3.2570 \(\int \frac{1}{(1-2 x)^{3/2} (2+3 x)^4 (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=166 \[ -\frac{36657025 \sqrt{1-2 x}}{332024 \sqrt{5 x+3}}-\frac{73435}{15092 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{6525}{392 \sqrt{1-2 x} (3 x+2) \sqrt{5 x+3}}+\frac{37}{28 \sqrt{1-2 x} (3 x+2)^2 \sqrt{5 x+3}}+\frac{1}{7 \sqrt{1-2 x} (3 x+2)^3 \sqrt{5 x+3}}+\frac{2079585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

[Out]

-73435/(15092*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (36657025*Sqrt[1 - 2*x])/(332024*Sqrt[3 + 5*x]) + 1/(7*Sqrt[1 - 2
*x]*(2 + 3*x)^3*Sqrt[3 + 5*x]) + 37/(28*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x]) + 6525/(392*Sqrt[1 - 2*x]*(2
+ 3*x)*Sqrt[3 + 5*x]) + (2079585*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*Sqrt[7])

________________________________________________________________________________________

Rubi [A]  time = 0.0578747, antiderivative size = 166, 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} \[ -\frac{36657025 \sqrt{1-2 x}}{332024 \sqrt{5 x+3}}-\frac{73435}{15092 \sqrt{1-2 x} \sqrt{5 x+3}}+\frac{6525}{392 \sqrt{1-2 x} (3 x+2) \sqrt{5 x+3}}+\frac{37}{28 \sqrt{1-2 x} (3 x+2)^2 \sqrt{5 x+3}}+\frac{1}{7 \sqrt{1-2 x} (3 x+2)^3 \sqrt{5 x+3}}+\frac{2079585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2744 \sqrt{7}} \]

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

-73435/(15092*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]) - (36657025*Sqrt[1 - 2*x])/(332024*Sqrt[3 + 5*x]) + 1/(7*Sqrt[1 - 2
*x]*(2 + 3*x)^3*Sqrt[3 + 5*x]) + 37/(28*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x]) + 6525/(392*Sqrt[1 - 2*x]*(2
+ 3*x)*Sqrt[3 + 5*x]) + (2079585*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/(2744*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)^4 (3+5 x)^{3/2}} \, dx &=\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}+\frac{1}{21} \int \frac{\frac{99}{2}-120 x}{(1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{3/2}} \, dx\\ &=\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}+\frac{37}{28 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{1}{294} \int \frac{\frac{14595}{4}-11655 x}{(1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{3/2}} \, dx\\ &=\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}+\frac{37}{28 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{6525}{392 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}+\frac{\int \frac{\frac{1198365}{8}-685125 x}{(1-2 x)^{3/2} (2+3 x) (3+5 x)^{3/2}} \, dx}{2058}\\ &=-\frac{73435}{15092 \sqrt{1-2 x} \sqrt{3+5 x}}+\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}+\frac{37}{28 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{6525}{392 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}-\frac{\int \frac{-\frac{98442645}{16}+\frac{23132025 x}{4}}{\sqrt{1-2 x} (2+3 x) (3+5 x)^{3/2}} \, dx}{79233}\\ &=-\frac{73435}{15092 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{36657025 \sqrt{1-2 x}}{332024 \sqrt{3+5 x}}+\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}+\frac{37}{28 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{6525}{392 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}+\frac{2 \int -\frac{5284225485}{32 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{871563}\\ &=-\frac{73435}{15092 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{36657025 \sqrt{1-2 x}}{332024 \sqrt{3+5 x}}+\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}+\frac{37}{28 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{6525}{392 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}-\frac{2079585 \int \frac{1}{\sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}} \, dx}{5488}\\ &=-\frac{73435}{15092 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{36657025 \sqrt{1-2 x}}{332024 \sqrt{3+5 x}}+\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}+\frac{37}{28 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{6525}{392 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}-\frac{2079585 \operatorname{Subst}\left (\int \frac{1}{-7-x^2} \, dx,x,\frac{\sqrt{1-2 x}}{\sqrt{3+5 x}}\right )}{2744}\\ &=-\frac{73435}{15092 \sqrt{1-2 x} \sqrt{3+5 x}}-\frac{36657025 \sqrt{1-2 x}}{332024 \sqrt{3+5 x}}+\frac{1}{7 \sqrt{1-2 x} (2+3 x)^3 \sqrt{3+5 x}}+\frac{37}{28 \sqrt{1-2 x} (2+3 x)^2 \sqrt{3+5 x}}+\frac{6525}{392 \sqrt{1-2 x} (2+3 x) \sqrt{3+5 x}}+\frac{2079585 \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{3+5 x}}\right )}{2744 \sqrt{7}}\\ \end{align*}

Mathematica [A]  time = 0.0824826, size = 84, normalized size = 0.51 \[ \frac{\frac{7 \left (1979479350 x^4+2925598635 x^3+622325745 x^2-723664682 x-283149136\right )}{\sqrt{1-2 x} (3 x+2)^3 \sqrt{5 x+3}}[phone] \sqrt{7} \tan ^{-1}\left (\frac{\sqrt{1-2 x}}{\sqrt{7} \sqrt{5 x+3}}\right )}{2324168} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^4*(3 + 5*x)^(3/2)),x]

[Out]

((7*(-283149136 - 723664682*x + 622325745*x^2 + 2925598635*x^3 + 1979479350*x^4))/(Sqrt[1 - 2*x]*(2 + 3*x)^3*S
qrt[3 + 5*x]) + 251629785*Sqrt[7]*ArcTan[Sqrt[1 - 2*x]/(Sqrt[7]*Sqrt[3 + 5*x])])/2324168

________________________________________________________________________________________

Maple [B]  time = 0.017, size = 305, normalized size = 1.8 \begin{align*} -{\frac{1}{4648336\, \left ( 2+3\,x \right ) ^{3} \left ( 2\,x-1 \right ) }\sqrt{1-2\,x} \left ( 67940041950\,\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}-11574970110\,\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}-25162978500\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) x+8712560430\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}-6039114840\,\sqrt{7}\arctan \left ( 1/14\,{\frac{ \left ( 37\,x+20 \right ) \sqrt{7}}{\sqrt{-10\,{x}^{2}-x+3}}} \right ) -10131305548\,x\sqrt{-10\,{x}^{2}-x+3}-3964087904\,\sqrt{-10\,{x}^{2}-x+3} \right ){\frac{1}{\sqrt{-10\,{x}^{2}-x+3}}}{\frac{1}{\sqrt{3+5\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(3/2),x)

[Out]

-1/4648336*(1-2*x)^(1/2)*(67940041950*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*7^(1/2)*x^5+142674088
095*7^(1/2)*arctan(1/14*(37*x+20)*7^(1/2)/(-10*x^2-x+3)^(1/2))*x^4+83792718405*7^(1/2)*arctan(1/14*(37*x+20)*7
^(1/2)/(-10*x^2-x+3)^(1/2))*x^3+27712710900*x^4*(-10*x^2-x+3)^(1/2)-11574970110*7^(1/2)*arctan(1/14*(37*x+20)*
7^(1/2)/(-10*x^2-x+3)^(1/2))*x^2+40958380890*x^3*(-10*x^2-x+3)^(1/2)-25162978500*7^(1/2)*arctan(1/14*(37*x+20)
*7^(1/2)/(-10*x^2-x+3)^(1/2))*x+8712560430*x^2*(-10*x^2-x+3)^(1/2)-6039114840*7^(1/2)*arctan(1/14*(37*x+20)*7^
(1/2)/(-10*x^2-x+3)^(1/2))-10131305548*x*(-10*x^2-x+3)^(1/2)-3964087904*(-10*x^2-x+3)^(1/2))/(2+3*x)^3/(2*x-1)
/(-10*x^2-x+3)^(1/2)/(3+5*x)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 4.09718, size = 285, normalized size = 1.72 \begin{align*} -\frac{2079585}{38416} \, \sqrt{7} \arcsin \left (\frac{37 \, x}{11 \,{\left | 3 \, x + 2 \right |}} + \frac{20}{11 \,{\left | 3 \, x + 2 \right |}}\right ) + \frac{36657025 \, x}{166012 \, \sqrt{-10 \, x^{2} - x + 3}} - \frac{38272595}{332024 \, \sqrt{-10 \, x^{2} - x + 3}} + \frac{1}{7 \,{\left (27 \, \sqrt{-10 \, x^{2} - x + 3} x^{3} + 54 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 36 \, \sqrt{-10 \, x^{2} - x + 3} x + 8 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{37}{28 \,{\left (9 \, \sqrt{-10 \, x^{2} - x + 3} x^{2} + 12 \, \sqrt{-10 \, x^{2} - x + 3} x + 4 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} + \frac{6525}{392 \,{\left (3 \, \sqrt{-10 \, x^{2} - x + 3} x + 2 \, \sqrt{-10 \, x^{2} - x + 3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

-2079585/38416*sqrt(7)*arcsin(37/11*x/abs(3*x + 2) + 20/11/abs(3*x + 2)) + 36657025/166012*x/sqrt(-10*x^2 - x
+ 3) - 38272595/332024/sqrt(-10*x^2 - x + 3) + 1/7/(27*sqrt(-10*x^2 - x + 3)*x^3 + 54*sqrt(-10*x^2 - x + 3)*x^
2 + 36*sqrt(-10*x^2 - x + 3)*x + 8*sqrt(-10*x^2 - x + 3)) + 37/28/(9*sqrt(-10*x^2 - x + 3)*x^2 + 12*sqrt(-10*x
^2 - x + 3)*x + 4*sqrt(-10*x^2 - x + 3)) + 6525/392/(3*sqrt(-10*x^2 - x + 3)*x + 2*sqrt(-10*x^2 - x + 3))

________________________________________________________________________________________

Fricas [A]  time = 1.574, size = 437, normalized size = 2.63 \begin{align*} \frac{251629785 \, \sqrt{7}{\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\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 (1979479350 \, x^{4} + 2925598635 \, x^{3} + 622325745 \, x^{2} - 723664682 \, x - 283149136\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{4648336 \,{\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

1/4648336*(251629785*sqrt(7)*(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)*arctan(1/14*sqrt(7)*(37*x + 2
0)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3)) - 14*(1979479350*x^4 + 2925598635*x^3 + 622325745*x^2 - 7236
64682*x - 283149136)*sqrt(5*x + 3)*sqrt(-2*x + 1))/(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)

________________________________________________________________________________________

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)**4/(3+5*x)**(3/2),x)

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Giac [B]  time = 4.70983, size = 544, normalized size = 3.28 \begin{align*} -\frac{415917}{76832} \, \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{625}{242} \, \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 )} - \frac{64 \, \sqrt{5} \sqrt{5 \, x + 3} \sqrt{-10 \, x + 5}}{1452605 \,{\left (2 \, x - 1\right )}} - \frac{297 \,{\left (37841 \, \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} + 16959040 \, \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} + 2009470400 \, \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 )}}{9604 \,{\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 )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(3/2)/(2+3*x)^4/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

-415917/76832*sqrt(70)*sqrt(10)*(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)))) - 625/242*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sq
rt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))) - 64/1452605*sqrt(5)*sqrt(5*x +
3)*sqrt(-10*x + 5)/(2*x - 1) - 297/9604*(37841*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 + 16959040*sqrt(10)*((sqrt(2)*sqrt(-10*x + 5) - sqrt(2
2))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^3 + 2009470400*sqrt(10)*((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22))))/(((sqrt(2)*sq
rt(-10*x + 5) - sqrt(22))/sqrt(5*x + 3) - 4*sqrt(5*x + 3)/(sqrt(2)*sqrt(-10*x + 5) - sqrt(22)))^2 + 280)^3