 \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}-2682163745327160 x^{8}+68611823622144 \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}-70641349671168 \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}-11086773017544216 x^{7}+7623535958016 \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 )-7849038852352 \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 )-18462351947377842 x^{6}-14880670165585224 x^{5}-4403137275106857 x^{4}+1855445492717208 x^{3}+1998778232441424 x^{2}+639405497204220 x +74455894574127\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{61322776515 \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)*(1946/8444007*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^6-145 118/59108049*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^5+3126842/965431467*(-30 *x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+73596464/11263367115*(-30*x^3-23*x^2+7* x+6)^(1/2)/(2/3+x)^3+3523482724/26281189935*(-30*x^3-23*x^2+7*x+6)^(1/2)/( 2/3+x)^2+245282464136/61322776515*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9) )^(1/2)+310600366136/429259435605*(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))+490564928272/429259435605*(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+15*x)^(1/2),1/35*70^(1/2)))-14/2302911*(-30*x^3-23 *x^2+7*x+6)^(1/2)/(2/3+x)^7)
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 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\frac {2 \, {\left (135 \, {\left (89405458177572 \, x^{6} + 360618554767050 \, x^{5} + 606171513555828 \, x^{4} + 543590753927373 \, x^{3} + 274263621177573 \, x^{2} + 73802680969881 \, x + 8272877174903\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 2083879950248 \, \sqrt {-30} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 5518855443060 \, \sqrt {-30} {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{2759524943175 \, {\left (2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128\right )}} \]
2/2759524943175*(135*(89405458177572*x^6 + 360618554767050*x^5 + 606171513 555828*x^4 + 543590753927373*x^3 + 274263621177573*x^2 + 73802680969881*x + 8272877174903)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 208387995024 8*sqrt(-30)*(2187*x^7 + 10206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 60 48*x^2 + 1344*x + 128)*weierstrassPInverse(1159/675, 38998/91125, x + 23/9 0) + 5518855443060*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, 38998/91 125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(2187*x^7 + 1 0206*x^6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128)
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {15}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {15}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{15/2}} \,d x \]