Integrand size = 28, antiderivative size = 249 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {2513419 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{1260}-\frac {107983}{252} \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {12871}{126} \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}-\frac {485}{18} \sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}-\frac {29 (2+3 x)^{7/2} (3+5 x)^{3/2}}{3 \sqrt {1-2 x}}+\frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {174654791 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{1080 \sqrt {35}}+\frac {2513419 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{540 \sqrt {35}} \] Output:
-2513419/1260*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)-107983/252*(1-2*x) ^(1/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2)-12871/126*(1-2*x)^(1/2)*(2+3*x)^(5/2)*( 3+5*x)^(1/2)-485/18*(1-2*x)^(1/2)*(2+3*x)^(7/2)*(3+5*x)^(1/2)-29/3*(2+3*x) ^(7/2)*(3+5*x)^(3/2)/(1-2*x)^(1/2)+1/3*(2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x) ^(3/2)-174654791/37800*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/ 2))*35^(1/2)+2513419/18900*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 = 8.96 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.53 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {30 \sqrt {2+3 x} \sqrt {3+5 x} \left (4641769-12151171 x+2783146 x^2+1279350 x^3+486900 x^4+94500 x^5\right )+174654791 i \sqrt {33-66 x} (-1+2 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-179908435 i \sqrt {33-66 x} (-1+2 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{37800 (1-2 x)^{3/2}} \] Input:
Integrate[((2 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/(1 - 2*x)^(5/2),x]
Output:
-1/37800*(30*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(4641769 - 12151171*x + 2783146*x ^2 + 1279350*x^3 + 486900*x^4 + 94500*x^5) + (174654791*I)*Sqrt[33 - 66*x] *(-1 + 2*x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (179908435*I)*Sq rt[33 - 66*x]*(-1 + 2*x)*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/(1 - 2*x)^(3/2)
Time = 0.32 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.13, number of steps used = 15, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.536, Rules used = {108, 27, 167, 27, 171, 27, 171, 25, 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.
| \(\displaystyle \int \frac {(3 x+2)^{7/2} (5 x+3)^{5/2}}{(1-2 x)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {(3 x+2)^{7/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{3} \int \frac {(3 x+2)^{5/2} (5 x+3)^{3/2} (180 x+113)}{2 (1-2 x)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {(3 x+2)^{7/2} (5 x+3)^{5/2}}{3 (1-2 x)^{3/2}}-\frac {1}{6} \int \frac {(3 x+2)^{5/2} (5 x+3)^{3/2} (180 x+113)}{(1-2 x)^{3/2}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{6} \left (-\frac {1}{11} \int -\frac {5 (3 x+2)^{3/2} (5 x+3)^{3/2} (6075 x+3847)}{\sqrt {1-2 x}}dx-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{11} \int \frac {(3 x+2)^{3/2} (5 x+3)^{3/2} (6075 x+3847)}{\sqrt {1-2 x}}dx-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{11} \left (-\frac {1}{45} \int -\frac {45 \sqrt {3 x+2} (5 x+3)^{3/2} (37662 x+24163)}{2 \sqrt {1-2 x}}dx-135 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{11} \left (\frac {1}{2} \int \frac {\sqrt {3 x+2} (5 x+3)^{3/2} (37662 x+24163)}{\sqrt {1-2 x}}dx-135 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{11} \left (\frac {1}{2} \left (-\frac {1}{35} \int -\frac {(5 x+3)^{3/2} (3930609 x+2576467)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {37662}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-135 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{11} \left (\frac {1}{2} \left (\frac {1}{35} \int \frac {(5 x+3)^{3/2} (3930609 x+2576467)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {37662}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-135 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{11} \left (\frac {1}{2} \left (\frac {1}{35} \left (-\frac {1}{15} \int -\frac {99 \sqrt {5 x+3} (5253644 x+3414227)}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {1310203}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {37662}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-135 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{11} \left (\frac {1}{2} \left (\frac {1}{35} \left (\frac {33}{10} \int \frac {\sqrt {5 x+3} (5253644 x+3414227)}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {1310203}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {37662}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-135 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{11} \left (\frac {1}{2} \left (\frac {1}{35} \left (\frac {33}{10} \left (-\frac {1}{9} \int -\frac {174654791 x+110571883}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5253644}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {1310203}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {37662}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-135 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{11} \left (\frac {1}{2} \left (\frac {1}{35} \left (\frac {33}{10} \left (\frac {1}{9} \int \frac {174654791 x+110571883}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {5253644}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {1310203}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {37662}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-135 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{11} \left (\frac {1}{2} \left (\frac {1}{35} \left (\frac {33}{10} \left (\frac {1}{9} \left (\frac {28895042}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {174654791}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {5253644}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {1310203}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {37662}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-135 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{11} \left (\frac {1}{2} \left (\frac {1}{35} \left (\frac {33}{10} \left (\frac {1}{9} \left (\frac {28895042}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {174654791}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {5253644}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {1310203}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {37662}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-135 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{6} \left (\frac {5}{11} \left (\frac {1}{2} \left (\frac {1}{35} \left (\frac {33}{10} \left (\frac {1}{9} \left (-\frac {5253644}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {174654791}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {5253644}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {1310203}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {37662}{35} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-135 \sqrt {1-2 x} (3 x+2)^{3/2} (5 x+3)^{5/2}\right )-\frac {406 (3 x+2)^{5/2} (5 x+3)^{5/2}}{11 \sqrt {1-2 x}}\right )+\frac {(5 x+3)^{5/2} (3 x+2)^{7/2}}{3 (1-2 x)^{3/2}}\) |
Input:
Int[((2 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/(1 - 2*x)^(5/2),x]
Output:
((2 + 3*x)^(7/2)*(3 + 5*x)^(5/2))/(3*(1 - 2*x)^(3/2)) + ((-406*(2 + 3*x)^( 5/2)*(3 + 5*x)^(5/2))/(11*Sqrt[1 - 2*x]) + (5*(-135*Sqrt[1 - 2*x]*(2 + 3*x )^(3/2)*(3 + 5*x)^(5/2) + ((-37662*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^( 5/2))/35 + ((-1310203*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5 + (33 *((-5253644*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((-174654791*Sq rt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (5253644*S qrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/9))/10)/35 )/2))/11)/6
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 + 1)*(c + d*x)^n*((e + f*x)^p/(b*(m + 1))) , x] - Simp[1/(b*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f* x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c , d, e, f}, x] && LtQ[m, -1] && GtQ[n, 0] && GtQ[p, 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[(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[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 = 0.74 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.95
| method | result | size |
| default | \(-\frac {\left (42525000 x^{7}+173370252 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-349309582 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+272970000 x^{6}-86685126 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+174654791 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+870250500 x^{5}+2069287200 x^{4}-3651350730 x^{3}-4336405140 x^{2}+458597550 x +835518420\right ) \sqrt {3+5 x}\, \sqrt {2+3 x}}{37800 \left (1-2 x \right )^{\frac {3}{2}} \left (15 x^{2}+19 x +6\right )}\) | \(237\) |
| elliptic | \(\frac {\sqrt {3+5 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {40825 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{112}-\frac {4474921 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5040}+\frac {110571883 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{52920 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {174654791 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{52920 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {1615 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{14}-\frac {75 x^{3} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4}+\frac {41503 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{384 \left (x -\frac {1}{2}\right )^{2}}+\frac {-\frac {1549625}{32} x^{2}-\frac {5888575}{96} x -\frac {309925}{16}}{\sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}\right )}{\left (15 x^{2}+19 x +6\right ) \sqrt {1-2 x}}\) | \(320\) |
Input:
int((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/37800*(42525000*x^7+173370252*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2 *70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-349309582*2^(1/2)* EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2) *(1-2*x)^(1/2)+272970000*x^6-86685126*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))+174654791*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))+870250500*x^5+2069287200*x^4-3651350730*x^3-4336405140*x^2 +458597550*x+835518420)*(3+5*x)^(1/2)*(2+3*x)^(1/2)/(1-2*x)^(3/2)/(15*x^2+ 19*x+6)
Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.43 \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=-\frac {2700 \, {\left (94500 \, x^{5} + 486900 \, x^{4} + 1279350 \, x^{3} + 2783146 \, x^{2} - 12151171 \, x + 4641769\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 5934409277 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 15718931190 \, \sqrt {-30} {\left (4 \, x^{2} - 4 \, x + 1\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{3402000 \, {\left (4 \, x^{2} - 4 \, x + 1\right )}} \] Input:
integrate((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(5/2),x, algorithm="fricas")
Output:
-1/3402000*(2700*(94500*x^5 + 486900*x^4 + 1279350*x^3 + 2783146*x^2 - 121 51171*x + 4641769)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 5934409277 *sqrt(-30)*(4*x^2 - 4*x + 1)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 15718931190*sqrt(-30)*(4*x^2 - 4*x + 1)*weierstrassZeta(1159/67 5, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(4 *x^2 - 4*x + 1)
Timed out. \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((2+3*x)**(7/2)*(3+5*x)**(5/2)/(1-2*x)**(5/2),x)
Output:
Timed out
\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(5/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2), x)
\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (3 \, x + 2\right )}^{\frac {7}{2}}}{{\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(5/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(5/2)*(3*x + 2)^(7/2)/(-2*x + 1)^(5/2), x)
Timed out. \[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}} \,d x \] Input:
int(((3*x + 2)^(7/2)*(5*x + 3)^(5/2))/(1 - 2*x)^(5/2),x)
Output:
int(((3*x + 2)^(7/2)*(5*x + 3)^(5/2))/(1 - 2*x)^(5/2), x)
\[ \int \frac {(2+3 x)^{7/2} (3+5 x)^{5/2}}{(1-2 x)^{5/2}} \, dx=\frac {-300510000 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{5}-1548342000 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}-4068333000 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}-8850404280 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}-35412907604 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x +25414914666 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-4845622181400 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right ) x^{2}+4845622181400 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right ) x -1211405545350 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right )+1938807080364 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right ) x^{2}-1938807080364 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right ) x +484701770091 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{120 x^{5}-28 x^{4}-90 x^{3}+27 x^{2}+17 x -6}d x \right )}{16027200 x^{2}-16027200 x +4006800} \] Input:
int((2+3*x)^(7/2)*(3+5*x)^(5/2)/(1-2*x)^(5/2),x)
Output:
( - 300510000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**5 - 15483420 00*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 4068333000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 8850404280*sqrt(3*x + 2)*sqrt( 5*x + 3)*sqrt( - 2*x + 1)*x**2 - 35412907604*sqrt(3*x + 2)*sqrt(5*x + 3)*s qrt( - 2*x + 1)*x + 25414914666*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) - 4845622181400*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2) /(120*x**5 - 28*x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x)*x**2 + 48456221814 00*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(120*x**5 - 28* x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x)*x - 1211405545350*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(120*x**5 - 28*x**4 - 90*x**3 + 27 *x**2 + 17*x - 6),x) + 1938807080364*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt ( - 2*x + 1))/(120*x**5 - 28*x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x)*x**2 - 1938807080364*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(120*x* *5 - 28*x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x)*x + 484701770091*int((sqrt (3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(120*x**5 - 28*x**4 - 90*x**3 + 27*x**2 + 17*x - 6),x))/(4006800*(4*x**2 - 4*x + 1))