Integrand size = 28, antiderivative size = 249 \[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\frac {902407192 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{760134375}+\frac {227099 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{433125}-\frac {247381 (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}}{450450}-\frac {5791 (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt {3+5 x}}{19305}-\frac {53}{715} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac {2}{65} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{5/2}-\frac {30660308017 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{651543750 \sqrt {35}}+\frac {880870681 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{651543750 \sqrt {35}} \] Output:
902407192/760134375*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+227099/43312 5*(1-2*x)^(3/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-247381/450450*(1-2*x)^(5/2)*(2 +3*x)^(1/2)*(3+5*x)^(1/2)-5791/19305*(1-2*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^( 1/2)-53/715*(1-2*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(3/2)+2/65*(1-2*x)^(5/2)*( 2+3*x)^(3/2)*(3+5*x)^(5/2)-30660308017/22804031250*EllipticE(1/11*55^(1/2) *(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+880870681/22804031250*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 = 7.45 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.45 \[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x} \left (1020785999+5290733520 x-3707642250 x^2-13684072500 x^3+5400675000 x^4+14033250000 x^5\right )+30660308017 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-31584251720 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{22804031250} \] Input:
Integrate[(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2),x]
Output:
(15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(1020785999 + 5290733520*x - 3707642250*x^2 - 13684072500*x^3 + 5400675000*x^4 + 14033250000*x^5) + (3 0660308017*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (3158 4251720*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/228040312 50
Time = 0.32 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.14, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {112, 27, 171, 27, 171, 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.
| \(\displaystyle \int (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{3/2} \, dx\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle \frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}-\frac {2}{65} \int -\frac {1}{2} (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2} (159 x+113)dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \int (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2} (159 x+113)dx+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {2}{165} \int \frac {3}{2} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2} (4159 x+4042)dx+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{55} \int \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2} (4159 x+4042)dx+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{55} \left (\frac {2}{135} \int \frac {(275335-230427 x) \sqrt {3 x+2} (5 x+3)^{3/2}}{2 \sqrt {1-2 x}}dx+\frac {8318}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{55} \left (\frac {1}{135} \int \frac {(275335-230427 x) \sqrt {3 x+2} (5 x+3)^{3/2}}{\sqrt {1-2 x}}dx+\frac {8318}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{55} \left (\frac {1}{135} \left (\frac {230427}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}-\frac {1}{35} \int -\frac {(5 x+3)^{3/2} (40768752 x+27716831)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {8318}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{70} \int \frac {(5 x+3)^{3/2} (40768752 x+27716831)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx+\frac {230427}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8318}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (-\frac {1}{15} \int -\frac {3 \sqrt {5 x+3} (923943703 x+599211849)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {13589584}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {230427}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8318}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (\frac {1}{5} \int \frac {\sqrt {5 x+3} (923943703 x+599211849)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {13589584}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {230427}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8318}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (\frac {1}{5} \left (-\frac {1}{9} \int -\frac {61320616034 x+38825045767}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {923943703}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {13589584}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {230427}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8318}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (\frac {1}{5} \left (\frac {1}{18} \int \frac {61320616034 x+38825045767}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {923943703}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {13589584}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {230427}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8318}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {10163380733}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {61320616034}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {923943703}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {13589584}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {230427}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8318}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (\frac {1}{5} \left (\frac {1}{18} \left (\frac {10163380733}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {61320616034}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {923943703}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {13589584}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {230427}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8318}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{65} \left (\frac {1}{55} \left (\frac {1}{135} \left (\frac {1}{70} \left (\frac {1}{5} \left (\frac {1}{18} \left (-\frac {1847887406}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {61320616034}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {923943703}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {13589584}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )+\frac {230427}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )+\frac {8318}{135} \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {106}{55} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )+\frac {2}{65} (1-2 x)^{5/2} (3 x+2)^{3/2} (5 x+3)^{5/2}\) |
Input:
Int[(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2),x]
Output:
(2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/65 + ((106*(1 - 2*x)^( 3/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(5/2))/55 + ((8318*Sqrt[1 - 2*x]*(2 + 3*x)^ (3/2)*(3 + 5*x)^(5/2))/135 + ((230427*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x )^(5/2))/35 + ((-13589584*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5 + ((-923943703*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-6132061603 4*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (18478 87406*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/18) /5)/70)/135)/55)/65
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 = 0.47 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.65
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (-12629925000000 x^{8}-14543550000000 x^{7}+11536182000000 x^{6}+30490142199 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-61320616034 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+16439014800000 x^{5}-4104920740500 x^{4}-7811051450400 x^{3}-260663905110 x^{2}+1166697093390 x +183741479820\right )}{45608062500 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(163\) |
| risch | \(-\frac {\left (14033250000 x^{5}+5400675000 x^{4}-13684072500 x^{3}-3707642250 x^{2}+5290733520 x +1020785999\right ) \sqrt {3+5 x}\, \left (-1+2 x \right ) \sqrt {2+3 x}\, \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{1520268750 \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \sqrt {1-2 x}}-\frac {\left (-\frac {38825045767 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{167229562500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {30660308017 \sqrt {66+110 x}\, \sqrt {10+15 x}\, \sqrt {-110 x +55}\, \left (\frac {\operatorname {EllipticE}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{15}-\frac {2 \operatorname {EllipticF}\left (\frac {\sqrt {66+110 x}}{11}, \frac {i \sqrt {66}}{2}\right )}{3}\right )}{83614781250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right ) \sqrt {\left (1-2 x \right ) \left (2+3 x \right ) \left (3+5 x \right )}}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(267\) |
| elliptic | \(\frac {\sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {58785928 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{16891875}+\frac {1020785999 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1520268750}+\frac {38825045767 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{63851287500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {30660308017 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{31925643750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {126757 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{51975}-\frac {57922 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6435}+\frac {508 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{4}}{143}+\frac {120 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}\, x^{5}}{13}\right )}{\left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}\) | \(312\) |
Input:
int((1-2*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/45608062500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-12629925000000* x^8-14543550000000*x^7+11536182000000*x^6+30490142199*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)) -61320616034*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))+16439014800000*x^5-4104920740500*x^4-781 1051450400*x^3-260663905110*x^2+1166697093390*x+183741479820)/(30*x^3+23*x ^2-7*x-6)
Time = 0.07 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.30 \[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\frac {1}{1520268750} \, {\left (14033250000 \, x^{5} + 5400675000 \, x^{4} - 13684072500 \, x^{3} - 3707642250 \, x^{2} + 5290733520 \, x + 1020785999\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - \frac {260484993781}{513090703125} \, \sqrt {-30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {30660308017}{22804031250} \, \sqrt {-30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(3/2),x, algorithm="fricas")
Output:
1/1520268750*(14033250000*x^5 + 5400675000*x^4 - 13684072500*x^3 - 3707642 250*x^2 + 5290733520*x + 1020785999)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 260484993781/513090703125*sqrt(-30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 30660308017/22804031250*sqrt(-30)*weierstrassZet a(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23 /90))
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\text {Timed out} \] Input:
integrate((1-2*x)**(5/2)*(2+3*x)**(3/2)*(3+5*x)**(3/2),x)
Output:
Timed out
\[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(3/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(3/2)*(3*x + 2)^(3/2)*(-2*x + 1)^(5/2), x)
\[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\int { {\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(3/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(3/2)*(3*x + 2)^(3/2)*(-2*x + 1)^(5/2), x)
Timed out. \[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\int {\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2} \,d x \] Input:
int((1 - 2*x)^(5/2)*(3*x + 2)^(3/2)*(5*x + 3)^(3/2),x)
Output:
int((1 - 2*x)^(5/2)*(3*x + 2)^(3/2)*(5*x + 3)^(3/2), x)
\[ \int (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2} \, dx=\frac {120 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{5}}{13}+\frac {508 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}}{143}-\frac {57922 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}}{6435}-\frac {126757 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{51975}+\frac {58785928 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x}{16891875}-\frac {26600852 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{129504375}+\frac {30660308017 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{777026250}-\frac {227900866 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{30 x^{3}+23 x^{2}-7 x -6}d x \right )}{14389375} \] Input:
int((1-2*x)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(3/2),x)
Output:
(7172550000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**5 + 2760345000 *sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 6994081500*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 1895017150*sqrt(3*x + 2)*sqrt(5* x + 3)*sqrt( - 2*x + 1)*x**2 + 2704152688*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt ( - 2*x + 1)*x - 159605112*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 30660308017*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) - 12306646764*int((sqrt(3*x + 2)*sqrt(5*x + 3)* sqrt( - 2*x + 1))/(30*x**3 + 23*x**2 - 7*x - 6),x))/777026250