Integrand size = 28, antiderivative size = 160 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\frac {14 \sqrt {1-2 x} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {256 \sqrt {1-2 x} \sqrt {3+5 x}}{45 (2+3 x)^{3/2}}+\frac {17804 \sqrt {1-2 x} \sqrt {3+5 x}}{315 \sqrt {2+3 x}}-\frac {17804}{315} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {536}{315} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right ) \]
-17804/945*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)- 536/945*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+14/ 15*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+256/45*(1-2*x)^(1/2)*(3+5*x)^ (1/2)/(2+3*x)^(3/2)+17804/315*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 6.86 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.60 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\frac {4}{945} \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (37547+109512 x+80118 x^2\right )}{2 (2+3 x)^{5/2}}+i \sqrt {33} \left (4451 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-4585 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right ) \]
(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(37547 + 109512*x + 80118*x^2))/(2*(2 + 3*x)^(5/2)) + I*Sqrt[33]*(4451*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33 ] - 4585*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/945
Time = 0.23 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.09, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {109, 169, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2}}{(3 x+2)^{7/2} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{15} \int \frac {86-95 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {14 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{15} \left (\frac {2}{21} \int \frac {7 (1057-640 x)}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {128 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \int \frac {1057-640 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {128 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {2}{7} \int \frac {5 (4451 x+2818)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {8902 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {128 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {10}{7} \int \frac {4451 x+2818}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {8902 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {128 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {10}{7} \left (\frac {737}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {4451}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {8902 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {128 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {10}{7} \left (\frac {737}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4451}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {8902 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {128 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{3} \left (\frac {10}{7} \left (-\frac {134}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {4451}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {8902 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {128 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {1-2 x} \sqrt {5 x+3}}{15 (3 x+2)^{5/2}}\) |
(14*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^(5/2)) + (2*((128*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(3/2)) + ((8902*Sqrt[1 - 2*x]*Sqrt[3 + 5* x])/(7*Sqrt[2 + 3*x]) + (10*((-4451*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]* Sqrt[1 - 2*x]], 35/33])/5 - (134*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqr t[1 - 2*x]], 35/33])/5))/7)/3))/15
3.28.35.3.1 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 + 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]
Leaf count of result is larger than twice the leaf count of optimal. \(242\) vs. \(2(116)=232\).
Time = 1.29 (sec) , antiderivative size = 243, normalized size of antiderivative = 1.52
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{405 \left (\frac {2}{3}+x \right )^{3}}+\frac {256 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{405 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {35608}{63} x^{2}-\frac {17804}{315} x +\frac {17804}{105}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {22544 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6615 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {35608 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{6615 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(243\) |
| default | \(-\frac {2 \left (77814 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-80118 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+103752 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-106824 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+34584 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-35608 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-2403540 x^{4}-3525714 x^{3}-733884 x^{2}+872967 x +337923\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{945 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {5}{2}}}\) | \(314\) |
(-(-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 )*(14/405*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+256/405*(-30*x^3-23*x^2+7 *x+6)^(1/2)/(2/3+x)^2+17804/945*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^ (1/2)+22544/6615*(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))+35608/6615*(1 0+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 = 108, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\frac {2 \, {\left (135 \, {\left (80118 \, x^{2} + 109512 \, x + 37547\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 151247 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 400590 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{42525 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \]
2/42525*(135*(80118*x^2 + 109512*x + 37547)*sqrt(5*x + 3)*sqrt(3*x + 2)*sq rt(-2*x + 1) - 151247*sqrt(-30)*(27*x^3 + 54*x^2 + 36*x + 8)*weierstrassPI nverse(1159/675, 38998/91125, x + 23/90) + 400590*sqrt(-30)*(27*x^3 + 54*x ^2 + 36*x + 8)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse( 1159/675, 38998/91125, x + 23/90)))/(27*x^3 + 54*x^2 + 36*x + 8)
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{\frac {7}{2}} \sqrt {5 x + 3}}\, dx \]
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \]
\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{7/2}\,\sqrt {5\,x+3}} \,d x \]