3.106 \(\int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{5/2}} \, dx\)
Optimal. Leaf size=288 \[ \frac{103964 \sqrt{5 x+7} \text{EllipticF}\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right ),-\frac{39}{23}\right )}{1918683 \sqrt{253} \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}+\frac{358120 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{2319687747 \sqrt{2 x-5}}-\frac{895300 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{2319687747 \sqrt{5 x+7}}-\frac{50 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{83421 (5 x+7)^{3/2}}-\frac{179060 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{59479173 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}} \]
[Out]
(-50*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(83421*(7 + 5*x)^(3/2)) - (895300*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x
]*Sqrt[1 + 4*x])/(2319687747*Sqrt[7 + 5*x]) + (358120*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(2319687747*S
qrt[-5 + 2*x]) - (179060*Sqrt[11/39]*Sqrt[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqr
t[1 + 4*x])/Sqrt[-5 + 2*x]], -23/39])/(59479173*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) + (103964*Sqrt[7 + 5*
x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(1918683*Sqrt[253]*Sqrt[-5 + 2*x]*Sqrt[(7
+ 5*x)/(5 - 2*x)])
________________________________________________________________________________________
Rubi [A] time = 0.302918, antiderivative size = 288, normalized size of antiderivative = 1.,
number of steps used = 8, number of rules used = 8, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.216, Rules used =
{172, 1599, 1602, 12, 170, 418, 176, 424} \[ \frac{358120 \sqrt{2-3 x} \sqrt{4 x+1} \sqrt{5 x+7}}{2319687747 \sqrt{2 x-5}}-\frac{895300 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{2319687747 \sqrt{5 x+7}}-\frac{50 \sqrt{2-3 x} \sqrt{2 x-5} \sqrt{4 x+1}}{83421 (5 x+7)^{3/2}}+\frac{103964 \sqrt{5 x+7} F\left (\tan ^{-1}\left (\frac{\sqrt{4 x+1}}{\sqrt{2} \sqrt{2-3 x}}\right )|-\frac{39}{23}\right )}{1918683 \sqrt{253} \sqrt{2 x-5} \sqrt{\frac{5 x+7}{5-2 x}}}-\frac{179060 \sqrt{\frac{11}{39}} \sqrt{2-3 x} \sqrt{\frac{5 x+7}{5-2 x}} E\left (\sin ^{-1}\left (\frac{\sqrt{\frac{39}{23}} \sqrt{4 x+1}}{\sqrt{2 x-5}}\right )|-\frac{23}{39}\right )}{59479173 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{5 x+7}} \]
Antiderivative was successfully verified.
[In]
Int[1/(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(5/2)),x]
[Out]
(-50*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x])/(83421*(7 + 5*x)^(3/2)) - (895300*Sqrt[2 - 3*x]*Sqrt[-5 + 2*x
]*Sqrt[1 + 4*x])/(2319687747*Sqrt[7 + 5*x]) + (358120*Sqrt[2 - 3*x]*Sqrt[1 + 4*x]*Sqrt[7 + 5*x])/(2319687747*S
qrt[-5 + 2*x]) - (179060*Sqrt[11/39]*Sqrt[2 - 3*x]*Sqrt[(7 + 5*x)/(5 - 2*x)]*EllipticE[ArcSin[(Sqrt[39/23]*Sqr
t[1 + 4*x])/Sqrt[-5 + 2*x]], -23/39])/(59479173*Sqrt[(2 - 3*x)/(5 - 2*x)]*Sqrt[7 + 5*x]) + (103964*Sqrt[7 + 5*
x]*EllipticF[ArcTan[Sqrt[1 + 4*x]/(Sqrt[2]*Sqrt[2 - 3*x])], -39/23])/(1918683*Sqrt[253]*Sqrt[-5 + 2*x]*Sqrt[(7
+ 5*x)/(5 - 2*x)])
Rule 172
Int[((a_.) + (b_.)*(x_))^(m_)/(Sqrt[(c_.) + (d_.)*(x_)]*Sqrt[(e_.) + (f_.)*(x_)]*Sqrt[(g_.) + (h_.)*(x_)]), x_
Symbol] :> Simp[(b^2*(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[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) - 2*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)*b^2*x^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IntegerQ[2*m] && LeQ[m, -2]
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{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{5/2}} \, dx &=-\frac{50 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{83421 (7+5 x)^{3/2}}+\frac{\int \frac{11928-4270 x}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} (7+5 x)^{3/2}} \, dx}{83421}\\ &=-\frac{50 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{83421 (7+5 x)^{3/2}}-\frac{895300 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{2319687747 \sqrt{7+5 x}}+\frac{\int \frac{41179978+16294460 x-21487200 x^2}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{2319687747}\\ &=-\frac{50 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{83421 (7+5 x)^{3/2}}-\frac{895300 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{2319687747 \sqrt{7+5 x}}+\frac{358120 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{2319687747 \sqrt{-5+2 x}}-\frac{\int -\frac{15083097120}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{556725059280}+\frac{1969660 \int \frac{\sqrt{2-3 x}}{(-5+2 x)^{3/2} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{59479173}\\ &=-\frac{50 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{83421 (7+5 x)^{3/2}}-\frac{895300 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{2319687747 \sqrt{7+5 x}}+\frac{358120 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{2319687747 \sqrt{-5+2 x}}+\frac{51982 \int \frac{1}{\sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x} \sqrt{7+5 x}} \, dx}{1918683}-\frac{\left (179060 \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 )}{59479173 \sqrt{-\frac{2-3 x}{-5+2 x}} \sqrt{7+5 x}}\\ &=-\frac{50 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{83421 (7+5 x)^{3/2}}-\frac{895300 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{2319687747 \sqrt{7+5 x}}+\frac{358120 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{2319687747 \sqrt{-5+2 x}}-\frac{179060 \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 )}{59479173 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{7+5 x}}+\frac{\left (51982 \sqrt{\frac{2}{253}} \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 )}{1918683 \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{2-3 x}}}\\ &=-\frac{50 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{83421 (7+5 x)^{3/2}}-\frac{895300 \sqrt{2-3 x} \sqrt{-5+2 x} \sqrt{1+4 x}}{2319687747 \sqrt{7+5 x}}+\frac{358120 \sqrt{2-3 x} \sqrt{1+4 x} \sqrt{7+5 x}}{2319687747 \sqrt{-5+2 x}}-\frac{179060 \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 )}{59479173 \sqrt{\frac{2-3 x}{5-2 x}} \sqrt{7+5 x}}+\frac{103964 \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 )}{1918683 \sqrt{253} \sqrt{-5+2 x} \sqrt{\frac{7+5 x}{5-2 x}}}\\ \end{align*}
Mathematica [A] time = 1.71931, size = 246, normalized size = 0.85 \[ -\frac{2 \sqrt{2 x-5} \sqrt{4 x+1} \left (-28819 \sqrt{682} (3 x-2) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} (5 x+7)^2 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right ),\frac{39}{62}\right )+1705 \sqrt{\frac{5 x+7}{3 x-2}} \left (608600 x^3-294854 x^2-2797991 x-671560\right )-984830 \sqrt{682} (3 x-2) \sqrt{\frac{8 x^2-18 x-5}{(2-3 x)^2}} (5 x+7)^2 E\left (\sin ^{-1}\left (\sqrt{\frac{31}{39}} \sqrt{\frac{2 x-5}{3 x-2}}\right )|\frac{39}{62}\right )\right )}{25516565217 \sqrt{2-3 x} (5 x+7)^{3/2} \sqrt{\frac{5 x+7}{3 x-2}} \left (8 x^2-18 x-5\right )} \]
Antiderivative was successfully verified.
[In]
Integrate[1/(Sqrt[2 - 3*x]*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(7 + 5*x)^(5/2)),x]
[Out]
(-2*Sqrt[-5 + 2*x]*Sqrt[1 + 4*x]*(1705*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-671560 - 2797991*x - 294854*x^2 + 608600*x
^3) - 984830*Sqrt[682]*(-2 + 3*x)*(7 + 5*x)^2*Sqrt[(-5 - 18*x + 8*x^2)/(2 - 3*x)^2]*EllipticE[ArcSin[Sqrt[31/3
9]*Sqrt[(-5 + 2*x)/(-2 + 3*x)]], 39/62] - 28819*Sqrt[682]*(-2 + 3*x)*(7 + 5*x)^2*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]))/(25516565217*Sqrt[2 - 3*x]*(7 + 5
*x)^(3/2)*Sqrt[(7 + 5*x)/(-2 + 3*x)]*(-5 - 18*x + 8*x^2))
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Maple [B] time = 0.031, size = 786, normalized size = 2.7 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(1/(7+5*x)^(5/2)/(2-3*x)^(1/2)/(2*x-5)^(1/2)/(4*x+1)^(1/2),x)
[Out]
2/25516565217*(50128960*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+78786400*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+95245024*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))+149694160*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*3
1^(1/2)*78^(1/2))+38223332*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))+60074630*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))+4386284*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))+6893810*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))+496006500*x^3-665223020*x^2-2040625895*x+1509107050)*(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{1}{{\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(1/(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(1/((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 (-\frac{\sqrt{5 \, x + 7} \sqrt{4 \, x + 1} \sqrt{2 \, x - 5} \sqrt{-3 \, x + 2}}{3000 \, x^{6} + 3850 \, x^{5} - 16485 \, x^{4} - 30943 \, x^{3} - 3325 \, x^{2} + 14553 \, x + 3430}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(1/(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(-sqrt(5*x + 7)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)/(3000*x^6 + 3850*x^5 - 16485*x^4 - 30943*x^
3 - 3325*x^2 + 14553*x + 3430), 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(1/(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 \frac{1}{{\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(1/(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(1/((5*x + 7)^(5/2)*sqrt(4*x + 1)*sqrt(2*x - 5)*sqrt(-3*x + 2)), x)