Integrand size = 28, antiderivative size = 191 \[ \int \frac {(2+3 x)^{9/2}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=-\frac {2 \sqrt {1-2 x} (2+3 x)^{7/2}}{165 (3+5 x)^{3/2}}-\frac {668 \sqrt {1-2 x} (2+3 x)^{5/2}}{9075 \sqrt {3+5 x}}-\frac {87476 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{378125}+\frac {403 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{75625}-\frac {6515539 \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{2268750}+\frac {87476 \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{1134375} \] Output:
-2/165*(1-2*x)^(1/2)*(2+3*x)^(7/2)/(3+5*x)^(3/2)-668/9075*(1-2*x)^(1/2)*(2 +3*x)^(5/2)/(3+5*x)^(1/2)-87476/378125*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x) ^(1/2)+403/75625*(1-2*x)^(1/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2)-6515539/1134375 0*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+87476/56 71875*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 = 7.88 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.54 \[ \int \frac {(2+3 x)^{9/2}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=\frac {-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x} \left (3365042+12517925 x+13721400 x^2+3675375 x^3\right )}{(3+5 x)^{3/2}}+6515539 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-6724865 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{11343750} \] Input:
Integrate[(2 + 3*x)^(9/2)/(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)),x]
Output:
((-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3365042 + 12517925*x + 13721400*x^2 + 3 675375*x^3))/(3 + 5*x)^(3/2) + (6515539*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sq rt[9 + 15*x]], -2/33] - (6724865*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/11343750
Time = 0.32 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.10, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {109, 27, 167, 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.
| \(\displaystyle \int \frac {(3 x+2)^{9/2}}{\sqrt {1-2 x} (5 x+3)^{5/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle -\frac {2}{165} \int -\frac {(3 x+2)^{5/2} (267 x+227)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{165} \int \frac {(3 x+2)^{5/2} (267 x+227)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{165} \left (\frac {2}{55} \int \frac {3 (3628-403 x) (3 x+2)^{3/2}}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {668 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{165} \left (\frac {3}{55} \int \frac {(3628-403 x) (3 x+2)^{3/2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {668 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{165} \left (\frac {3}{55} \left (\frac {403}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}-\frac {1}{25} \int -\frac {3 \sqrt {3 x+2} (174952 x+117575)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {668 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{165} \left (\frac {3}{55} \left (\frac {3}{50} \int \frac {\sqrt {3 x+2} (174952 x+117575)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {403}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {668 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{165} \left (\frac {3}{55} \left (\frac {3}{50} \left (-\frac {1}{15} \int -\frac {6515539 x+4139582}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {174952}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {403}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {668 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {1}{165} \left (\frac {3}{55} \left (\frac {3}{50} \left (\frac {1}{15} \int \frac {6515539 x+4139582}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {174952}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {403}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {668 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{165} \left (\frac {3}{55} \left (\frac {3}{50} \left (\frac {1}{15} \left (\frac {1151293}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {6515539}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {174952}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {403}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {668 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{165} \left (\frac {3}{55} \left (\frac {3}{50} \left (\frac {1}{15} \left (\frac {1151293}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {6515539}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {174952}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {403}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {668 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{165} \left (\frac {3}{55} \left (\frac {3}{50} \left (\frac {1}{15} \left (-\frac {209326}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {6515539}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {174952}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {403}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {668 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )-\frac {2 \sqrt {1-2 x} (3 x+2)^{7/2}}{165 (5 x+3)^{3/2}}\) |
Input:
Int[(2 + 3*x)^(9/2)/(Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)),x]
Output:
(-2*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2))/(165*(3 + 5*x)^(3/2)) + ((-668*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(55*Sqrt[3 + 5*x]) + (3*((403*Sqrt[1 - 2*x]*(2 + 3* x)^(3/2)*Sqrt[3 + 5*x])/25 + (3*((-174952*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt [3 + 5*x])/15 + ((-6515539*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (209326*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/15))/50))/55)/165
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[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 = 0.65 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.18
| method | result | size |
| default | \(-\frac {\left (17269395 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-32577695 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+10361637 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-19546617 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+220522500 x^{5}+860037750 x^{4}+814782000 x^{3}+52653770 x^{2}-216708080 x -67300840\right ) \sqrt {1-2 x}\, \sqrt {2+3 x}}{11343750 \left (6 x^{2}+x -2\right ) \left (3+5 x \right )^{\frac {3}{2}}}\) | \(225\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {81 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{625}-\frac {1026 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3125}+\frac {2069791 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{7940625 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {6515539 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{15881250 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{515625 \left (x +\frac {3}{5}\right )^{2}}-\frac {866 \left (-30 x^{2}-5 x +10\right )}{1134375 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(264\) |
Input:
int((2+3*x)^(9/2)/(1-2*x)^(1/2)/(3+5*x)^(5/2),x,method=_RETURNVERBOSE)
Output:
-1/11343750*(17269395*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)-32577695*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)+10361637*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1 /7*(28+42*x)^(1/2),1/2*70^(1/2))-19546617*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))+220522500*x ^5+860037750*x^4+814782000*x^3+52653770*x^2-216708080*x-67300840)*(1-2*x)^ (1/2)*(2+3*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)
Time = 0.07 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.51 \[ \int \frac {(2+3 x)^{9/2}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=-\frac {900 \, {\left (3675375 \, x^{3} + 13721400 \, x^{2} + 12517925 \, x + 3365042\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 222704983 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 586398510 \, \sqrt {-30} {\left (25 \, x^{2} + 30 \, x + 9\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{1020937500 \, {\left (25 \, x^{2} + 30 \, x + 9\right )}} \] Input:
integrate((2+3*x)^(9/2)/(1-2*x)^(1/2)/(3+5*x)^(5/2),x, algorithm="fricas")
Output:
-1/1020937500*(900*(3675375*x^3 + 13721400*x^2 + 12517925*x + 3365042)*sqr t(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 222704983*sqrt(-30)*(25*x^2 + 30 *x + 9)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 586398510* sqrt(-30)*(25*x^2 + 30*x + 9)*weierstrassZeta(1159/675, 38998/91125, weier strassPInverse(1159/675, 38998/91125, x + 23/90)))/(25*x^2 + 30*x + 9)
Timed out. \[ \int \frac {(2+3 x)^{9/2}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=\text {Timed out} \] Input:
integrate((2+3*x)**(9/2)/(1-2*x)**(1/2)/(3+5*x)**(5/2),x)
Output:
Timed out
\[ \int \frac {(2+3 x)^{9/2}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}} \,d x } \] Input:
integrate((2+3*x)^(9/2)/(1-2*x)^(1/2)/(3+5*x)^(5/2),x, algorithm="maxima")
Output:
integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*sqrt(-2*x + 1)), x)
\[ \int \frac {(2+3 x)^{9/2}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {5}{2}} \sqrt {-2 \, x + 1}} \,d x } \] Input:
integrate((2+3*x)^(9/2)/(1-2*x)^(1/2)/(3+5*x)^(5/2),x, algorithm="giac")
Output:
integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*sqrt(-2*x + 1)), x)
Timed out. \[ \int \frac {(2+3 x)^{9/2}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{9/2}}{\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{5/2}} \,d x \] Input:
int((3*x + 2)^(9/2)/((1 - 2*x)^(1/2)*(5*x + 3)^(5/2)),x)
Output:
int((3*x + 2)^(9/2)/((1 - 2*x)^(1/2)*(5*x + 3)^(5/2)), x)
\[ \int \frac {(2+3 x)^{9/2}}{\sqrt {1-2 x} (3+5 x)^{5/2}} \, dx=\frac {-210600 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}-786240 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}-1650636 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x -719526 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+120150 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x^{2}+144180 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x +43254 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right )-2790925 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x^{2}-3349110 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right ) x -1004733 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{750 x^{5}+1475 x^{4}+785 x^{3}-153 x^{2}-243 x -54}d x \right )}{1625000 x^{2}+1950000 x +585000} \] Input:
int((2+3*x)^(9/2)/(1-2*x)^(1/2)/(3+5*x)^(5/2),x)
Output:
( - 210600*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 786240*sqrt (3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 - 1650636*sqrt(3*x + 2)*sqrt (5*x + 3)*sqrt( - 2*x + 1)*x - 719526*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 120150*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 + 144180 *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 + 43254*int((sqrt(3*x + 2)*s qrt(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) - 2790925*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 - 3349110*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 - 1004733*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))/(65000*(25*x**2 + 30*x + 9))