Integrand size = 28, antiderivative size = 127 \[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {74}{125} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {3}{25} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {5161 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1250}-\frac {857 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{625 \sqrt {33}} \]
-5161/3750*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)- 857/20625*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-3 /25*(2+3*x)^(3/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-74/125*(1-2*x)^(1/2)*(2+3*x) ^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 3.30 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.73 \[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\frac {-330 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} (104+45 x)+56771 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-58485 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{41250} \]
(-330*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(104 + 45*x) + (56771*I)*S qrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (58485*I)*Sqrt[33]*E llipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/41250
Time = 0.23 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.08, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {113, 27, 171, 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.
| \(\displaystyle \int \frac {(3 x+2)^{5/2}}{\sqrt {1-2 x} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 113 |
| \(\displaystyle -\frac {1}{25} \int -\frac {\sqrt {3 x+2} (444 x+275)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {3}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{50} \int \frac {\sqrt {3 x+2} (444 x+275)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {3}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{50} \left (-\frac {1}{15} \int -\frac {3 (5161 x+3268)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {148}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {3}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{50} \left (\frac {1}{5} \int \frac {5161 x+3268}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {148}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {3}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{50} \left (\frac {1}{5} \left (\frac {857}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {5161}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {148}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {3}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{50} \left (\frac {1}{5} \left (\frac {857}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5161}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {148}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {3}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{50} \left (\frac {1}{5} \left (-\frac {1714 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5 \sqrt {33}}-\frac {5161}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {148}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {3}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\) |
(-3*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/25 + ((-148*Sqrt[1 - 2*x] *Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5 + ((-5161*Sqrt[11/3]*EllipticE[ArcSin[Sqrt [3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (1714*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5*Sqrt[33]))/5)/50
3.29.60.3.1 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 )/(d*f*(m + n + p + 1))), x] + Simp[1/(d*f*(m + n + p + 1)) Int[(a + b*x) ^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] & & GtQ[m, 1] && NeQ[m + n + p + 1, 0] && 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[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 ) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 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]
Time = 1.36 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.14
| method | result | size |
| default | \(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (5013 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-5161 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+40500 x^{4}+124650 x^{3}+62310 x^{2}-29940 x -18720\right )}{3750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(145\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {9 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{25}-\frac {104 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{125}+\frac {3268 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{13125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5161 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{13125 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(206\) |
| risch | \(\frac {\left (104+45 x \right ) \left (-1+2 x \right ) \sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{125 \sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}+\frac {\left (\frac {1634 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{6875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {5161 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {E\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 F\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{13750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(246\) |
-1/3750*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(5013*5^(1/2)*(2+3*x)^(1 /2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70 ^(1/2))-5161*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1/2)*El lipticE((10+15*x)^(1/2),1/35*70^(1/2))+40500*x^4+124650*x^3+62310*x^2-2994 0*x-18720)/(30*x^3+23*x^2-7*x-6)
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.07 (sec) , antiderivative size = 54, normalized size of antiderivative = 0.43 \[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=-\frac {1}{125} \, {\left (45 \, x + 104\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {175417}{337500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {5161}{3750} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \]
-1/125*(45*x + 104)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 175417/33 7500*sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 516 1/3750*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInvers e(1159/675, 38998/91125, x + 23/90))
\[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\int \frac {\left (3 x + 2\right )^{\frac {5}{2}}}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \]
\[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}} \,d x } \]
\[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{5/2}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx=\int \frac {{\left (3\,x+2\right )}^{5/2}}{\sqrt {1-2\,x}\,\sqrt {5\,x+3}} \,d x \]