Integrand size = 28, antiderivative size = 129 \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=\frac {4 \sqrt {2+3 x}}{77 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {370 \sqrt {1-2 x} \sqrt {2+3 x}}{847 \sqrt {3+5 x}}+\frac {74}{121} \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {8}{121} \sqrt {\frac {5}{7}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:
4/77*(2+3*x)^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)-370/847*(1-2*x)^(1/2)*(2+3* x)^(1/2)/(3+5*x)^(1/2)+74/847*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1 155^(1/2))*35^(1/2)-8/847*EllipticF(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^ (1/2))*35^(1/2)
Result contains complex when optimal does not.
Time = 5.51 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.91 \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=\frac {2 \sqrt {2+3 x} \sqrt {3+5 x} (-163+370 x)-74 i \sqrt {33-66 x} (3+5 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+70 i \sqrt {33-66 x} (3+5 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{847 \sqrt {1-2 x} (3+5 x)} \] Input:
Integrate[1/((1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)),x]
Output:
(2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-163 + 370*x) - (74*I)*Sqrt[33 - 66*x]*(3 + 5*x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (70*I)*Sqrt[33 - 66*x ]*(3 + 5*x)*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(847*Sqrt[1 - 2*x ]*(3 + 5*x))
Time = 0.22 (sec) , antiderivative size = 139, 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 = {115, 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.
| \(\displaystyle \int \frac {1}{(1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 115 |
| \(\displaystyle \frac {4 \sqrt {3 x+2}}{77 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {2}{77} \int -\frac {5 (6 x+11)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{77} \int \frac {6 x+11}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {4 \sqrt {3 x+2}}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{77} \left (-\frac {2}{11} \int \frac {3 (37 x+20)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {74 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {4 \sqrt {3 x+2}}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{77} \left (-\frac {6}{11} \int \frac {37 x+20}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {74 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {4 \sqrt {3 x+2}}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {5}{77} \left (-\frac {6}{11} \left (\frac {37}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {11}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {74 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {4 \sqrt {3 x+2}}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {5}{77} \left (-\frac {6}{11} \left (-\frac {11}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {37}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {74 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {4 \sqrt {3 x+2}}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {5}{77} \left (-\frac {6}{11} \left (\frac {2}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {37}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {74 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {4 \sqrt {3 x+2}}{77 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
Input:
Int[1/((1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)),x]
Output:
(4*Sqrt[2 + 3*x])/(77*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + (5*((-74*Sqrt[1 - 2*x ]*Sqrt[2 + 3*x])/(11*Sqrt[3 + 5*x]) - (6*((-37*Sqrt[11/3]*EllipticE[ArcSin [Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (2*Sqrt[11/3]*EllipticF[ArcSin[Sqrt [3/7]*Sqrt[1 - 2*x]], 35/33])/5))/11))/77
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 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[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*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] && 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[(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]
Time = 0.62 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.03
| method | result | size |
| default | \(-\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (33 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+37 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1110 x^{2}+251 x -326\right )}{847 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(133\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (-\frac {163}{8470}+\frac {37 x}{847}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}-\frac {200 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5929 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {370 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{5929 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(201\) |
Input:
int(1/(1-2*x)^(3/2)/(2+3*x)^(1/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-2/847*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(33*2^(1/2)*(2+3*x)^(1/2) *(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+ 37*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42 *x)^(1/2),1/2*70^(1/2))+1110*x^2+251*x-326)/(30*x^3+23*x^2-7*x-6)
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.64 \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=-\frac {90 \, {\left (370 \, x - 163\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 949 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 3330 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{38115 \, {\left (10 \, x^{2} + x - 3\right )}} \] Input:
integrate(1/(1-2*x)^(3/2)/(2+3*x)^(1/2)/(3+5*x)^(3/2),x, algorithm="fricas ")
Output:
-1/38115*(90*(370*x - 163)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 94 9*sqrt(-30)*(10*x^2 + x - 3)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 3330*sqrt(-30)*(10*x^2 + x - 3)*weierstrassZeta(1159/675, 38998 /91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(10*x^2 + x - 3)
\[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=\int \frac {1}{\left (1 - 2 x\right )^{\frac {3}{2}} \sqrt {3 x + 2} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \] Input:
integrate(1/(1-2*x)**(3/2)/(2+3*x)**(1/2)/(3+5*x)**(3/2),x)
Output:
Integral(1/((1 - 2*x)**(3/2)*sqrt(3*x + 2)*(5*x + 3)**(3/2)), x)
\[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(1-2*x)^(3/2)/(2+3*x)^(1/2)/(3+5*x)^(3/2),x, algorithm="maxima ")
Output:
integrate(1/((5*x + 3)^(3/2)*sqrt(3*x + 2)*(-2*x + 1)^(3/2)), x)
\[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=\int { \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate(1/(1-2*x)^(3/2)/(2+3*x)^(1/2)/(3+5*x)^(3/2),x, algorithm="giac")
Output:
integrate(1/((5*x + 3)^(3/2)*sqrt(3*x + 2)*(-2*x + 1)^(3/2)), x)
Timed out. \[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=\int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \] Input:
int(1/((1 - 2*x)^(3/2)*(3*x + 2)^(1/2)*(5*x + 3)^(3/2)),x)
Output:
int(1/((1 - 2*x)^(3/2)*(3*x + 2)^(1/2)*(5*x + 3)^(3/2)), x)
\[ \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{300 x^{5}+260 x^{4}-137 x^{3}-136 x^{2}+15 x +18}d x \] Input:
int(1/(1-2*x)^(3/2)/(2+3*x)^(1/2)/(3+5*x)^(3/2),x)
Output:
int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(300*x**5 + 260*x**4 - 137*x**3 - 136*x**2 + 15*x + 18),x)