Integrand size = 28, antiderivative size = 249 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{13/2}} \, dx=\frac {940 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{9/2}}-\frac {251590 \sqrt {1-2 x} \sqrt {3+5 x}}{2139291 (2+3 x)^{7/2}}-\frac {362666 \sqrt {1-2 x} \sqrt {3+5 x}}{14975037 (2+3 x)^{5/2}}+\frac {11460644 \sqrt {1-2 x} \sqrt {3+5 x}}{104825259 (2+3 x)^{3/2}}+\frac {924247516 \sqrt {1-2 x} \sqrt {3+5 x}}{733776813 \sqrt {2+3 x}}+\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{231 (2+3 x)^{11/2}}-\frac {924247516 E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{66706983 \sqrt {33}}-\frac {31704544 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{66706983 \sqrt {33}} \]
-924247516/2201330439*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2) )*33^(1/2)-31704544/2201330439*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1 155^(1/2))*33^(1/2)+2/231*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2)+940/4 3659*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)-251590/2139291*(1-2*x)^(1/2 )*(3+5*x)^(1/2)/(2+3*x)^(7/2)-362666/14975037*(1-2*x)^(1/2)*(3+5*x)^(1/2)/ (2+3*x)^(5/2)+11460644/104825259*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2) +924247516/733776813*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)
Result contains complex when optimal does not.
Time = 8.73 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.45 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{13/2}} \, dx=\frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (15211411193+113962415157 x+340525216341 x^2+507518001945 x^3+377569336554 x^4+112296073194 x^5\right )}{2 (2+3 x)^{11/2}}+i \sqrt {33} \left (231061879 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-238988015 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{2201330439} \]
(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(15211411193 + 113962415157*x + 3405252 16341*x^2 + 507518001945*x^3 + 377569336554*x^4 + 112296073194*x^5))/(2*(2 + 3*x)^(11/2)) + I*Sqrt[33]*(231061879*EllipticE[I*ArcSinh[Sqrt[9 + 15*x] ], -2/33] - 238988015*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/22013 30439
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 = {109, 27, 167, 27, 169, 27, 169, 27, 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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{\sqrt {1-2 x} (3 x+2)^{13/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}-\frac {2}{231} \int -\frac {5 \sqrt {5 x+3} (371 x+216)}{2 \sqrt {1-2 x} (3 x+2)^{11/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{231} \int \frac {\sqrt {5 x+3} (371 x+216)}{\sqrt {1-2 x} (3 x+2)^{11/2}}dx+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {5}{231} \left (\frac {2}{189} \int \frac {110285 x+65137}{2 \sqrt {1-2 x} (3 x+2)^{9/2} \sqrt {5 x+3}}dx+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{231} \left (\frac {1}{189} \int \frac {110285 x+65137}{\sqrt {1-2 x} (3 x+2)^{9/2} \sqrt {5 x+3}}dx+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{231} \left (\frac {1}{189} \left (\frac {2}{49} \int \frac {1257950 x+778189}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {50318 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{231} \left (\frac {1}{189} \left (\frac {1}{49} \int \frac {1257950 x+778189}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {50318 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{231} \left (\frac {1}{189} \left (\frac {1}{49} \left (\frac {2}{35} \int \frac {3 (906665 x+1559497)}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {362666 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {50318 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{231} \left (\frac {1}{189} \left (\frac {1}{49} \left (\frac {6}{35} \int \frac {906665 x+1559497}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {362666 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {50318 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{231} \left (\frac {1}{189} \left (\frac {1}{49} \left (\frac {6}{35} \left (\frac {2}{21} \int \frac {57919553-28651610 x}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {5730322 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {362666 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {50318 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{231} \left (\frac {1}{189} \left (\frac {1}{49} \left (\frac {6}{35} \left (\frac {1}{21} \int \frac {57919553-28651610 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {5730322 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {362666 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {50318 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{231} \left (\frac {1}{189} \left (\frac {1}{49} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {2}{7} \int \frac {5 (231061879 x+147355877)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {462123758 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {5730322 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {362666 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {50318 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{231} \left (\frac {1}{189} \left (\frac {1}{49} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \int \frac {231061879 x+147355877}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {462123758 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {5730322 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {362666 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {50318 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {5}{231} \left (\frac {1}{189} \left (\frac {1}{49} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {43593748}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {231061879}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {462123758 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {5730322 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {362666 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {50318 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {5}{231} \left (\frac {1}{189} \left (\frac {1}{49} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (\frac {43593748}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {231061879}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {462123758 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {5730322 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {362666 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {50318 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {5}{231} \left (\frac {1}{189} \left (\frac {1}{49} \left (\frac {6}{35} \left (\frac {1}{21} \left (\frac {10}{7} \left (-\frac {7926136}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {231061879}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {462123758 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {5730322 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {362666 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {50318 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {188 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{231 (3 x+2)^{11/2}}\) |
(2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(231*(2 + 3*x)^(11/2)) + (5*((188*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(189*(2 + 3*x)^(9/2)) + ((-50318*Sqrt[1 - 2*x]*Sqrt [3 + 5*x])/(49*(2 + 3*x)^(7/2)) + ((-362666*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/( 35*(2 + 3*x)^(5/2)) + (6*((5730322*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3 *x)^(3/2)) + ((462123758*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x]) + (10*((-231061879*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/ 33])/5 - (7926136*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35 /33])/5))/7)/21))/35)/49)/189))/231
3.29.40.3.1 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*((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]
Time = 1.38 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.27
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{505197 \left (\frac {2}{3}+x \right )^{6}}+\frac {1570 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{10609137 \left (\frac {2}{3}+x \right )^{5}}-\frac {251590 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{173282571 \left (\frac {2}{3}+x \right )^{4}}-\frac {362666 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{404325999 \left (\frac {2}{3}+x \right )^{3}}+\frac {11460644 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{943427331 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {9242475160}{733776813} x^{2}-\frac {924247516}{733776813} x +\frac {924247516}{244592271}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {1178847016 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{15409313073 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1848495032 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{15409313073 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(315\) |
| default | \(-\frac {2 \left (109511136702 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-112296073194 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+365037122340 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-374320243980 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+486716163120 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-499093658640 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+324477442080 \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}-332729105760 \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}+108159147360 \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}-110909701920 \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}-3368882195820 x^{7}+14421219648 \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 )-14787960256 \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 )-11663968316202 x^{6}-15347583409266 x^{5}-8340186467079 x^{4}+127213913772 x^{3}+2266497365808 x^{2}+980027502834 x +136902700737\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{2201330439 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\) | \(599\) |
(-(-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 )*(-2/505197*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^6+1570/10609137*(-30*x^3 -23*x^2+7*x+6)^(1/2)/(2/3+x)^5-251590/173282571*(-30*x^3-23*x^2+7*x+6)^(1/ 2)/(2/3+x)^4-362666/404325999*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+11460 644/943427331*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+924247516/2201330439* (-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+1178847016/15409313073*(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))+1848495032/15409313073*(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)*(-7/6*E llipticE((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 = 0.08 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.67 \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{13/2}} \, dx=\frac {2 \, {\left (135 \, {\left (112296073194 \, x^{5} + 377569336554 \, x^{4} + 507518001945 \, x^{3} + 340525216341 \, x^{2} + 113962415157 \, x + 15211411193\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 7947605713 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 20795569110 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{99059869755 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \]
2/99059869755*(135*(112296073194*x^5 + 377569336554*x^4 + 507518001945*x^3 + 340525216341*x^2 + 113962415157*x + 15211411193)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 7947605713*sqrt(-30)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*weierstrassPInverse(1159/675, 38998/9 1125, x + 23/90) + 20795569110*sqrt(-30)*(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*weierstrassZeta(1159/675, 38998/91125, w eierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 576*x + 64)
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{13/2}} \, dx=\text {Timed out} \]
\[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{13/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
\[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{13/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}} \sqrt {-2 \, x + 1}} \,d x } \]
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{\sqrt {1-2 x} (2+3 x)^{13/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{13/2}} \,d x \]