\(\int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^3} \, dx\) [41]

Optimal result

Integrand size = 35, antiderivative size = 227 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^3} \, dx=-\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{556140 (7+5 x)}-\frac {8953 \sqrt {11} \sqrt {-5+2 x} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1390350 \sqrt {5-2 x}}+\frac {397 \sqrt {\frac {3}{22}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right ),\frac {1}{3}\right )}{89125 \sqrt {-5+2 x}}-\frac {14832503 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{287339000 \sqrt {11} \sqrt {-5+2 x}} \]

[Out]

397/1960750*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)-14832503/
3160729000*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)-
8953/1390350*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)-1/10*(
2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^2+8953/556140*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+
5*x)

Rubi [A] (verified)

Time = 0.20 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.314, Rules used = {166, 1618, 1621, 174, 552, 551, 164, 115, 114, 122, 120} \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^3} \, dx=\frac {397 \sqrt {\frac {3}{22}} \sqrt {5-2 x} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{11}} \sqrt {4 x+1}\right ),\frac {1}{3}\right )}{89125 \sqrt {2 x-5}}-\frac {8953 \sqrt {11} \sqrt {2 x-5} E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1390350 \sqrt {5-2 x}}-\frac {14832503 \sqrt {5-2 x} \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )}{287339000 \sqrt {11} \sqrt {2 x-5}}+\frac {8953 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{556140 (5 x+7)}-\frac {\sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{10 (5 x+7)^2} \]

[In]

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

[Out]

-1/10*(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(7 + 5*x)^2 + (8953*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4
*x])/(556140*(7 + 5*x)) - (8953*Sqrt[11]*Sqrt[-5 + 2*x]*EllipticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(
1390350*Sqrt[5 - 2*x]) + (397*Sqrt[3/22]*Sqrt[5 - 2*x]*EllipticF[ArcSin[Sqrt[3/11]*Sqrt[1 + 4*x]], 1/3])/(8912
5*Sqrt[-5 + 2*x]) - (14832503*Sqrt[5 - 2*x]*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2])/(287
339000*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 166

Int[((a_.) + (b_.)*(x_))^(m_)*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)], x_Sy
mbol] :> Simp[(a + b*x)^(m + 1)*Sqrt[c + d*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(b*(m + 1))), x] - Dist[1/(2*b*(m +
1)), Int[((a + b*x)^(m + 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[d*e*g + c*f*g + c*e*h + 2*(d*f*
g + d*e*h + c*f*h)*x + 3*d*f*h*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] && Lt
Q[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 1618

Int[(((a_.) + (b_.)*(x_))^(m_)*((A_.) + (B_.)*(x_) + (C_.)*(x_)^2))/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_
.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> 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] - Dist[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 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 {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {1}{20} \int \frac {-21+140 x-72 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^2} \, dx \\ & = -\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{556140 (7+5 x)}+\frac {\int \frac {-106729-199200 x+214872 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{1112280} \\ & = -\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{556140 (7+5 x)}+\frac {\int \frac {-\frac {2500104}{25}+\frac {214872 x}{5}}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{1112280}+\frac {14832503 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)} \, dx}{27807000} \\ & = -\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{556140 (7+5 x)}+\frac {1191 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}} \, dx}{178250}+\frac {8953 \int \frac {\sqrt {-5+2 x}}{\sqrt {2-3 x} \sqrt {1+4 x}} \, dx}{463450}-\frac {14832503 \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 )}{13903500} \\ & = -\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{556140 (7+5 x)}+\frac {\left (1191 \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}{89125 \sqrt {22} \sqrt {-5+2 x}}-\frac {\left (14832503 \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 )}{4634500 \sqrt {33} \sqrt {-5+2 x}}+\frac {\left (8953 \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}{463450 \sqrt {5-2 x}} \\ & = -\frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{10 (7+5 x)^2}+\frac {8953 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{556140 (7+5 x)}-\frac {8953 \sqrt {11} \sqrt {-5+2 x} E\left (\sin ^{-1}\left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )}{1390350 \sqrt {5-2 x}}+\frac {397 \sqrt {\frac {3}{22}} \sqrt {5-2 x} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{11}} \sqrt {1+4 x}\right )|\frac {1}{3}\right )}{89125 \sqrt {-5+2 x}}-\frac {14832503 \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 )}{287339000 \sqrt {11} \sqrt {-5+2 x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 5.77 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{(7+5 x)^3} \, dx=\frac {\sqrt {-5+2 x} \left (\frac {17050 \sqrt {2-3 x} \sqrt {1+4 x} (7057+44765 x)}{(7+5 x)^2}+\frac {\sqrt {11} \left (-61059460 E\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right )|-\frac {1}{2}\right )+5759676 \operatorname {EllipticF}\left (\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )[phone] \operatorname {EllipticPi}\left (\frac {55}{124},\arcsin \left (\frac {2 \sqrt {2-3 x}}{\sqrt {11}}\right ),-\frac {1}{2}\right )\right )}{\sqrt {5-2 x}}\right )}{9482187000} \]

[In]

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

[Out]

(Sqrt[-5 + 2*x]*((17050*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*(7057 + 44765*x))/(7 + 5*x)^2 + (Sqrt[11]*(-61059460*Ellip
ticE[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] + 5759676*EllipticF[ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2] +
44497509*EllipticPi[55/124, ArcSin[(2*Sqrt[2 - 3*x])/Sqrt[11]], -1/2]))/Sqrt[5 - 2*x]))/9482187000

Maple [A] (verified)

Time = 1.64 (sec) , antiderivative size = 273, normalized size of antiderivative = 1.20

method	result	size
elliptic	\(\frac {\sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (-\frac {\sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{10 \left (7+5 x \right )^{2}}+\frac {8953 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}{556140 \left (7+5 x \right )}-\frac {104171 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{140193625 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {8953 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \left (-\frac {11 E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{12}+\frac {2 F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )}{3}\right )}{28038725 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {14832503 \sqrt {11+44 x}\, \sqrt {22-33 x}\, \sqrt {110-44 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )}{19346720250 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(273\)
risch	\(-\frac {\left (-2+3 x \right ) \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (7057+44765 x \right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{556140 \left (7+5 x \right )^{2} \sqrt {-\left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \sqrt {2-3 x}}-\frac {\left (-\frac {104171 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{420580875 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {8953 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \left (-\frac {11 E\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{6}+\frac {5 F\left (\frac {2 \sqrt {22-33 x}}{11}, \frac {i \sqrt {2}}{2}\right )}{2}\right )}{84116175 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}+\frac {14832503 \sqrt {22-33 x}\, \sqrt {-66 x +165}\, \sqrt {33+132 x}\, \Pi \left (\frac {2 \sqrt {22-33 x}}{11}, \frac {55}{124}, \frac {i \sqrt {2}}{2}\right )}{104304057000 \sqrt {-24 x^{3}+70 x^{2}-21 x -10}}\right ) \sqrt {\left (2-3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}}\)	\(311\)
default	\(\frac {\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (512860900 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}[phone] \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x^{2}-741625150 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x^{2}[phone] \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x [phone] \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right ) x -2076550420 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right ) x [phone] \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, F\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )[phone] \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, E\left (\frac {\sqrt {11+44 x}}{11}, \sqrt {3}\right )-1453585294 \sqrt {1+4 x}\, \sqrt {2-3 x}\, \sqrt {22}\, \sqrt {5-2 x}\, \Pi \left (\frac {\sqrt {11+44 x}}{11}, -\frac {55}{23}, \sqrt {3}\right )[phone] x^{4}-18748451150 x^{3}[phone] x^{2}[phone] x [phone]\right )}{3517585500 \left (24 x^{3}-70 x^{2}+21 x +10\right ) \left (7+5 x \right )^{2}}\)	\(434\)

[In]

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

[Out]

(-(-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)*(-1/10/(7+5*x)^2*(-24*x^3+70*x^2
-21*x-10)^(1/2)+8953/556140/(7+5*x)*(-24*x^3+70*x^2-21*x-10)^(1/2)-104171/140193625*(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))+8953/28038725*(1
1+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)))[phone]/19346720250*(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)))

Fricas [F]

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

[In]

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

[Out]

integral(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(125*x^3 + 525*x^2 + 735*x + 343), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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