\(\int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx\) [105]

Optimal result

Integrand size = 37, antiderivative size = 195 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\frac {10 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5-2 x}{7+5 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{22}} \sqrt {1+4 x}}{\sqrt {7+5 x}}\right )|\frac {62}{39}\right )}{713 \sqrt {-5+2 x} \sqrt {\frac {2-3 x}{7+5 x}}}+\frac {2 \sqrt {\frac {3}{143}} (2-3 x) \sqrt {\frac {5-2 x}{2-3 x}} \sqrt {-\frac {1+4 x}{2-3 x}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {\frac {11}{23}} \sqrt {7+5 x}}{\sqrt {2-3 x}}\right ),-\frac {23}{39}\right )}{31 \sqrt {-5+2 x} \sqrt {1+4 x}} \]

[Out]

2/4433*(2-3*x)*EllipticF(1/23*253^(1/2)*(7+5*x)^(1/2)/(2-3*x)^(1/2),1/39*I*897^(1/2))*429^(1/2)*((5-2*x)/(2-3*
x))^(1/2)*((-1-4*x)/(2-3*x))^(1/2)/(-5+2*x)^(1/2)/(1+4*x)^(1/2)+10/27807*EllipticE(1/22*858^(1/2)*(1+4*x)^(1/2
)/(7+5*x)^(1/2),1/39*2418^(1/2))*429^(1/2)*(2-3*x)^(1/2)*((5-2*x)/(7+5*x))^(1/2)/(-5+2*x)^(1/2)/((2-3*x)/(7+5*
x))^(1/2)

Rubi [A] (verified)

Time = 0.13 (sec) , antiderivative size = 270, normalized size of antiderivative = 1.38, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.189, Rules used = {177, 176, 429, 182, 433, 506, 422} \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=\frac {6 \sqrt {5 x+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{31 \sqrt {253} \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}-\frac {5 \sqrt {\frac {22}{31}} \sqrt {4 x+1} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {\frac {31}{11}} \sqrt {2 x-5}}{\sqrt {5 x+7}}\right ),\frac {39}{62}\right )}{1209 \sqrt {2-3 x} \sqrt {-\frac {4 x+1}{2-3 x}}}+\frac {10 \sqrt {\frac {22}{31}} \sqrt {4 x+1} E\left (\arctan \left (\frac {\sqrt {\frac {31}{11}} \sqrt {2 x-5}}{\sqrt {5 x+7}}\right )|\frac {39}{62}\right )}{897 \sqrt {2-3 x} \sqrt {-\frac {4 x+1}{2-3 x}}}-\frac {10 \sqrt {2 x-5} \sqrt {4 x+1}}{897 \sqrt {2-3 x} \sqrt {5 x+7}} \]

[In]

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

[Out]

(-10*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(897*Sqrt[2 - 3*x]*Sqrt[7 + 5*x]) + (10*Sqrt[22/31]*Sqrt[1 + 4*x]*EllipticE
[ArcTan[(Sqrt[31/11]*Sqrt[-5 + 2*x])/Sqrt[7 + 5*x]], 39/62])/(897*Sqrt[2 - 3*x]*Sqrt[-((1 + 4*x)/(2 - 3*x))])
+ (6*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(31*Sqrt[253]*Sqrt[-5 + 2
*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) - (5*Sqrt[22/31]*Sqrt[1 + 4*x]*EllipticF[ArcTan[(Sqrt[31/11]*Sqrt[-5 + 2*x])/Sq
rt[7 + 5*x]], 39/62])/(1209*Sqrt[2 - 3*x]*Sqrt[-((1 + 4*x)/(2 - 3*x))])

Rule 176

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

Rule 177

Int[1/(((a_.) + (b_.)*(x_))^(3/2)*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]),
x_Symbol] :> Dist[-d/(b*c - a*d), Int[1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] + D
ist[b/(b*c - a*d), Int[Sqrt[c + d*x]/((a + b*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x], x] /; FreeQ[{a, b, c,
d, e, f, g, h}, x]

Rule 182

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

Rule 422

Int[Sqrt[(a_) + (b_.)*(x_)^2]/((c_) + (d_.)*(x_)^2)^(3/2), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(c*Rt[d/c, 2]*Sq
rt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticE[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /;
FreeQ[{a, b, c, d}, x] && PosQ[b/a] && PosQ[d/c]

Rule 429

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[(Sqrt[a + b*x^2]/(a*Rt[d/c, 2]*
Sqrt[c + d*x^2]*Sqrt[c*((a + b*x^2)/(a*(c + d*x^2)))]))*EllipticF[ArcTan[Rt[d/c, 2]*x], 1 - b*(c/(a*d))], x] /
; FreeQ[{a, b, c, d}, x] && PosQ[d/c] && PosQ[b/a] &&  !SimplerSqrtQ[b/a, d/c]

Rule 433

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Dist[a, Int[1/(Sqrt[a + b*x^2]*Sqrt[c +
d*x^2]), x], x] + Dist[b, Int[x^2/(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d}, x] && PosQ[
d/c] && PosQ[b/a]

Rule 506

Int[(x_)^2/(Sqrt[(a_) + (b_.)*(x_)^2]*Sqrt[(c_) + (d_.)*(x_)^2]), x_Symbol] :> Simp[x*(Sqrt[a + b*x^2]/(b*Sqrt
[c + d*x^2])), x] - Dist[c/b, Int[Sqrt[a + b*x^2]/(c + d*x^2)^(3/2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b
*c - a*d, 0] && PosQ[b/a] && PosQ[d/c] &&  !SimplerSqrtQ[b/a, d/c]

Rubi steps \begin{align*} \text {integral}& = \frac {3}{31} \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx+\frac {5}{31} \int \frac {\sqrt {2-3 x}}{\sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx \\ & = \frac {\left (5 \sqrt {2} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{7+5 x}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {31 x^2}{11}}}{\sqrt {1+\frac {23 x^2}{22}}} \, dx,x,\frac {\sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )}{1209 \sqrt {1+4 x} \sqrt {-\frac {2-3 x}{7+5 x}}}+\frac {\left (3 \sqrt {\frac {2}{253}} \sqrt {-\frac {-5+2 x}{2-3 x}} \sqrt {7+5 x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{2}} \sqrt {1+\frac {31 x^2}{23}}} \, dx,x,\frac {\sqrt {1+4 x}}{\sqrt {2-3 x}}\right )}{31 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}} \\ & = \frac {6 \sqrt {7+5 x} F\left (\tan ^{-1}\left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{31 \sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}+\frac {\left (5 \sqrt {2} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{7+5 x}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {23 x^2}{22}} \sqrt {1+\frac {31 x^2}{11}}} \, dx,x,\frac {\sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )}{1209 \sqrt {1+4 x} \sqrt {-\frac {2-3 x}{7+5 x}}}+\frac {\left (5 \sqrt {2} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{7+5 x}}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {1+\frac {23 x^2}{22}} \sqrt {1+\frac {31 x^2}{11}}} \, dx,x,\frac {\sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )}{429 \sqrt {1+4 x} \sqrt {-\frac {2-3 x}{7+5 x}}} \\ & = -\frac {10 \sqrt {-5+2 x} \sqrt {1+4 x}}{897 \sqrt {2-3 x} \sqrt {7+5 x}}+\frac {6 \sqrt {7+5 x} F\left (\tan ^{-1}\left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{31 \sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {5 \sqrt {\frac {22}{31}} \sqrt {1+4 x} F\left (\tan ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{1209 \sqrt {2-3 x} \sqrt {-\frac {1+4 x}{2-3 x}}}-\frac {\left (10 \sqrt {2} \sqrt {2-3 x} \sqrt {\frac {1+4 x}{7+5 x}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {23 x^2}{22}}}{\left (1+\frac {31 x^2}{11}\right )^{3/2}} \, dx,x,\frac {\sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )}{897 \sqrt {1+4 x} \sqrt {-\frac {2-3 x}{7+5 x}}} \\ & = -\frac {10 \sqrt {-5+2 x} \sqrt {1+4 x}}{897 \sqrt {2-3 x} \sqrt {7+5 x}}+\frac {10 \sqrt {\frac {22}{31}} \sqrt {1+4 x} E\left (\tan ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{897 \sqrt {2-3 x} \sqrt {-\frac {1+4 x}{2-3 x}}}+\frac {6 \sqrt {7+5 x} F\left (\tan ^{-1}\left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right )|-\frac {39}{23}\right )}{31 \sqrt {253} \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}}-\frac {5 \sqrt {\frac {22}{31}} \sqrt {1+4 x} F\left (\tan ^{-1}\left (\frac {\sqrt {\frac {31}{11}} \sqrt {-5+2 x}}{\sqrt {7+5 x}}\right )|\frac {39}{62}\right )}{1209 \sqrt {2-3 x} \sqrt {-\frac {1+4 x}{2-3 x}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 18.16 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.22 \[ \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx=-\frac {2 \sqrt {-5+2 x} \sqrt {1+4 x} \left (1705 \sqrt {\frac {7+5 x}{-2+3 x}} \left (-5-18 x+8 x^2\right )-55 \sqrt {682} \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \left (-14+11 x+15 x^2\right ) E\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right )|\frac {39}{62}\right )-23 \sqrt {682} \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \left (-14+11 x+15 x^2\right ) \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right ),\frac {39}{62}\right )\right )}{305877 \sqrt {2-3 x} \sqrt {7+5 x} \sqrt {\frac {7+5 x}{-2+3 x}} \left (-5-18 x+8 x^2\right )} \]

[In]

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

[Out]

(-2*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(1705*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8*x^2) - 55*Sqrt[682]*Sqrt[(-5
- 18*x + 8*x^2)/(2 - 3*x)^2]*(-14 + 11*x + 15*x^2)*EllipticE[ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]],
39/62] - 23*Sqrt[682]*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*(-14 + 11*x + 15*x^2)*EllipticF[ArcSin[Sqrt[31/39]
*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62]))/(305877*Sqrt[2 - 3*x]*Sqrt[7 + 5*x]*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 1
8*x + 8*x^2))

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 434 vs. \(2 (155 ) = 310\).

Time = 1.62 (sec) , antiderivative size = 435, normalized size of antiderivative = 2.23

method	result	size
elliptic	\(\frac {\sqrt {-\left (7+5 x \right ) \left (-2+3 x \right ) \left (-5+2 x \right ) \left (1+4 x \right )}\, \left (\frac {7252 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{8505521739 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}+\frac {140 \sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {2 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{3}-\frac {31 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{15}\right )}{654270903 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}-\frac {200 \left (\left (x +\frac {7}{5}\right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )-\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}\, \left (-\frac {2}{3}+x \right )^{2} \sqrt {806}\, \sqrt {\frac {x -\frac {5}{2}}{-\frac {2}{3}+x}}\, \sqrt {2139}\, \sqrt {\frac {x +\frac {1}{4}}{-\frac {2}{3}+x}}\, \left (\frac {181 F\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{341}-\frac {117 E\left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, \frac {i \sqrt {897}}{39}\right )}{62}+\frac {91 \Pi \left (\frac {\sqrt {-\frac {3795 \left (x +\frac {7}{5}\right )}{-\frac {2}{3}+x}}}{69}, -\frac {69}{55}, \frac {i \sqrt {897}}{39}\right )}{55}\right )}{80730}\right )}{9269 \sqrt {-30 \left (x +\frac {7}{5}\right ) \left (-\frac {2}{3}+x \right ) \left (x -\frac {5}{2}\right ) \left (x +\frac {1}{4}\right )}}-\frac {10 \left (-120 x^{3}+350 x^{2}-105 x -50\right )}{27807 \sqrt {\left (x +\frac {7}{5}\right ) \left (-120 x^{3}+350 x^{2}-105 x -50\right )}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}}\)	\(435\)
default	\(-\frac {2 \sqrt {7+5 x}\, \sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \left (1116 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x^{2} F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-495 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x^{2} E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-1488 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )+660 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, x E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )+496 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, F\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-220 \sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}\, \sqrt {13}\, \sqrt {3}\, \sqrt {\frac {-5+2 x}{-2+3 x}}\, \sqrt {23}\, \sqrt {\frac {1+4 x}{-2+3 x}}\, E\left (\frac {\sqrt {-\frac {253 \left (7+5 x \right )}{-2+3 x}}}{23}, \frac {i \sqrt {897}}{39}\right )-313720 x^{2}+705870 x +196075\right )}{7035171 \left (120 x^{4}-182 x^{3}-385 x^{2}+197 x +70\right )}\)	\(563\)

[In]

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

[Out]

(-(7+5*x)*(-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)/(7+5*x)^(1/2)*(7252/8505
521739*(-3795*(x+7/5)/(-2/3+x))^(1/2)*(-2/3+x)^2*806^(1/2)*((x-5/2)/(-2/3+x))^(1/2)*2139^(1/2)*((x+1/4)/(-2/3+
x))^(1/2)/(-30*(x+7/5)*(-2/3+x)*(x-5/2)*(x+1/4))^(1/2)*EllipticF(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),1/39*I*89
7^(1/2))+140/654270903*(-3795*(x+7/5)/(-2/3+x))^(1/2)*(-2/3+x)^2*806^(1/2)*((x-5/2)/(-2/3+x))^(1/2)*2139^(1/2)
*((x+1/4)/(-2/3+x))^(1/2)/(-30*(x+7/5)*(-2/3+x)*(x-5/2)*(x+1/4))^(1/2)*(2/3*EllipticF(1/69*(-3795*(x+7/5)/(-2/
3+x))^(1/2),1/39*I*897^(1/2))-31/15*EllipticPi(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),-69/55,1/39*I*897^(1/2)))-2
00/9269*((x+7/5)*(x-5/2)*(x+1/4)-1/80730*(-3795*(x+7/5)/(-2/3+x))^(1/2)*(-2/3+x)^2*806^(1/2)*((x-5/2)/(-2/3+x)
)^(1/2)*2139^(1/2)*((x+1/4)/(-2/3+x))^(1/2)*(181/341*EllipticF(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^
(1/2))-117/62*EllipticE(1/69*(-3795*(x+7/5)/(-2/3+x))^(1/2),1/39*I*897^(1/2))+91/55*EllipticPi(1/69*(-3795*(x+
7/5)/(-2/3+x))^(1/2),-69/55,1/39*I*897^(1/2))))/(-30*(x+7/5)*(-2/3+x)*(x-5/2)*(x+1/4))^(1/2)-10/27807*(-120*x^
3+350*x^2-105*x-50)/((x+7/5)*(-120*x^3+350*x^2-105*x-50))^(1/2))

Fricas [F]

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

[In]

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

[Out]

integral(-sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(600*x^5 - 70*x^4 - 3199*x^3 - 1710*x^2 + 1
729*x + 490), x)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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