3.2477 \(\int \frac{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac{694229 \sqrt{1-2 x} \sqrt{5 x+3}}{921984 (3 x+2)}+\frac{6107 \sqrt{1-2 x} \sqrt{5 x+3}}{65856 (3 x+2)^2}-\frac{73 \sqrt{1-2 x} \sqrt{5 x+3}}{11760 (3 x+2)^3}-\frac{367 \sqrt{1-2 x} \sqrt{5 x+3}}{5880 (3 x+2)^4}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{105 (3 x+2)^5}-\frac{2664057 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{307328 \sqrt{7}} \]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(105*(2 + 3*x)^5) - (367*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5880*(2 + 3*x)^4) - (73*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(11760*(2 + 3*x)^3) + (6107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(65856*(2 + 3*x)^2) + (69
4229*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(921984*(2 + 3*x)) - (2664057*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])
/(307328*Sqrt[7])

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Rubi [A]  time = 0.0617117, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {98, 151, 12, 93, 204} \[ \frac{694229 \sqrt{1-2 x} \sqrt{5 x+3}}{921984 (3 x+2)}+\frac{6107 \sqrt{1-2 x} \sqrt{5 x+3}}{65856 (3 x+2)^2}-\frac{73 \sqrt{1-2 x} \sqrt{5 x+3}}{11760 (3 x+2)^3}-\frac{367 \sqrt{1-2 x} \sqrt{5 x+3}}{5880 (3 x+2)^4}+\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{105 (3 x+2)^5}-\frac{2664057 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{307328 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[(3 + 5*x)^(3/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^6),x]

[Out]

(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(105*(2 + 3*x)^5) - (367*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5880*(2 + 3*x)^4) - (73*S
qrt[1 - 2*x]*Sqrt[3 + 5*x])/(11760*(2 + 3*x)^3) + (6107*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(65856*(2 + 3*x)^2) + (69
4229*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(921984*(2 + 3*x)) - (2664057*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])
/(307328*Sqrt[7])

Rule 98

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

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 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{(3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^6} \, dx &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^5}-\frac{1}{105} \int \frac{-\frac{991}{2}-835 x}{\sqrt{1-2 x} (2+3 x)^5 \sqrt{3+5 x}} \, dx\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^5}-\frac{367 \sqrt{1-2 x} \sqrt{3+5 x}}{5880 (2+3 x)^4}-\frac{\int \frac{-\frac{14169}{4}-5505 x}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx}{2940}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^5}-\frac{367 \sqrt{1-2 x} \sqrt{3+5 x}}{5880 (2+3 x)^4}-\frac{73 \sqrt{1-2 x} \sqrt{3+5 x}}{11760 (2+3 x)^3}-\frac{\int \frac{-\frac{254625}{8}-7665 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{61740}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^5}-\frac{367 \sqrt{1-2 x} \sqrt{3+5 x}}{5880 (2+3 x)^4}-\frac{73 \sqrt{1-2 x} \sqrt{3+5 x}}{11760 (2+3 x)^3}+\frac{6107 \sqrt{1-2 x} \sqrt{3+5 x}}{65856 (2+3 x)^2}-\frac{\int \frac{-\frac{15748215}{16}+\frac{3206175 x}{4}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{864360}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^5}-\frac{367 \sqrt{1-2 x} \sqrt{3+5 x}}{5880 (2+3 x)^4}-\frac{73 \sqrt{1-2 x} \sqrt{3+5 x}}{11760 (2+3 x)^3}+\frac{6107 \sqrt{1-2 x} \sqrt{3+5 x}}{65856 (2+3 x)^2}+\frac{694229 \sqrt{1-2 x} \sqrt{3+5 x}}{921984 (2+3 x)}-\frac{\int -\frac{839177955}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{6050520}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^5}-\frac{367 \sqrt{1-2 x} \sqrt{3+5 x}}{5880 (2+3 x)^4}-\frac{73 \sqrt{1-2 x} \sqrt{3+5 x}}{11760 (2+3 x)^3}+\frac{6107 \sqrt{1-2 x} \sqrt{3+5 x}}{65856 (2+3 x)^2}+\frac{694229 \sqrt{1-2 x} \sqrt{3+5 x}}{921984 (2+3 x)}+\frac{2664057 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{614656}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^5}-\frac{367 \sqrt{1-2 x} \sqrt{3+5 x}}{5880 (2+3 x)^4}-\frac{73 \sqrt{1-2 x} \sqrt{3+5 x}}{11760 (2+3 x)^3}+\frac{6107 \sqrt{1-2 x} \sqrt{3+5 x}}{65856 (2+3 x)^2}+\frac{694229 \sqrt{1-2 x} \sqrt{3+5 x}}{921984 (2+3 x)}+\frac{2664057 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{307328}\\ &=\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{105 (2+3 x)^5}-\frac{367 \sqrt{1-2 x} \sqrt{3+5 x}}{5880 (2+3 x)^4}-\frac{73 \sqrt{1-2 x} \sqrt{3+5 x}}{11760 (2+3 x)^3}+\frac{6107 \sqrt{1-2 x} \sqrt{3+5 x}}{65856 (2+3 x)^2}+\frac{694229 \sqrt{1-2 x} \sqrt{3+5 x}}{921984 (2+3 x)}-\frac{2664057 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{307328 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0673839, size = 84, normalized size = 0.47 \[ \frac{\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (93720915 x^4+253769850 x^3+257531412 x^2+115804328 x+19437408\right )}{(3 x+2)^5}-13320285 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{10756480} \]

Antiderivative was successfully verified.

[In]

Integrate[(3 + 5*x)^(3/2)/(Sqrt[1 - 2*x]*(2 + 3*x)^6),x]

[Out]

((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(19437408 + 115804328*x + 257531412*x^2 + 253769850*x^3 + 93720915*x^4))/(2 +
3*x)^5 - 13320285*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/10756480

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Maple [B]  time = 0.013, size = 298, normalized size = 1.7 \begin{align*}{\frac{1}{21512960\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 3236829255\,\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}[phone]\,\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}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+3605439768\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((3+5*x)^(3/2)/(2+3*x)^6/(1-2*x)^(1/2),x)

[Out]

1/21512960*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(3236829255*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*
x^5+10789430850*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+14385907800*7^(1/2)*arctan(1/14
*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+1312092810*x^4*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*
(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+3552777900*x^3*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(
37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+3605439768*x^2*(-10*x^2-x+3)^(1/2)[phone]*7^(1/2)*arctan(1/14*(37*x
+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))[phone]*x*(-10*x^2-x+3)^(1/2)[phone]*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3
)^(1/2)/(2+3*x)^5

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Maxima [A]  time = 2.43909, size = 248, normalized size = 1.38 \begin{align*} \frac{2664057}{4302592} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{\sqrt{-10 \, x^{2} - x + 3}}{105 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} - \frac{367 \, \sqrt{-10 \, x^{2} - x + 3}}{5880 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} - \frac{73 \, \sqrt{-10 \, x^{2} - x + 3}}{11760 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{6107 \, \sqrt{-10 \, x^{2} - x + 3}}{65856 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} + \frac{694229 \, \sqrt{-10 \, x^{2} - x + 3}}{921984 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(3/2)/(2+3*x)^6/(1-2*x)^(1/2),x, algorithm="maxima")

[Out]

2664057/4302592*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 1/105*sqrt(-10*x^2 - x + 3)/(243*x
^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) - 367/5880*sqrt(-10*x^2 - x + 3)/(81*x^4 + 216*x^3 + 216*x^2 +
96*x + 16) - 73/11760*sqrt(-10*x^2 - x + 3)/(27*x^3 + 54*x^2 + 36*x + 8) + 6107/65856*sqrt(-10*x^2 - x + 3)/(
9*x^2 + 12*x + 4) + 694229/921984*sqrt(-10*x^2 - x + 3)/(3*x + 2)

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Fricas [A]  time = 1.92459, size = 439, normalized size = 2.44 \begin{align*} -\frac{13320285 \, \sqrt{7}{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\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 (93720915 \, x^{4} + 253769850 \, x^{3} + 257531412 \, x^{2} + 115804328 \, x + 19437408\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{21512960 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(3/2)/(2+3*x)^6/(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

-1/21512960*(13320285*sqrt(7)*(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)*arctan(1/14*sqrt(7)*(37*x
+ 20)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(93720915*x^4 + 253769850*x^3 + 257531412*x^2 + 1158
04328*x + 19437408)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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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((3+5*x)**(3/2)/(2+3*x)**6/(1-2*x)**(1/2),x)

[Out]

Exception raised: ValueError

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Giac [B]  time = 2.94597, size = 594, normalized size = 3.3 \begin{align*} \frac{2664057}{43025920} \, \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{121 \,{\left (22017 \, \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 )}^{9} + 28768880 \, \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} - 9856573440 \, \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} - 2123818368000 \, \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} - 133530503680000 \, \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 )}}{153664 \,{\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 )}^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(3/2)/(2+3*x)^6/(1-2*x)^(1/2),x, algorithm="giac")

[Out]

2664057/43025920*sqrt(70)*sqrt(10)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 121/153664*(22017*sqrt(10)*((sqrt(2)*sqrt(-1
0*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^9 + 28768880*sqrt(1
0)*((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 - 9856573440*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 - 2123818368000*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 - 133530503680000*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)^5