Integrand size = 28, antiderivative size = 158 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{15 (2+3 x)^{5/2}}+\frac {1736 \sqrt {1-2 x} \sqrt {3+5 x}}{135 (2+3 x)^{3/2}}+\frac {16564 \sqrt {1-2 x} \sqrt {3+5 x}}{135 \sqrt {2+3 x}}-\frac {16564}{81} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )+\frac {3352 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{81 \sqrt {35}} \] Output:
14/15*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+1736/135*(1-2*x)^(1/2)*(3+ 5*x)^(1/2)/(2+3*x)^(3/2)+16564/135*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/ 2)-16564/405*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/ 2)+3352/2835*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 = 7.71 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.61 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\frac {4}{405} \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (34927+101862 x+74538 x^2\right )}{2 (2+3 x)^{5/2}}+i \sqrt {33} \left (4141 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-4265 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right ) \] Input:
Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(7/2)*Sqrt[3 + 5*x]),x]
Output:
(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(34927 + 101862*x + 74538*x^2))/(2*(2 + 3*x)^(5/2)) + I*Sqrt[33]*(4141*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33 ] - 4265*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/405
Time = 0.23 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.08, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {109, 167, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2}}{(3 x+2)^{7/2} \sqrt {5 x+3}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2}{15} \int \frac {(128-25 x) \sqrt {1-2 x}}{(3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {2}{15} \left (\frac {868 \sqrt {1-2 x} \sqrt {5 x+3}}{9 (3 x+2)^{3/2}}-\frac {2}{9} \int -\frac {6869-4190 x}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{9} \int \frac {6869-4190 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {868 \sqrt {1-2 x} \sqrt {5 x+3}}{9 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{9} \left (\frac {2}{7} \int \frac {35 (4141 x+2621)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {8282 \sqrt {1-2 x} \sqrt {5 x+3}}{\sqrt {3 x+2}}\right )+\frac {868 \sqrt {1-2 x} \sqrt {5 x+3}}{9 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{9} \left (10 \int \frac {4141 x+2621}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {8282 \sqrt {1-2 x} \sqrt {5 x+3}}{\sqrt {3 x+2}}\right )+\frac {868 \sqrt {1-2 x} \sqrt {5 x+3}}{9 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{9} \left (10 \left (\frac {682}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {4141}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {8282 \sqrt {1-2 x} \sqrt {5 x+3}}{\sqrt {3 x+2}}\right )+\frac {868 \sqrt {1-2 x} \sqrt {5 x+3}}{9 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{9} \left (10 \left (\frac {682}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {4141}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {8282 \sqrt {1-2 x} \sqrt {5 x+3}}{\sqrt {3 x+2}}\right )+\frac {868 \sqrt {1-2 x} \sqrt {5 x+3}}{9 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {2}{15} \left (\frac {1}{9} \left (10 \left (-\frac {124}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {4141}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {8282 \sqrt {1-2 x} \sqrt {5 x+3}}{\sqrt {3 x+2}}\right )+\frac {868 \sqrt {1-2 x} \sqrt {5 x+3}}{9 (3 x+2)^{3/2}}\right )+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{15 (3 x+2)^{5/2}}\) |
Input:
Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(7/2)*Sqrt[3 + 5*x]),x]
Output:
(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(15*(2 + 3*x)^(5/2)) + (2*((868*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(9*(2 + 3*x)^(3/2)) + ((8282*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/Sqrt[2 + 3*x] + 10*((-4141*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqr t[1 - 2*x]], 35/33])/5 - (124*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5))/9))/15
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[(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. \(248\) vs. \(2(116)=232\).
Time = 0.44 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.58
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1215 \left (\frac {2}{3}+x \right )^{3}}+\frac {1652 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1215 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {33128}{27} x^{2}-\frac {16564}{135} x +\frac {16564}{45}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {10484 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{567 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {16564 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{567 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(249\) |
| default | \(-\frac {2 \left (36828 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-74538 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+49104 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-99384 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+16368 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-33128 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2236140 x^{4}-3279474 x^{3}-682554 x^{2}+811977 x +314343\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{405 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {5}{2}}}\) | \(308\) |
Input:
int((1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2),x,method=_RETURNVERBOSE)
Output:
(-(3+5*x)*(-1+2*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 )*(98/1215*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+1652/1215*(-30*x^3-23*x^ 2+7*x+6)^(1/2)/(2/3+x)^2+16564/405*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9 ))^(1/2)+10484/567*(28+42*x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^ 3-23*x^2+7*x+6)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+16564/56 7*(28+42*x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^3-23*x^2+7*x+6)^( 1/2)*(-1/15*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-3/5*EllipticF(1/7* (28+42*x)^(1/2),1/2*70^(1/2))))
Time = 0.07 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.68 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\frac {2 \, {\left (135 \, {\left (74538 \, x^{2} + 101862 \, x + 34927\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 140647 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 372690 \, \sqrt {-30} {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{18225 \, {\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2),x, algorithm="fricas")
Output:
2/18225*(135*(74538*x^2 + 101862*x + 34927)*sqrt(5*x + 3)*sqrt(3*x + 2)*sq rt(-2*x + 1) - 140647*sqrt(-30)*(27*x^3 + 54*x^2 + 36*x + 8)*weierstrassPI nverse(1159/675, 38998/91125, x + 23/90) + 372690*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)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {5}{2}}}{\left (3 x + 2\right )^{\frac {7}{2}} \sqrt {5 x + 3}}\, dx \] Input:
integrate((1-2*x)**(5/2)/(2+3*x)**(7/2)/(3+5*x)**(1/2),x)
Output:
Integral((1 - 2*x)**(5/2)/((3*x + 2)**(7/2)*sqrt(5*x + 3)), x)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2),x, algorithm="maxima")
Output:
integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(7/2)), x)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {7}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2),x, algorithm="giac")
Output:
integrate((-2*x + 1)^(5/2)/(sqrt(5*x + 3)*(3*x + 2)^(7/2)), x)
Timed out. \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}}{{\left (3\,x+2\right )}^{7/2}\,\sqrt {5\,x+3}} \,d x \] Input:
int((1 - 2*x)^(5/2)/((3*x + 2)^(7/2)*(5*x + 3)^(1/2)),x)
Output:
int((1 - 2*x)^(5/2)/((3*x + 2)^(7/2)*(5*x + 3)^(1/2)), x)
\[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx=\frac {-112 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x +222 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+313578 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x^{3}+627156 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x^{2}+418104 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x +92912 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right )-164673 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x^{3}-329346 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x^{2}-219564 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right ) x -48792 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{810 x^{6}+2241 x^{5}+2133 x^{4}+528 x^{3}-392 x^{2}-272 x -48}d x \right )}{5670 x^{3}+11340 x^{2}+7560 x +1680} \] Input:
int((1-2*x)^(5/2)/(2+3*x)^(7/2)/(3+5*x)^(1/2),x)
Output:
( - 112*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 222*sqrt(3*x + 2) *sqrt(5*x + 3)*sqrt( - 2*x + 1) + 313578*int((sqrt(3*x + 2)*sqrt(5*x + 3)* sqrt( - 2*x + 1)*x**2)/(810*x**6 + 2241*x**5 + 2133*x**4 + 528*x**3 - 392* x**2 - 272*x - 48),x)*x**3 + 627156*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(810*x**6 + 2241*x**5 + 2133*x**4 + 528*x**3 - 392*x**2 - 272*x - 48),x)*x**2 + 418104*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2* x + 1)*x**2)/(810*x**6 + 2241*x**5 + 2133*x**4 + 528*x**3 - 392*x**2 - 272 *x - 48),x)*x + 92912*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x* *2)/(810*x**6 + 2241*x**5 + 2133*x**4 + 528*x**3 - 392*x**2 - 272*x - 48), x) - 164673*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(810*x**6 + 2241*x**5 + 2133*x**4 + 528*x**3 - 392*x**2 - 272*x - 48),x)*x**3 - 32934 6*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(810*x**6 + 2241*x**5 + 2133*x**4 + 528*x**3 - 392*x**2 - 272*x - 48),x)*x**2 - 219564*int((sqr t(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(810*x**6 + 2241*x**5 + 2133*x* *4 + 528*x**3 - 392*x**2 - 272*x - 48),x)*x - 48792*int((sqrt(3*x + 2)*sqr t(5*x + 3)*sqrt( - 2*x + 1))/(810*x**6 + 2241*x**5 + 2133*x**4 + 528*x**3 - 392*x**2 - 272*x - 48),x))/(210*(27*x**3 + 54*x**2 + 36*x + 8))