Integrand size = 28, antiderivative size = 186 \[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {29293}{875} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2517}{350} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}+\frac {12}{7} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}+\frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{\sqrt {1-2 x}}+\frac {4071079 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{17500}+\frac {673523 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{8750 \sqrt {33}} \]
4071079/52500*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/ 2)+673523/288750*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^ (1/2)+(2+3*x)^(7/2)*(3+5*x)^(1/2)/(1-2*x)^(1/2)+2517/350*(2+3*x)^(3/2)*(1- 2*x)^(1/2)*(3+5*x)^(1/2)+12/7*(2+3*x)^(5/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)+29 293/875*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 6.52 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.60 \[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {-330 \sqrt {2+3 x} \sqrt {3+5 x} \left (-109756+54757 x+26010 x^2+6750 x^3\right )-44781869 i \sqrt {33-66 x} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+46128915 i \sqrt {33-66 x} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{577500 \sqrt {1-2 x}} \]
(-330*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-109756 + 54757*x + 26010*x^2 + 6750*x^ 3) - (44781869*I)*Sqrt[33 - 66*x]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/ 33] + (46128915*I)*Sqrt[33 - 66*x]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2 /33])/(577500*Sqrt[1 - 2*x])
Time = 0.27 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.11, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {108, 27, 171, 27, 171, 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.
| \(\displaystyle \int \frac {(3 x+2)^{7/2} \sqrt {5 x+3}}{(1-2 x)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(3 x+2)^{7/2} \sqrt {5 x+3}}{\sqrt {1-2 x}}-\int \frac {(3 x+2)^{5/2} (120 x+73)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^{7/2} \sqrt {5 x+3}}{\sqrt {1-2 x}}-\frac {1}{2} \int \frac {(3 x+2)^{5/2} (120 x+73)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{35} \int -\frac {5 (3 x+2)^{3/2} (2517 x+1538)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {24}{7} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {24}{7} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}-\frac {1}{7} \int \frac {(3 x+2)^{3/2} (2517 x+1538)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{25} \int -\frac {\sqrt {3 x+2} (351516 x+216725)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {2517}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {24}{7} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {2517}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}-\frac {1}{50} \int \frac {\sqrt {3 x+2} (351516 x+216725)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )+\frac {24}{7} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{50} \left (\frac {1}{15} \int -\frac {3 (4071079 x+2577352)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {117172}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2517}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {24}{7} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{50} \left (\frac {117172}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {1}{5} \int \frac {4071079 x+2577352}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {2517}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {24}{7} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{50} \left (\frac {1}{5} \left (-\frac {673523}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4071079}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {117172}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2517}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {24}{7} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{50} \left (\frac {1}{5} \left (\frac {4071079}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {673523}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {117172}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2517}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {24}{7} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{2} \left (\frac {1}{7} \left (\frac {1}{50} \left (\frac {1}{5} \left (\frac {1347046 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5 \sqrt {33}}+\frac {4071079}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {117172}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2517}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {24}{7} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )+\frac {\sqrt {5 x+3} (3 x+2)^{7/2}}{\sqrt {1-2 x}}\) |
((2 + 3*x)^(7/2)*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] + ((24*Sqrt[1 - 2*x]*(2 + 3* x)^(5/2)*Sqrt[3 + 5*x])/7 + ((2517*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x])/25 + ((117172*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5 + ((40710 79*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (1347 046*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5*Sqrt[33]))/5)/50 )/7)/2
3.29.91.3.1 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/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 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[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 = 4.56 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.81
| method | result | size |
| default | \(\frac {\sqrt {2+3 x}\, \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (3953907 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-4071079 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+3037500 x^{5}+15552000 x^{4}+40681350 x^{3}-13496910 x^{2}-52704660 x -19756080\right )}{1575000 x^{3}+1207500 x^{2}-367500 x -315000}\) | \(150\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {5877 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{700}+\frac {138899 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{7000}-\frac {1288676 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{91875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4071079 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{183750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {27 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{14}-\frac {343 \left (-30 x^{2}-38 x -12\right )}{16 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
1/52500*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(3953907*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))-4071079*5^(1/2)*(2+3*x)^(1/2)*7^(1/2)*(1-2*x)^(1/2)*(-3-5*x)^(1 /2)*EllipticE((10+15*x)^(1/2),1/35*70^(1/2))+3037500*x^5+15552000*x^4+4068 1350*x^3-13496910*x^2-52704660*x-19756080)/(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 = 83, normalized size of antiderivative = 0.45 \[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\frac {2700 \, {\left (6750 \, x^{3} + 26010 \, x^{2} + 54757 \, x - 109756\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 138326863 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 366397110 \, \sqrt {-30} {\left (2 \, x - 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{4725000 \, {\left (2 \, x - 1\right )}} \]
1/4725000*(2700*(6750*x^3 + 26010*x^2 + 54757*x - 109756)*sqrt(5*x + 3)*sq rt(3*x + 2)*sqrt(-2*x + 1) + 138326863*sqrt(-30)*(2*x - 1)*weierstrassPInv erse(1159/675, 38998/91125, x + 23/90) - 366397110*sqrt(-30)*(2*x - 1)*wei erstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91 125, x + 23/90)))/(2*x - 1)
Timed out. \[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\int { \frac {\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{7/2} \sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{7/2}\,\sqrt {5\,x+3}}{{\left (1-2\,x\right )}^{3/2}} \,d x \]