3.1.48 \(\int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)} \, dx\) [48]

3.1.48.1 Optimal result

Integrand size = 35, antiderivative size = 151 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)} \, dx=\frac {2 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{5 \sqrt {5-2 x}}-\frac {41 \sqrt {\frac {2}{33}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{25 \sqrt {-5+2 x}}+\frac {69 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{25 \sqrt {11} \sqrt {-5+2 x}} \]

output

-41/825*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)+69/275*EllipticPi(2/11*(2-3*x)^(1/2)*11^(1/2),55/12 
4,1/2*I*2^(1/2))*(5-2*x)^(1/2)*11^(1/2)/(-5+2*x)^(1/2)+2/5*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 
)
 

3.1.48.2 Mathematica [A] (verified)

Time = 2.58 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)} \, dx=\frac {\sqrt {5-2 x} \left (-110 E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )+41 \operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )+69 \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )\right )}{25 \sqrt {-55+22 x}} \]

input

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

output

(Sqrt[5 - 2*x]*(-110*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] + 
 41*EllipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] + 69*EllipticPi[55 
/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))/(25*Sqrt[-55 + 22*x])
 

3.1.48.3 Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.07, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.343, Rules used = {181, 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.

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

\(\Big \downarrow \) 181

\(\displaystyle \frac {1}{25} \int \frac {109-60 x}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {713}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx\)

\(\Big \downarrow \) 176

\(\displaystyle \frac {1}{25} \left (-41 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-30 \int \frac {\sqrt {2 x-5}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx\right )-\frac {713}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx\)

\(\Big \downarrow \) 124

\(\displaystyle \frac {1}{25} \left (-\frac {30 \sqrt {2 x-5} \int \frac {\sqrt {5-2 x}}{\sqrt {2-3 x} \sqrt {4 x+1}}dx}{\sqrt {5-2 x}}-41 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx\right )-\frac {713}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx\)

\(\Big \downarrow \) 123

\(\displaystyle \frac {1}{25} \left (-41 \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}dx-\frac {5 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )-\frac {713}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx\)

\(\Big \downarrow \) 131

\(\displaystyle \frac {1}{25} \left (-\frac {41 \sqrt {\frac {2}{11}} \sqrt {5-2 x} \int \frac {\sqrt {\frac {11}{2}}}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{\sqrt {2 x-5}}-\frac {5 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )-\frac {713}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {1}{25} \left (-\frac {41 \sqrt {5-2 x} \int \frac {1}{\sqrt {2-3 x} \sqrt {5-2 x} \sqrt {4 x+1}}dx}{\sqrt {2 x-5}}-\frac {5 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )-\frac {713}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx\)

\(\Big \downarrow \) 129

\(\displaystyle \frac {1}{25} \left (-\frac {41 \sqrt {\frac {2}{33}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{\sqrt {2 x-5}}-\frac {5 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )-\frac {713}{25} \int \frac {1}{\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} (5 x+7)}dx\)

\(\Big \downarrow \) 186

\(\displaystyle \frac {1426}{25} \int \frac {3}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {-2 (2-3 x)-11}}d\sqrt {2-3 x}+\frac {1}{25} \left (-\frac {41 \sqrt {\frac {2}{33}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{\sqrt {2 x-5}}-\frac {5 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4278}{25} \int \frac {1}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {-2 (2-3 x)-11}}d\sqrt {2-3 x}+\frac {1}{25} \left (-\frac {41 \sqrt {\frac {2}{33}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{\sqrt {2 x-5}}-\frac {5 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )\)

\(\Big \downarrow \) 413

\(\displaystyle \frac {4278 \sqrt {2 (2-3 x)+11} \int \frac {\sqrt {11}}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {2 (2-3 x)+11}}d\sqrt {2-3 x}}{25 \sqrt {11} \sqrt {-2 (2-3 x)-11}}+\frac {1}{25} \left (-\frac {41 \sqrt {\frac {2}{33}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{\sqrt {2 x-5}}-\frac {5 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {4278 \sqrt {2 (2-3 x)+11} \int \frac {1}{(31-5 (2-3 x)) \sqrt {11-4 (2-3 x)} \sqrt {2 (2-3 x)+11}}d\sqrt {2-3 x}}{25 \sqrt {-2 (2-3 x)-11}}+\frac {1}{25} \left (-\frac {41 \sqrt {\frac {2}{33}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{\sqrt {2 x-5}}-\frac {5 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )\)

\(\Big \downarrow \) 412

\(\displaystyle \frac {69 \sqrt {2 (2-3 x)+11} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{25 \sqrt {11} \sqrt {-2 (2-3 x)-11}}+\frac {1}{25} \left (-\frac {41 \sqrt {\frac {2}{33}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{\sqrt {2 x-5}}-\frac {5 \sqrt {66} \sqrt {2 x-5} E\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right )|\frac {1}{3}\right )}{\sqrt {5-2 x}}\right )\)

input

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

output

((-5*Sqrt[66]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1 
/3])/Sqrt[5 - 2*x] - (41*Sqrt[2/33]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/ 
11]*Sqrt[1 + 4*x]], 1/3])/Sqrt[-5 + 2*x])/25 + (69*Sqrt[11 + 2*(2 - 3*x)]* 
EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(25*Sqrt[11] 
*Sqrt[-11 - 2*(2 - 3*x)])
 

3.1.48.3.1 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 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 181

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

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]
 

3.1.48.4 Maple [A] (verified)

Time = 1.58 (sec) , antiderivative size = 67, normalized size of antiderivative = 0.44

method	result	size
default	\(\frac {\left (69 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+55 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-124 \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )\right ) \sqrt {5-2 x}\, \sqrt {22}}{275 \sqrt {-5+2 x}}\)	\(67\)
elliptic	\(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {109 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3025 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {12 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{605 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}-\frac {124 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )}{3025 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(221\)

input

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

output

1/275*(69*EllipticF(1/11*(11+44*x)^(1/2),3^(1/2))+55*EllipticE(1/11*(11+44 
*x)^(1/2),3^(1/2))-124*EllipticPi(1/11*(11+44*x)^(1/2),-55/23,3^(1/2)))*(5 
-2*x)^(1/2)*22^(1/2)/(-5+2*x)^(1/2)
 

3.1.48.5 Fricas [F]

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

input

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

output

integral(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(10*x^2 - 11*x - 35), 
x)
 

3.1.48.6 Sympy [F]

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

input

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

output

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

3.1.48.7 Maxima [F]

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

input

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

output

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

3.1.48.8 Giac [F]

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

input

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

output

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

3.1.48.9 Mupad [F(-1)]

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

input

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

output

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