\(\int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx\) [1312]

Optimal result

Integrand size = 28, antiderivative size = 127 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx=-\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{3 \sqrt {2+3 x}}-\frac {16}{27} \sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}+\frac {494}{81} \sqrt {\frac {7}{5}} E\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {46}{81} \sqrt {35} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right ),\frac {33}{35}\right ) \] Output:

-2/3*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)-16/27*(1-2*x)^(1/2)*(2+3*x) 
^(1/2)*(3+5*x)^(1/2)+494/405*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*11 
55^(1/2))*35^(1/2)-46/81*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 = 5.52 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.73 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx=\frac {2}{405} \left (-\frac {15 \sqrt {1-2 x} \sqrt {3+5 x} (25+6 x)}{\sqrt {2+3 x}}-247 i \sqrt {33} E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )+140 i \sqrt {33} \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right ) \] Input:

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

Output:

(2*((-15*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(25 + 6*x))/Sqrt[2 + 3*x] - (247*I)*S 
qrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] + (140*I)*Sqrt[33]*Ell 
ipticF[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/405
 

Rubi [A] (verified)

Time = 0.21 (sec) , antiderivative size = 139, normalized size of antiderivative = 1.09, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {108, 27, 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.

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

\(\Big \downarrow \) 108

\(\displaystyle \frac {2}{3} \int -\frac {\sqrt {1-2 x} (40 x+13)}{2 \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{3 \sqrt {3 x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {1}{3} \int \frac {\sqrt {1-2 x} (40 x+13)}{\sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{3 \sqrt {3 x+2}}\)

\(\Big \downarrow \) 171

\(\displaystyle \frac {1}{3} \left (-\frac {2}{45} \int \frac {5 (494 x+61)}{2 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {16}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{3 \sqrt {3 x+2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{3} \left (-\frac {1}{9} \int \frac {494 x+61}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {16}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{3 \sqrt {3 x+2}}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{3} \left (\frac {1}{9} \left (\frac {1177}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx-\frac {494}{5} \int \frac {\sqrt {5 x+3}}{\sqrt {1-2 x} \sqrt {3 x+2}}dx\right )-\frac {16}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{3 \sqrt {3 x+2}}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{3} \left (\frac {1}{9} \left (\frac {1177}{5} \int \frac {1}{\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}}dx+\frac {494}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\right )-\frac {16}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{3 \sqrt {3 x+2}}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{3} \left (\frac {1}{9} \left (\frac {494}{5} \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )-\frac {214}{5} \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )\right )-\frac {16}{9} \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}\right )-\frac {2 (1-2 x)^{3/2} \sqrt {5 x+3}}{3 \sqrt {3 x+2}}\)

Input:

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

Output:

(-2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(3*Sqrt[2 + 3*x]) + ((-16*Sqrt[1 - 2*x] 
*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9 + ((494*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3 
/7]*Sqrt[1 - 2*x]], 35/33])/5 - (214*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7] 
*Sqrt[1 - 2*x]], 35/33])/5)/9)/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 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 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 = 0.28 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.09

method	result	size
default	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (3531 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+494 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+1800 x^{3}+7680 x^{2}+210 x -2250\right )}{405 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(138\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {14 \left (-30 x^{2}-3 x +9\right )}{27 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {61 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{567 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {494 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\operatorname {EllipticE}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \operatorname {EllipticF}\left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{567 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {4 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{27}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(220\)

Input:

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

Output:

-1/405*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(3531*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) 
)+494*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))+1800*x^3+7680*x^2+210*x-2250)/(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 \[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx=-\frac {2 \, {\left (675 \, {\left (6 \, x + 25\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} + 1468 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 11115 \, \sqrt {-30} {\left (3 \, x + 2\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{18225 \, {\left (3 \, x + 2\right )}} \] Input:

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

Output:

-2/18225*(675*(6*x + 25)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1) + 1468 
*sqrt(-30)*(3*x + 2)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) 
 + 11115*sqrt(-30)*(3*x + 2)*weierstrassZeta(1159/675, 38998/91125, weiers 
trassPInverse(1159/675, 38998/91125, x + 23/90)))/(3*x + 2)
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F]

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

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {(1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{3/2}} \, dx=\frac {-160 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x +114 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}+24150 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{90 x^{4}+129 x^{3}+25 x^{2}-32 x -12}d x \right ) x +16100 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, x^{2}}{90 x^{4}+129 x^{3}+25 x^{2}-32 x -12}d x \right )-8001 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{90 x^{4}+129 x^{3}+25 x^{2}-32 x -12}d x \right ) x -5334 \left (\int \frac {\sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}}{90 x^{4}+129 x^{3}+25 x^{2}-32 x -12}d x \right )}{1080 x +720} \] Input:

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

Output:

( - 160*sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x + 114*sqrt(3*x + 2) 
*sqrt(5*x + 3)*sqrt( - 2*x + 1) + 24150*int((sqrt(3*x + 2)*sqrt(5*x + 3)*s 
qrt( - 2*x + 1)*x**2)/(90*x**4 + 129*x**3 + 25*x**2 - 32*x - 12),x)*x + 16 
100*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1)*x**2)/(90*x**4 + 129 
*x**3 + 25*x**2 - 32*x - 12),x) - 8001*int((sqrt(3*x + 2)*sqrt(5*x + 3)*sq 
rt( - 2*x + 1))/(90*x**4 + 129*x**3 + 25*x**2 - 32*x - 12),x)*x - 5334*int 
((sqrt(3*x + 2)*sqrt(5*x + 3)*sqrt( - 2*x + 1))/(90*x**4 + 129*x**3 + 25*x 
**2 - 32*x - 12),x))/(360*(3*x + 2))