Integrand size = 28, antiderivative size = 191 \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=-\frac {37 \sqrt {1-2 x} (2+3 x)^{5/2}}{605 \sqrt {3+5 x}}+\frac {7 (2+3 x)^{7/2}}{11 \sqrt {1-2 x} \sqrt {3+5 x}}+\frac {502941 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{151250}+\frac {10851 \sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}}{15125}+\frac {2911577 \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{75625}-\frac {167647 \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{151250} \] Output:
-37/605*(1-2*x)^(1/2)*(2+3*x)^(5/2)/(3+5*x)^(1/2)+7/11*(2+3*x)^(7/2)/(1-2* x)^(1/2)/(3+5*x)^(1/2)+502941/151250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^( 1/2)+10851/15125*(1-2*x)^(1/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2)+2911577/378125* EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-167647/756 250*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 = 7.23 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.67 \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {-5 \sqrt {2+3 x} \sqrt {3+5 x} \left (-2892883-3684629 x+2188890 x^2+490050 x^3\right )-5823154 i \sqrt {33-66 x} (3+5 x) E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+5998265 i \sqrt {33-66 x} (3+5 x) \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{756250 \sqrt {1-2 x} (3+5 x)} \] Input:
Integrate[(2 + 3*x)^(9/2)/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)),x]
Output:
(-5*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(-2892883 - 3684629*x + 2188890*x^2 + 4900 50*x^3) - (5823154*I)*Sqrt[33 - 66*x]*(3 + 5*x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (5998265*I)*Sqrt[33 - 66*x]*(3 + 5*x)*EllipticF[I*ArcS inh[Sqrt[9 + 15*x]], -2/33])/(756250*Sqrt[1 - 2*x]*(3 + 5*x))
Time = 0.29 (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, 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.
| \(\displaystyle \int \frac {(3 x+2)^{9/2}}{(1-2 x)^{3/2} (5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 109 |
| \(\displaystyle \frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {1}{11} \int \frac {(3 x+2)^{5/2} (624 x+367)}{2 \sqrt {1-2 x} (5 x+3)^{3/2}}dx\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {1}{22} \int \frac {(3 x+2)^{5/2} (624 x+367)}{\sqrt {1-2 x} (5 x+3)^{3/2}}dx\) |
| \(\Big \downarrow \) 167 |
| \(\displaystyle \frac {1}{22} \left (-\frac {2}{55} \int \frac {3 (3 x+2)^{3/2} (7234 x+4391)}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {74 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (-\frac {3}{55} \int \frac {(3 x+2)^{3/2} (7234 x+4391)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {74 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{22} \left (-\frac {3}{55} \left (-\frac {1}{25} \int -\frac {3 \sqrt {3 x+2} (167647 x+103325)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {7234}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {74 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (-\frac {3}{55} \left (\frac {3}{25} \int \frac {\sqrt {3 x+2} (167647 x+103325)}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {7234}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {74 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {1}{22} \left (-\frac {3}{55} \left (\frac {3}{25} \left (-\frac {1}{15} \int -\frac {11646308 x+7373029}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {167647}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7234}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {74 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {1}{22} \left (-\frac {3}{55} \left (\frac {3}{25} \left (\frac {1}{30} \int \frac {11646308 x+7373029}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {167647}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7234}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {74 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {1}{22} \left (-\frac {3}{55} \left (\frac {3}{25} \left (\frac {1}{30} \left (\frac {1926221}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {11646308}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {167647}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7234}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {74 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {1}{22} \left (-\frac {3}{55} \left (\frac {3}{25} \left (\frac {1}{30} \left (\frac {1926221}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {11646308}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {167647}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7234}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {74 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {1}{22} \left (-\frac {3}{55} \left (\frac {3}{25} \left (\frac {1}{30} \left (-\frac {350222}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {11646308}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {167647}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {7234}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}\right )-\frac {74 \sqrt {1-2 x} (3 x+2)^{5/2}}{55 \sqrt {5 x+3}}\right )+\frac {7 (3 x+2)^{7/2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\) |
Input:
Int[(2 + 3*x)^(9/2)/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)),x]
Output:
(7*(2 + 3*x)^(7/2))/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + ((-74*Sqrt[1 - 2*x] *(2 + 3*x)^(5/2))/(55*Sqrt[3 + 5*x]) - (3*((-7234*Sqrt[1 - 2*x]*(2 + 3*x)^ (3/2)*Sqrt[3 + 5*x])/25 + (3*((-167647*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/15 + ((-11646308*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2* x]], 35/33])/5 - (350222*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2* x]], 35/33])/5)/30))/25))/55)/22
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 = 0.68 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.75
| method | result | size |
| default | \(\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (5778663 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-11646308 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+14701500 x^{4}+75467700 x^{3}-66761070 x^{2}-160479070 x -57857660\right )}{45375000 x^{3}+34787500 x^{2}-10587500 x -9075000}\) | \(143\) |
| elliptic | \(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (\frac {900367}{1210000}+\frac {1500641 x}{1210000}\right )}{\sqrt {\left (-\frac {3}{10}+x^{2}+\frac {1}{10} x \right ) \left (-20-30 x \right )}}+\frac {81 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{250}+\frac {3537 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2500}-\frac {7373029 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{2117500 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2911577 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{529375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\) | \(240\) |
Input:
int((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
1/1512500*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(5778663*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))-11646308*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*Ellipt icE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+14701500*x^4+75467700*x^3-66761070*x ^2-160479070*x-57857660)/(30*x^3+23*x^2-7*x-6)
Time = 0.10 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.48 \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {450 \, {\left (490050 \, x^{3} + 2188890 \, x^{2} - 3684629 \, x - 2892883\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 197853763 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 524083860 \, \sqrt {-30} {\left (10 \, x^{2} + x - 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{68062500 \, {\left (10 \, x^{2} + x - 3\right )}} \] Input:
integrate((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="fricas")
Output:
1/68062500*(450*(490050*x^3 + 2188890*x^2 - 3684629*x - 2892883)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 197853763*sqrt(-30)*(10*x^2 + x - 3)*w eierstrassPInverse(1159/675, 38998/91125, x + 23/90) - 524083860*sqrt(-30) *(10*x^2 + x - 3)*weierstrassZeta(1159/675, 38998/91125, weierstrassPInver se(1159/675, 38998/91125, x + 23/90)))/(10*x^2 + x - 3)
Timed out. \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((2+3*x)**(9/2)/(1-2*x)**(3/2)/(3+5*x)**(3/2),x)
Output:
Timed out
\[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="maxima")
Output:
integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)), x)
\[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {9}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x, algorithm="giac")
Output:
integrate((3*x + 2)^(9/2)/((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)), x)
Timed out. \[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\int \frac {{\left (3\,x+2\right )}^{9/2}}{{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \] Input:
int((3*x + 2)^(9/2)/((1 - 2*x)^(3/2)*(5*x + 3)^(3/2)),x)
Output:
int((3*x + 2)^(9/2)/((1 - 2*x)^(3/2)*(5*x + 3)^(3/2)), x)
\[ \int \frac {(2+3 x)^{9/2}}{(1-2 x)^{3/2} (3+5 x)^{3/2}} \, dx=\frac {64800 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}+289440 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}+1052784 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x -1071654 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+19773180 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{300 x^{5}+260 x^{4}-137 x^{3}-136 x^{2}+15 x +18}d x \right ) x^{2}+1977318 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{300 x^{5}+260 x^{4}-137 x^{3}-136 x^{2}+15 x +18}d x \right ) x -5931954 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{300 x^{5}+260 x^{4}-137 x^{3}-136 x^{2}+15 x +18}d x \right )+10626070 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{300 x^{5}+260 x^{4}-137 x^{3}-136 x^{2}+15 x +18}d x \right ) x^{2}+1062607 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{300 x^{5}+260 x^{4}-137 x^{3}-136 x^{2}+15 x +18}d x \right ) x -3187821 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{300 x^{5}+260 x^{4}-137 x^{3}-136 x^{2}+15 x +18}d x \right )}{200000 x^{2}+20000 x -60000} \] Input:
int((2+3*x)^(9/2)/(1-2*x)^(3/2)/(3+5*x)^(3/2),x)
Output:
(64800*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 + 289440*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 1052784*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 1071654*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 19773180*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/( 300*x**5 + 260*x**4 - 137*x**3 - 136*x**2 + 15*x + 18),x)*x**2 + 1977318*i nt((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(300*x**5 + 260*x** 4 - 137*x**3 - 136*x**2 + 15*x + 18),x)*x - 5931954*int((sqrt(3*x + 2)*sqr t(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(300*x**5 + 260*x**4 - 137*x**3 - 136*x* *2 + 15*x + 18),x) + 10626070*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(300*x**5 + 260*x**4 - 137*x**3 - 136*x**2 + 15*x + 18),x)*x**2 + 1 062607*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(300*x**5 + 260* x**4 - 137*x**3 - 136*x**2 + 15*x + 18),x)*x - 3187821*int((sqrt(3*x + 2)* sqrt(5*x + 3)*sqrt( - 2*x + 1))/(300*x**5 + 260*x**4 - 137*x**3 - 136*x**2 + 15*x + 18),x))/(20000*(10*x**2 + x - 3))