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

Optimal. Leaf size=471 \[ -\frac{245264762213 \sqrt{\frac{11}{23}} \sqrt{5 x+7} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right ),-\frac{39}{23}\right )}{99532800 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{1}{25} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{7/2}-\frac{427 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{5/2}}{2400}-\frac{83363 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{3/2}}{34560}-\frac{70489981 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7}}{1658880}-\frac{1450582567 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{3686400 \sqrt{2 x-5}}+\frac{1450582567 \sqrt{\frac{143}{3}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{2457600 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}-\frac{57691792727443 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{497664000 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

[Out]

(-1450582567*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(3686400*Sqrt[-5 + 2*x]) - (70489981*Sqrt[2 - 3*x]*Sqr
t[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/1658880 - (83363*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)
^(3/2))/34560 - (427*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(5/2))/2400 + (Sqrt[2 - 3*x]*Sqrt[-5
+ 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(7/2))/25 + (1450582567*Sqrt[143/3]*Sqrt[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*El
lipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sqrt[-5 + 2*x]], -23/39])/(2457600*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7
+ 5*x]) - (245264762213*Sqrt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -3
9/23])/(99532800*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) - (57691792727443*(2 - 3*x)*Sqrt[(5 - 2*x)/(2 - 3*x
)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55, ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])
/(497664000*Sqrt[429]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])

________________________________________________________________________________________

Rubi [A]  time = 0.665089, antiderivative size = 471, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 10, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.27, Rules used = {161, 1600, 1602, 1598, 170, 418, 165, 537, 176, 424} \[ \frac{1}{25} \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{7/2}-\frac{427 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{5/2}}{2400}-\frac{83363 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} (5 x+7)^{3/2}}{34560}-\frac{70489981 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7}}{1658880}-\frac{1450582567 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{3686400 \sqrt{2 x-5}}-\frac{245264762213 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{99532800 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{1450582567 \sqrt{\frac{143}{3}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{2457600 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}-\frac{57691792727443 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{497664000 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1450582567*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(3686400*Sqrt[-5 + 2*x]) - (70489981*Sqrt[2 - 3*x]*Sqr
t[-5 + 2*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/1658880 - (83363*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)
^(3/2))/34560 - (427*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(5/2))/2400 + (Sqrt[2 - 3*x]*Sqrt[-5
+ 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(7/2))/25 + (1450582567*Sqrt[143/3]*Sqrt[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*El
lipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sqrt[-5 + 2*x]], -23/39])/(2457600*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7
+ 5*x]) - (245264762213*Sqrt[11/23]*Sqrt[7 + 5*x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -3
9/23])/(99532800*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) - (57691792727443*(2 - 3*x)*Sqrt[(5 - 2*x)/(2 - 3*x
)]*Sqrt[-((1 + 4*x)/(2 - 3*x))]*EllipticPi[-69/55, ArcSin[(Sqrt[11/23]*Sqrt[7 + 5*x])/Sqrt[2 - 3*x]], -23/39])
/(497664000*Sqrt[429]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])

Rule 161

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*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])/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g +
h*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m}, x] && IntegerQ[2*m] &&  !LtQ[m, -1]

Rule 1600

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[(2*C*(a + b*x)^m*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h
*x])/(d*f*h*(2*m + 3)), x] + Dist[1/(d*f*h*(2*m + 3)), Int[((a + b*x)^(m - 1)/(Sqrt[c + d*x]*Sqrt[e + f*x]*Sqr
t[g + h*x]))*Simp[a*A*d*f*h*(2*m + 3) - C*(a*(d*e*g + c*f*g + c*e*h) + 2*b*c*e*g*m) + ((A*b + a*B)*d*f*h*(2*m
+ 3) - C*(2*a*(d*f*g + d*e*h + c*f*h) + b*(2*m + 1)*(d*e*g + c*f*g + c*e*h)))*x + (b*B*d*f*h*(2*m + 3) + 2*C*(
a*d*f*h*m - b*(m + 1)*(d*f*g + d*e*h + c*f*h)))*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, A, B, C}, x]
&& IntegerQ[2*m] && GtQ[m, 0]

Rule 1602

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*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])/(Sqrt[a + b*x]*Sqrt[c + d*x]*Sqrt[e + f*x]*Sqrt[g + h*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 1598

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

Rule 170

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 418

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

Rule 165

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

Rule 537

Int[1/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]*Sqrt[(e_) + (f_.)*(x_)^2]), x_Symbol] :> Simp[(1*Ellipt
icPi[(b*c)/(a*d), ArcSin[Rt[-(d/c), 2]*x], (c*f)/(d*e)])/(a*Sqrt[c]*Sqrt[e]*Rt[-(d/c), 2]), 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 176

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 424

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

Rubi steps

\begin{align*} \int \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{5/2} \, dx &=\frac{1}{25} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{7/2}+\frac{1}{50} \int \frac{(7+5 x)^{5/2} \left (-3-1190 x+854 x^2\right )}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx\\ &=-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{5/2}}{2400}+\frac{1}{25} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{7/2}-\frac{\int \frac{(7+5 x)^{3/2} \left (343070+1303340 x-1667260 x^2\right )}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{9600}\\ &=-\frac{83363 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{3/2}}{34560}-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{5/2}}{2400}+\frac{1}{25} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{7/2}+\frac{\int \frac{\sqrt{7+5 x} \left (-840990780-627111520 x+2819599240 x^2\right )}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}} \, dx}{1382400}\\ &=-\frac{70489981 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}}{1658880}-\frac{83363 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{3/2}}{34560}-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{5/2}}{2400}+\frac{1}{25} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{7/2}-\frac{\int \frac{1120606854440-1345996898960 x-3133258344720 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{132710400}\\ &=-\frac{1450582567 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{3686400 \sqrt{-5+2 x}}-\frac{70489981 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}}{1658880}-\frac{83363 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{3/2}}{34560}-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{5/2}}{2400}+\frac{1}{25} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{7/2}+\frac{\int \frac{-1027194164487840+893292274489440 x}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{31850496000}-\frac{207433307081 \int \frac{\sqrt{2-3 x}}{(-5+2 x)^{3/2} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{2457600}\\ &=-\frac{1450582567 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{3686400 \sqrt{-5+2 x}}-\frac{70489981 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}}{1658880}-\frac{83363 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{3/2}}{34560}-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{5/2}}{2400}+\frac{1}{25} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{7/2}-\frac{1861025571853 \int \frac{\sqrt{2-3 x}}{\sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{199065600}-\frac{2697912384343 \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{199065600}+\frac{\left (18857573371 \sqrt{\frac{11}{23}} \sqrt{2-3 x} \sqrt{-\frac{7+5 x}{-5+2 x}}\right ) \operatorname{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 )}{2457600 \sqrt{-\frac{2-3 x}{-5+2 x}} \sqrt{7+5 x}}\\ &=-\frac{1450582567 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{3686400 \sqrt{-5+2 x}}-\frac{70489981 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}}{1658880}-\frac{83363 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{3/2}}{34560}-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{5/2}}{2400}+\frac{1}{25} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{7/2}+\frac{1450582567 \sqrt{\frac{143}{3}} \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 )}{2457600 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{7+5 x}}-\frac{\left (57691792727443 (2-3 x) \sqrt{-\frac{-5+2 x}{2-3 x}} \sqrt{-\frac{1+4 x}{2-3 x}}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{1-\frac{11 x^2}{23}} \sqrt{1+\frac{11 x^2}{39}} \left (5+3 x^2\right )} \, dx,x,\frac{\sqrt{7+5 x}}{\sqrt{2-3 x}}\right )}{99532800 \sqrt{897} \sqrt{-5+2 x} \sqrt{1+4 x}}-\frac{\left (245264762213 \sqrt{\frac{11}{46}} \sqrt{-\frac{-5+2 x}{2-3 x}} \sqrt{7+5 x}\right ) \operatorname{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 )}{99532800 \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{2-3 x}}}\\ &=-\frac{1450582567 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{3686400 \sqrt{-5+2 x}}-\frac{70489981 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}}{1658880}-\frac{83363 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{3/2}}{34560}-\frac{427 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{5/2}}{2400}+\frac{1}{25} \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{7/2}+\frac{1450582567 \sqrt{\frac{143}{3}} \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 )}{2457600 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{7+5 x}}-\frac{245264762213 \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 )}{99532800 \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{5-2 x}}}-\frac{57691792727443 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{1+4 x}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{7+5 x}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{497664000 \sqrt{429} \sqrt{-5+2 x} \sqrt{1+4 x}}\\ \end{align*}

Mathematica [A]  time = 4.38093, size = 350, normalized size = 0.74 \[ -\frac{\sqrt{2 x-5} \sqrt{4 x+1} \left (-62507925572 \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right ),\frac{39}{62}\right )[phone] \sqrt{682} \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \left (15 x^2+11 x-14\right ) E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+\sqrt{\frac{5 x+7}{3 x-2}} \left (6 \left (39813120000 x^7+71414784000 x^6-288728294400 x^5-849459145920 x^4-795166559320 x^3+2861488598626 x^2+5225923788019 x+1118234665415\right )-60033082963 \sqrt{682} (2-3 x)^2 \sqrt{\frac{4 x+1}{3 x-2}} \sqrt{\frac{10 x^2-11 x-35}{(2-3 x)^2}} \Pi \left (\frac{117}{62};\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )\right )\right )}{398131200 \sqrt{2-3 x} \sqrt{5 x+7} \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

-(Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(78331458618*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] - 62507925572*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]
+ Sqrt[(7 + 5*x)/(-2 + 3*x)]*(6*(1118234665415 + 5225923788019*x + 2861488598626*x^2 - 795166559320*x^3 - 8494
59145920*x^4 - 288728294400*x^5 + 71414784000*x^6 + 39813120000*x^7) - 60033082963*Sqrt[682]*(2 - 3*x)^2*Sqrt[
(1 + 4*x)/(-2 + 3*x)]*Sqrt[(-35 - 11*x + 10*x^2)/(2 - 3*x)^2]*EllipticPi[117/62, ArcSin[Sqrt[31/39]*Sqrt[(-5 +
2*x)/(-2 + 3*x)]], 39/62])))/(398131200*Sqrt[2 - 3*x]*Sqrt[7 + 5*x]*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8
*x^2))

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Maple [B]  time = 0.074, size = 895, normalized size = 1.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/284663808000*(7+5*x)^(1/2)*(2-3*x)^(1/2)*(2*x-5)^(1/2)*(4*x+1)^(1/2)*(62433731183120*11^(1/2)*((7+5*x)/(4*x+
1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((-2+3*x)/(4*x+1))^(1/2)*x^2*EllipticF(1/31*31^(1/2)*11^(1/
2)*((7+5*x)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))-684857410441904*11^(1/2)*((7+5*x)/(4*x+1))^(1/2)*3^(1/2)*13
^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((-2+3*x)/(4*x+1))^(1/2)*x^2*EllipticPi(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(4*x+1)
)^(1/2),124/55,1/39*31^(1/2)*78^(1/2))-896111886589920*11^(1/2)*((7+5*x)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x
-5)/(4*x+1))^(1/2)*((-2+3*x)/(4*x+1))^(1/2)*x^2*EllipticE(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(4*x+1))^(1/2),1/39*
31^(1/2)*78^(1/2))[phone]*11^(1/2)*((7+5*x)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((
-2+3*x)/(4*x+1))^(1/2)*x*EllipticF(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))-3424
28705220952*11^(1/2)*((7+5*x)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((-2+3*x)/(4*x+1))^(1/2)
*x*EllipticPi(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(4*x+1))^(1/2),124/55,1/39*31^(1/2)*78^(1/2))-448055943294960*11
^(1/2)*((7+5*x)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((-2+3*x)/(4*x+1))^(1/2)*x*EllipticE(1
/31*31^(1/2)*11^(1/2)*((7+5*x)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))[phone]*11^(1/2)*((7+5*x)/(4*x+1))
^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^(1/2)*((-2+3*x)/(4*x+1))^(1/2)*EllipticF(1/31*31^(1/2)*11^(1/2)*((7+
5*x)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))-42803588152619*11^(1/2)*((7+5*x)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*(
(2*x-5)/(4*x+1))^(1/2)*((-2+3*x)/(4*x+1))^(1/2)*EllipticPi(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(4*x+1))^(1/2),124/
55,1/39*31^(1/2)*78^(1/2))-56006992911870*11^(1/2)*((7+5*x)/(4*x+1))^(1/2)*3^(1/2)*13^(1/2)*((2*x-5)/(4*x+1))^
(1/2)*((-2+3*x)/(4*x+1))^(1/2)*EllipticE(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2)
)[phone]*x^7+306369423360000*x^6-1238644382976000*x^5-3644179735996800*x^4-3411264539482800*x^3+18436
555308411240*x^2+15642366908265240*x-16765465556439600)/(120*x^4-182*x^3-385*x^2+197*x+70)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (5 \, x + 7\right )}^{\frac{5}{2}} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (25 \, x^{2} + 70 \, x + 49\right )} \sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((25*x^2 + 70*x + 49)*sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (5 \, x + 7\right )}^{\frac{5}{2}} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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