Integrand size = 28, antiderivative size = 160 \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {37 \sqrt {1-2 x} (2+3 x)^{3/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{5/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {2388 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{3025}+\frac {55019 \sqrt {\frac {3}{11}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2750}+\frac {823 \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1375} \]
55019/30250*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2) +823/15125*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+ 7/11*(2+3*x)^(5/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)-37/605*(2+3*x)^(3/2)*(1-2*x )^(1/2)/(3+5*x)^(1/2)+2388/3025*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)
Result contains complex when optimal does not.
Time = 7.16 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.77 \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {10 \sqrt {2+3 x} \sqrt {3+5 x} \left (14494+20897 x-5445 x^2\right )-55019 i \sqrt {33-66 x} (3+5 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+56665 i \sqrt {33-66 x} (3+5 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{30250 \sqrt {1-2 x} (3+5 x)} \]
(10*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(14494 + 20897*x - 5445*x^2) - (55019*I)*S qrt[33 - 66*x]*(3 + 5*x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (56 665*I)*Sqrt[33 - 66*x]*(3 + 5*x)*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/3 3])/(30250*Sqrt[1 - 2*x]*(3 + 5*x))
Time = 0.24 (sec) , antiderivative size = 175, 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 = {109, 27, 167, 27, 171, 25, 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.
| \(\displaystyle \int \frac {(3 x+2)^{7/2}}{(1-2 x)^{3/2} (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {1}{11} \int \frac {(3 x+2)^{3/2} (414 x+241)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {1}{22} \int \frac {(3 x+2)^{3/2} (414 x+241)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (-\frac {2}{55} \int \frac {9 \sqrt {3 x+2} (1592 x+975)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {74 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (-\frac {9}{55} \int \frac {\sqrt {3 x+2} (1592 x+975)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {74 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{22} \left (-\frac {9}{55} \left (-\frac {1}{15} \int -\frac {55019 x+34822}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1592}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {74 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{22} \left (-\frac {9}{55} \left (\frac {1}{15} \int \frac {55019 x+34822}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1592}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {74 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{22} \left (-\frac {9}{55} \left (\frac {1}{15} \left (\frac {9053}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {55019}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {1592}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {74 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{22} \left (-\frac {9}{55} \left (\frac {1}{15} \left (\frac {9053}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {55019}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1592}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {74 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{22} \left (-\frac {9}{55} \left (\frac {1}{15} \left (-\frac {1646}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {55019}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {1592}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {74 \sqrt {1-2 x} (3 x+2)^{3/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{5/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
(7*(2 + 3*x)^(5/2))/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + ((-74*Sqrt[1 - 2*x] *(2 + 3*x)^(3/2))/(55*Sqrt[3 + 5*x]) - (9*((-1592*Sqrt[1 - 2*x]*Sqrt[2 + 3 *x]*Sqrt[3 + 5*x])/15 + ((-55019*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqr t[1 - 2*x]], 35/33])/5 - (1646*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[ 1 - 2*x]], 35/33])/5)/15))/55)/22
3.30.27.3.1 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[((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 = 1.35 (sec) , antiderivative size = 140, normalized size of antiderivative = 0.88
| method | result | size |
| default | \(\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (53427 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-55019 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )+163350 x^{3}-518010 x^{2}-852760 x -289880\right )}{907500 x^{3}+695750 x^{2}-211750 x -181500}\) | \(140\) |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (\frac {25721}{121000}+\frac {42883 x}{121000}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}+\frac {9 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{50}-\frac {34822 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{105875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {55019 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{105875 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(214\) |
1/30250*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(53427*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*7 0^(1/2))-55019*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))+163350*x^3-518010*x^2-852760*x-28 9880)/(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 = 87, normalized size of antiderivative = 0.54 \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {900 \, {\left (5445 \, x^{2} - 20897 \, x - 14494\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 1868543 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 4951710 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{2722500 \, {\left (10 \, x^{2} + x - 3\right )}} \]
1/2722500*(900*(5445*x^2 - 20897*x - 14494)*sqrt(5*x + 3)*sqrt(3*x + 2)*sq rt(-2*x + 1) + 1868543*sqrt(-30)*(10*x^2 + x - 3)*weierstrassPInverse(1159 /675, 38998/91125, x + 23/90) - 4951710*sqrt(-30)*(10*x^2 + x - 3)*weierst rassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(10*x^2 + x - 3)
Timed out. \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(2+3 x)^{7/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{7/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]