Integrand size = 28, antiderivative size = 218 \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\frac {24433447 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{35083125}+\frac {79967 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{259875}-\frac {11236 (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}}{31185}-\frac {163 (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2}}{1485}+\frac {2}{55} (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{5/2}-\frac {829177897 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{30071250 \sqrt {35}}+\frac {11437073 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{15035625 \sqrt {35}} \] Output:
24433447/35083125*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+79967/259875*( 1-2*x)^(3/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-11236/31185*(1-2*x)^(5/2)*(2+3*x) ^(1/2)*(3+5*x)^(1/2)-163/1485*(1-2*x)^(5/2)*(2+3*x)^(1/2)*(3+5*x)^(3/2)+2/ 55*(1-2*x)^(5/2)*(2+3*x)^(1/2)*(3+5*x)^(5/2)-829177897/1052493750*Elliptic E(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+11437073/526246875 *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 = 6.27 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.50 \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\frac {30 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (22517617+48272535 x-95024250 x^2-51502500 x^3+127575000 x^4\right )+829177897 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-855170645 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{1052493750} \] Input:
Integrate[(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2),x]
Output:
(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(22517617 + 48272535*x - 950 24250*x^2 - 51502500*x^3 + 127575000*x^4) + (829177897*I)*Sqrt[33]*Ellipti cE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (855170645*I)*Sqrt[33]*EllipticF[I* ArcSinh[Sqrt[9 + 15*x]], -2/33])/1052493750
Time = 0.32 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {112, 27, 171, 27, 171, 27, 171, 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.
\(\displaystyle \int (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{3/2} \, dx\) |
\(\Big \downarrow \) 112 |
\(\displaystyle \frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {2}{55} \int -\frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (163 x+111)}{2 \sqrt {3 x+2}}dx\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{55} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (163 x+111)}{\sqrt {3 x+2}}dx+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {1}{55} \left (\frac {2}{135} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (15181 x+11236)}{2 \sqrt {3 x+2}}dx+\frac {326}{135} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{55} \left (\frac {1}{135} \int \frac {\sqrt {1-2 x} (5 x+3)^{3/2} (15181 x+11236)}{\sqrt {3 x+2}}dx+\frac {326}{135} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {1}{55} \left (\frac {1}{135} \left (\frac {2}{105} \int \frac {(5 x+3)^{3/2} (236391 x+466273)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx+\frac {30362}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {326}{135} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{105} \int \frac {(5 x+3)^{3/2} (236391 x+466273)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx+\frac {30362}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {326}{135} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{105} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (25992748 x+16115709)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {78797}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {30362}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {326}{135} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {1}{10} \int \frac {\sqrt {5 x+3} (25992748 x+16115709)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {78797}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {30362}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {326}{135} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {1}{10} \left (-\frac {1}{9} \int -\frac {829177897 x+526098761}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {25992748}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {78797}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {30362}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {326}{135} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {1}{10} \left (\frac {1}{9} \int \frac {829177897 x+526098761}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {25992748}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {78797}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {30362}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {326}{135} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 176 |
\(\displaystyle \frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {142960114}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {829177897}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {25992748}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {78797}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {30362}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {326}{135} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {142960114}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {829177897}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {25992748}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {78797}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {30362}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {326}{135} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
\(\Big \downarrow \) 129 |
\(\displaystyle \frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{105} \left (\frac {1}{10} \left (\frac {1}{9} \left (-\frac {25992748}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {829177897}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {25992748}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {78797}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {30362}{105} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {326}{135} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {2}{55} (1-2 x)^{5/2} \sqrt {3 x+2} (5 x+3)^{5/2}\) |
Input:
Int[(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2),x]
Output:
(2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/55 + ((326*(1 - 2*x)^(3/ 2)*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/135 + ((30362*Sqrt[1 - 2*x]*Sqrt[2 + 3*x ]*(3 + 5*x)^(5/2))/105 + ((-78797*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3 /2))/5 + ((-25992748*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-829 177897*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - ( 25992748*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/ 9)/10)/105)/135)/55
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 = 0.40 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.72
method | result | size |
default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-114817500000 x^{7}-41674500000 x^{6}+428880342 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-829177897 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+147849300000 x^{5}+34269426000 x^{4}-82799446950 x^{3}-22504288380 x^{2}+13417755870 x +4053171060\right )}{1052493750 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(158\) |
risch | \(-\frac {\left (127575000 x^{4}-51502500 x^{3}-95024250 x^{2}+48272535 x +22517617\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{35083125 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {526098761 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3859143750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {829177897 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{3859143750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(262\) |
elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {1072723 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{779625}+\frac {22517617 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35083125}+\frac {526098761 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{1473491250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {829177897 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1473491250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {84466 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{31185}-\frac {436 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{297}+\frac {40 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{11}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(278\) |
Input:
int((1-2*x)^(5/2)*(2+3*x)^(1/2)*(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/1052493750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-114817500000*x^7 -41674500000*x^6+428880342*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))-829177897*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))+147849300000*x^5+34269426000*x^4-82799446950*x^3-22504288380*x^2+1341 7755870*x+4053171060)/(30*x^3+23*x^2-7*x-6)
Time = 0.08 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.32 \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\frac {1}{35083125} \, {\left (127575000 \, x^{4} - 51502500 \, x^{3} - 95024250 \, x^{2} + 48272535 \, x + 22517617\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {28277796859}{94724437500} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {829177897}{1052493750} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(1/2)*(3+5*x)^(3/2),x, algorithm="fricas")
Output:
1/35083125*(127575000*x^4 - 51502500*x^3 - 95024250*x^2 + 48272535*x + 225 17617)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 28277796859/9472443750 0*sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 829177 897/1052493750*sqrt(-30)*weierstrassZeta(1159/675, 38998/91125, weierstras sPInverse(1159/675, 38998/91125, x + 23/90))
Timed out. \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\text {Timed out} \] Input:
integrate((1-2*x)**(5/2)*(2+3*x)**(1/2)*(3+5*x)**(3/2),x)
Output:
Timed out
\[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(1/2)*(3+5*x)^(3/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(3/2)*sqrt(3*x + 2)*(-2*x + 1)^(5/2), x)
\[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(1/2)*(3+5*x)^(3/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(3/2)*sqrt(3*x + 2)*(-2*x + 1)^(5/2), x)
Timed out. \[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{3/2} \,d x \] Input:
int((1 - 2*x)^(5/2)*(3*x + 2)^(1/2)*(5*x + 3)^(3/2),x)
Output:
int((1 - 2*x)^(5/2)*(3*x + 2)^(1/2)*(5*x + 3)^(3/2), x)
\[ \int (1-2 x)^{5/2} \sqrt {2+3 x} (3+5 x)^{3/2} \, dx=\frac {40 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{11}-\frac {436 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{297}-\frac {84466 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{31185}+\frac {1072723 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{779625}+\frac {1530611 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{11954250}+\frac {829177897 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{35862750}-\frac {24695299 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{2656500} \] Input:
int((1-2*x)^(5/2)*(2+3*x)^(1/2)*(3+5*x)^(3/2),x)
Output:
(260820000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 105294000*s qrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 194271800*sqrt(3*x + 2) *sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 98690516*sqrt(3*x + 2)*sqrt(5*x + 3 )*sqrt( - 2*x + 1)*x + 9183666*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1 ) + 1658355794*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) - 666773073*int((sqrt(3*x + 2)*sqrt(5*x + 3) *sqrt( - 2*x + 1))/(30*x**3 + 23*x**2 - 7*x - 6),x))/71725500