\(\int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx\) [1597]

Optimal result

Integrand size = 28, antiderivative size = 156 \[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {7 \sqrt {2+3 x}}{33 (1-2 x)^{3/2} \sqrt {3+5 x}}+\frac {8 \sqrt {2+3 x}}{363 \sqrt {1-2 x} \sqrt {3+5 x}}-\frac {245 \sqrt {1-2 x} \sqrt {2+3 x}}{3993 \sqrt {3+5 x}}+\frac {49 \sqrt {35} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )}{3993}-\frac {362 \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right )}{3993 \sqrt {35}} \] Output:

7/33*(2+3*x)^(1/2)/(1-2*x)^(3/2)/(3+5*x)^(1/2)+8/363*(2+3*x)^(1/2)/(1-2*x) 
^(1/2)/(3+5*x)^(1/2)-245/3993*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)+49 
/3993*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-362/ 
139755*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 = 6.74 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.60 \[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=\frac {-\frac {2 \sqrt {2+3 x} \left (-345-402 x+490 x^2\right )}{(1-2 x)^{3/2} \sqrt {3+5 x}}-i \sqrt {33} \left (49 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-41 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{3993} \] Input:

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

Output:

((-2*Sqrt[2 + 3*x]*(-345 - 402*x + 490*x^2))/((1 - 2*x)^(3/2)*Sqrt[3 + 5*x 
]) - I*Sqrt[33]*(49*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 41*Ellip 
ticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/3993
 

Rubi [A] (verified)

Time = 0.25 (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 = {109, 27, 169, 27, 169, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}-\frac {1}{33} \int -\frac {18 x+19}{2 (1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{66} \int \frac {18 x+19}{(1-2 x)^{3/2} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{66} \left (\frac {16 \sqrt {3 x+2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}-\frac {2}{77} \int -\frac {7 (120 x+121)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{66} \left (\frac {1}{11} \int \frac {120 x+121}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx+\frac {16 \sqrt {3 x+2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {1}{66} \left (\frac {1}{11} \left (-\frac {2}{11} \int \frac {3 (245 x+103)}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {490 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {16 \sqrt {3 x+2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{66} \left (\frac {1}{11} \left (-\frac {6}{11} \int \frac {245 x+103}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {490 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {16 \sqrt {3 x+2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{66} \left (\frac {1}{11} \left (-\frac {6}{11} \left (49 \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx-44 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx\right )-\frac {490 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {16 \sqrt {3 x+2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{66} \left (\frac {1}{11} \left (-\frac {6}{11} \left (-44 \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-49 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {490 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {16 \sqrt {3 x+2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{66} \left (\frac {1}{11} \left (-\frac {6}{11} \left (8 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-49 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {490 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )+\frac {16 \sqrt {3 x+2}}{11 \sqrt {1-2 x} \sqrt {5 x+3}}\right )+\frac {7 \sqrt {3 x+2}}{33 (1-2 x)^{3/2} \sqrt {5 x+3}}\)

Input:

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

Output:

(7*Sqrt[2 + 3*x])/(33*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x]) + ((16*Sqrt[2 + 3*x]) 
/(11*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) + ((-490*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1 
1*Sqrt[3 + 5*x]) - (6*(-49*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 
2*x]], 35/33] + 8*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35 
/33]))/11)/11)/66
 

Defintions of rubi rules used

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 109

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])
 

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 169

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 + 1)*((e + f*x)^(p + 1)/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + S 
imp[1/((m + 1)*(b*c - a*d)*(b*e - a*f))   Int[(a + b*x)^(m + 1)*(c + d*x)^n 
*(e + f*x)^p*Simp[(a*d*f*g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a* 
h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p + 3)*x, x], x], 
 x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && 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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(252\) vs. \(2(116)=232\).

Time = 0.75 (sec) , antiderivative size = 253, normalized size of antiderivative = 1.62

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {7 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{726 \left (x -\frac {1}{2}\right )^{2}}-\frac {43 \left (-30 x^{2}-38 x -12\right )}{3993 \sqrt {\left (x -\frac {1}{2}\right ) \left (-30 x^{2}-38 x -12\right )}}-\frac {103 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{27951 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {35 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{3993 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{1331 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(253\)
default	\(-\frac {\left (5544 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, x +2058 \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, x -2772 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right )-1029 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+61740 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{3}-9492 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{2}-77238 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x +869400 x^{3}+666540 x^{2}-202860 x -173880\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}\, \sqrt {2+3 x}}{83853 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \left (-1+2 x \right )^{2} \left (15 x^{2}+19 x +6\right )}\)	\(359\)

Input:

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

Output:

(-(3+5*x)*(-1+2*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2 
)*(7/726*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x-1/2)^2-43/3993*(-30*x^2-38*x-12)/ 
((x-1/2)*(-30*x^2-38*x-12))^(1/2)-103/27951*(28+42*x)^(1/2)*(-15*x-9)^(1/2 
)*(21-42*x)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*EllipticF(1/7*(28+42*x)^(1/ 
2),1/2*70^(1/2))-35/3993*(28+42*x)^(1/2)*(-15*x-9)^(1/2)*(21-42*x)^(1/2)/( 
-30*x^3-23*x^2+7*x+6)^(1/2)*(-1/15*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1 
/2))-3/5*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2)))-2/1331*(-30*x^2-5*x+ 
10)/((x+3/5)*(-30*x^2-5*x+10))^(1/2))
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^{3/2}}{(1-2 x)^{5/2} (3+5 x)^{3/2}} \, dx=-\frac {180 \, {\left (490 \, x^{2} - 402 \, x - 345\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 727 \, \sqrt {-30} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 4410 \, \sqrt {-30} {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{359370 \, {\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )}} \] Input:

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

Output:

-1/359370*(180*(490*x^2 - 402*x - 345)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2 
*x + 1) - 727*sqrt(-30)*(20*x^3 - 8*x^2 - 7*x + 3)*weierstrassPInverse(115 
9/675, 38998/91125, x + 23/90) + 4410*sqrt(-30)*(20*x^3 - 8*x^2 - 7*x + 3) 
*weierstrassZeta(1159/675, 38998/91125, weierstrassPInverse(1159/675, 3899 
8/91125, x + 23/90)))/(20*x^3 - 8*x^2 - 7*x + 3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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