Integrand size = 28, antiderivative size = 249 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=-\frac {74 (1-2 x)^{3/2} \sqrt {3+5 x}}{2079 (2+3 x)^{9/2}}+\frac {4222 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{3/2}}+\frac {1446357824 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}-\frac {1446357824 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{6417873 \sqrt {35}}+\frac {41599832 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{6417873 \sqrt {35}} \] Output:
-74/2079*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)+4222/43659*(1-2*x)^(1/2 )*(3+5*x)^(1/2)/(2+3*x)^(7/2)+442076/1528065*(1-2*x)^(1/2)*(3+5*x)^(1/2)/( 2+3*x)^(5/2)+20799916/10696455*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+1 446357824/74875185*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)-2/33*(1-2*x)^ (3/2)*(3+5*x)^(3/2)/(2+3*x)^(11/2)-1446357824/224625555*EllipticE(1/11*55^ (1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+41599832/224625555*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 = 7.55 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.45 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=\frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (24398176891+180988667568 x+537061687749 x^2+797050394730 x^3+591671694906 x^4+175732475616 x^5\right )}{4 (2+3 x)^{11/2}}+i \sqrt {33} \left (180794728 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-186236855 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{224625555} \] Input:
Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(13/2),x]
Output:
(8*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(24398176891 + 180988667568*x + 5370616 87749*x^2 + 797050394730*x^3 + 591671694906*x^4 + 175732475616*x^5))/(4*(2 + 3*x)^(11/2)) + I*Sqrt[33]*(180794728*EllipticE[I*ArcSinh[Sqrt[9 + 15*x] ], -2/33] - 186236855*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/22462 5555
Time = 0.32 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.14, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {108, 27, 167, 167, 27, 169, 27, 169, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^{13/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{33} \int -\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (20 x+1)}{2 (3 x+2)^{11/2}}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {1}{11} \int \frac {\sqrt {1-2 x} \sqrt {5 x+3} (20 x+1)}{(3 x+2)^{11/2}}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{27} \int \frac {(576-745 x) \sqrt {5 x+3}}{\sqrt {1-2 x} (3 x+2)^{9/2}}dx+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{27} \left (\frac {2}{147} \int \frac {22423-21625 x}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {6436 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{27} \left (\frac {1}{147} \int \frac {22423-21625 x}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {6436 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{27} \left (\frac {1}{147} \left (\frac {2}{35} \int \frac {3 (996533-1105190 x)}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {221038 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {6436 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{27} \left (\frac {1}{147} \left (\frac {3}{35} \int \frac {996533-1105190 x}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx+\frac {221038 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {6436 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{27} \left (\frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \int \frac {42931646-25999895 x}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx+\frac {10399958 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {221038 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {6436 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{27} \left (\frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {2}{7} \int \frac {5 (361589456 x+228926353)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {361589456 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {10399958 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {221038 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {6436 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{27} \left (\frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \int \frac {361589456 x+228926353}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {361589456 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {10399958 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {221038 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {6436 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{27} \left (\frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {59863397}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {361589456}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {361589456 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {10399958 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {221038 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {6436 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{27} \left (\frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {59863397}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {361589456}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {361589456 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {10399958 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {221038 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {6436 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{11} \left (\frac {2}{27} \left (\frac {1}{147} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (-\frac {10884254}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {361589456}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {361589456 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )+\frac {10399958 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )+\frac {221038 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {6436 \sqrt {1-2 x} \sqrt {5 x+3}}{147 (3 x+2)^{7/2}}\right )+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}\) |
Input:
Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(13/2),x]
Output:
(-2*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(33*(2 + 3*x)^(11/2)) + ((74*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(27*(2 + 3*x)^(9/2)) + (2*((-6436*Sqrt[1 - 2*x]*Sqr t[3 + 5*x])/(147*(2 + 3*x)^(7/2)) + ((221038*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/ (35*(2 + 3*x)^(5/2)) + (3*((10399958*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(3/2)) + (2*((361589456*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x ]) + (5*((-361589456*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (10884254*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x] ], 35/33])/5))/7))/21))/35)/147))/27)/11
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[(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 = 0.53 (sec) , antiderivative size = 332, normalized size of antiderivative = 1.33
| method | result | size |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{6}}-\frac {296 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{216513 \left (\frac {2}{3}+x \right )^{5}}+\frac {8198 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3536379 \left (\frac {2}{3}+x \right )^{4}}+\frac {442076 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{41257755 \left (\frac {2}{3}+x \right )^{3}}+\frac {20799916 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{96268095 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {2892715648}{14975037} x^{2}-\frac {1446357824}{74875185} x +\frac {1446357824}{24958395}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {915705412 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{314475777 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {1446357824 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{314475777 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(332\) |
| default | \(-\frac {2 \left (87280832826 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-175732475616 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+290936109420 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-585774918720 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+387914812560 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-781033224960 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+258609875040 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-520688816640 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-5271974268480 x^{7}+86203291680 \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}-173562938880 \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}-18277348274028 x^{6}+11493772224 \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 )-23141725184 \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 )-24104934646074 x^{5}-13177956562506 x^{4}+132608462283 x^{3}+3558643880307 x^{2}+1555703477439 x +219583592019\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{224625555 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {11}{2}}}\) | \(587\) |
Input:
int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(13/2),x,method=_RETURNVERBOSE)
Output:
-(1-2*x)^(1/2)*(3+5*x)^(1/2)*(-(3+5*x)*(-1+2*x)*(2+3*x))^(1/2)/(10*x^2+x-3 )/(2+3*x)^(1/2)*(14/216513*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^6-296/2165 13*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^5+8198/3536379*(-30*x^3-23*x^2+7*x +6)^(1/2)/(2/3+x)^4+442076/41257755*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3 +20799916/96268095*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+1446357824/22462 5555*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+915705412/314475777*( 28+42*x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2 )*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+1446357824/314475777*(28+42* x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-1/ 15*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-3/5*EllipticF(1/7*(28+42*x) ^(1/2),1/2*70^(1/2))))
Time = 0.12 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=\frac {2 \, {\left (135 \, {\left (175732475616 \, x^{5} + 591671694906 \, x^{4} + 797050394730 \, x^{3} + 537061687749 \, x^{2} + 180988667568 \, x + 24398176891\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 12286814282 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 32543051040 \, \sqrt {-30} {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{10108149975 \, {\left (729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64\right )}} \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(13/2),x, algorithm="fricas" )
Output:
2/10108149975*(135*(175732475616*x^5 + 591671694906*x^4 + 797050394730*x^3 + 537061687749*x^2 + 180988667568*x + 24398176891)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 12286814282*sqrt(-30)*(729*x^6 + 2916*x^5 + 4860*x^ 4 + 4320*x^3 + 2160*x^2 + 576*x + 64)*weierstrassPInverse(1159/675, 38998/ 91125, x + 23/90) + 32543051040*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, weierstrassPInverse(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. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=\text {Timed out} \] Input:
integrate((1-2*x)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**(13/2),x)
Output:
Timed out
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(13/2),x, algorithm="maxima" )
Output:
integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(13/2), x)
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}} \,d x } \] Input:
integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(13/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(13/2), x)
Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{13/2}} \,d x \] Input:
int(((1 - 2*x)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^(13/2),x)
Output:
int(((1 - 2*x)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^(13/2), x)
\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx =\text {Too large to display} \] Input:
int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(13/2),x)
Output:
(700*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 440*sqrt(3*x + 2) *sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 1518*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) - 154420425*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)* x**2)/(21870*x**9 + 104247*x**8 + 207765*x**7 + 216594*x**6 + 112644*x**5 + 7560*x**4 - 25872*x**3 - 15520*x**2 - 3904*x - 384),x)*x**6 - 617681700* int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(21870*x**9 + 1042 47*x**8 + 207765*x**7 + 216594*x**6 + 112644*x**5 + 7560*x**4 - 25872*x**3 - 15520*x**2 - 3904*x - 384),x)*x**5 - 1029469500*int((sqrt(3*x + 2)*sqrt (5*x + 3)*sqrt( - 2*x + 1)*x**2)/(21870*x**9 + 104247*x**8 + 207765*x**7 + 216594*x**6 + 112644*x**5 + 7560*x**4 - 25872*x**3 - 15520*x**2 - 3904*x - 384),x)*x**4 - 915084000*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(21870*x**9 + 104247*x**8 + 207765*x**7 + 216594*x**6 + 112644*x* *5 + 7560*x**4 - 25872*x**3 - 15520*x**2 - 3904*x - 384),x)*x**3 - 4575420 00*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(21870*x**9 + 1 04247*x**8 + 207765*x**7 + 216594*x**6 + 112644*x**5 + 7560*x**4 - 25872*x **3 - 15520*x**2 - 3904*x - 384),x)*x**2 - 122011200*int((sqrt(3*x + 2)*sq rt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(21870*x**9 + 104247*x**8 + 207765*x**7 + 216594*x**6 + 112644*x**5 + 7560*x**4 - 25872*x**3 - 15520*x**2 - 3904* x - 384),x)*x - 13556800*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) *x**2)/(21870*x**9 + 104247*x**8 + 207765*x**7 + 216594*x**6 + 112644*x...