3.2560 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^5 \sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=173 \[ -\frac{7986105 \sqrt{5 x+3}}{845152 \sqrt{1-2 x}}+\frac{698295 \sqrt{5 x+3}}{21952 \sqrt{1-2 x} (3 x+2)}+\frac{6621 \sqrt{5 x+3}}{1568 \sqrt{1-2 x} (3 x+2)^2}+\frac{263 \sqrt{5 x+3}}{392 \sqrt{1-2 x} (3 x+2)^3}+\frac{3 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)^4}-\frac{24922335 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664 \sqrt{7}} \]

[Out]

(-7986105*Sqrt[3 + 5*x])/(845152*Sqrt[1 - 2*x]) + (3*Sqrt[3 + 5*x])/(28*Sqrt[1 - 2*x]*(2 + 3*x)^4) + (263*Sqrt
[3 + 5*x])/(392*Sqrt[1 - 2*x]*(2 + 3*x)^3) + (6621*Sqrt[3 + 5*x])/(1568*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (698295*S
qrt[3 + 5*x])/(21952*Sqrt[1 - 2*x]*(2 + 3*x)) - (24922335*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1536
64*Sqrt[7])

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Rubi [A]  time = 0.0604215, antiderivative size = 173, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {103, 151, 152, 12, 93, 204} \[ -\frac{7986105 \sqrt{5 x+3}}{845152 \sqrt{1-2 x}}+\frac{698295 \sqrt{5 x+3}}{21952 \sqrt{1-2 x} (3 x+2)}+\frac{6621 \sqrt{5 x+3}}{1568 \sqrt{1-2 x} (3 x+2)^2}+\frac{263 \sqrt{5 x+3}}{392 \sqrt{1-2 x} (3 x+2)^3}+\frac{3 \sqrt{5 x+3}}{28 \sqrt{1-2 x} (3 x+2)^4}-\frac{24922335 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{153664 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7986105*Sqrt[3 + 5*x])/(845152*Sqrt[1 - 2*x]) + (3*Sqrt[3 + 5*x])/(28*Sqrt[1 - 2*x]*(2 + 3*x)^4) + (263*Sqrt
[3 + 5*x])/(392*Sqrt[1 - 2*x]*(2 + 3*x)^3) + (6621*Sqrt[3 + 5*x])/(1568*Sqrt[1 - 2*x]*(2 + 3*x)^2) + (698295*S
qrt[3 + 5*x])/(21952*Sqrt[1 - 2*x]*(2 + 3*x)) - (24922335*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(1536
64*Sqrt[7])

Rule 103

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(a +
b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)), x] + Dist[1/((m + 1)*(b*
c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegerQ[m] && (IntegerQ[n] || IntegersQ[2*n, 2*p])

Rule 151

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegerQ[m]

Rule 152

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[((b*g - a*h)*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*
f)), x] + Dist[1/((m + 1)*(b*c - a*d)*(b*e - a*f)), Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[(a*d*f*
g - b*(d*e + c*f)*g + b*c*e*h)*(m + 1) - (b*g - a*h)*(d*e*(n + 1) + c*f*(p + 1)) - d*f*(b*g - a*h)*(m + n + p
+ 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && LtQ[m, -1] && IntegersQ[2*m, 2*n, 2*p]

Rule 12

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

Rule 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 204

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> -Simp[ArcTan[(Rt[-b, 2]*x)/Rt[-a, 2]]/(Rt[-a, 2]*Rt[-b, 2]), x] /
; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{1}{(1-2 x)^{3/2} (2+3 x)^5 \sqrt{3+5 x}} \, dx &=\frac{3 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^4}+\frac{1}{28} \int \frac{\frac{103}{2}-120 x}{(1-2 x)^{3/2} (2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=\frac{3 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^4}+\frac{263 \sqrt{3+5 x}}{392 \sqrt{1-2 x} (2+3 x)^3}+\frac{1}{588} \int \frac{\frac{14787}{4}-11835 x}{(1-2 x)^{3/2} (2+3 x)^3 \sqrt{3+5 x}} \, dx\\ &=\frac{3 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^4}+\frac{263 \sqrt{3+5 x}}{392 \sqrt{1-2 x} (2+3 x)^3}+\frac{6621 \sqrt{3+5 x}}{1568 \sqrt{1-2 x} (2+3 x)^2}+\frac{\int \frac{\frac{1180305}{8}-695205 x}{(1-2 x)^{3/2} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{8232}\\ &=\frac{3 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^4}+\frac{263 \sqrt{3+5 x}}{392 \sqrt{1-2 x} (2+3 x)^3}+\frac{6621 \sqrt{3+5 x}}{1568 \sqrt{1-2 x} (2+3 x)^2}+\frac{698295 \sqrt{3+5 x}}{21952 \sqrt{1-2 x} (2+3 x)}+\frac{\int \frac{-\frac{21066255}{16}-\frac{73320975 x}{4}}{(1-2 x)^{3/2} (2+3 x) \sqrt{3+5 x}} \, dx}{57624}\\ &=-\frac{7986105 \sqrt{3+5 x}}{845152 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^4}+\frac{263 \sqrt{3+5 x}}{392 \sqrt{1-2 x} (2+3 x)^3}+\frac{6621 \sqrt{3+5 x}}{1568 \sqrt{1-2 x} (2+3 x)^2}+\frac{698295 \sqrt{3+5 x}}{21952 \sqrt{1-2 x} (2+3 x)}-\frac{\int -\frac{5757059385}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{2218524}\\ &=-\frac{7986105 \sqrt{3+5 x}}{845152 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^4}+\frac{263 \sqrt{3+5 x}}{392 \sqrt{1-2 x} (2+3 x)^3}+\frac{6621 \sqrt{3+5 x}}{1568 \sqrt{1-2 x} (2+3 x)^2}+\frac{698295 \sqrt{3+5 x}}{21952 \sqrt{1-2 x} (2+3 x)}+\frac{24922335 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{307328}\\ &=-\frac{7986105 \sqrt{3+5 x}}{845152 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^4}+\frac{263 \sqrt{3+5 x}}{392 \sqrt{1-2 x} (2+3 x)^3}+\frac{6621 \sqrt{3+5 x}}{1568 \sqrt{1-2 x} (2+3 x)^2}+\frac{698295 \sqrt{3+5 x}}{21952 \sqrt{1-2 x} (2+3 x)}+\frac{24922335 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{153664}\\ &=-\frac{7986105 \sqrt{3+5 x}}{845152 \sqrt{1-2 x}}+\frac{3 \sqrt{3+5 x}}{28 \sqrt{1-2 x} (2+3 x)^4}+\frac{263 \sqrt{3+5 x}}{392 \sqrt{1-2 x} (2+3 x)^3}+\frac{6621 \sqrt{3+5 x}}{1568 \sqrt{1-2 x} (2+3 x)^2}+\frac{698295 \sqrt{3+5 x}}{21952 \sqrt{1-2 x} (2+3 x)}-\frac{24922335 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{153664 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0696951, size = 95, normalized size = 0.55 \[ \frac{-7 \sqrt{5 x+3} \left (1293749010 x^4+1998242055 x^3+482249808 x^2-491393004 x-205593328\right )-274145685 \sqrt{7-14 x} (3 x+2)^4 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{11832128 \sqrt{1-2 x} (3 x+2)^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-7*Sqrt[3 + 5*x]*(-205593328 - 491393004*x + 482249808*x^2 + 1998242055*x^3 + 1293749010*x^4) - 274145685*Sqr
t[7 - 14*x]*(2 + 3*x)^4*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(11832128*Sqrt[1 - 2*x]*(2 + 3*x)^4)

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Maple [B]  time = 0.015, size = 305, normalized size = 1.8 \begin{align*}{\frac{1}{23664256\, \left ( 2+3\,x \right ) ^{4} \left ( 2\,x-1 \right ) } \left ( 44411600970\,\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) \sqrt{7}{x}^{5}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{4}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{3}[phone]\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-6579496440\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ){x}^{2}[phone]\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}-17545323840\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+6751497312\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-4386330960\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -6879502056\,x\sqrt{-10\,{x}^{2}-x+3}-2878306592\,\sqrt{-10\,{x}^{2}-x+3} \right ) \sqrt{3+5\,x}\sqrt{1-2\,x}{\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/23664256*(44411600970*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+96225135435*7^(1/2)*arc
tan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+59215467960*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))*x^3+18112486140*x^4*(-10*x^2-x+3)^(1/2)-6579496440*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2
-x+3)^(1/2))*x^2+27975388770*x^3*(-10*x^2-x+3)^(1/2)-17545323840*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^
2-x+3)^(1/2))*x+6751497312*x^2*(-10*x^2-x+3)^(1/2)-4386330960*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x
+3)^(1/2))-6879502056*x*(-10*x^2-x+3)^(1/2)-2878306592*(-10*x^2-x+3)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(2+3*x
)^4/(2*x-1)/(-10*x^2-x+3)^(1/2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.82248, size = 437, normalized size = 2.53 \begin{align*} -\frac{274145685 \, \sqrt{7}{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )} \arctan \left (\frac{\sqrt{7}{\left (37 \, x + 20\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{14 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) - 14 \,{\left (1293749010 \, x^{4} + 1998242055 \, x^{3} + 482249808 \, x^{2} - 491393004 \, x - 205593328\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{23664256 \,{\left (162 \, x^{5} + 351 \, x^{4} + 216 \, x^{3} - 24 \, x^{2} - 64 \, x - 16\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/23664256*(274145685*sqrt(7)*(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)*arctan(1/14*sqrt(7)*(37*x +
20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1293749010*x^4 + 1998242055*x^3 + 482249808*x^2 - 491
393004*x - 205593328)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(162*x^5 + 351*x^4 + 216*x^3 - 24*x^2 - 64*x - 16)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Giac [B]  time = 3.93325, size = 547, normalized size = 3.16 \begin{align*} \frac{4984467}{4302592} \, \sqrt{70} \sqrt{10}{\left (\pi + 2 \, \arctan \left (-\frac{\sqrt{70} \sqrt{5 \, x + 3}{\left (\frac{{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}^{2}}{5 \, x + 3} - 4\right )}}{140 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}\right )\right )} - \frac{64 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{924385 \,{\left (2 \, x - 1\right )}} + \frac{99 \,{\left (4411181 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{7} + 2388710520 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{5} + 506212728000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{3} + 38676680000000 \, \sqrt{10}{\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}\right )}}{537824 \,{\left ({\left (\frac{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}{\sqrt{5 \, x + 3}} - \frac{4 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}^{2} + 280\right )}^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4984467/4302592*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sqr
t(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 64/924385*sqrt(5)*sqrt(5*x + 3)*sqrt(-10*x +
5)/(2*x - 1) + 99/537824*(4411181*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x +
3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^7 + 2388710520*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*
x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^5 + 506212728000*sqrt(10)*((sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 38676680000000*sqrt
(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)
)))/(((sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)
))^2 + 280)^4