Integrand size = 28, antiderivative size = 125 \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=-\frac {448 \sqrt {2+3 x} \sqrt {3+5 x}}{363 \sqrt {1-2 x}}+\frac {7 (2+3 x)^{3/2} \sqrt {3+5 x}}{33 (1-2 x)^{3/2}}-\frac {4451 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{110 \sqrt {33}}-\frac {67 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{55 \sqrt {33}} \]
-4451/3630*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)- 67/1815*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+7/3 3*(2+3*x)^(3/2)*(3+5*x)^(1/2)/(1-2*x)^(3/2)-448/363*(2+3*x)^(1/2)*(3+5*x)^ (1/2)/(1-2*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.51 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.89 \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {490 \sqrt {2+3 x} \sqrt {3+5 x} (-6+23 x)-4451 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+4585 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{3630 (1-2 x)^{3/2}} \]
(490*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-6 + 23*x) - (4451*I)*Sqrt[33 - 66*x]*(- 1 + 2*x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (4585*I)*Sqrt[33 - 66*x]*(-1 + 2*x)*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(3630*(1 - 2 *x)^(3/2))
Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.11, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {109, 27, 167, 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.
| \(\displaystyle \int \frac {(3 x+2)^{5/2}}{(1-2 x)^{5/2} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^{3/2} \sqrt {5 x+3}}{33 (1-2 x)^{3/2}}-\frac {1}{33} \int \frac {\sqrt {3 x+2} (402 x+247)}{2 (1-2 x)^{3/2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^{3/2} \sqrt {5 x+3}}{33 (1-2 x)^{3/2}}-\frac {1}{66} \int \frac {\sqrt {3 x+2} (402 x+247)}{(1-2 x)^{3/2} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{66} \left (-\frac {1}{11} \int -\frac {3 (4451 x+2818)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {896 \sqrt {3 x+2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{3/2}}{33 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{66} \left (\frac {3}{11} \int \frac {4451 x+2818}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {896 \sqrt {3 x+2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{3/2}}{33 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{66} \left (\frac {3}{11} \left (\frac {737}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {4451}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {896 \sqrt {3 x+2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{3/2}}{33 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{66} \left (\frac {3}{11} \left (\frac {737}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4451}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {896 \sqrt {3 x+2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{3/2}}{33 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{66} \left (\frac {3}{11} \left (-\frac {134}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {4451}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {896 \sqrt {3 x+2} \sqrt {5 x+3}}{11 \sqrt {1-2 x}}\right )+\frac {7 \sqrt {5 x+3} (3 x+2)^{3/2}}{33 (1-2 x)^{3/2}}\) |
(7*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/(33*(1 - 2*x)^(3/2)) + ((-896*Sqrt[2 + 3 *x]*Sqrt[3 + 5*x])/(11*Sqrt[1 - 2*x]) + (3*((-4451*Sqrt[11/3]*EllipticE[Ar cSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (134*Sqrt[11/3]*EllipticF[ArcSi n[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5))/11)/66
3.30.73.3.1 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*c - a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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[((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. \(218\) vs. \(2(93)=186\).
Time = 1.39 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.75
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {49 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{264 \left (x -\frac {1}{2}\right )^{2}}+\frac {-\frac {5635}{242} x^{2}-\frac {21413}{726} x -\frac {1127}{121}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {2818 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{12705 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {4451 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{12705 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(219\) |
| default | \(-\frac {\left (8646 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-8902 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-4323 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+4451 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-169050 x^{3}-170030 x^{2}-11760 x +17640\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}}{3630 \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\) | \(228\) |
(-(-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 )*(49/264*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x-1/2)^2+1127/1452*(-30*x^2-38*x-1 2)/((x-1/2)*(-30*x^2-38*x-12))^(1/2)+2818/12705*(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))+4451/12705*(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 = 88, normalized size of antiderivative = 0.70 \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\frac {44100 \, {\left (23 \, x - 6\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 151247 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 400590 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{326700 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \]
1/326700*(44100*(23*x - 6)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 15 1247*sqrt(-30)*(4*x^2 - 4*x + 1)*weierstrassPInverse(1159/675, 38998/91125 , x + 23/90) + 400590*sqrt(-30)*(4*x^2 - 4*x + 1)*weierstrassZeta(1159/675 , 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(4* x^2 - 4*x + 1)
\[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {\left (3 x + 2\right )^{\frac {5}{2}}}{\left (1 - 2 x\right )^{\frac {5}{2}} \sqrt {5 x + 3}}\, dx \]
\[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
\[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{5/2}}{(1-2 x)^{5/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}} \,d x \]