Integrand size = 28, antiderivative size = 187 \[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=-\frac {29357 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{17010}-\frac {223}{945} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}-\frac {31}{945} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2}+\frac {2}{45} \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{7/2}-\frac {488149 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{3645 \sqrt {35}}+\frac {28109 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{7290 \sqrt {35}} \] Output:
-29357/17010*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-223/945*(1-2*x)^(1/ 2)*(2+3*x)^(1/2)*(3+5*x)^(3/2)-31/945*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^ (5/2)+2/45*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(7/2)-488149/127575*Ellipti cE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+28109/255150*Elli pticF(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 = 3.23 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.55 \[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (-26009+65250 x+156150 x^2+94500 x^3\right )+976298 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1005655 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{255150} \] Input:
Integrate[Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2),x]
Output:
(15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-26009 + 65250*x + 156150*x ^2 + 94500*x^3) + (976298*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (1005655*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] )/255150
Time = 0.30 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {112, 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.
| \(\displaystyle \int \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}-\frac {2}{45} \int -\frac {(5 x+3)^{5/2} (31 x+23)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{45} \int \frac {(5 x+3)^{5/2} (31 x+23)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx+\frac {2}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{45} \left (-\frac {1}{21} \int -\frac {5 (5 x+3)^{3/2} (1338 x+871)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {31}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{45} \left (\frac {5}{42} \int \frac {(5 x+3)^{3/2} (1338 x+871)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {31}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{45} \left (\frac {5}{42} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (29357 x+19086)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {446}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {31}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{45} \left (\frac {5}{42} \left (\frac {1}{5} \int \frac {\sqrt {5 x+3} (29357 x+19086)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {446}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {31}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{45} \left (\frac {5}{42} \left (\frac {1}{5} \left (-\frac {1}{9} \int -\frac {1952596 x+1236143}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {29357}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {446}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {31}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{45} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \int \frac {1952596 x+1236143}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {29357}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {446}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {31}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{45} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {322927}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {1952596}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {29357}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {446}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {31}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{45} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {322927}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {1952596}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {29357}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {446}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {31}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{45} \left (\frac {5}{42} \left (\frac {1}{5} \left (\frac {1}{18} \left (-\frac {58714}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {1952596}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {29357}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {446}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {31}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{45} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{7/2}\) |
Input:
Int[Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2),x]
Output:
(2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/45 + ((-31*Sqrt[1 - 2*x]*S qrt[2 + 3*x]*(3 + 5*x)^(5/2))/21 + (5*((-446*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*( 3 + 5*x)^(3/2))/5 + ((-29357*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-1952596*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33]) /5 - (58714*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/ 5)/18)/5))/42)/45
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*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 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 = 0.41 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.82
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-85050000 x^{6}+968781 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-1952596 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-205740000 x^{5}-146623500 x^{4}+28187100 x^{3}+59755710 x^{2}+6283110 x -4681620\right )}{510300 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(153\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {725 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{189}-\frac {26009 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{17010}+\frac {1236143 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{714420 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {488149 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{178605 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1735 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{189}+\frac {50 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{9}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(256\) |
| risch | \(-\frac {\left (94500 x^{3}+156150 x^{2}+65250 x -26009\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{17010 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {1236143 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{1871100 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {488149 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{467775 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(257\) |
Input:
int((1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/510300*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-85050000*x^6+968781* 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))-1952596*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))-205740000*x^5-146623500* x^4+28187100*x^3+59755710*x^2+6283110*x-4681620)/(30*x^3+23*x^2-7*x-6)
Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.34 \[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\frac {1}{17010} \, {\left (94500 \, x^{3} + 156150 \, x^{2} + 65250 \, x - 26009\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {33171581}{22963500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {488149}{127575} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \] Input:
integrate((1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(5/2),x, algorithm="fricas")
Output:
1/17010*(94500*x^3 + 156150*x^2 + 65250*x - 26009)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 33171581/22963500*sqrt(-30)*weierstrassPInverse(1159 /675, 38998/91125, x + 23/90) + 488149/127575*sqrt(-30)*weierstrassZeta(11 59/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90) )
\[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\int \sqrt {1 - 2 x} \sqrt {3 x + 2} \left (5 x + 3\right )^{\frac {5}{2}}\, dx \] Input:
integrate((1-2*x)**(1/2)*(2+3*x)**(1/2)*(3+5*x)**(5/2),x)
Output:
Integral(sqrt(1 - 2*x)*sqrt(3*x + 2)*(5*x + 3)**(5/2), x)
\[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \,d x } \] Input:
integrate((1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(5/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(5/2)*sqrt(3*x + 2)*sqrt(-2*x + 1), x)
\[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} \,d x } \] Input:
integrate((1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(5/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(5/2)*sqrt(3*x + 2)*sqrt(-2*x + 1), x)
Timed out. \[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\int \sqrt {1-2\,x}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{5/2} \,d x \] Input:
int((1 - 2*x)^(1/2)*(3*x + 2)^(1/2)*(5*x + 3)^(5/2),x)
Output:
int((1 - 2*x)^(1/2)*(3*x + 2)^(1/2)*(5*x + 3)^(5/2), x)
\[ \int \sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{5/2} \, dx=\frac {50 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{9}+\frac {1735 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{189}+\frac {725 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{189}-\frac {11663 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{2898}+\frac {488149 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{4347}-\frac {29025 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{644} \] Input:
int((1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(5/2),x)
Output:
(96600*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 159620*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 66700*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 69978*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1 ) + 1952596*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(30*x* *3 + 23*x**2 - 7*x - 6),x) - 783675*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(30*x**3 + 23*x**2 - 7*x - 6),x))/17388