Integrand size = 28, antiderivative size = 218 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=-\frac {2 (1-2 x)^{5/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac {62 (1-2 x)^{5/2} (2+3 x)^{3/2}}{165 \sqrt {3+5 x}}-\frac {942 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}-\frac {842 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{144375}+\frac {566 (1-2 x)^{5/2} \sqrt {2+3 x} \sqrt {3+5 x}}{1925}+\frac {49321 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{9375 \sqrt {35}}-\frac {33778 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{9375 \sqrt {35}} \] Output:
-2/15*(1-2*x)^(5/2)*(2+3*x)^(5/2)/(3+5*x)^(3/2)-62/165*(1-2*x)^(5/2)*(2+3* x)^(3/2)/(3+5*x)^(1/2)-942/21875*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2) -842/144375*(1-2*x)^(3/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+566/1925*(1-2*x)^(5/ 2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)+49321/328125*EllipticE(1/11*55^(1/2)*(1-2*x )^(1/2),1/35*1155^(1/2))*35^(1/2)-33778/328125*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 = 8.41 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.50 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=\frac {\frac {10 \sqrt {1-2 x} \sqrt {2+3 x} \left (-19087-23425 x-41025 x^2-47250 x^3+67500 x^4\right )}{(3+5 x)^{3/2}}-49321 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+16485 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{328125} \] Input:
Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^(5/2))/(3 + 5*x)^(5/2),x]
Output:
((10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-19087 - 23425*x - 41025*x^2 - 47250*x^3 + 67500*x^4))/(3 + 5*x)^(3/2) - (49321*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sq rt[9 + 15*x]], -2/33] + (16485*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15 *x]], -2/33])/328125
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, 27, 171, 27, 171, 27, 171, 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.
\(\displaystyle \int \frac {(1-2 x)^{5/2} (3 x+2)^{5/2}}{(5 x+3)^{5/2}} \, dx\) |
\(\Big \downarrow \) 108 |
\(\displaystyle \frac {2}{15} \int -\frac {5 (1-2 x)^{3/2} (3 x+2)^{3/2} (12 x+1)}{2 (5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle -\frac {1}{3} \int \frac {(1-2 x)^{3/2} (3 x+2)^{3/2} (12 x+1)}{(5 x+3)^{3/2}}dx-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 167 |
\(\displaystyle \frac {1}{3} \left (-\frac {2}{5} \int \frac {3 \sqrt {1-2 x} (3 x+2)^{3/2} (213 x+2)}{\sqrt {5 x+3}}dx-\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (-\frac {6}{5} \int \frac {\sqrt {1-2 x} (3 x+2)^{3/2} (213 x+2)}{\sqrt {5 x+3}}dx-\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {1}{3} \left (-\frac {6}{5} \left (\frac {2}{105} \int -\frac {3 (2983-11433 x) (3 x+2)^{3/2}}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {142}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (-\frac {6}{5} \left (\frac {142}{35} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}-\frac {1}{35} \int \frac {(2983-11433 x) (3 x+2)^{3/2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {1}{3} \left (-\frac {6}{5} \left (\frac {1}{35} \left (\frac {1}{25} \int -\frac {(12475-101334 x) \sqrt {3 x+2}}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {11433}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )+\frac {142}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (-\frac {6}{5} \left (\frac {1}{35} \left (-\frac {1}{50} \int \frac {(12475-101334 x) \sqrt {3 x+2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {11433}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )+\frac {142}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 171 |
\(\displaystyle \frac {1}{3} \left (-\frac {6}{5} \left (\frac {1}{35} \left (\frac {1}{50} \left (\frac {1}{15} \int -\frac {3 (6527-49321 x)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {33778}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {11433}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )+\frac {142}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {1}{3} \left (-\frac {6}{5} \left (\frac {1}{35} \left (\frac {1}{50} \left (-\frac {1}{5} \int \frac {6527-49321 x}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {33778}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {11433}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )+\frac {142}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 176 |
\(\displaystyle \frac {1}{3} \left (-\frac {6}{5} \left (\frac {1}{35} \left (\frac {1}{50} \left (\frac {1}{5} \left (\frac {49321}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-\frac {180598}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {33778}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {11433}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )+\frac {142}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 123 |
\(\displaystyle \frac {1}{3} \left (-\frac {6}{5} \left (\frac {1}{35} \left (\frac {1}{50} \left (\frac {1}{5} \left (-\frac {180598}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {49321}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {33778}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {11433}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )+\frac {142}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
\(\Big \downarrow \) 129 |
\(\displaystyle \frac {1}{3} \left (-\frac {6}{5} \left (\frac {1}{35} \left (\frac {1}{50} \left (\frac {1}{5} \left (\frac {32836}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {49321}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {33778}{5} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {11433}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )+\frac {142}{35} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{5/2}\right )-\frac {62 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\right )-\frac {2 (1-2 x)^{5/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}\) |
Input:
Int[((1 - 2*x)^(5/2)*(2 + 3*x)^(5/2))/(3 + 5*x)^(5/2),x]
Output:
(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(5/2))/(15*(3 + 5*x)^(3/2)) + ((-62*(1 - 2*x )^(3/2)*(2 + 3*x)^(5/2))/(5*Sqrt[3 + 5*x]) - (6*((142*Sqrt[1 - 2*x]*(2 + 3 *x)^(5/2)*Sqrt[3 + 5*x])/35 + ((-11433*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[ 3 + 5*x])/25 + ((-33778*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/5 + ((- 49321*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 + (3 2836*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/5)/5 0)/35))/5)/3
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 = 0.60 (sec) , antiderivative size = 230, normalized size of antiderivative = 1.06
method | result | size |
default | \(-\frac {\left (2708970 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+246605 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4050000 x^{6}+1625382 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+147963 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+2160000 x^{5}+4284000 x^{4}+870750 x^{3}+558970 x^{2}-277630 x -381740\right ) \sqrt {2+3 x}\, \sqrt {1-2 x}}{328125 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(230\) |
elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {684 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4375}+\frac {2362 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21875}+\frac {6527 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{459375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {49321 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{459375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {72 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{875}-\frac {242 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{234375 \left (x +\frac {3}{5}\right )^{2}}-\frac {4774 \left (-30 x^{2}-5 x +10\right )}{46875 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\) | \(297\) |
Input:
int((1-2*x)^(5/2)*(2+3*x)^(5/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/328125*(2708970*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)+246605*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)-40 50000*x^6+1625382*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ellip ticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+147963*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))+2160000 *x^5+4284000*x^4+870750*x^3+558970*x^2-277630*x-381740)*(2+3*x)^(1/2)*(1-2 *x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)
Time = 0.09 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.47 \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=\frac {900 \, {\left (67500 \, x^{4} - 47250 \, x^{3} - 41025 \, x^{2} - 23425 \, x - 19087\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 1721813 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 4438890 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{29531250 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
1/29531250*(900*(67500*x^4 - 47250*x^3 - 41025*x^2 - 23425*x - 19087)*sqrt (5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) - 1721813*sqrt(-30)*(25*x^2 + 30*x + 9)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 4438890*sqrt( -30)*(25*x^2 + 30*x + 9)*weierstrassZeta(1159/675, 38998/91125, weierstras sPInverse(1159/675, 38998/91125, x + 23/90)))/(25*x^2 + 30*x + 9)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((1-2*x)**(5/2)*(2+3*x)**(5/2)/(3+5*x)**(5/2),x)
Output:
Timed out
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
integrate((3*x + 2)^(5/2)*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2), x)
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}}} \,d x } \] Input:
integrate((1-2*x)^(5/2)*(2+3*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")
Output:
integrate((3*x + 2)^(5/2)*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2), x)
Timed out. \[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (3\,x+2\right )}^{5/2}}{{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:
int(((1 - 2*x)^(5/2)*(3*x + 2)^(5/2))/(5*x + 3)^(5/2),x)
Output:
int(((1 - 2*x)^(5/2)*(3*x + 2)^(5/2))/(5*x + 3)^(5/2), x)
\[ \int \frac {(1-2 x)^{5/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx=\frac {468000 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{4}-327600 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}-284440 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}+692484 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x -558606 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-586980350 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x^{2}-704376420 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x -211312926 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right )+288797575 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x^{2}+346557090 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x +103967127 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right )}{5687500 x^{2}+6825000 x +2047500} \] Input:
int((1-2*x)^(5/2)*(2+3*x)^(5/2)/(3+5*x)^(5/2),x)
Output:
(468000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**4 - 327600*sqrt(3* x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 284440*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 692484*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2 *x + 1)*x - 558606*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) - 58698035 0*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(750*x**5 + 1475 *x**4 + 785*x**3 - 153*x**2 - 243*x - 54),x)*x**2 - 704376420*int((sqrt(3* x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(750*x**5 + 1475*x**4 + 785*x* *3 - 153*x**2 - 243*x - 54),x)*x - 211312926*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(750*x**5 + 1475*x**4 + 785*x**3 - 153*x**2 - 2 43*x - 54),x) + 288797575*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1 ))/(750*x**5 + 1475*x**4 + 785*x**3 - 153*x**2 - 243*x - 54),x)*x**2 + 346 557090*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(750*x**5 + 1475 *x**4 + 785*x**3 - 153*x**2 - 243*x - 54),x)*x + 103967127*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(750*x**5 + 1475*x**4 + 785*x**3 - 153 *x**2 - 243*x - 54),x))/(227500*(25*x**2 + 30*x + 9))