Integrand size = 28, antiderivative size = 187 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \sqrt {1-2 x}}{7 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {428 \sqrt {1-2 x}}{49 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {94420 \sqrt {1-2 x} \sqrt {2+3 x}}{1617 (3+5 x)^{3/2}}+\frac {6277760 \sqrt {1-2 x} \sqrt {2+3 x}}{17787 \sqrt {3+5 x}}-\frac {1255552 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{539 \sqrt {33}}-\frac {37768 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{539 \sqrt {33}} \]
-1255552/17787*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1 /2)-37768/17787*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^( 1/2)+2/7*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2)+428/49*(1-2*x)^(1/2)/(3 +5*x)^(3/2)/(2+3*x)^(1/2)-94420/1617*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^( 3/2)+6277760/17787*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.81 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.52 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \left (\frac {\sqrt {1-2 x} \left (35747225+169778606 x+268408770 x^2+141249600 x^3\right )}{(2+3 x)^{3/2} (3+5 x)^{3/2}}+4 i \sqrt {33} \left (156944 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-161665 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{17787} \]
(2*((Sqrt[1 - 2*x]*(35747225 + 169778606*x + 268408770*x^2 + 141249600*x^3 ))/((2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (4*I)*Sqrt[33]*(156944*EllipticE[I* ArcSinh[Sqrt[9 + 15*x]], -2/33] - 161665*EllipticF[I*ArcSinh[Sqrt[9 + 15*x ]], -2/33])))/17787
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.13, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {115, 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.
| \(\displaystyle \int \frac {1}{\sqrt {1-2 x} (3 x+2)^{5/2} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {2}{21} \int \frac {3 (19-25 x)}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{7} \int \frac {19-25 x}{\sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{7} \left (\frac {2}{7} \int \frac {5 (559-642 x)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {214 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{7} \left (\frac {5}{7} \int \frac {559-642 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {214 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{7} \left (\frac {5}{7} \left (-\frac {2}{33} \int \frac {22891-14163 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {9442 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {214 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{7} \left (\frac {5}{7} \left (-\frac {2}{33} \left (-\frac {2}{11} \int \frac {3 (313888 x+198719)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {313888 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {9442 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {214 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{7} \left (\frac {5}{7} \left (-\frac {2}{33} \left (-\frac {3}{11} \int \frac {313888 x+198719}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {313888 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {9442 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {214 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2}{7} \left (\frac {5}{7} \left (-\frac {2}{33} \left (-\frac {3}{11} \left (\frac {51931}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {313888}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {313888 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {9442 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {214 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2}{7} \left (\frac {5}{7} \left (-\frac {2}{33} \left (-\frac {3}{11} \left (\frac {51931}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {313888}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {313888 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {9442 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {214 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {2}{7} \left (\frac {5}{7} \left (-\frac {2}{33} \left (-\frac {3}{11} \left (-\frac {9442}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {313888}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {313888 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {9442 \sqrt {1-2 x} \sqrt {3 x+2}}{33 (5 x+3)^{3/2}}\right )+\frac {214 \sqrt {1-2 x}}{7 \sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {2 \sqrt {1-2 x}}{7 (3 x+2)^{3/2} (5 x+3)^{3/2}}\) |
(2*Sqrt[1 - 2*x])/(7*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (2*((214*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + (5*((-9442*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(33*(3 + 5*x)^(3/2)) - (2*((-313888*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/( 11*Sqrt[3 + 5*x]) - (3*((-313888*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqr t[1 - 2*x]], 35/33])/5 - (9442*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[ 1 - 2*x]], 35/33])/5))/11))/33))/7))/7
3.29.82.3.1 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 = 1.33 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.22
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {1294}{51975}-\frac {136 x}{3465}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right )^{2}}-\frac {2 \left (15-30 x \right ) \left (-\frac {5966174}{266805}-\frac {627776 x}{17787}\right )}{\sqrt {\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right ) \left (15-30 x \right )}}+\frac {1589752 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{124509 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2511104 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{124509 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(228\) |
| default | \(-\frac {2 \sqrt {1-2 x}\, \left (9145620 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-9416640 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+11584452 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-11927744 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+3658248 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-3766656 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-282499200 x^{4}-395567940 x^{3}-71148442 x^{2}+98284156 x +35747225\right )}{17787 \left (2+3 x \right )^{\frac {3}{2}} \left (3+5 x \right )^{\frac {3}{2}} \left (-1+2 x \right )}\) | \(311\) |
(-(-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 )*((-1294/51975-136/3465*x)*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x^2+19/15*x+2/5) ^2-2*(15-30*x)*(-5966174/266805-627776/17787*x)/((x^2+19/15*x+2/5)*(15-30* x))^(1/2)+1589752/124509*(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))+25111 04/124509*(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 = 0.08 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.68 \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (45 \, {\left (141249600 \, x^{3} + 268408770 \, x^{2} + 169778606 \, x + 35747225\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 10665286 \, \sqrt {-30} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 28249920 \, \sqrt {-30} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{800415 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \]
2/800415*(45*(141249600*x^3 + 268408770*x^2 + 169778606*x + 35747225)*sqrt (5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 10665286*sqrt(-30)*(225*x^4 + 570 *x^3 + 541*x^2 + 228*x + 36)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 28249920*sqrt(-30)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*w eierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/ 91125, x + 23/90)))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\text {Timed out} \]
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
\[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
Timed out. \[ \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}} \,d x \]