\(\int \frac {(2+3 x)^{3/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx\) [1460]

Optimal result

Integrand size = 28, antiderivative size = 98 \[ \int \frac {(2+3 x)^{3/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{55 \sqrt {3+5 x}}-\frac {31}{55} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {2}{55} \sqrt {\frac {7}{5}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:

-2/55*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)-31/275*EllipticE(1/11*55^( 
1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-2/275*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 = 5.54 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.90 \[ \int \frac {(2+3 x)^{3/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\frac {1}{275} \left (-\frac {10 \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}}+31 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-35 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \] Input:

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

Output:

((-10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/Sqrt[3 + 5*x] + (31*I)*Sqrt[33]*Ellipti 
cE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - (35*I)*Sqrt[33]*EllipticF[I*ArcSinh 
[Sqrt[9 + 15*x]], -2/33])/275
 

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {109, 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.

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

\(\Big \downarrow \) 109

\(\displaystyle -\frac {2}{55} \int -\frac {3 (31 x+23)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{55 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {3}{55} \int \frac {31 x+23}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{55 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {3}{55} \left (\frac {22}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {31}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{55 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {3}{55} \left (\frac {22}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {31}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{55 \sqrt {5 x+3}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {3}{55} \left (-\frac {4}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {31}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{55 \sqrt {5 x+3}}\)

Input:

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

Output:

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(55*Sqrt[3 + 5*x]) + (3*((-31*Sqrt[11/3]* 
EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5 - (4*Sqrt[11/3]*Ellip 
ticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5))/55
 

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 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 = 0.53 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.36

method	result	size
default	\(-\frac {\sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (66 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-31 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+60 x^{2}+10 x -20\right )}{275 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(133\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {23 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{385 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {31 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{385 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{275 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(201\)

Input:

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

Output:

-1/275*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(66*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))- 
31*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))+60*x^2+10*x-20)/(30*x^3+23*x^2-7*x-6)
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.69 \[ \int \frac {(2+3 x)^{3/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=-\frac {1357 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) - 2790 \, \sqrt {-30} {\left (5 \, x + 3\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) + 900 \, \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{24750 \, {\left (5 \, x + 3\right )}} \] Input:

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

Output:

-1/24750*(1357*sqrt(-30)*(5*x + 3)*weierstrassPInverse(1159/675, 38998/911 
25, x + 23/90) - 2790*sqrt(-30)*(5*x + 3)*weierstrassZeta(1159/675, 38998/ 
91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90)) + 900*sqrt(5 
*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1))/(5*x + 3)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(2+3 x)^{3/2}}{\sqrt {1-2 x} (3+5 x)^{3/2}} \, dx=\frac {-2 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+15 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right ) x +9 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right )+5 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right ) x +3 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{150 x^{4}+205 x^{3}+34 x^{2}-51 x -18}d x \right )}{15 x +9} \] Input:

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

Output:

( - 2*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 15*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 + 9*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) + 5*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 + 3*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))/(3*(5*x + 3))