3.82 \(\int \frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{(7+5 x)^{5/2}} \, dx\)
Optimal. Leaf size=391 \[ -\frac{496 \sqrt{\frac{11}{23}} \sqrt{5 x+7} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right ),-\frac{39}{23}\right )}{1725 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{35812 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{2085525 \sqrt{2 x-5}}+\frac{17906 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{417105 \sqrt{5 x+7}}-\frac{2 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{15 (5 x+7)^{3/2}}+\frac{17906 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{53475 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{496 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{125 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]
[Out]
(-2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(15*(7 + 5*x)^(3/2)) + (17906*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqr
t[1 + 4*x])/(417105*Sqrt[7 + 5*x]) - (35812*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(2085525*Sqrt[-5 + 2*x]
) + (17906*Sqrt[11/39]*Sqrt[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sq
rt[-5 + 2*x]], -23/39])/(53475*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) - (496*Sqrt[11/23]*Sqrt[7 + 5*x]*Ellip
ticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(1725*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) +
(496*(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])/(125*Sqrt[429]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])
________________________________________________________________________________________
Rubi [A] time = 0.429386, antiderivative size = 391, normalized size of antiderivative = 1.,
number of steps used = 10, number of rules used = 10, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.27, Rules used
= {160, 1604, 1602, 1598, 170, 418, 165, 537, 176, 424} \[ -\frac{35812 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{2085525 \sqrt{2 x-5}}+\frac{17906 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{417105 \sqrt{5 x+7}}-\frac{2 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{15 (5 x+7)^{3/2}}-\frac{496 \sqrt{\frac{11}{23}} \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{1725 \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{17906 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{53475 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}}+\frac{496 (2-3 x) \sqrt{\frac{5-2 x}{2-3 x}} \sqrt{-\frac{4 x+1}{2-3 x}} \Pi \left (-\frac{69}{55};\sin ^{-1}\left (\frac{\sqrt{\frac{11}{23}} \sqrt{5 x+7}}{\sqrt{2-3 x}}\right )|-\frac{23}{39}\right )}{125 \sqrt{429} \sqrt{2 x-5} \sqrt{4 x+1}} \]
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]
(-2*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(15*(7 + 5*x)^(3/2)) + (17906*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqr
t[1 + 4*x])/(417105*Sqrt[7 + 5*x]) - (35812*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(2085525*Sqrt[-5 + 2*x]
) + (17906*Sqrt[11/39]*Sqrt[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqrt[1 + 4*x])/Sq
rt[-5 + 2*x]], -23/39])/(53475*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) - (496*Sqrt[11/23]*Sqrt[7 + 5*x]*Ellip
ticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(1725*Sqrt[-5 + 2*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]) +
(496*(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])/(125*Sqrt[429]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])
Rule 160
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)*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])/(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] && Lt
Q[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 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 \frac{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{(7+5 x)^{5/2}} \, dx &=-\frac{2 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{15 (7+5 x)^{3/2}}+\frac{1}{15} \int \frac{-21+140 x-72 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{3/2}} \, dx\\ &=-\frac{2 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{15 (7+5 x)^{3/2}}+\frac{17906 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{417105 \sqrt{7+5 x}}+\frac{\int \frac{40642-726310 x+429744 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{417105}\\ &=-\frac{2 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{15 (7+5 x)^{3/2}}+\frac{17906 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{417105 \sqrt{7+5 x}}-\frac{35812 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{2085525 \sqrt{-5+2 x}}-\frac{\int \frac{94243968+96100992 x}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{100105200}-\frac{196966 \int \frac{\sqrt{2-3 x}}{(-5+2 x)^{3/2} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{53475}\\ &=-\frac{2 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{15 (7+5 x)^{3/2}}+\frac{17906 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{417105 \sqrt{7+5 x}}-\frac{35812 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{2085525 \sqrt{-5+2 x}}+\frac{8}{25} \int \frac{\sqrt{2-3 x}}{\sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx-\frac{2728 \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{1725}+\frac{\left (17906 \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 )}{53475 \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}}{15 (7+5 x)^{3/2}}+\frac{17906 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{417105 \sqrt{7+5 x}}-\frac{35812 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{2085525 \sqrt{-5+2 x}}+\frac{17906 \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 )}{53475 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{7+5 x}}+\frac{\left (496 (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 )}{25 \sqrt{897} \sqrt{-5+2 x} \sqrt{1+4 x}}-\frac{\left (248 \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 )}{1725 \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}}{15 (7+5 x)^{3/2}}+\frac{17906 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{417105 \sqrt{7+5 x}}-\frac{35812 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{2085525 \sqrt{-5+2 x}}+\frac{17906 \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 )}{53475 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{7+5 x}}-\frac{496 \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 )}{1725 \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{5-2 x}}}+\frac{496 (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 )}{125 \sqrt{429} \sqrt{-5+2 x} \sqrt{1+4 x}}\\ \end{align*}
Mathematica [A] time = 2.97431, size = 366, normalized size = 0.94 \[ \frac{\sqrt{2 x-5} \sqrt{4 x+1} \sqrt{5 x+7} \left (\frac{30 \sqrt{6-9 x} (44765 x+34864)}{(5 x+7)^2}-\frac{2 \left (-37053 \sqrt{682} (3 x-2) (5 x+7) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right ),\frac{39}{62}\right )+80577 \sqrt{682} (3 x-2) (5 x+7) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )+\sqrt{\frac{5 x+7}{3 x-2}} \left (48438 \sqrt{682} (2-3 x)^2 \sqrt{\frac{4 x+1}{3 x-2}} \sqrt{\frac{10 x^2-11 x-35}{(2-3 x)^2}} \Pi \left (\frac{117}{62};\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )-241731 \left (40 x^3-34 x^2-151 x-35\right )\right )\right )}{\sqrt{3} (2-3 x)^{3/2} \left (\frac{5 x+7}{3 x-2}\right )^{3/2} \left (-8 x^2+18 x+5\right )}\right )}{6256575 \sqrt{3}} \]
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]*Sqrt[7 + 5*x]*((30*Sqrt[6 - 9*x]*(34864 + 44765*x))/(7 + 5*x)^2 - (2*(80577*Sqrt
[682]*(-2 + 3*x)*(7 + 5*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] - 37053*Sqrt[682]*(-2 + 3*x)*(7 + 5*x)*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticF[Ar
cSin[Sqrt[31/39]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] + Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-241731*(-35 - 151*x - 34*
x^2 + 40*x^3) + 48438*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])))/(Sqrt[3]*(2 - 3*x)^(3/2)*((7 +
5*x)/(-2 + 3*x))^(3/2)*(5 + 18*x - 8*x^2))))/(6256575*Sqrt[3])
________________________________________________________________________________________
Maple [B] time = 0.04, size = 1149, normalized size = 2.9 \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)*(2*x-5)^(1/2)*(4*x+1)^(1/2)/(7+5*x)^(5/2),x)
[Out]
-2/114703875*(39393200*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^3-23614560*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))*x^3-24389200*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+74847080*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))-44867664*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))-46339480*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))+30037315*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))-18006102*11^(1/2)*((7+5*x)/(4*x+1))^(1/2)*3^(1/2)*1
3^(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))-18596765*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)*7
8^(1/2))+3446905*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))-2066274*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)*1
1^(1/2)*((7+5*x)/(4*x+1))^(1/2),124/55,1/39*31^(1/2)*78^(1/2))-2134055*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))+523292550*x^3-1135538635*x^2-779434810*x+869257400)*(4*x+1)^(1/2)*(2*x-5)^(1/2)
*(2-3*x)^(1/2)/(120*x^4-182*x^3-385*x^2+197*x+70)/(7+5*x)^(1/2)
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(5/2),x, algorithm="maxima")
[Out]
integrate(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(5*x + 7)^(5/2), 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}}{125 \, x^{3} + 525 \, x^{2} + 735 \, x + 343}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(5/2),x, algorithm="fricas")
[Out]
integral(sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(125*x^3 + 525*x^2 + 735*x + 343), 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)*(-5+2*x)**(1/2)*(1+4*x)**(1/2)/(7+5*x)**(5/2),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{2 \, x - 5} \sqrt{-3 \, x + 2}}{{\left (5 \, x + 7\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2-3*x)^(1/2)*(-5+2*x)^(1/2)*(1+4*x)^(1/2)/(7+5*x)^(5/2),x, algorithm="giac")
[Out]
integrate(sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(5*x + 7)^(5/2), x)