Integrand size = 28, antiderivative size = 280 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\frac {4}{231 (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {632}{5929 \sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}}-\frac {3606 \sqrt {1-2 x}}{207515 (2+3 x)^{5/2} (3+5 x)^{3/2}}+\frac {649224 \sqrt {1-2 x}}{1452605 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {140700876 \sqrt {1-2 x}}{10168235 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {6208896328 \sqrt {1-2 x} \sqrt {2+3 x}}{67110351 (3+5 x)^{3/2}}+\frac {412810345784 \sqrt {1-2 x} \sqrt {2+3 x}}{738213861 \sqrt {3+5 x}}-\frac {412810345784 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{111850585 \sqrt {33}}-\frac {12417792656 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{111850585 \sqrt {33}} \]
4/231/(1-2*x)^(3/2)/(2+3*x)^(5/2)/(3+5*x)^(3/2)-412810345784/3691069305*El lipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-12417792656/3 691069305*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+6 32/5929/(2+3*x)^(5/2)/(3+5*x)^(3/2)/(1-2*x)^(1/2)-3606/207515*(1-2*x)^(1/2 )/(2+3*x)^(5/2)/(3+5*x)^(3/2)+649224/1452605*(1-2*x)^(1/2)/(2+3*x)^(3/2)/( 3+5*x)^(3/2)+140700876/10168235*(1-2*x)^(1/2)/(3+5*x)^(3/2)/(2+3*x)^(1/2)- 6208896328/67110351*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)+412810345784 /738213861*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.67 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.40 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {23506658680609+52875828155808 x-149619576926754 x^2-430611138612568 x^3+84649478011164 x^4+873229924799280 x^5+557293966808400 x^6}{(1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{3/2}}+4 i \sqrt {33} \left (51601293223 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-53153517305 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{3691069305} \]
(2*((23506658680609 + 52875828155808*x - 149619576926754*x^2 - 43061113861 2568*x^3 + 84649478011164*x^4 + 873229924799280*x^5 + 557293966808400*x^6) /((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (4*I)*Sqrt[33]*(51601 293223*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 53153517305*EllipticF [I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/3691069305
Time = 0.34 (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 = {115, 27, 169, 27, 169, 169, 27, 169, 27, 169, 27, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{5/2} (3 x+2)^{7/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac {2}{231} \int -\frac {3 (110 x+103)}{2 (1-2 x)^{3/2} (3 x+2)^{7/2} (5 x+3)^{5/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \int \frac {110 x+103}{(1-2 x)^{3/2} (3 x+2)^{7/2} (5 x+3)^{5/2}}dx+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}-\frac {2}{77} \int -\frac {42660 x+27839}{2 \sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}}dx\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \int \frac {42660 x+27839}{\sqrt {1-2 x} (3 x+2)^{7/2} (5 x+3)^{5/2}}dx+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{35} \int \frac {63105 x+204376}{\sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}}dx-\frac {3606 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {2}{21} \int \frac {3 (6314873-8115300 x)}{2 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {324612 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )-\frac {3606 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \int \frac {6314873-8115300 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {324612 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )-\frac {3606 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {2}{7} \int \frac {5 (91907014-105525657 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {70350438 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {324612 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )-\frac {3606 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \int \frac {91907014-105525657 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {70350438 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {324612 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )-\frac {3606 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {2}{33} \int \frac {7526255297-4656672246 x}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {1552224082 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {70350438 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {324612 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )-\frac {3606 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (-\frac {1}{33} \int \frac {7526255297-4656672246 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {1552224082 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {70350438 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {324612 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )-\frac {3606 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {2}{11} \int \frac {3 (51601293223 x+32668222424)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {103202586446 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {1552224082 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {70350438 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {324612 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )-\frac {3606 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \int \frac {51601293223 x+32668222424}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {103202586446 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {1552224082 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {70350438 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {324612 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )-\frac {3606 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \left (\frac {8537232451}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {51601293223}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {103202586446 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {1552224082 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {70350438 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {324612 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )-\frac {3606 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \left (\frac {8537232451}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {51601293223}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {103202586446 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {1552224082 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {70350438 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {324612 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )-\frac {3606 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{77} \left (\frac {1}{77} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {1}{33} \left (\frac {6}{11} \left (-\frac {1552224082}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {51601293223}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {103202586446 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {1552224082 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {70350438 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {324612 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\right )-\frac {3606 \sqrt {1-2 x}}{35 (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {632}{77 \sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{3/2}}\right )+\frac {4}{231 (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{3/2}}\) |
4/(231*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (632/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + ((-3606*Sqrt[1 - 2*x])/(35*(2 + 3*x)^(5/2)*(3 + 5*x)^(3/2)) + (2*((324612*Sqrt[1 - 2*x])/(7*(2 + 3*x)^(3/2 )*(3 + 5*x)^(3/2)) + ((70350438*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*(3 + 5*x)^ (3/2)) + (10*((-1552224082*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(33*(3 + 5*x)^(3/2 )) + ((103202586446*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) + (6*( (-51601293223*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33] )/5 - (1552224082*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35 /33])/5))/11)/33))/7)/7))/35)/77)/77
3.30.100.3.1 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[b*(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] + Simp[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*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2 *n, 2*p]
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 + 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 = 1.40 (sec) , antiderivative size = 304, normalized size of antiderivative = 1.09
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {7503029}{43578150}+\frac {1500641 x}{4357815}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right )^{2}}-\frac {2 \left (-20-30 x \right ) \left (\frac {92281045511}{7382138610}-\frac {18225070049 x}{738213861}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}+\frac {18 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{3}}+\frac {1332 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {31495068}{16807} x^{2}-\frac {15747534}{84035} x +\frac {47242602}{84035}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {522691558784 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{25837485135 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {825620691568 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{25837485135 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(304\) |
| default | \(\frac {2 \sqrt {1-2 x}\, \left (18576465560280 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-18041822445240 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+26626267303068 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-25859945504844 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+5160129322300 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-5011617345900 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-6604965532544 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+6414870202752 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-2476862074704 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+2405576326032 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+557293966808400 x^{6}+873229924799280 x^{5}+84649478011164 x^{4}-430611138612568 x^{3}-149619576926754 x^{2}+52875828155808 x +23506658680609\right )}{3691069305 \left (2+3 x \right )^{\frac {5}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )^{2}}\) | \(501\) |
(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 )*((-7503029/43578150+1500641/4357815*x)*(-30*x^3-23*x^2+7*x+6)^(1/2)/(-3/ 10+x^2+1/10*x)^2-2*(-20-30*x)*(92281045511/7382138610-18225070049/73821386 1*x)/((-3/10+x^2+1/10*x)*(-20-30*x))^(1/2)+18/1715*(-30*x^3-23*x^2+7*x+6)^ (1/2)/(2/3+x)^3+1332/1715*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+5249178/8 4035*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+522691558784/25837485 135*(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))+825620691568/25837485135*( 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))))
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 \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (557293966808400 \, x^{6} + 873229924799280 \, x^{5} + 84649478011164 \, x^{4} - 430611138612568 \, x^{3} - 149619576926754 \, x^{2} + 52875828155808 \, x + 23506658680609\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 3506620548062 \, \sqrt {-30} {\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 9288232780140 \, \sqrt {-30} {\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{166098118725 \, {\left (2700 \, x^{7} + 5940 \, x^{6} + 3087 \, x^{5} - 1828 \, x^{4} - 2045 \, x^{3} - 202 \, x^{2} + 276 \, x + 72\right )}} \]
2/166098118725*(45*(557293966808400*x^6 + 873229924799280*x^5 + 8464947801 1164*x^4 - 430611138612568*x^3 - 149619576926754*x^2 + 52875828155808*x + 23506658680609)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 3506620548062 *sqrt(-30)*(2700*x^7 + 5940*x^6 + 3087*x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 + 276*x + 72)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 928 8232780140*sqrt(-30)*(2700*x^7 + 5940*x^6 + 3087*x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 + 276*x + 72)*weierstrassZeta(1159/675, 38998/91125, weierstras sPInverse(1159/675, 38998/91125, x + 23/90)))/(2700*x^7 + 5940*x^6 + 3087* x^5 - 1828*x^4 - 2045*x^3 - 202*x^2 + 276*x + 72)
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {1}{(1-2 x)^{5/2} (2+3 x)^{7/2} (3+5 x)^{5/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]