3.2272 \(\int \frac{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^6} \, dx\)

Optimal. Leaf size=180 \[ \frac{1466281 \sqrt{1-2 x} \sqrt{5 x+3}}{131712 (3 x+2)}+\frac{14023 \sqrt{1-2 x} \sqrt{5 x+3}}{9408 (3 x+2)^2}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3}}{1680 (3 x+2)^3}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{840 (3 x+2)^4}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^5}-\frac{5591773 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(840*(2 + 3*x)^4) + (403*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x])/(1680*(2 + 3*x)^3) + (14023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(9408*(2 + 3*x)^2) + (1466
281*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(131712*(2 + 3*x)) - (5591773*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/
(43904*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0658313, 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 = {97, 151, 12, 93, 204} \[ \frac{1466281 \sqrt{1-2 x} \sqrt{5 x+3}}{131712 (3 x+2)}+\frac{14023 \sqrt{1-2 x} \sqrt{5 x+3}}{9408 (3 x+2)^2}+\frac{403 \sqrt{1-2 x} \sqrt{5 x+3}}{1680 (3 x+2)^3}+\frac{37 \sqrt{1-2 x} \sqrt{5 x+3}}{840 (3 x+2)^4}-\frac{\sqrt{1-2 x} \sqrt{5 x+3}}{15 (3 x+2)^5}-\frac{5591773 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{43904 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(15*(2 + 3*x)^5) + (37*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(840*(2 + 3*x)^4) + (403*Sq
rt[1 - 2*x]*Sqrt[3 + 5*x])/(1680*(2 + 3*x)^3) + (14023*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(9408*(2 + 3*x)^2) + (1466
281*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(131712*(2 + 3*x)) - (5591773*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/
(43904*Sqrt[7])

Rule 97

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a + b
*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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{\sqrt{1-2 x} \sqrt{3+5 x}}{(2+3 x)^6} \, dx &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{1}{15} \int \frac{-\frac{1}{2}-10 x}{\sqrt{1-2 x} (2+3 x)^5 \sqrt{3+5 x}} \, dx\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{840 (2+3 x)^4}+\frac{1}{420} \int \frac{\frac{1341}{4}-555 x}{\sqrt{1-2 x} (2+3 x)^4 \sqrt{3+5 x}} \, dx\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{840 (2+3 x)^4}+\frac{403 \sqrt{1-2 x} \sqrt{3+5 x}}{1680 (2+3 x)^3}+\frac{\int \frac{\frac{265125}{8}-42315 x}{\sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}} \, dx}{8820}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{840 (2+3 x)^4}+\frac{403 \sqrt{1-2 x} \sqrt{3+5 x}}{1680 (2+3 x)^3}+\frac{14023 \sqrt{1-2 x} \sqrt{3+5 x}}{9408 (2+3 x)^2}+\frac{\int \frac{\frac{31687635}{16}-\frac{7362075 x}{4}}{\sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}} \, dx}{123480}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{840 (2+3 x)^4}+\frac{403 \sqrt{1-2 x} \sqrt{3+5 x}}{1680 (2+3 x)^3}+\frac{14023 \sqrt{1-2 x} \sqrt{3+5 x}}{9408 (2+3 x)^2}+\frac{1466281 \sqrt{1-2 x} \sqrt{3+5 x}}{131712 (2+3 x)}+\frac{\int \frac{1761408495}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{864360}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{840 (2+3 x)^4}+\frac{403 \sqrt{1-2 x} \sqrt{3+5 x}}{1680 (2+3 x)^3}+\frac{14023 \sqrt{1-2 x} \sqrt{3+5 x}}{9408 (2+3 x)^2}+\frac{1466281 \sqrt{1-2 x} \sqrt{3+5 x}}{131712 (2+3 x)}+\frac{5591773 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{87808}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{840 (2+3 x)^4}+\frac{403 \sqrt{1-2 x} \sqrt{3+5 x}}{1680 (2+3 x)^3}+\frac{14023 \sqrt{1-2 x} \sqrt{3+5 x}}{9408 (2+3 x)^2}+\frac{1466281 \sqrt{1-2 x} \sqrt{3+5 x}}{131712 (2+3 x)}+\frac{5591773 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{43904}\\ &=-\frac{\sqrt{1-2 x} \sqrt{3+5 x}}{15 (2+3 x)^5}+\frac{37 \sqrt{1-2 x} \sqrt{3+5 x}}{840 (2+3 x)^4}+\frac{403 \sqrt{1-2 x} \sqrt{3+5 x}}{1680 (2+3 x)^3}+\frac{14023 \sqrt{1-2 x} \sqrt{3+5 x}}{9408 (2+3 x)^2}+\frac{1466281 \sqrt{1-2 x} \sqrt{3+5 x}}{131712 (2+3 x)}-\frac{5591773 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{43904 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.134089, size = 135, normalized size = 0.75 \[ \frac{1}{35} \left (\frac{5 \left (\frac{7 \sqrt{1-2 x} \sqrt{5 x+3} \left (4398843 x^2+6119462 x+2067760\right )}{(3 x+2)^3}-5591773 \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )\right )}{43904}+\frac{111 (1-2 x)^{3/2} (5 x+3)^{3/2}}{8 (3 x+2)^4}+\frac{3 (1-2 x)^{3/2} (5 x+3)^{3/2}}{(3 x+2)^5}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

((3*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(2 + 3*x)^5 + (111*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(8*(2 + 3*x)^4) + (5*
((7*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(2067760 + 6119462*x + 4398843*x^2))/(2 + 3*x)^3 - 5591773*Sqrt[7]*ArcTan[Sqrt
[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])]))/43904)/35

________________________________________________________________________________________

Maple [B]  time = 0.013, size = 298, normalized size = 1.7 \begin{align*}{\frac{1}{3073280\, \left ( 2+3\,x \right ) ^{5}}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 6794004195\,\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+7644056952\,{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((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^6,x)

[Out]

1/3073280*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(6794004195*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x
^5+22646680650*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+30195574200*7^(1/2)*arctan(1/14*
(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+2771271090*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+7513739100*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+7644056952*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

________________________________________________________________________________________

Maxima [A]  time = 2.17888, size = 267, normalized size = 1.48 \begin{align*} \frac{5591773}{614656} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{231065}{32928} \, \sqrt{-10 \, x^{2} - x + 3} + \frac{3 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{35 \,{\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} + \frac{111 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{280 \,{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} + \frac{1305 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{784 \,{\left (27 \, x^{3} + 54 \, x^{2} + 36 \, x + 8\right )}} + \frac{138639 \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}}}{21952 \,{\left (9 \, x^{2} + 12 \, x + 4\right )}} - \frac{1709881 \, \sqrt{-10 \, x^{2} - x + 3}}{131712 \,{\left (3 \, x + 2\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

5591773/614656*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 231065/32928*sqrt(-10*x^2 - x + 3)
+ 3/35*(-10*x^2 - x + 3)^(3/2)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32) + 111/280*(-10*x^2 - x +
3)^(3/2)/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16) + 1305/784*(-10*x^2 - x + 3)^(3/2)/(27*x^3 + 54*x^2 + 36*x +
8) + 138639/21952*(-10*x^2 - x + 3)^(3/2)/(9*x^2 + 12*x + 4) - 1709881/131712*sqrt(-10*x^2 - x + 3)/(3*x + 2)

________________________________________________________________________________________

Fricas [A]  time = 1.8903, size = 439, normalized size = 2.44 \begin{align*} -\frac{27958865 \, \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 (197947935 \, x^{4} + 536695650 \, x^{3} + 546004068 \, x^{2} + 247045192 \, x + 41933792\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{3073280 \,{\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((1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^6,x, algorithm="fricas")

[Out]

-1/3073280*(27958865*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*(197947935*x^4 + 536695650*x^3 + 546004068*x^2 + 2470
45192*x + 41933792)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{1 - 2 x} \sqrt{5 x + 3}}{\left (3 x + 2\right )^{6}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 4.41085, size = 582, normalized size = 3.23 \begin{align*} \frac{121}{6146560} \, \sqrt{5}{\left (46213 \, \sqrt{70} \sqrt{2}{\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{280 \, \sqrt{2}{\left (46213 \,{\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} - 85961680 \,{\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} - 30665564160 \,{\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} - 4732042112000 \,{\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} - \frac{284050977280000 \,{\left (\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}\right )}}{\sqrt{5 \, x + 3}} + \frac{1136203909120000 \, \sqrt{5 \, x + 3}}{\sqrt{2} \sqrt{-10 \, x + 5} - \sqrt{22}}\right )}}{{\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}}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

121/6146560*sqrt(5)*(46213*sqrt(70)*sqrt(2)*(pi + 2*arctan(-1/140*sqrt(70)*sqrt(5*x + 3)*((sqrt(2)*sqrt(-10*x
+ 5) - sqrt(22))^2/(5*x + 3) - 4)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))) - 280*sqrt(2)*(46213*((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)))^9 - 85961680*((sqr
t(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 - 306
65564160*((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 - 4732042112000*((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 - 284050977280000*(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) + 11362039091200
00*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)