Integrand size = 28, antiderivative size = 158 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{35 (2+3 x)^{5/2}}+\frac {18 \sqrt {1-2 x} \sqrt {3+5 x}}{245 (2+3 x)^{3/2}}+\frac {1752 \sqrt {1-2 x} \sqrt {3+5 x}}{1715 \sqrt {2+3 x}}-\frac {584 \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1715}-\frac {68 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1715} \]
-68/5145*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-58 4/1715*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/35 *(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+18/245*(1-2*x)^(1/2)*(3+5*x)^(1 /2)/(2+3*x)^(3/2)+1752/1715*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 3.32 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.61 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (3581+10701 x+7884 x^2\right )}{2 (2+3 x)^{5/2}}+i \sqrt {33} \left (438 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-455 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{5145} \]
(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(3581 + 10701*x + 7884*x^2))/(2*(2 + 3* x)^(5/2)) + I*Sqrt[33]*(438*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 455*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/5145
Time = 0.25 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {110, 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.
| \(\displaystyle \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^{7/2}} \, dx\) |
| \(\Big \downarrow \) 110 |
| \(\displaystyle \frac {2}{35} \int \frac {30 x+29}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} \int \frac {30 x+29}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{21} \int \frac {3 (116-45 x)}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{7} \int \frac {116-45 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{7} \left (\frac {2}{7} \int \frac {5 (876 x+563)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \int \frac {876 x+563}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \left (\frac {187}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {876}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \left (\frac {187}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {292}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{35} \left (\frac {2}{7} \left (\frac {5}{7} \left (-\frac {34}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {292}{5} \sqrt {33} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {876 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {18 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\) |
(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35*(2 + 3*x)^(5/2)) + ((18*Sqrt[1 - 2*x] *Sqrt[3 + 5*x])/(7*(2 + 3*x)^(3/2)) + (2*((876*Sqrt[1 - 2*x]*Sqrt[3 + 5*x] )/(7*Sqrt[2 + 3*x]) + (5*((-292*Sqrt[33]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (34*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2 *x]], 35/33])/5))/7))/7)/35
3.29.21.3.1 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 + 1)/((m + 1)*(b*e - a*f))), x] - Simp[1/((m + 1)*(b*e - a*f)) Int[(a + b*x)^(m + 1) *(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && Gt Q[n, 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 + 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]
Leaf count of result is larger than twice the leaf count of optimal. \(242\) vs. \(2(116)=232\).
Time = 1.38 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.54
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{945 \left (\frac {2}{3}+x \right )^{3}}+\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{245 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {3504}{343} x^{2}-\frac {1752}{1715} x +\frac {5256}{1715}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {2252 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{36015 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1168 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{12005 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(243\) |
| default | \(-\frac {2 \left (2574 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-2628 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+3432 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-3504 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+1144 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-1168 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-78840 x^{4}-114894 x^{3}-22859 x^{2}+28522 x +10743\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{1715 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {5}{2}}}\) | \(314\) |
(-(-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 )*(-2/945*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+2/245*(-30*x^3-23*x^2+7*x +6)^(1/2)/(2/3+x)^2+584/1715*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/ 2)+2252/36015*(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))+1168/12005*(10+1 5*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 = 108, normalized size of antiderivative = 0.68 \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\frac {2 \, {\left (45 \, {\left (7884 \, x^{2} + 10701 \, x + 3581\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 5087 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 13140 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{77175 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
2/77175*(45*(7884*x^2 + 10701*x + 3581)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(- 2*x + 1) - 5087*sqrt(-30)*(27*x^3 + 54*x^2 + 36*x + 8)*weierstrassPInverse (1159/675, 38998/91125, x + 23/90) + 13140*sqrt(-30)*(27*x^3 + 54*x^2 + 36 *x + 8)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/67 5, 38998/91125, x + 23/90)))/(27*x^3 + 54*x^2 + 36*x + 8)
\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\int \frac {\sqrt {5 x + 3}}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{\frac {7}{2}}}\, dx \]
\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
\[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3}}{{\left (3 \, x + 2\right )}^{\frac {7}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
Timed out. \[ \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx=\int \frac {\sqrt {5\,x+3}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{7/2}} \,d x \]