\(\int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{9/2}} \, dx\) [92]

Optimal result

Integrand size = 37, antiderivative size = 370 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{9/2}} \, dx=\frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1956607901151813 \sqrt {7+5 x}}+\frac {16377776536 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1956607901151813 \sqrt {-5+2 x}}-\frac {8188888268 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{50169433362867 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {258506776 \sqrt {\frac {11}{23}} \sqrt {7+5 x} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {1+4 x}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{1618368818157 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}} \]

[Out]

2/273*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(7/2)+98/1807455*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)
^(1/2)/(7+5*x)^(5/2)-3217468/50259901185*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(3/2)-40944441340/
1956607901151813*(2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(1/2)[phone]/1956607901151813*(2-3*x)
^(1/2)*(1+4*x)^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)[phone]/37222482817611*(1/(4+2*(1+4*x)/(2-3*x)))^(1/2)*(4+
2*(1+4*x)/(2-3*x))^(1/2)*EllipticF((1+4*x)^(1/2)*2^(1/2)/(2-3*x)^(1/2)/(4+2*(1+4*x)/(2-3*x))^(1/2),1/23*I*897^
(1/2))*253^(1/2)*(7+5*x)^(1/2)/(-5+2*x)^(1/2)/((7+5*x)/(5-2*x))^(1/2)-8188888268/1956607901151813*EllipticE(1/
23*897^(1/2)*(1+4*x)^(1/2)/(-5+2*x)^(1/2),1/39*I*897^(1/2))*429^(1/2)*(2-3*x)^(1/2)*((7+5*x)/(5-2*x))^(1/2)/((
2-3*x)/(5-2*x))^(1/2)/(7+5*x)^(1/2)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 370, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.243, Rules used = {170, 1618, 1613, 1616, 12, 176, 429, 182, 435} \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{9/2}} \, dx=-\frac {8188888268 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {5 x+7}{5-2 x}} E\left (\arcsin \left (\frac {\sqrt {\frac {39}{23}} \sqrt {4 x+1}}{\sqrt {2 x-5}}\right )|-\frac {23}{39}\right )}{50169433362867 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {5 x+7}}+\frac {258506776 \sqrt {\frac {11}{23}} \sqrt {5 x+7} \operatorname {EllipticF}\left (\arctan \left (\frac {\sqrt {4 x+1}}{\sqrt {2} \sqrt {2-3 x}}\right ),-\frac {39}{23}\right )}{1618368818157 \sqrt {2 x-5} \sqrt {\frac {5 x+7}{5-2 x}}}+\frac {16377776536 \sqrt {2-3 x} \sqrt {4 x+1} \sqrt {5 x+7}}{1956607901151813 \sqrt {2 x-5}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1956607901151813 \sqrt {5 x+7}}-\frac {3217468 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{50259901185 (5 x+7)^{3/2}}+\frac {98 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{1807455 (5 x+7)^{5/2}}+\frac {2 \sqrt {2-3 x} \sqrt {2 x-5} \sqrt {4 x+1}}{273 (5 x+7)^{7/2}} \]

[In]

Int[(Sqrt[2 - 3*x]*Sqrt[1 + 4*x])/(Sqrt[-5 + 2*x]*(7 + 5*x)^(9/2)),x]

[Out]

(2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(273*(7 + 5*x)^(7/2)) + (98*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1
+ 4*x])/(1807455*(7 + 5*x)^(5/2)) - (3217468*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(50259901185*(7 + 5*
x)^(3/2)) - (40944441340*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(1956607901151813*Sqrt[7 + 5*x]) + (16377
776536*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(1956607901151813*Sqrt[-5 + 2*x]) - (8188888268*Sqrt[11/39]*
Sqrt[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sqrt[-5 + 2*x]], -23/39])
/(50169433362867*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) + (258506776*Sqrt[11/23]*Sqrt[7 + 5*x]*EllipticF[Arc
Tan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(1618368818157*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 170

Int[(((a_.) + (b_.)*(x_))^(m_)*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)])/Sqrt[(c_.) + (d_.)*(x_)], x_
Symbol] :> 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] - Dist
[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 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 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 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 435

Int[Sqrt[(a_) + (b_.)*(x_)^2]/Sqrt[(c_) + (d_.)*(x_)^2], x_Symbol] :> Simp[(Sqrt[a]/(Sqrt[c]*Rt[-d/c, 2]))*Ell
ipticE[ArcSin[Rt[-d/c, 2]*x], b*(c/(a*d))], x] /; FreeQ[{a, b, c, d}, x] && NegQ[d/c] && GtQ[c, 0] && GtQ[a, 0
]

Rule 1613

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

Rule 1616

Int[((A_.) + (B_.)*(x_) + (C_.)*(x_)^2)/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*
(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_Symbol] :> Simp[C*Sqrt[a + b*x]*Sqrt[e + f*x]*(Sqrt[g + h*x]/(b*f*h*Sqrt[c
+ d*x])), x] + (Dist[1/(2*b*d*f*h), Int[(1/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*x]))*Simp[2*A
*b*d*f*h - C*(b*d*e*g + a*c*f*h) + (2*b*B*d*f*h - C*(a*d*f*h + b*(d*f*g + d*e*h + c*f*h)))*x, x], x], x] + Dis
t[C*(d*e - c*f)*((d*g - c*h)/(2*b*d*f*h)), Int[Sqrt[a + b*x]/((c + d*x)^(3/2)*Sqrt[e + f*x]*Sqrt[g + h*x]), x]
, x]) /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x]

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]

Rubi steps \begin{align*} \text {integral}& = \frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}-\frac {1}{273} \int \frac {-49+70 x+96 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{7/2}} \, dx \\ & = \frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {\int \frac {-958104+2280510 x+49392 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{5/2}} \, dx}{37956555} \\ & = \frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {\int \frac {-11461434930+18134687340 x}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} (7+5 x)^{3/2}} \, dx}{3166373774655} \\ & = \frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1956607901151813 \sqrt {7+5 x}}-\frac {\int \frac {-32763839696280-33533497457460 x+44219996647200 x^2}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{88047355551831585} \\ & = \frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1956607901151813 \sqrt {7+5 x}}+\frac {16377776536 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1956607901151813 \sqrt {-5+2 x}}+\frac {\int \frac {18564560715729600}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{21131365332439580400}+\frac {90077770948 \int \frac {\sqrt {2-3 x}}{(-5+2 x)^{3/2} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{50169433362867} \\ & = \frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1956607901151813 \sqrt {7+5 x}}+\frac {16377776536 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1956607901151813 \sqrt {-5+2 x}}+\frac {1421787268 \int \frac {1}{\sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x}} \, dx}{1618368818157}-\frac {\left (8188888268 \sqrt {\frac {11}{23}} \sqrt {2-3 x} \sqrt {-\frac {7+5 x}{-5+2 x}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+x^2}}{\sqrt {1-\frac {39 x^2}{23}}} \, dx,x,\frac {\sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )}{50169433362867 \sqrt {-\frac {2-3 x}{-5+2 x}} \sqrt {7+5 x}} \\ & = \frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1956607901151813 \sqrt {7+5 x}}+\frac {16377776536 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1956607901151813 \sqrt {-5+2 x}}-\frac {8188888268 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{50169433362867 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {\left (129253388 \sqrt {\frac {22}{23}} \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 )}{1618368818157 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{2-3 x}}} \\ & = \frac {2 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{273 (7+5 x)^{7/2}}+\frac {98 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1807455 (7+5 x)^{5/2}}-\frac {3217468 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{50259901185 (7+5 x)^{3/2}}-\frac {40944441340 \sqrt {2-3 x} \sqrt {-5+2 x} \sqrt {1+4 x}}{1956607901151813 \sqrt {7+5 x}}+\frac {16377776536 \sqrt {2-3 x} \sqrt {1+4 x} \sqrt {7+5 x}}{1956607901151813 \sqrt {-5+2 x}}-\frac {8188888268 \sqrt {\frac {11}{39}} \sqrt {2-3 x} \sqrt {\frac {7+5 x}{5-2 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {39}{23}} \sqrt {1+4 x}}{\sqrt {-5+2 x}}\right )|-\frac {23}{39}\right )}{50169433362867 \sqrt {\frac {2-3 x}{5-2 x}} \sqrt {7+5 x}}+\frac {258506776 \sqrt {\frac {11}{23}} \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 )}{1618368818157 \sqrt {-5+2 x} \sqrt {\frac {7+5 x}{5-2 x}}} \\ \end{align*}

Mathematica [A] (verified)

Time = 27.32 (sec) , antiderivative size = 258, normalized size of antiderivative = 0.70 \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{9/2}} \, dx=\frac {2 \sqrt {-5+2 x} \sqrt {1+4 x} \sqrt {7+5 x} \left (\frac {(-2+3 x) \left (2552362046246+19165803061167 x+12313608173580 x^2+2559027583750 x^3\right )}{(7+5 x)^4}-\frac {22 \left (558333291 \sqrt {\frac {7+5 x}{-2+3 x}} \left (-5-18 x+8 x^2\right )-186111097 \sqrt {682} (-2+3 x) \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} E\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right )|\frac {39}{62}\right )[phone] \sqrt {682} (-2+3 x) \sqrt {\frac {-5-18 x+8 x^2}{(2-3 x)^2}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {31}{39}} \sqrt {\frac {-5+2 x}{-2+3 x}}\right ),\frac {39}{62}\right )\right )}{\sqrt {\frac {7+5 x}{-2+3 x}} \left (-5-18 x+8 x^2\right )}\right )}{1956607901151813 \sqrt {2-3 x}} \]

[In]

Integrate[(Sqrt[2 - 3*x]*Sqrt[1 + 4*x])/(Sqrt[-5 + 2*x]*(7 + 5*x)^(9/2)),x]

[Out]

(2*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x]*(((-2 + 3*x)*(2552362046246 + 19165803061167*x + 12313608173580*
x^2 + 2559027583750*x^3))/(7 + 5*x)^4 - (22*(558333291*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8*x^2) - 186111
097*Sqrt[682]*(-2 + 3*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticE[ArcSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-
2 + 3*x)]], 39/62] + 71545594*Sqrt[682]*(-2 + 3*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticF[ArcSin[Sqrt
[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62]))/(Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8*x^2))))/(19566079011
51813*Sqrt[2 - 3*x])

Maple [A] (verified)

Time = 1.63 (sec) , antiderivative size = 522, normalized size of antiderivative = 1.41

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 {2 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{170625 \left (x +\frac {7}{5}\right )^{4}}+\frac {98 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{225931875 \left (x +\frac {7}{5}\right )^{3}}-\frac {3217468 \sqrt {-120 x^{4}+182 x^{3}+385 x^{2}-197 x -70}}{1256497529625 \left (x +\frac {7}{5}\right )^{2}}-\frac {8188888268 \left (-120 x^{3}+350 x^{2}-105 x -50\right )}{1956607901151813 \sqrt {\left (x +\frac {7}{5}\right ) \left (-120 x^{3}+350 x^{2}-105 x -50\right )}}+\frac {18911307184 \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 )}{7772485129618352013 \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 {1488888776 \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 )}{597883471509104001 \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 {163777765360 \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 )}{652202633717271 \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 )}}\right )}{\sqrt {2-3 x}\, \sqrt {-5+2 x}\, \sqrt {1+4 x}\, \sqrt {7+5 x}}\)	\(522\)
default	\(\text {Expression too large to display}\)	\(1088\)

[In]

int((2-3*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(9/2)/(-5+2*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)*(2/170625*
(-120*x^4+182*x^3+385*x^2-197*x-70)^(1/2)/(x+7/5)^4+98/225931875*(-120*x^4+182*x^3+385*x^2-197*x-70)^(1/2)/(x+
7/5)^3-3217468/1256497529625*(-120*x^4+182*x^3+385*x^2-197*x-70)^(1/2)/(x+7/5)^2-8188888268/1956607901151813*(
-120*x^3+350*x^2-105*x-50)/((x+7/5)*(-120*x^3+350*x^2-105*x-50))^(1/2)[phone]/7772485129618352013*(-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*897^(1/2))+14888
88776/597883471509104001*(-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)))
-163777765360/652202633717271*((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*Elli
pticPi(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))

Fricas [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{9/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {9}{2}} \sqrt {2 \, x - 5}} \,d x } \]

[In]

integrate((2-3*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(9/2)/(-5+2*x)^(1/2),x, algorithm="fricas")

[Out]

integral(sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(6250*x^6 + 28125*x^5 + 13125*x^4 - 134750*x
^3 - 308700*x^2 - 266511*x - 84035), x)

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate((2-3*x)**(1/2)*(1+4*x)**(1/2)/(7+5*x)**(9/2)/(-5+2*x)**(1/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{9/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {9}{2}} \sqrt {2 \, x - 5}} \,d x } \]

[In]

integrate((2-3*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(9/2)/(-5+2*x)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(4*x + 1)*sqrt(-3*x + 2)/((5*x + 7)^(9/2)*sqrt(2*x - 5)), x)

Giac [F]

\[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{9/2}} \, dx=\int { \frac {\sqrt {4 \, x + 1} \sqrt {-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac {9}{2}} \sqrt {2 \, x - 5}} \,d x } \]

[In]

integrate((2-3*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(9/2)/(-5+2*x)^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(4*x + 1)*sqrt(-3*x + 2)/((5*x + 7)^(9/2)*sqrt(2*x - 5)), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2-3 x} \sqrt {1+4 x}}{\sqrt {-5+2 x} (7+5 x)^{9/2}} \, dx=\int \frac {\sqrt {2-3\,x}\,\sqrt {4\,x+1}}{\sqrt {2\,x-5}\,{\left (5\,x+7\right )}^{9/2}} \,d x \]

[In]

int(((2 - 3*x)^(1/2)*(4*x + 1)^(1/2))/((2*x - 5)^(1/2)*(5*x + 7)^(9/2)),x)

[Out]

int(((2 - 3*x)^(1/2)*(4*x + 1)^(1/2))/((2*x - 5)^(1/2)*(5*x + 7)^(9/2)), x)