Integrand size = 28, antiderivative size = 187 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{5 \sqrt {3+5 x}}+\frac {3034 \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}}{21875}-\frac {27 (1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}}{4375}+\frac {48}{175} (1-2 x)^{3/2} (2+3 x)^{3/2} \sqrt {3+5 x}-\frac {47342 E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{9375 \sqrt {35}}-\frac {2719 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{9375 \sqrt {35}} \] Output:
-2/5*(1-2*x)^(3/2)*(2+3*x)^(5/2)/(3+5*x)^(1/2)+3034/21875*(1-2*x)^(1/2)*(2 +3*x)^(1/2)*(3+5*x)^(1/2)-27/4375*(1-2*x)^(3/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2 )+48/175*(1-2*x)^(3/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2)-47342/328125*EllipticE( 1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-2719/328125*Elliptic F(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 = 7.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.55 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx=\frac {-\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} \left (-9697-22305 x+5400 x^2+22500 x^3\right )}{\sqrt {3+5 x}}+47342 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-53095 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )}{328125} \] Input:
Integrate[((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2))/(3 + 5*x)^(3/2),x]
Output:
((-15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-9697 - 22305*x + 5400*x^2 + 22500*x^3) )/Sqrt[3 + 5*x] + (47342*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (53095*I)*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33])/3 28125
Time = 0.27 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.13, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.393, Rules used = {108, 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.
| \(\displaystyle \int \frac {(1-2 x)^{3/2} (3 x+2)^{5/2}}{(5 x+3)^{3/2}} \, dx\) |
| \(\Big \downarrow \) 108 |
| \(\displaystyle \frac {2}{5} \int \frac {3 (1-16 x) \sqrt {1-2 x} (3 x+2)^{3/2}}{2 \sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{5} \int \frac {(1-16 x) \sqrt {1-2 x} (3 x+2)^{3/2}}{\sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {3}{5} \left (\frac {2}{105} \int \frac {(793-2818 x) (3 x+2)^{3/2}}{2 \sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {32}{105} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{5} \left (\frac {1}{105} \int \frac {(793-2818 x) (3 x+2)^{3/2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx-\frac {32}{105} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {3}{5} \left (\frac {1}{105} \left (\frac {2818}{25} \sqrt {1-2 x} (3 x+2)^{3/2} \sqrt {5 x+3}-\frac {1}{25} \int -\frac {3 (1475-2719 x) \sqrt {3 x+2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx\right )-\frac {32}{105} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{5} \left (\frac {1}{105} \left (\frac {3}{25} \int \frac {(1475-2719 x) \sqrt {3 x+2}}{\sqrt {1-2 x} \sqrt {5 x+3}}dx+\frac {2818}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {32}{105} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 171 |
| \(\displaystyle \frac {3}{5} \left (\frac {1}{105} \left (\frac {3}{25} \left (\frac {2719}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {1}{15} \int -\frac {94684 x+69467}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {2818}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {32}{105} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {3}{5} \left (\frac {1}{105} \left (\frac {3}{25} \left (\frac {1}{30} \int \frac {94684 x+69467}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2719}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2818}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {32}{105} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 176 |
| \(\displaystyle \frac {3}{5} \left (\frac {1}{105} \left (\frac {3}{25} \left (\frac {1}{30} \left (\frac {63283}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {94684}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {2719}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2818}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {32}{105} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 123 |
| \(\displaystyle \frac {3}{5} \left (\frac {1}{105} \left (\frac {3}{25} \left (\frac {1}{30} \left (\frac {63283}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {94684}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {2719}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2818}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {32}{105} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\) |
| \(\Big \downarrow \) 129 |
| \(\displaystyle \frac {3}{5} \left (\frac {1}{105} \left (\frac {3}{25} \left (\frac {1}{30} \left (-\frac {11506}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {94684}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {2719}{15} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )+\frac {2818}{25} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^{3/2}\right )-\frac {32}{105} \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{5 \sqrt {5 x+3}}\) |
Input:
Int[((1 - 2*x)^(3/2)*(2 + 3*x)^(5/2))/(3 + 5*x)^(3/2),x]
Output:
(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2))/(5*Sqrt[3 + 5*x]) + (3*((-32*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/105 + ((2818*Sqrt[1 - 2*x]*(2 + 3*x)^ (3/2)*Sqrt[3 + 5*x])/25 + (3*((2719*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5 *x])/15 + ((-94684*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3 5/33])/5 - (11506*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35 /33])/5)/30))/25)/105))/5
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[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 = 148, normalized size of antiderivative = 0.79
| method | result | size |
| default | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (189849 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-94684 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+4050000 x^{5}+1647000 x^{4}-5202900 x^{3}-2738610 x^{2}+1047390 x +581820\right )}{656250 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\) | \(148\) |
| elliptic | \(-\frac {\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {324 x \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{4375}+\frac {3489 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{21875}+\frac {69467 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{918750 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {47342 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{459375 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {36 x^{2} \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{175}-\frac {22 \left (-30 x^{2}-5 x +10\right )}{3125 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\left (6 x^{2}+x -2\right ) \sqrt {3+5 x}}\) | \(273\) |
Input:
int((1-2*x)^(3/2)*(2+3*x)^(5/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)
Output:
-1/656250*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(189849*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))-94684*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))+4050000*x^5+1647000*x^4-5202900*x^3-2738 610*x^2+1047390*x+581820)/(30*x^3+23*x^2-7*x-6)
Time = 0.08 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.44 \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx=-\frac {1350 \, {\left (22500 \, x^{3} + 5400 \, x^{2} - 22305 \, x - 9697\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 2037149 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 4260780 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{29531250 \, {\left (5 \, x + 3\right )}} \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="fricas")
Output:
-1/29531250*(1350*(22500*x^3 + 5400*x^2 - 22305*x - 9697)*sqrt(5*x + 3)*sq rt(3*x + 2)*sqrt(-2*x + 1) + 2037149*sqrt(-30)*(5*x + 3)*weierstrassPInver se(1159/675, 38998/91125, x + 23/90) - 4260780*sqrt(-30)*(5*x + 3)*weierst rassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(5*x + 3)
Timed out. \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx=\text {Timed out} \] Input:
integrate((1-2*x)**(3/2)*(2+3*x)**(5/2)/(3+5*x)**(3/2),x)
Output:
Timed out
\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="maxima")
Output:
integrate((3*x + 2)^(5/2)*(-2*x + 1)^(3/2)/(5*x + 3)^(3/2), x)
\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx=\int { \frac {{\left (3 \, x + 2\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (5 \, x + 3\right )}^{\frac {3}{2}}} \,d x } \] Input:
integrate((1-2*x)^(3/2)*(2+3*x)^(5/2)/(3+5*x)^(3/2),x, algorithm="giac")
Output:
integrate((3*x + 2)^(5/2)*(-2*x + 1)^(3/2)/(5*x + 3)^(3/2), x)
Timed out. \[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{5/2}}{{\left (5\,x+3\right )}^{3/2}} \,d x \] Input:
int(((1 - 2*x)^(3/2)*(3*x + 2)^(5/2))/(5*x + 3)^(3/2),x)
Output:
int(((1 - 2*x)^(3/2)*(3*x + 2)^(5/2))/(5*x + 3)^(3/2), x)
\[ \int \frac {(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx=\frac {-324000 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{3}-77760 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}+321192 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x -112918 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+1898490 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right ) x +1139094 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right )+705595 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right ) x +423357 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right )}{1575000 x +945000} \] Input:
int((1-2*x)^(3/2)*(2+3*x)^(5/2)/(3+5*x)^(3/2),x)
Output:
( - 324000*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**3 - 77760*sqrt( 3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 321192*sqrt(3*x + 2)*sqrt(5 *x + 3)*sqrt( - 2*x + 1)*x - 112918*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2* x + 1) + 1898490*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/( 150*x**4 + 205*x**3 + 34*x**2 - 51*x - 18),x)*x + 1139094*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(150*x**4 + 205*x**3 + 34*x**2 - 5 1*x - 18),x) + 705595*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/( 150*x**4 + 205*x**3 + 34*x**2 - 51*x - 18),x)*x + 423357*int((sqrt(3*x + 2 )*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(150*x**4 + 205*x**3 + 34*x**2 - 51*x - 18),x))/(315000*(5*x + 3))