3.92 \(\int \frac{\sqrt{2-3 x} \sqrt{1+4 x}}{\sqrt{-5+2 x} (7+5 x)^{9/2}} \, dx\)
Optimal. Leaf size=370 \[ \frac{258506776 \sqrt{\frac{11}{23}} \sqrt{5 x+7} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right ),-\frac{39}{23}\right )}{1618368818157 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{16377776536 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{1956607901151813 \sqrt{2 x-5}}-\frac{40944441340 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1956607901151813 \sqrt{5 x+7}}-\frac{3217468 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{50259901185 (5 x+7)^{3/2}}+\frac{98 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1807455 (5 x+7)^{5/2}}+\frac{2 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{273 (5 x+7)^{7/2}}-\frac{8188888268 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{50169433362867 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}} \]
[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)])
________________________________________________________________________________________
Rubi [A] time = 0.521292, antiderivative size = 370, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 9, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.243, Rules used
= {164, 1604, 1599, 1602, 12, 170, 418, 176, 424} \[ \frac{16377776536 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{1956607901151813 \sqrt{2 x-5}}-\frac{40944441340 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1956607901151813 \sqrt{5 x+7}}-\frac{3217468 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{50259901185 (5 x+7)^{3/2}}+\frac{98 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{1807455 (5 x+7)^{5/2}}+\frac{2 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{273 (5 x+7)^{7/2}}+\frac{258506776 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{1618368818157 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{8188888268 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{50169433362867 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}} \]
Antiderivative was successfully verified.
[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 164
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 1604
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 1599
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 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 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[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 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 \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{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 ) \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 )}{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 ) \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 )}{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] time = 2.19595, size = 258, normalized size = 0.7 \[ \frac{2 \sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7} \left (\frac{(3 x-2) \left (2559027583750 x^3+12313608173580 x^2+19165803061167 x+2552362046246\right )}{(5 x+7)^4}-\frac{22 \left (71545594 \sqrt{682} (3 x-2) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right ),\frac{39}{62}\right )+558333291 \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )-186111097 \sqrt{682} (3 x-2) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )\right )}{\sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )}\right )}{1956607901151813 \sqrt{2-3 x}} \]
Antiderivative was successfully verified.
[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 [B] time = 0.031, size = 1160, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2-3*x)^(1/2)*(4*x+1)^(1/2)/(7+5*x)^(9/2)/(2*x-5)^(1/2),x)
[Out]
-2/1956607901151813*(175178212000*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))*x^5-81
88888268000*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))*x^5+823337596400*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))*x^4-38487774859600*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))*x^4+1408870770010*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))*x^3-65859133895390*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))*x^3+1
041697237658*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))-48695224085662*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))+304722499774*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))-14244571142186*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*3
1^(1/2)*78^(1/2))+30043063358*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))-1404394337
962*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)*Ellipti
cE(1/31*31^(1/2)*11^(1/2)*((7+5*x)/(4*x+1))^(1/2),1/39*31^(1/2)*78^(1/2))-33052545587580*x^5-83732628367442*x^
4+43651554581534*x^3+1041927172311711*x^2-131120048990980*x-367706794166900)*(2*x-5)^(1/2)*(4*x+1)^(1/2)*(2-3*
x)^(1/2)/(120*x^4-182*x^3-385*x^2+197*x+70)/(7+5*x)^(5/2)
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac{9}{2}} \sqrt{2 \, x - 5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\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\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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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((2-3*x)**(1/2)*(1+4*x)**(1/2)/(7+5*x)**(9/2)/(-5+2*x)**(1/2),x)
[Out]
Timed out
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{4 \, x + 1} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac{9}{2}} \sqrt{2 \, x - 5}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)