\(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx\) [1421]

Optimal result

Integrand size = 28, antiderivative size = 191 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {14 (1-2 x)^{3/2}}{9 (2+3 x)^{3/2} (3+5 x)^{3/2}}+\frac {308 \sqrt {1-2 x}}{3 \sqrt {2+3 x} (3+5 x)^{3/2}}-\frac {6116 \sqrt {1-2 x} \sqrt {2+3 x}}{9 (3+5 x)^{3/2}}+\frac {36968 \sqrt {1-2 x} \sqrt {2+3 x}}{9 \sqrt {3+5 x}}-\frac {36968}{9} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )+\frac {1064}{9} \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:

14/9*(1-2*x)^(3/2)/(2+3*x)^(3/2)/(3+5*x)^(3/2)+308/3*(1-2*x)^(1/2)/(2+3*x) 
^(1/2)/(3+5*x)^(3/2)-6116/9*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(3/2)+3696 
8/9*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)-36968/45*EllipticE(1/11*55^( 
1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)+1064/45*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 = 7.94 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.52 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2}{45} \left (\frac {15 \sqrt {1-2 x} \left (70169+333260 x+526862 x^2+277260 x^3\right )}{(2+3 x)^{3/2} (3+5 x)^{3/2}}+4 i \sqrt {33} \left (4621 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-4760 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right ) \] Input:

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

Output:

(2*((15*Sqrt[1 - 2*x]*(70169 + 333260*x + 526862*x^2 + 277260*x^3))/((2 + 
3*x)^(3/2)*(3 + 5*x)^(3/2)) + (4*I)*Sqrt[33]*(4621*EllipticE[I*ArcSinh[Sqr 
t[9 + 15*x]], -2/33] - 4760*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))) 
/45
 

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.07, 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, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle \frac {2}{9} \int \frac {33 (5-3 x) \sqrt {1-2 x}}{(3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {22}{3} \int \frac {(5-3 x) \sqrt {1-2 x}}{(3 x+2)^{3/2} (5 x+3)^{5/2}}dx+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 167

\(\displaystyle \frac {22}{3} \left (\frac {14 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}-\frac {2}{3} \int -\frac {3 (181-208 x)}{2 \sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {22}{3} \left (\int \frac {181-208 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{5/2}}dx+\frac {14 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {22}{3} \left (-\frac {2}{33} \int \frac {11 (674-417 x)}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {22}{3} \left (-\frac {2}{3} \int \frac {674-417 x}{\sqrt {1-2 x} \sqrt {3 x+2} (5 x+3)^{3/2}}dx-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 169

\(\displaystyle \frac {22}{3} \left (-\frac {2}{3} \left (-\frac {2}{11} \int \frac {3 (9242 x+5851)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {9242 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {22}{3} \left (-\frac {2}{3} \left (-\frac {3}{11} \int \frac {9242 x+5851}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {9242 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {22}{3} \left (-\frac {2}{3} \left (-\frac {3}{11} \left (\frac {1529}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {9242}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {9242 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {22}{3} \left (-\frac {2}{3} \left (-\frac {3}{11} \left (\frac {1529}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {9242}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {9242 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {22}{3} \left (-\frac {2}{3} \left (-\frac {3}{11} \left (-\frac {278}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {9242}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {9242 \sqrt {1-2 x} \sqrt {3 x+2}}{11 \sqrt {5 x+3}}\right )-\frac {278 \sqrt {1-2 x} \sqrt {3 x+2}}{3 (5 x+3)^{3/2}}+\frac {14 \sqrt {1-2 x}}{\sqrt {3 x+2} (5 x+3)^{3/2}}\right )+\frac {14 (1-2 x)^{3/2}}{9 (3 x+2)^{3/2} (5 x+3)^{3/2}}\)

Input:

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

Output:

(14*(1 - 2*x)^(3/2))/(9*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2)) + (22*((14*Sqrt[1 
 - 2*x])/(Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (278*Sqrt[1 - 2*x]*Sqrt[2 + 3*x 
])/(3*(3 + 5*x)^(3/2)) - (2*((-9242*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(11*Sqrt[ 
3 + 5*x]) - (3*((-9242*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x] 
], 35/33])/5 - (278*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 
35/33])/5))/11))/3))/3
 

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 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 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 [A] (verified)

Time = 0.78 (sec) , antiderivative size = 234, normalized size of antiderivative = 1.23

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\left (-\frac {974}{3375}-\frac {4628 x}{10125}\right ) \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right )^{2}}-\frac {2 \left (15-30 x \right ) \left (-\frac {175666}{675}-\frac {18484 x}{45}\right )}{\sqrt {\left (x^{2}+\frac {19}{15} x +\frac {2}{5}\right ) \left (15-30 x \right )}}+\frac {23404 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{63 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {36968 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{63 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(234\)
default	\(\frac {2 \left (-132619410-154722645 x [phone] x^{3}[phone] x^{2}[phone] \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{4}[phone] \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{2}-60770430 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x -1155924 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right )+2328984 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] \left (-30 x^{3}-23 x^{2}+7 x +6\right ) x^{3}-2889810 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2}+5822460 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2}-3660426 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, x +7375116 \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, x \right ) \sqrt {1-2 x}}{945 \left (-30 x^{3}-23 x^{2}+7 x +6\right ) \left (-1+2 x \right ) \left (3+5 x \right )^{\frac {3}{2}} \left (2+3 x \right )^{\frac {3}{2}}}\)	\(485\)

Input:

int((1-2*x)^(5/2)/(2+3*x)^(5/2)/(3+5*x)^(5/2),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 
)*((-974/3375-4628/10125*x)*(-30*x^3-23*x^2+7*x+6)^(1/2)/(x^2+19/15*x+2/5) 
^2-2*(15-30*x)*(-175666/675-18484/45*x)/((x^2+19/15*x+2/5)*(15-30*x))^(1/2 
)+23404/63*(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))+36968/63*(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))))
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{5/2}}{(2+3 x)^{5/2} (3+5 x)^{5/2}} \, dx=\frac {2 \, {\left (675 \, {\left (277260 \, x^{3} + 526862 \, x^{2} + 333260 \, x + 70169\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 314024 \, \sqrt {-30} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 831780 \, \sqrt {-30} {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{2025 \, {\left (225 \, x^{4} + 570 \, x^{3} + 541 \, x^{2} + 228 \, x + 36\right )}} \] Input:

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

Output:

2/2025*(675*(277260*x^3 + 526862*x^2 + 333260*x + 70169)*sqrt(5*x + 3)*sqr 
t(3*x + 2)*sqrt(-2*x + 1) - 314024*sqrt(-30)*(225*x^4 + 570*x^3 + 541*x^2 
+ 228*x + 36)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 8317 
80*sqrt(-30)*(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)*weierstrassZeta(11 
59/675, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90) 
))/(225*x^4 + 570*x^3 + 541*x^2 + 228*x + 36)
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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