3.1.50 \(\int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx\) [50]

3.1.50.1 Optimal result

Integrand size = 35, antiderivative size = 225 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{78 (7+5 x)^2}-\frac {361 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{481988 (7+5 x)}+\frac {361 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1204970 \sqrt {5-2 x}}-\frac {6101 \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{231725 \sqrt {66} \sqrt {-5+2 x}}-\frac {6655867 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{747081400 \sqrt {11} \sqrt {-5+2 x}} \]

output

-6655867/8217895400*EllipticPi(2/11*(2-3*x)^(1/2)*11^(1/2),55/124,1/2*I*2^ 
(1/2))*(5-2*x)^(1/2)*11^(1/2)/(-5+2*x)^(1/2)-6101/15293850*EllipticF(1/11* 
33^(1/2)*(1+4*x)^(1/2),1/3*3^(1/2))*66^(1/2)*(5-2*x)^(1/2)/(-5+2*x)^(1/2)+ 
361/1204970*EllipticE(2/11*(2-3*x)^(1/2)*11^(1/2),1/2*I*2^(1/2))*11^(1/2)* 
(-5+2*x)^(1/2)/(5-2*x)^(1/2)+1/78*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/ 
2)/(7+5*x)^2-361/481988*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)
 

3.1.50.2 Mathematica [A] (verified)

Time = 5.29 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.60 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\frac {-\frac {17050 \sqrt {2-3 x} (-5+2 x) \sqrt {1+4 x} (-10957+5415 x)}{(7+5 x)^2}-3 \sqrt {55-22 x} \left (2462020 E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )-9834812 \operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )+6655867 \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )\right )}{24653686200 \sqrt {-5+2 x}} \]

input

Integrate[(Sqrt[2 - 3*x]*Sqrt[1 + 4*x])/(Sqrt[-5 + 2*x]*(7 + 5*x)^3),x]
 

output

((-17050*Sqrt[2 - 3*x]*(-5 + 2*x)*Sqrt[1 + 4*x]*(-10957 + 5415*x))/(7 + 5* 
x)^2 - 3*Sqrt[55 - 22*x]*(2462020*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[ 
11]], -1/2] - 9834812*EllipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] 
+ 6655867*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))/( 
24653686200*Sqrt[-5 + 2*x])
 

3.1.50.3 Rubi [A] (verified)

Time = 0.61 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.09, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.457, Rules used = {182, 25, 2107, 27, 2110, 176, 124, 123, 131, 27, 129, 186, 27, 413, 27, 412}

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.

\(\displaystyle \int \frac {\sqrt {2-3 x} \sqrt {4 x+1}}{\sqrt {2 x-5} (5 x+7)^3} \, dx\)

\(\Big \downarrow \) 182

\(\displaystyle \frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}-\frac {1}{156} \int -\frac {-24 x^2-100 x+37}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2}dx\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {1}{156} \int \frac {-24 x^2-100 x+37}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)^2}dx+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 2107

\(\displaystyle \frac {1}{156} \left (\frac {\int \frac {3 \left (-25992 x^2-161760 x+90715\right )}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{55614}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{156} \left (\frac {\int \frac {-25992 x^2-161760 x+90715}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 2110

\(\displaystyle \frac {1}{156} \left (\frac {\int \frac {-\frac {25992 x}{5}-\frac {626856}{25}}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {6655867}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{156} \left (\frac {-\frac {951756}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {12996}{5} \int \frac {\sqrt {2 x-5}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx+\frac {6655867}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {1}{156} \left (\frac {-\frac {12996 \sqrt {2 x-5} \int \frac {\sqrt {5-2 x}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx}{5 \sqrt {5-2 x}}-\frac {951756}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {6655867}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{156} \left (\frac {-\frac {951756}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx+\frac {6655867}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx-\frac {2166 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {1}{156} \left (\frac {-\frac {951756 \sqrt {\frac {2}{11}} \sqrt {5-2 x} \int \frac {\sqrt {\frac {11}{2}}}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{25 \sqrt {2 x-5}}+\frac {6655867}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx-\frac {2166 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{156} \left (\frac {-\frac {951756 \sqrt {5-2 x} \int \frac {1}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{25 \sqrt {2 x-5}}+\frac {6655867}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx-\frac {2166 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{156} \left (\frac {\frac {6655867}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx-\frac {317252 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{25 \sqrt {2 x-5}}-\frac {2166 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 186

\(\displaystyle \frac {1}{156} \left (\frac {-\frac {13311734}{25} \int \frac {3}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {-2 (2-3 x)-11}}d\sqrt {2-3 x}-\frac {317252 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{25 \sqrt {2 x-5}}-\frac {2166 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{156} \left (\frac {-\frac {39935202}{25} \int \frac {1}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {-2 (2-3 x)-11}}d\sqrt {2-3 x}-\frac {317252 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{25 \sqrt {2 x-5}}-\frac {2166 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {1}{156} \left (\frac {-\frac {39935202 \sqrt {2 (2-3 x)+11} \int \frac {\sqrt {11}}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {2 (2-3 x)+11}}d\sqrt {2-3 x}}{25 \sqrt {11} \sqrt {-2 (2-3 x)-11}}-\frac {317252 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{25 \sqrt {2 x-5}}-\frac {2166 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{156} \left (\frac {-\frac {39935202 \sqrt {2 (2-3 x)+11} \int \frac {1}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {2 (2-3 x)+11}}d\sqrt {2-3 x}}{25 \sqrt {-2 (2-3 x)-11}}-\frac {317252 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{25 \sqrt {2 x-5}}-\frac {2166 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {1}{156} \left (\frac {-\frac {317252 \sqrt {\frac {6}{11}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{25 \sqrt {2 x-5}}-\frac {2166 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{5 \sqrt {5-2 x}}-\frac {19967601 \sqrt {2 (2-3 x)+11} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{775 \sqrt {11} \sqrt {-2 (2-3 x)-11}}}{18538}-\frac {1083 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{9269 (5 x+7)}\right )+\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{78 (5 x+7)^2}\)

input

Int[(Sqrt[2 - 3*x]*Sqrt[1 + 4*x])/(Sqrt[-5 + 2*x]*(7 + 5*x)^3),x]
 

output

(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(78*(7 + 5*x)^2) + ((-1083*Sq 
rt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(9269*(7 + 5*x)) + ((-2166*Sqrt[ 
66]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(5*Sq 
rt[5 - 2*x]) - (317252*Sqrt[6/11]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11 
]*Sqrt[1 + 4*x]], 1/3])/(25*Sqrt[-5 + 2*x]) - (19967601*Sqrt[11 + 2*(2 - 3 
*x)]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(775*Sq 
rt[11]*Sqrt[-11 - 2*(2 - 3*x)]))/18538)/156
 

3.1.50.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 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 124

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_ 
)]), x_] :> Simp[Sqrt[e + f*x]*(Sqrt[b*((c + d*x)/(b*c - a*d))]/(Sqrt[c + d 
*x]*Sqrt[b*((e + f*x)/(b*e - a*f))]))   Int[Sqrt[b*(e/(b*e - a*f)) + b*f*(x 
/(b*e - a*f))]/(Sqrt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))] 
), x], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !(GtQ[b/(b*c - a*d), 0] && Gt 
Q[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 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 131

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x 
_)]), x_] :> Simp[Sqrt[b*((c + d*x)/(b*c - a*d))]/Sqrt[c + d*x]   Int[1/(Sq 
rt[a + b*x]*Sqrt[b*(c/(b*c - a*d)) + b*d*(x/(b*c - a*d))]*Sqrt[e + f*x]), x 
], x] /; FreeQ[{a, b, c, d, e, f}, x] &&  !GtQ[(b*c - a*d)/b, 0] && Simpler 
Q[a + b*x, c + d*x] && SimplerQ[a + b*x, e + f*x]
 

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]
 

rule 182

Int[(((a_.) + (b_.)*(x_))^(m_)*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)* 
(x_)])/Sqrt[(c_.) + (d_.)*(x_)], x_] :> Simp[(a + b*x)^(m + 1)*Sqrt[c + d*x 
]*Sqrt[e + f*x]*(Sqrt[g + h*x]/((m + 1)*(b*c - a*d))), x] - Simp[1/(2*(m + 
1)*(b*c - a*d))   Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[ 
g + h*x]))*Simp[c*(f*g + e*h) + d*e*g*(2*m + 3) + 2*(c*f*h + d*(m + 2)*(f*g 
 + e*h))*x + d*f*h*(2*m + 5)*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, 
 h, m}, x] && IntegerQ[2*m] && LtQ[m, -1]
 

rule 186

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_ 
)]*Sqrt[(g_.) + (h_.)*(x_)]), x_] :> Simp[-2   Subst[Int[1/(Simp[b*c - a*d 
- b*x^2, x]*Sqrt[Simp[(d*e - c*f)/d + f*(x^2/d), x]]*Sqrt[Simp[(d*g - c*h)/ 
d + h*(x^2/d), x]]), x], x, Sqrt[c + d*x]], x] /; FreeQ[{a, b, c, d, e, f, 
g, h}, x] && GtQ[(d*e - c*f)/d, 0]
 

rule 412

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[(1/(a*Sqrt[c]*Sqrt[e]*Rt[-d/c, 2]))*EllipticPi[b* 
(c/(a*d)), ArcSin[Rt[-d/c, 2]*x], c*(f/(d*e))], x] /; FreeQ[{a, b, c, d, e, 
 f}, x] &&  !GtQ[d/c, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !( !GtQ[f/e, 0] && S 
implerSqrtQ[-f/e, -d/c])
 

rule 413

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x 
_)^2]), x_Symbol] :> Simp[Sqrt[1 + (d/c)*x^2]/Sqrt[c + d*x^2]   Int[1/((a + 
 b*x^2)*Sqrt[1 + (d/c)*x^2]*Sqrt[e + f*x^2]), x], x] /; FreeQ[{a, b, c, d, 
e, f}, x] &&  !GtQ[c, 0]
 

rule 2107

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[( 
c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sy 
mbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[ 
e + f*x]*(Sqrt[g + h*x]/((m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))), x] 
- Simp[1/(2*(m + 1)*(b*c - a*d)*(b*e - a*f)*(b*g - a*h))   Int[((a + b*x)^( 
m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[A*(2*a^2*d*f*h*(m 
+ 1) - 2*a*b*(m + 1)*(d*f*g + d*e*h + c*f*h) + b^2*(2*m + 3)*(d*e*g + c*f*g 
 + c*e*h)) - (b*B - a*C)*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*(m + 1)) - 
2*((A*b - a*B)*(a*d*f*h*(m + 1) - b*(m + 2)*(d*f*g + d*e*h + c*f*h)) - C*(a 
^2*(d*f*g + d*e*h + c*f*h) - b^2*c*e*g*(m + 1) + a*b*(m + 1)*(d*e*g + c*f*g 
 + c*e*h)))*x + d*f*h*(2*m + 5)*(A*b^2 - a*b*B + a^2*C)*x^2, x], x], x] /; 
FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x] && IntegerQ[2*m] && LtQ[m, -1]
 

rule 2110

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f 
_.)*(x_))^(p_.)*((g_.) + (h_.)*(x_))^(q_.), x_Symbol] :> Simp[PolynomialRem 
ainder[Px, a + b*x, x]   Int[(a + b*x)^m*(c + d*x)^n*(e + f*x)^p*(g + h*x)^ 
q, x], x] + Int[PolynomialQuotient[Px, a + b*x, x]*(a + b*x)^(m + 1)*(c + d 
*x)^n*(e + f*x)^p*(g + h*x)^q, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n, p 
, q}, x] && PolyQ[Px, x] && EqQ[m, -1]
 

3.1.50.4 Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.21

method	result	size
elliptic	\(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {\sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{78 \left (7+5 x \right )^{2}}-\frac {361 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{481988 \left (7+5 x \right )}-\frac {26119 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{364503425 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {1083 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{72900685 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {6655867 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )}{50301472650 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(273\)
risch	\(\frac {\left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (-10957+5415 x \right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{1445964 \left (7+5 x \right )^{2} \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {26119 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{1093510275 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {361 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{72900685 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {6655867 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \Pi \left (\frac {2 \sqrt {22-33 x}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )}{271190548200 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(311\)
default	\(\frac {\sqrt {2-3 x}\, \sqrt {1+4 x}\, \sqrt {-5+2 x}\, \left (205130100 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}-34249875 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}-332793350 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x^{2}[phone] \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -95899650 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -931821380 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x [phone] \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-67129755 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-652274966 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )-821997000 x^{4}[phone] x^{3}-5570459125 x^{2}[phone] x [phone]\right )}{9145722300 \left (24 x^{3}-70 x^{2}+21 x +10\right ) \left (7+5 x \right )^{2}}\)	\(434\)

input

int((2-3*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^3/(-5+2*x)^(1/2),x,method=_RETURNV 
ERBOSE)
 

output

(-(-2+3*x)*(-5+2*x)*(1+4*x))^(1/2)/(2-3*x)^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1 
/2)*(1/78/(7+5*x)^2*(-24*x^3+70*x^2-21*x-10)^(1/2)-361/481988/(7+5*x)*(-24 
*x^3+70*x^2-21*x-10)^(1/2)-26119/364503425*(11+44*x)^(1/2)*(22-33*x)^(1/2) 
*(110-44*x)^(1/2)/(-24*x^3+70*x^2-21*x-10)^(1/2)*EllipticF(1/11*(11+44*x)^ 
(1/2),3^(1/2))-1083/72900685*(11+44*x)^(1/2)*(22-33*x)^(1/2)*(110-44*x)^(1 
/2)/(-24*x^3+70*x^2-21*x-10)^(1/2)*(-11/12*EllipticE(1/11*(11+44*x)^(1/2), 
3^(1/2))+2/3*EllipticF(1/11*(11+44*x)^(1/2),3^(1/2)))+6655867/50301472650* 
(11+44*x)^(1/2)*(22-33*x)^(1/2)*(110-44*x)^(1/2)/(-24*x^3+70*x^2-21*x-10)^ 
(1/2)*EllipticPi(1/11*(11+44*x)^(1/2),-55/23,3^(1/2)))
 

3.1.50.5 Fricas [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {2 \, x - 5}} \,d x } \]

input

integrate((2-3*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^3/(-5+2*x)^(1/2),x, algorith 
m="fricas")
 

output

integral(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(250*x^4 + 425*x^3 - 1 
155*x^2 - 2989*x - 1715), x)
 

3.1.50.6 Sympy [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\int \frac {\sqrt {2 - 3 x} \sqrt {4 x + 1}}{\sqrt {2 x - 5} \left (5 x + 7\right )^{3}}\, dx \]

input

integrate((2-3*x)**(1/2)*(1+4*x)**(1/2)/(7+5*x)**3/(-5+2*x)**(1/2),x)
 

output

Integral(sqrt(2 - 3*x)*sqrt(4*x + 1)/(sqrt(2*x - 5)*(5*x + 7)**3), x)
 

3.1.50.7 Maxima [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {2 \, x - 5}} \,d x } \]

input

integrate((2-3*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^3/(-5+2*x)^(1/2),x, algorith 
m="maxima")
 

output

integrate(sqrt(4*x + 1)*sqrt(-3*x + 2)/((5*x + 7)^3*sqrt(2*x - 5)), x)
 

3.1.50.8 Giac [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{3} \sqrt {2 \, x - 5}} \,d x } \]

input

integrate((2-3*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^3/(-5+2*x)^(1/2),x, algorith 
m="giac")
 

output

integrate(sqrt(4*x + 1)*sqrt(-3*x + 2)/((5*x + 7)^3*sqrt(2*x - 5)), x)
 

3.1.50.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^3} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}}{\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^3} \,d x \]

input

int(((2 - 3*x)^(1/2)*(4*x + 1)^(1/2))/((2*x - 5)^(1/2)*(5*x + 7)^3),x)
 

output

int(((2 - 3*x)^(1/2)*(4*x + 1)^(1/2))/((2*x - 5)^(1/2)*(5*x + 7)^3), x)