Integrand size = 28, antiderivative size = 218 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\frac {18118 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{5103}+\frac {250}{189} (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {2350}{567} (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {370 (1-2 x)^{5/2} (3+5 x)^{3/2}}{189 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{9 (2+3 x)^{3/2}}-\frac {452399 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{2187 \sqrt {35}}+\frac {153452 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{2187 \sqrt {35}} \] Output:
18118/5103*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+250/189*(1-2*x)^(3/2) *(2+3*x)^(1/2)*(3+5*x)^(1/2)+2350/567*(1-2*x)^(5/2)*(2+3*x)^(1/2)*(3+5*x)^ (1/2)-370/189*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(1/2)-2/9*(1-2*x)^(5/2)* (3+5*x)^(5/2)/(2+3*x)^(3/2)-452399/76545*EllipticE(1/11*55^(1/2)*(1-2*x)^( 1/2),1/35*1155^(1/2))*35^(1/2)+153452/76545*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 = 8.72 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.50 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\frac {\frac {30 \sqrt {1-2 x} \sqrt {3+5 x} \left (56963+108285 x+5949 x^2-25110 x^3+24300 x^4\right )}{(2+3 x)^{3/2}}+452399 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-317065 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{76545} \] Input:
Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(5/2),x]
Output:
((30*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(56963 + 108285*x + 5949*x^2 - 25110*x^3 + 24300*x^4))/(2 + 3*x)^(3/2) + (452399*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sq rt[9 + 15*x]], -2/33] - (317065*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 1 5*x]], -2/33])/76545
Time = 0.30 (sec) , antiderivative size = 247, normalized size of antiderivative = 1.13, 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, 25, 171, 27, 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.
| \(\displaystyle \int \frac {(1-2 x)^{5/2} (5 x+3)^{5/2}}{(3 x+2)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{9} \int -\frac {5 (1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{2 (3 x+2)^{3/2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{9} \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2} (20 x+1)}{(3 x+2)^{3/2}}dx-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle -\frac {5}{9} \left (-\frac {2}{3} \int -\frac {(47-1315 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{\sqrt {3 x+2}}dx-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5}{9} \left (\frac {2}{3} \int \frac {(47-1315 x) \sqrt {1-2 x} (5 x+3)^{3/2}}{\sqrt {3 x+2}}dx-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{9} \left (\frac {2}{3} \left (\frac {2}{105} \int \frac {5 (13348-46947 x) (5 x+3)^{3/2}}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {526}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{9} \left (\frac {2}{3} \left (\frac {1}{21} \int \frac {(13348-46947 x) (5 x+3)^{3/2}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {526}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{9} \left (\frac {2}{3} \left (\frac {1}{21} \left (\frac {15649}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {1}{15} \int \frac {3 (22083-135334 x) \sqrt {5 x+3}}{2 \sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {526}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {5}{9} \left (\frac {2}{3} \left (\frac {1}{21} \left (\frac {15649}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}-\frac {1}{10} \int \frac {(22083-135334 x) \sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {526}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle -\frac {5}{9} \left (\frac {2}{3} \left (\frac {1}{21} \left (\frac {1}{10} \left (\frac {1}{9} \int -\frac {452399 x+122572}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {135334}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {15649}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {526}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {5}{9} \left (\frac {2}{3} \left (\frac {1}{21} \left (\frac {1}{10} \left (-\frac {1}{9} \int \frac {452399 x+122572}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {135334}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {15649}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {526}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle -\frac {5}{9} \left (\frac {2}{3} \left (\frac {1}{21} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {744337}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {452399}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {135334}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {15649}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {526}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle -\frac {5}{9} \left (\frac {2}{3} \left (\frac {1}{21} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {744337}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {452399}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {135334}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {15649}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {526}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle -\frac {5}{9} \left (\frac {2}{3} \left (\frac {1}{21} \left (\frac {1}{10} \left (\frac {1}{9} \left (\frac {452399}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {135334}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )\right )-\frac {135334}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {15649}{5} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}\right )-\frac {526}{21} \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}\right )-\frac {74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{3 \sqrt {3 x+2}}\right )-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{9 (3 x+2)^{3/2}}\) |
Input:
Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(5/2),x]
Output:
(-2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(9*(2 + 3*x)^(3/2)) - (5*((-74*(1 - 2 *x)^(3/2)*(3 + 5*x)^(5/2))/(3*Sqrt[2 + 3*x]) + (2*((-526*Sqrt[1 - 2*x]*Sqr t[2 + 3*x]*(3 + 5*x)^(5/2))/21 + ((15649*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2))/5 + ((-135334*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ( (452399*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (135334*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/9 )/10)/21))/3))/9
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 = 0.38 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.06
| method | result | size |
| default | \(\frac {\left (6699033 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1357197 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+7290000 x^{6}+4466022 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+904798 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-6804000 x^{5}-1155600 x^{4}+34923870 x^{3}+19802040 x^{2}-8036760 x -5126670\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{76545 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {3}{2}}}\) | \(230\) |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {1420 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{567}+\frac {15962 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{5103}+\frac {122572 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{107163 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {452399 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{107163 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {200 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{189}-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6561 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {36260}{729} x^{2}-\frac {3626}{729} x +\frac {3626}{243}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\) | \(297\) |
Input:
int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
1/76545*(6699033*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)+1357197*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)+729 0000*x^6+4466022*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ellipt icF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+904798*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))-6804000* x^5-1155600*x^4+34923870*x^3+19802040*x^2-8036760*x-5126670)*(3+5*x)^(1/2) *(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2)
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\frac {2700 \, {\left (24300 \, x^{4} - 25110 \, x^{3} + 5949 \, x^{2} + 108285 \, x + 56963\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 626303 \, \sqrt {-30} {\left (9 \, x^{2} + 12 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 40715910 \, \sqrt {-30} {\left (9 \, x^{2} + 12 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{6889050 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(5/2),x, algorithm="fricas")
Output:
1/6889050*(2700*(24300*x^4 - 25110*x^3 + 5949*x^2 + 108285*x + 56963)*sqrt (5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 626303*sqrt(-30)*(9*x^2 + 12*x + 4)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 40715910*sqrt(- 30)*(9*x^2 + 12*x + 4)*weierstrassZeta(1159/675, 38998/91125, weierstrassP Inverse(1159/675, 38998/91125, x + 23/90)))/(9*x^2 + 12*x + 4)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**(5/2),x)
Output:
Timed out
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(5/2),x, algorithm="maxima")
Output:
integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(5/2), x)
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(5/2),x, algorithm="giac")
Output:
integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(5/2), x)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{5/2}} \,d x \] Input:
int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^(5/2),x)
Output:
int(((1 - 2*x)^(5/2)*(5*x + 3)^(5/2))/(3*x + 2)^(5/2), x)
\[ \int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{5/2}} \, dx=\frac {1836000 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}-1897200 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}+449480 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}-2072844 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x +1537950 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+1097318250 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x^{2}+1463091000 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x +487697000 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right )-359481591 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x^{2}-479308788 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x -159769596 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right )}{1735020 x^{2}+2313360 x +771120} \] Input:
int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(5/2),x)
Output:
(1836000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 1897200*sqrt( 3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 449480*sqrt(3*x + 2)*sqrt(5 *x + 3)*sqrt( - 2*x + 1)*x**2 - 2072844*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 1537950*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 1097 318250*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(270*x**5 + 567*x**4 + 333*x**3 - 46*x**2 - 100*x - 24),x)*x**2 + 1463091000*int((sqr t(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(270*x**5 + 567*x**4 + 333 *x**3 - 46*x**2 - 100*x - 24),x)*x + 487697000*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(270*x**5 + 567*x**4 + 333*x**3 - 46*x**2 - 1 00*x - 24),x) - 359481591*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1 ))/(270*x**5 + 567*x**4 + 333*x**3 - 46*x**2 - 100*x - 24),x)*x**2 - 47930 8788*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(270*x**5 + 567*x* *4 + 333*x**3 - 46*x**2 - 100*x - 24),x)*x - 159769596*int((sqrt(3*x + 2)* sqrt(5*x + 3)*sqrt( - 2*x + 1))/(270*x**5 + 567*x**4 + 333*x**3 - 46*x**2 - 100*x - 24),x))/(192780*(9*x**2 + 12*x + 4))