\(\int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{7+5 x} \, dx\) [39]

Optimal result

Integrand size = 35, antiderivative size = 182 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{7+5 x} \, dx=\frac {2}{15} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {427 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{225 \sqrt {5-2 x}}-\frac {1253 \sqrt {\frac {2}{33}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{375 \sqrt {-5+2 x}}-\frac {2691 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{125 \sqrt {11} \sqrt {-5+2 x}} \]

[Out]

-1253/12375*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)-2691/1375
*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)-427/225*El
lipticE(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)+2/15*(2-3*x)^(1/2)*(-
5+2*x)^(1/2)*(1+4*x)^(1/2)

Rubi [A] (verified)

Time = 0.15 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {167, 1621, 174, 552, 551, 164, 115, 114, 122, 120} \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{7+5 x} \, dx=-\frac {1253 \sqrt {\frac {2}{33}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{375 \sqrt {2 x-5}}-\frac {427 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{225 \sqrt {5-2 x}}-\frac {2691 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{125 \sqrt {11} \sqrt {2 x-5}}+\frac {2}{15} \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1} \]

[In]

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

[Out]

(2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/15 - (427*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 -
3*x])/Sqrt[11]], -1/2])/(225*Sqrt[5 - 2*x]) - (1253*Sqrt[2/33]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[
1 + 4*x]], 1/3])/(375*Sqrt[-5 + 2*x]) - (2691*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[1
1]], -1/2])/(125*Sqrt[11]*Sqrt[-5 + 2*x])

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> 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] &&  !LtQ[-(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 115

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[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] && GtQ[b/(b*e - a*f), 0]) &&  !LtQ[-(b*c - a*d)/d, 0]

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> 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] && Po
sQ[-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 122

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[b*((c
+ d*x)/(b*c - a*d))]/Sqrt[c + d*x], Int[1/(Sqrt[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] && SimplerQ[a + b*x, c + d*x] && Si
mplerQ[a + b*x, e + f*x]

Rule 164

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(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] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rule 167

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

Rule 174

Int[1/(((a_.) + (b_.)*(x_))*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Sym
bol] :> Dist[-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 551

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1/(a*Sqr
t[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] && SimplerSqrtQ[-f/e, -d/c])

Rule 552

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Dist[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 1621

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.)*((g_.) + (h_.)*(x_)
)^(q_.), x_Symbol] :> Dist[PolynomialRemainder[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]

Rubi steps \begin{align*} \text {integral}& = \frac {2}{15} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{15} \int \frac {-3-1190 x+854 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx \\ & = \frac {2}{15} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}+\frac {1}{15} \int \frac {-\frac {11928}{25}+\frac {854 x}{5}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx+\frac {27807}{125} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx \\ & = \frac {2}{15} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {1253}{375} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx+\frac {427}{75} \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx-\frac {55614}{125} \text {Subst}\left (\int \frac {1}{\left (31-5 x^2\right ) \sqrt {\frac {11}{3}-\frac {4 x^2}{3}} \sqrt {-\frac {11}{3}-\frac {2 x^2}{3}}} \, dx,x,\sqrt {2-3 x}\right ) \\ & = \frac {2}{15} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {\left (1253 \sqrt {\frac {2}{11}} \sqrt {5-2 x}\right ) \int \frac {1}{\sqrt {2-3 x} \sqrt {\frac {10}{11}-\frac {4 x}{11}} \sqrt {1+4 x}} \, dx}{375 \sqrt {-5+2 x}}-\frac {\left (55614 \sqrt {\frac {3}{11}} \sqrt {5-2 x}\right ) \text {Subst}\left (\int \frac {1}{\left (31-5 x^2\right ) \sqrt {\frac {11}{3}-\frac {4 x^2}{3}} \sqrt {1+\frac {2 x^2}{11}}} \, dx,x,\sqrt {2-3 x}\right )}{125 \sqrt {-5+2 x}}+\frac {\left (427 \sqrt {-5+2 x}\right ) \int \frac {\sqrt {\frac {15}{11}-\frac {6 x}{11}}}{\sqrt {2-3 x} \sqrt {\frac {3}{11}+\frac {12 x}{11}}} \, dx}{75 \sqrt {5-2 x}} \\ & = \frac {2}{15} \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}-\frac {427 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{225 \sqrt {5-2 x}}-\frac {1253 \sqrt {\frac {2}{33}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{375 \sqrt {-5+2 x}}-\frac {2691 \sqrt {5-2 x} \Pi \left (\frac {55}{124};\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{125 \sqrt {11} \sqrt {-5+2 x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.29 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.76 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{7+5 x} \, dx=\frac {\sqrt {-5+2 x} \left (1650 \sqrt {2-3 x} \sqrt {5-2 x} \sqrt {1+4 x}-23485 \sqrt {11} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )-3759 \sqrt {11} \operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )+24219 \sqrt {11} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )\right )}{12375 \sqrt {5-2 x}} \]

[In]

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

[Out]

(Sqrt[-5 + 2*x]*(1650*Sqrt[2 - 3*x]*Sqrt[5 - 2*x]*Sqrt[1 + 4*x] - 23485*Sqrt[11]*EllipticE[ArcSin[(2*Sqrt[2 -
3*x])/Sqrt[11]], -1/2] - 3759*Sqrt[11]*EllipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] + 24219*Sqrt[11]*El
lipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))/(12375*Sqrt[5 - 2*x])

Maple [A] (verified)

Time = 1.84 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.96

method	result	size
default	\(\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (54488 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )+23485 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-87048 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )+79200 x^{3}-231000 x^{2}+69300 x +33000\right )}{594000 x^{3}-1732500 x^{2}+519750 x +247500}\)	\(174\)
elliptic	\(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {2 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{15}-\frac {3976 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{15125 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {854 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{9075 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {4836 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )}{15125 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(240\)
risch	\(-\frac {2 \left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{15 \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {3976 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{45375 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {854 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{27225 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {897 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \Pi \left (\frac {2 \sqrt {22-33 x}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )}{15125 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(299\)

[In]

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

[Out]

1/24750*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)*(54488*(1+4*x)^(1/2)*(2-3*x)^(1/2)*22^(1/2)*(5-2*x)^(1/2)*E
llipticF(1/11*(11+44*x)^(1/2),3^(1/2))+23485*(1+4*x)^(1/2)*(2-3*x)^(1/2)*22^(1/2)*(5-2*x)^(1/2)*EllipticE(1/11
*(11+44*x)^(1/2),3^(1/2))-87048*(1+4*x)^(1/2)*(2-3*x)^(1/2)*22^(1/2)*(5-2*x)^(1/2)*EllipticPi(1/11*(11+44*x)^(
1/2),-55/23,3^(1/2))+79200*x^3-231000*x^2+69300*x+33000)/(24*x^3-70*x^2+21*x+10)

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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