Integrand size = 28, antiderivative size = 253 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx=\frac {220 \sqrt {3+5 x}}{49 \sqrt {1-2 x} (2+3 x)^{7/2}}-\frac {4545 \sqrt {1-2 x} \sqrt {3+5 x}}{2401 (2+3 x)^{7/2}}-\frac {11433 \sqrt {1-2 x} \sqrt {3+5 x}}{16807 (2+3 x)^{5/2}}-\frac {33778 \sqrt {1-2 x} \sqrt {3+5 x}}{117649 (2+3 x)^{3/2}}-\frac {98642 \sqrt {1-2 x} \sqrt {3+5 x}}{823543 \sqrt {2+3 x}}+\frac {11 (3+5 x)^{3/2}}{21 (1-2 x)^{3/2} (2+3 x)^{7/2}}+\frac {98642 \sqrt {\frac {5}{7}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{352947}-\frac {67556 \sqrt {\frac {5}{7}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{352947} \] Output:
220/49*(3+5*x)^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(7/2)-4545/2401*(1-2*x)^(1/2)*( 3+5*x)^(1/2)/(2+3*x)^(7/2)-11433/16807*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x) ^(5/2)-33778/117649*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)-98642/823543 *(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)+11/21*(3+5*x)^(3/2)/(1-2*x)^(3/ 2)/(2+3*x)^(7/2)+98642/2470629*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35* 1155^(1/2))*35^(1/2)-67556/2470629*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 = 8.60 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.43 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx=\frac {2 \left (\frac {\sqrt {3+5 x} \left (866085+6524789 x+10746933 x^2-7681599 x^3-28748088 x^4-15980004 x^5\right )}{(1-2 x)^{3/2} (2+3 x)^{7/2}}-i \sqrt {33} \left (49321 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-16485 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{2470629} \] Input:
Integrate[(3 + 5*x)^(5/2)/((1 - 2*x)^(5/2)*(2 + 3*x)^(9/2)),x]
Output:
(2*((Sqrt[3 + 5*x]*(866085 + 6524789*x + 10746933*x^2 - 7681599*x^3 - 2874 8088*x^4 - 15980004*x^5))/((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)) - I*Sqrt[33]*( 49321*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 16485*EllipticF[I*ArcS inh[Sqrt[9 + 15*x]], -2/33])))/2470629
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, 169, 27, 169, 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.
| \(\displaystyle \int \frac {(5 x+3)^{5/2}}{(1-2 x)^{5/2} (3 x+2)^{9/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}-\frac {1}{21} \int -\frac {15 \sqrt {5 x+3} (42 x+23)}{2 (1-2 x)^{3/2} (3 x+2)^{9/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \int \frac {\sqrt {5 x+3} (42 x+23)}{(1-2 x)^{3/2} (3 x+2)^{9/2}}dx+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \int \frac {3885 x+2287}{\sqrt {1-2 x} (3 x+2)^{9/2} \sqrt {5 x+3}}dx+\frac {88 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {2}{49} \int \frac {45450 x+26489}{2 \sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {1818 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {88 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{49} \int \frac {45450 x+26489}{\sqrt {1-2 x} (3 x+2)^{7/2} \sqrt {5 x+3}}dx-\frac {1818 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {88 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{49} \left (\frac {2}{35} \int \frac {171495 x+97441}{\sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}}dx-\frac {22866 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1818 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {88 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{49} \left (\frac {2}{35} \left (\frac {2}{21} \int \frac {3 (168890 x+96153)}{2 \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {33778 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {22866 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1818 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {88 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{49} \left (\frac {2}{35} \left (\frac {1}{7} \int \frac {168890 x+96153}{\sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}}dx-\frac {33778 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {22866 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1818 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {88 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 169 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{49} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {2}{7} \int \frac {5 (6527-49321 x)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {98642 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {33778 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {22866 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1818 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {88 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{49} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \int \frac {6527-49321 x}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {98642 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {33778 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {22866 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1818 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {88 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{49} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {180598}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {49321}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {98642 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {33778 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {22866 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1818 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {88 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{49} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {180598}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {49321}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {98642 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {33778 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {22866 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1818 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {88 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {5}{14} \left (\frac {1}{7} \left (\frac {1}{49} \left (\frac {2}{35} \left (\frac {1}{7} \left (\frac {10}{7} \left (\frac {49321}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {32836}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )\right )-\frac {98642 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}}\right )-\frac {33778 \sqrt {1-2 x} \sqrt {5 x+3}}{7 (3 x+2)^{3/2}}\right )-\frac {22866 \sqrt {1-2 x} \sqrt {5 x+3}}{35 (3 x+2)^{5/2}}\right )-\frac {1818 \sqrt {1-2 x} \sqrt {5 x+3}}{49 (3 x+2)^{7/2}}\right )+\frac {88 \sqrt {5 x+3}}{7 \sqrt {1-2 x} (3 x+2)^{7/2}}\right )+\frac {11 (5 x+3)^{3/2}}{21 (1-2 x)^{3/2} (3 x+2)^{7/2}}\) |
Input:
Int[(3 + 5*x)^(5/2)/((1 - 2*x)^(5/2)*(2 + 3*x)^(9/2)),x]
Output:
(11*(3 + 5*x)^(3/2))/(21*(1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)) + (5*((88*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) + ((-1818*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(49*(2 + 3*x)^(7/2)) + ((-22866*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35*( 2 + 3*x)^(5/2)) + (2*((-33778*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*(2 + 3*x)^(3 /2)) + ((-98642*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x]) + (10*((493 21*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (3283 6*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5))/7)/7)) /35)/49)/7))/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 = 1.07 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.28
| method | result | size |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{50421 \left (x -\frac {1}{2}\right )^{2}}-\frac {1364 \left (-30 x^{2}-38 x -12\right )}{50421 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}+\frac {65270 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{17294403 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {493210 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{17294403 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{194481 \left (\frac {2}{3}+x \right )^{4}}+\frac {194 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{453789 \left (\frac {2}{3}+x \right )^{3}}-\frac {1814 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{352947 \left (\frac {2}{3}+x \right )^{2}}-\frac {10602 \left (-30 x^{2}-3 x +9\right )}{823543 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(325\) |
| default | \(-\frac {2 \left (-327380130-381943485 x +1677900420 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{6}+1636900650 x^{3}+1254957165 x^{2}+2617697439 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{4}-1362159624 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{2}-502000632 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x +18643338 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \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 ) x^{2}+614394396 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \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 ) x^{4}+55930014 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \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 ) x^{4}-20714820 \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, x -227553480 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, x +921591594 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \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 ) x^{3}+83895021 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \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 ) x^{3}+204798132 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \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 ) x^{2}+4025289492 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{5}-91021392 \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 ) \left (-30 x^{3}-23 x^{2}+7 x +6\right )-8285928 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-644487228 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{3}\right ) \sqrt {1-2 x}}{51883209 \sqrt {3+5 x}\, \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \left (-1+2 x \right )^{2} \left (2+3 x \right )^{\frac {7}{2}}}\) | \(761\) |
Input:
int((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^(9/2),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 )*(242/50421*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x-1/2)^2-1364/50421*(-30*x^2-38 *x-12)/((x-1/2)*(-30*x^2-38*x-12))^(1/2)+65270/17294403*(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))-493210/17294403*(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+4 2*x)^(1/2),1/2*70^(1/2))-3/5*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2)))- 2/194481*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^4+194/453789*(-30*x^3-23*x^2 +7*x+6)^(1/2)/(2/3+x)^3-1814/352947*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2 -10602/823543*(-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 \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx=-\frac {90 \, {\left (15980004 \, x^{5} + 28748088 \, x^{4} + 7681599 \, x^{3} - 10746933 \, x^{2} - 6524789 \, x - 866085\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 1721813 \, \sqrt {-30} {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 4438890 \, \sqrt {-30} {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{111178305 \, {\left (324 \, x^{6} + 540 \, x^{5} + 81 \, x^{4} - 264 \, x^{3} - 104 \, x^{2} + 32 \, x + 16\right )}} \] Input:
integrate((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^(9/2),x, algorithm="fricas")
Output:
-1/111178305*(90*(15980004*x^5 + 28748088*x^4 + 7681599*x^3 - 10746933*x^2 - 6524789*x - 866085)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 172181 3*sqrt(-30)*(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*w eierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 4438890*sqrt(-30)*( 324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)*weierstrassZet a(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23 /90)))/(324*x^6 + 540*x^5 + 81*x^4 - 264*x^3 - 104*x^2 + 32*x + 16)
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((3+5*x)**(5/2)/(1-2*x)**(5/2)/(2+3*x)**(9/2),x)
Output:
Timed out
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^(9/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(5/2)/((3*x + 2)^(9/2)*(-2*x + 1)^(5/2)), x)
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {9}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^(9/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(5/2)/((3*x + 2)^(9/2)*(-2*x + 1)^(5/2)), x)
Timed out. \[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx=\int \frac {{\left (5\,x+3\right )}^{5/2}}{{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{9/2}} \,d x \] Input:
int((5*x + 3)^(5/2)/((1 - 2*x)^(5/2)*(3*x + 2)^(9/2)),x)
Output:
int((5*x + 3)^(5/2)/((1 - 2*x)^(5/2)*(3*x + 2)^(9/2)), x)
\[ \int \frac {(3+5 x)^{5/2}}{(1-2 x)^{5/2} (2+3 x)^{9/2}} \, dx =\text {Too large to display} \] Input:
int((3+5*x)^(5/2)/(1-2*x)^(5/2)/(2+3*x)^(9/2),x)
Output:
(1300*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 2298*sqrt(3*x + 2)* sqrt(5*x + 3)*sqrt( - 2*x + 1) + 38598120*int((sqrt(3*x + 2)*sqrt(5*x + 3) *sqrt( - 2*x + 1)*x**2)/(9720*x**9 + 23652*x**8 + 12582*x**7 - 11781*x**6 - 12999*x**5 - 70*x**4 + 3528*x**3 + 768*x**2 - 304*x - 96),x)*x**6 + 6433 0200*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(9720*x**9 + 23652*x**8 + 12582*x**7 - 11781*x**6 - 12999*x**5 - 70*x**4 + 3528*x**3 + 768*x**2 - 304*x - 96),x)*x**5 + 9649530*int((sqrt(3*x + 2)*sqrt(5*x + 3)* sqrt( - 2*x + 1)*x**2)/(9720*x**9 + 23652*x**8 + 12582*x**7 - 11781*x**6 - 12999*x**5 - 70*x**4 + 3528*x**3 + 768*x**2 - 304*x - 96),x)*x**4 - 31450 320*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(9720*x**9 + 2 3652*x**8 + 12582*x**7 - 11781*x**6 - 12999*x**5 - 70*x**4 + 3528*x**3 + 7 68*x**2 - 304*x - 96),x)*x**3 - 12389520*int((sqrt(3*x + 2)*sqrt(5*x + 3)* sqrt( - 2*x + 1)*x**2)/(9720*x**9 + 23652*x**8 + 12582*x**7 - 11781*x**6 - 12999*x**5 - 70*x**4 + 3528*x**3 + 768*x**2 - 304*x - 96),x)*x**2 + 38121 60*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(9720*x**9 + 23 652*x**8 + 12582*x**7 - 11781*x**6 - 12999*x**5 - 70*x**4 + 3528*x**3 + 76 8*x**2 - 304*x - 96),x)*x + 1906080*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(9720*x**9 + 23652*x**8 + 12582*x**7 - 11781*x**6 - 1299 9*x**5 - 70*x**4 + 3528*x**3 + 768*x**2 - 304*x - 96),x) - 13354308*int((s qrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(9720*x**9 + 23652*x**8 + ...