Integrand size = 28, antiderivative size = 253 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\frac {295 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 (2+3 x)^{9/2}}-\frac {67345 \sqrt {1-2 x} \sqrt {3+5 x}}{64827 (2+3 x)^{7/2}}-\frac {167228 \sqrt {1-2 x} \sqrt {3+5 x}}{453789 (2+3 x)^{5/2}}-\frac {392998 \sqrt {1-2 x} \sqrt {3+5 x}}{3176523 (2+3 x)^{3/2}}+\frac {6036028 \sqrt {1-2 x} \sqrt {3+5 x}}{22235661 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{7 \sqrt {1-2 x} (2+3 x)^{9/2}}-\frac {6036028 \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{9529569}-\frac {785996 \sqrt {\frac {5}{7}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{9529569} \] Output:
295/1323*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)-67345/64827*(1-2*x)^(1/ 2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)-167228/453789*(1-2*x)^(1/2)*(3+5*x)^(1/2)/( 2+3*x)^(5/2)-392998/3176523*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+6036 028/22235661*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)+11/7*(3+5*x)^(3/2)/ (1-2*x)^(1/2)/(2+3*x)^(9/2)-6036028/66706983*EllipticE(1/11*55^(1/2)*(1-2* x)^(1/2),1/35*1155^(1/2))*35^(1/2)-785996/66706983*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 = 7.83 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.44 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\frac {4 \left (-\frac {3 \sqrt {3+5 x} \left (-52688263-243200677 x-227945505 x^2+466728543 x^3+985046292 x^4+488918268 x^5\right )}{2 \sqrt {1-2 x} (2+3 x)^{9/2}}+i \sqrt {33} \left (1509007 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-1808870 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{66706983} \] Input:
Integrate[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^(11/2)),x]
Output:
(4*((-3*Sqrt[3 + 5*x]*(-52688263 - 243200677*x - 227945505*x^2 + 466728543 *x^3 + 985046292*x^4 + 488918268*x^5))/(2*Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)) + I*Sqrt[33]*(1509007*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 1808870 *EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])))/66706983
Time = 0.31 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.464, Rules used = {109, 27, 167, 27, 169, 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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{(1-2 x)^{3/2} (3 x+2)^{11/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}-\frac {1}{7} \int -\frac {5 \sqrt {5 x+3} (196 x+111)}{2 \sqrt {1-2 x} (3 x+2)^{11/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \int \frac {\sqrt {5 x+3} (196 x+111)}{\sqrt {1-2 x} (3 x+2)^{11/2}}dx+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {5}{14} \left (\frac {2}{189} \int \frac {57610 x+33917}{2 \sqrt {1-2 x} (3 x+2)^{9/2} \sqrt {5 x+3}}dx+\frac {118 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{189} \int \frac {57610 x+33917}{\sqrt {1-2 x} (3 x+2)^{9/2} \sqrt {5 x+3}}dx+\frac {118 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{189} \left (\frac {2}{49} \int \frac {336725 x+196612}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {26938 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {118 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{189} \left (\frac {2}{49} \left (\frac {2}{35} \int \frac {3 (836140 x+491927)}{2 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {26938 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {118 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{189} \left (\frac {2}{49} \left (\frac {3}{35} \int \frac {836140 x+491927}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {26938 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {118 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{189} \left (\frac {2}{49} \left (\frac {3}{35} \left (\frac {2}{21} \int \frac {982495 x+1157999}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {392998 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {26938 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {118 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{189} \left (\frac {2}{49} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {2}{7} \int \frac {5 (3018014 x+2470507)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {3018014 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {392998 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {26938 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {118 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{189} \left (\frac {2}{49} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \int \frac {3018014 x+2470507}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {3018014 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {392998 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {26938 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {118 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{189} \left (\frac {2}{49} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {3298493}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {3018014}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {3018014 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {392998 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {26938 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {118 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{189} \left (\frac {2}{49} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (\frac {3298493}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {3018014}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {3018014 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {392998 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {26938 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {118 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{189} \left (\frac {2}{49} \left (\frac {3}{35} \left (\frac {2}{21} \left (\frac {5}{7} \left (-\frac {599726}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {3018014}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {3018014 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {392998 \sqrt {1-2 x} \sqrt {5 x+3}}{21 (3 x+2)^{3/2}}\right )-\frac {167228 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {26938 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {118 \sqrt {1-2 x} \sqrt {5 x+3}}{189 (3 x+2)^{9/2}}\right )+\frac {11 (5 x+3)^{3/2}}{7 \sqrt {1-2 x} (3 x+2)^{9/2}}\) |
Input:
Int[(3 + 5*x)^(5/2)/((1 - 2*x)^(3/2)*(2 + 3*x)^(11/2)),x]
Output:
(11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(9/2)) + (5*((118*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(189*(2 + 3*x)^(9/2)) + ((-26938*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(49*(2 + 3*x)^(7/2)) + (2*((-167228*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/ (35*(2 + 3*x)^(5/2)) + (3*((-392998*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(3/2)) + (2*((3018014*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x]) + (5*((-3018014*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/3 3])/5 - (599726*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/3 3])/5))/7))/21))/35))/49)/189))/14
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 = 1.01 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.28
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {968 \left (-30 x^{2}-38 x -12\right )}{117649 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {24705070 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{466948881 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {30180140 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{466948881 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{321489 \left (\frac {2}{3}+x \right )^{5}}+\frac {1262 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5250987 \left (\frac {2}{3}+x \right )^{4}}-\frac {30014 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{12252303 \left (\frac {2}{3}+x \right )^{3}}-\frac {118570 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{28588707 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {65848840}{22235661} x^{2}-\frac {6584884}{22235661} x +\frac {6584884}{7411887}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(325\) |
| default | \(-\frac {2 \sqrt {3+5 x}\, \sqrt {1-2 x}\, \left (801533799 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-244459134 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2137423464 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-651891024 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+2137423464 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-651891024 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+949965984 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-289729344 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-7333774020 x^{6}+158327664 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-48288224 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-19175958792 x^{5}-15866344773 x^{4}-781374312 x^{3}+5699519700 x^{2}+2979130038 x +474194367\right )}{66706983 \left (2+3 x \right )^{\frac {9}{2}} \left (10 x^{2}+x -3\right )}\) | \(494\) |
Input:
int((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(11/2),x,method=_RETURNVERBOSE)
Output:
(-(3+5*x)*(-1+2*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 )*(-968/117649*(-30*x^2-38*x-12)/((x-1/2)*(-30*x^2-38*x-12))^(1/2)+2470507 0/466948881*(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))+30180140/466948 881*(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)))-2/321489*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/ 3+x)^5+1262/5250987*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4-30014/12252303* (-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3-118570/28588707*(-30*x^3-23*x^2+7*x +6)^(1/2)/(2/3+x)^2+6584884/66706983*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x +9))^(1/2))
Time = 0.08 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.66 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\frac {2 \, {\left (135 \, {\left (488918268 \, x^{5} + 985046292 \, x^{4} + 466728543 \, x^{3} - 227945505 \, x^{2} - 243200677 \, x - 52688263\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 76465654 \, \sqrt {-30} {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 135810630 \, \sqrt {-30} {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{3001814235 \, {\left (486 \, x^{6} + 1377 \, x^{5} + 1350 \, x^{4} + 360 \, x^{3} - 240 \, x^{2} - 176 \, x - 32\right )}} \] Input:
integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(11/2),x, algorithm="fricas" )
Output:
2/3001814235*(135*(488918268*x^5 + 985046292*x^4 + 466728543*x^3 - 2279455 05*x^2 - 243200677*x - 52688263)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1 ) - 76465654*sqrt(-30)*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 1358 10630*sqrt(-30)*(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240*x^2 - 176*x - 32)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675 , 38998/91125, x + 23/90)))/(486*x^6 + 1377*x^5 + 1350*x^4 + 360*x^3 - 240 *x^2 - 176*x - 32)
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\text {Timed out} \] Input:
integrate((3+5*x)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**(11/2),x)
Output:
Timed out
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(11/2),x, algorithm="maxima" )
Output:
integrate((5*x + 3)^(5/2)/((3*x + 2)^(11/2)*(-2*x + 1)^(3/2)), x)
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(11/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(5/2)/((3*x + 2)^(11/2)*(-2*x + 1)^(3/2)), x)
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{11/2}} \,d x \] Input:
int((5*x + 3)^(5/2)/((1 - 2*x)^(3/2)*(3*x + 2)^(11/2)),x)
Output:
int((5*x + 3)^(5/2)/((1 - 2*x)^(3/2)*(3*x + 2)^(11/2)), x)
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{11/2}} \, dx =\text {Too large to display} \] Input:
int((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(11/2),x)
Output:
( - 200*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 906*sqrt(3*x + 2) *sqrt(5*x + 3)*sqrt( - 2*x + 1) - 41752260*int((sqrt(3*x + 2)*sqrt(5*x + 3 )*sqrt( - 2*x + 1)*x**2)/(14580*x**9 + 52488*x**8 + 68769*x**7 + 29295*x** 6 - 16632*x**5 - 21420*x**4 - 5488*x**3 + 1936*x**2 + 1280*x + 192),x)*x** 6 - 118298070*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(145 80*x**9 + 52488*x**8 + 68769*x**7 + 29295*x**6 - 16632*x**5 - 21420*x**4 - 5488*x**3 + 1936*x**2 + 1280*x + 192),x)*x**5 - 115978500*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(14580*x**9 + 52488*x**8 + 68769* x**7 + 29295*x**6 - 16632*x**5 - 21420*x**4 - 5488*x**3 + 1936*x**2 + 1280 *x + 192),x)*x**4 - 30927600*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(14580*x**9 + 52488*x**8 + 68769*x**7 + 29295*x**6 - 16632*x**5 - 21420*x**4 - 5488*x**3 + 1936*x**2 + 1280*x + 192),x)*x**3 + 20618400*i nt((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(14580*x**9 + 52488 *x**8 + 68769*x**7 + 29295*x**6 - 16632*x**5 - 21420*x**4 - 5488*x**3 + 19 36*x**2 + 1280*x + 192),x)*x**2 + 15120160*int((sqrt(3*x + 2)*sqrt(5*x + 3 )*sqrt( - 2*x + 1)*x**2)/(14580*x**9 + 52488*x**8 + 68769*x**7 + 29295*x** 6 - 16632*x**5 - 21420*x**4 - 5488*x**3 + 1936*x**2 + 1280*x + 192),x)*x + 2749120*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(14580*x* *9 + 52488*x**8 + 68769*x**7 + 29295*x**6 - 16632*x**5 - 21420*x**4 - 5488 *x**3 + 1936*x**2 + 1280*x + 192),x) + 14610618*int((sqrt(3*x + 2)*sqrt...