\(\int \frac {(1-2 x)^{5/2}}{\sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\) [1409]

Optimal result

Integrand size = 28, antiderivative size = 129 \[ \int \frac {(1-2 x)^{5/2}}{\sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=-\frac {22 (1-2 x)^{3/2} \sqrt {2+3 x}}{5 \sqrt {3+5 x}}-\frac {388}{225} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {5594}{675} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )+\frac {808}{675} \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:

-22/5*(1-2*x)^(3/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)-388/225*(1-2*x)^(1/2)*(2+3 
*x)^(1/2)*(3+5*x)^(1/2)+5594/3375*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/ 
35*1155^(1/2))*35^(1/2)+808/3375*EllipticF(1/11*55^(1/2)*(1-2*x)^(1/2),1/3 
5*1155^(1/2))*35^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 5.32 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.72 \[ \int \frac {(1-2 x)^{5/2}}{\sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=\frac {2 \left (\frac {15 \sqrt {1-2 x} \sqrt {2+3 x} (-1077+20 x)}{\sqrt {3+5 x}}-2797 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+3395 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )}{3375} \] Input:

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

Output:

(2*((15*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-1077 + 20*x))/Sqrt[3 + 5*x] - (2797* 
I)*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (3395*I)*Sqrt[33 
]*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/3375
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {109, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle -\frac {2}{5} \int \frac {\sqrt {1-2 x} (97 x+67)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {22 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 171

\(\displaystyle -\frac {2}{5} \left (\frac {2}{45} \int \frac {2797 x+2336}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {194}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {22 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {2}{5} \left (\frac {1}{45} \int \frac {2797 x+2336}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {194}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {22 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 176

\(\displaystyle -\frac {2}{5} \left (\frac {1}{45} \left (\frac {3289}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {2797}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )+\frac {194}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {22 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 123

\(\displaystyle -\frac {2}{5} \left (\frac {1}{45} \left (\frac {3289}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2797}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {194}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {22 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 129

\(\displaystyle -\frac {2}{5} \left (\frac {1}{45} \left (-\frac {598}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {2797}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )+\frac {194}{45} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {22 \sqrt {3 x+2} (1-2 x)^{3/2}}{5 \sqrt {5 x+3}}\)

Input:

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

Output:

(-22*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x])/(5*Sqrt[3 + 5*x]) - (2*((194*Sqrt[1 - 
2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/45 + ((-2797*Sqrt[11/3]*EllipticE[ArcSin 
[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (598*Sqrt[11/3]*EllipticF[ArcSin[Sq 
rt[3/7]*Sqrt[1 - 2*x]], 35/33])/5)/45))/5
 

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 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 = 0.31 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.07

method	result	size
default	\(\frac {2 \sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}\, \left (9867 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-2797 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1800 x^{3}-96630 x^{2}-16755 x +32310\right )}{3375 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(138\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {8 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{225}-\frac {4672 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{4725 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {5594 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{4725 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {242 \left (-30 x^{2}-5 x +10\right )}{125 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(220\)

Input:

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

Output:

2/3375*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(9867*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) 
)-2797*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(2 
8+42*x)^(1/2),1/2*70^(1/2))+1800*x^3-96630*x^2-16755*x+32310)/(30*x^3+23*x 
^2-7*x-6)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.57 \[ \int \frac {(1-2 x)^{5/2}}{\sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=\frac {1350 \, {\left (20 \, x - 1077\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 145909 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 251730 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )}{151875 \, {\left (5 \, x + 3\right )}} \] Input:

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

Output:

1/151875*(1350*(20*x - 1077)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 
145909*sqrt(-30)*(5*x + 3)*weierstrassPInverse(1159/675, 38998/91125, x + 
23/90) - 251730*sqrt(-30)*(5*x + 3)*weierstrassZeta(1159/675, 38998/91125, 
 weierstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(5*x + 3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(1-2 x)^{5/2}}{\sqrt {2+3 x} (3+5 x)^{3/2}} \, dx=\frac {144 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x +274 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}-98670 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right ) x -59202 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right )-8635 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right ) x -5181 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right )}{4050 x +2430} \] Input:

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

Output:

(144*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 274*sqrt(3*x + 2)*sq 
rt(5*x + 3)*sqrt( - 2*x + 1) - 98670*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 - 5920 
2*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) - 8635*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqr 
t( - 2*x + 1))/(150*x**4 + 205*x**3 + 34*x**2 - 51*x - 18),x)*x - 5181*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))/(810*(5*x + 3))