\(\int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx\) [1322]

Optimal result

Integrand size = 28, antiderivative size = 156 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx=-\frac {74 (1-2 x)^{3/2} \sqrt {3+5 x}}{63 \sqrt {2+3 x}}-\frac {724}{567} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{9 (2+3 x)^{3/2}}+\frac {592}{243} \sqrt {35} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {8774 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{243 \sqrt {35}} \] Output:

-74/63*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)-724/567*(1-2*x)^(1/2)*(2+ 
3*x)^(1/2)*(3+5*x)^(1/2)-2/9*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(3/2)+592 
/243*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-8774/ 
8505*EllipticF(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 8.01 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.63 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx=\frac {2}{243} \left (-\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (329+564 x+90 x^2\right )}{(2+3 x)^{3/2}}-296 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+181 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \] Input:

Integrate[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(5/2),x]
 

Output:

(2*((-3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(329 + 564*x + 90*x^2))/(2 + 3*x)^(3/2 
) - (296*I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (181*I) 
*Sqrt[33]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/243
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.10, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.321, Rules used = {108, 27, 167, 25, 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.

\(\displaystyle \int \frac {(1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^{5/2}} \, dx\)

\(\Big \downarrow \) 108

\(\displaystyle \frac {2}{9} \int -\frac {3 \sqrt {1-2 x} \sqrt {5 x+3} (20 x+1)}{2 (3 x+2)^{3/2}}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{3} \int \frac {\sqrt {1-2 x} \sqrt {5 x+3} (20 x+1)}{(3 x+2)^{3/2}}dx-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {1}{3} \left (\frac {2}{3} \int -\frac {(84-575 x) \sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 \sqrt {3 x+2}}\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{3} \left (\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \int \frac {(84-575 x) \sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{3} \left (\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \left (\frac {575}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-\frac {1}{9} \int -\frac {2960 x+511}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \left (\frac {1}{18} \int \frac {2960 x+511}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {575}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{3} \left (\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \left (\frac {1}{18} \left (592 \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-1265 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )+\frac {575}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{3} \left (\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \left (\frac {1}{18} \left (-1265 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-592 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {575}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{3} \left (\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{3 \sqrt {3 x+2}}-\frac {2}{3} \left (\frac {1}{18} \left (230 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-592 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {575}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )\right )-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{9 (3 x+2)^{3/2}}\)

Input:

Int[((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^(5/2),x]
 

Output:

(-2*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(9*(2 + 3*x)^(3/2)) + ((74*Sqrt[1 - 2 
*x]*(3 + 5*x)^(3/2))/(3*Sqrt[2 + 3*x]) - (2*((575*Sqrt[1 - 2*x]*Sqrt[2 + 3 
*x]*Sqrt[3 + 5*x])/9 + (-592*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 
- 2*x]], 35/33] + 230*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]] 
, 35/33])/18))/3)/3
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 108

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])
 

rule 123

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])
 

rule 129

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]))
 

rule 167

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]
 

rule 171

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]
 

rule 176

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]
 

Maple [A] (verified)

Time = 0.38 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.41

method	result	size
default	\(-\frac {\left (11385 \sqrt {2}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+1776 \sqrt {2}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+7590 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1184 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+5400 x^{4}+34380 x^{3}+21504 x^{2}-8178 x -5922\right ) \sqrt {1-2 x}\, \sqrt {3+5 x}}{243 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {3}{2}}}\)	\(220\)
elliptic	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{729 \left (\frac {2}{3}+x \right )^{2}}-\frac {296 \left (-30 x^{2}-3 x +9\right )}{243 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {73 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{243 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2960 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{1701 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {20 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{81}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\)	\(255\)

Input:

int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(5/2),x,method=_RETURNVERBOSE)
 

Output:

-1/243*(11385*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)+1776*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)+7590*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))+1184*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*E 
llipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+5400*x^4+34380*x^3+21504*x^2-81 
78*x-5922)*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(3/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.60 \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx=-\frac {270 \, {\left (90 \, x^{2} + 564 \, x + 329\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 2209 \, \sqrt {-30} {\left (9 \, x^{2} + 12 \, x + 4\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 26640 \, \sqrt {-30} {\left (9 \, x^{2} + 12 \, x + 4\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{10935 \, {\left (9 \, x^{2} + 12 \, x + 4\right )}} \] Input:

integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(5/2),x, algorithm="fricas")
 

Output:

-1/10935*(270*(90*x^2 + 564*x + 329)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x 
 + 1) + 2209*sqrt(-30)*(9*x^2 + 12*x + 4)*weierstrassPInverse(1159/675, 38 
998/91125, x + 23/90) + 26640*sqrt(-30)*(9*x^2 + 12*x + 4)*weierstrassZeta 
(1159/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/ 
90)))/(9*x^2 + 12*x + 4)
 

Sympy [F]

\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx=\int \frac {\left (1 - 2 x\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}{\left (3 x + 2\right )^{\frac {5}{2}}}\, dx \] Input:

integrate((1-2*x)**(3/2)*(3+5*x)**(3/2)/(2+3*x)**(5/2),x)
 

Output:

Integral((1 - 2*x)**(3/2)*(5*x + 3)**(3/2)/(3*x + 2)**(5/2), x)
 

Maxima [F]

\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(5/2),x, algorithm="maxima")
 

Output:

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(5/2), x)
 

Giac [F]

\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx=\int { \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {5}{2}}} \,d x } \] Input:

integrate((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(5/2),x, algorithm="giac")
 

Output:

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(3*x + 2)^(5/2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{5/2}} \,d x \] Input:

int(((1 - 2*x)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^(5/2),x)
                                                                                    
                                                                                    
 

Output:

int(((1 - 2*x)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^(5/2), x)
 

Reduce [F]

\[ \int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{5/2}} \, dx=\frac {-340 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}+1224 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x -726 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-455877 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x^{2}-607836 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x -202612 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right )+148446 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x^{2}+197928 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right ) x +65976 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{270 x^{5}+567 x^{4}+333 x^{3}-46 x^{2}-100 x -24}d x \right )}{1377 x^{2}+1836 x +612} \] Input:

int((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(5/2),x)
 

Output:

( - 340*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2 + 1224*sqrt(3*x 
+ 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x - 726*sqrt(3*x + 2)*sqrt(5*x + 3)*sq 
rt( - 2*x + 1) - 455877*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 - 60 
7836*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(270*x**5 + 5 
67*x**4 + 333*x**3 - 46*x**2 - 100*x - 24),x)*x - 202612*int((sqrt(3*x + 2 
)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(270*x**5 + 567*x**4 + 333*x**3 - 4 
6*x**2 - 100*x - 24),x) + 148446*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 
+ 197928*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(270*x**5 + 56 
7*x**4 + 333*x**3 - 46*x**2 - 100*x - 24),x)*x + 65976*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))/(153*(9*x**2 + 12*x + 4))