3.2411 \(\int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=216 \[ -\frac{1}{30} (3 x+2)^2 (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac{526103 (5 x+3)^{5/2} (1-2 x)^{7/2}}{768000}-\frac{5787133 (5 x+3)^{3/2} (1-2 x)^{7/2}}{3072000}-\frac{(5 x+3)^{7/2} (170940 x+245011) (1-2 x)^{7/2}}{672000}-\frac{63658463 \sqrt{5 x+3} (1-2 x)^{7/2}}{16384000}+\frac{700243093 \sqrt{5 x+3} (1-2 x)^{5/2}}{491520000}+\frac{7702674023 \sqrt{5 x+3} (1-2 x)^{3/2}}{1966080000}+\frac{84729414253 \sqrt{5 x+3} \sqrt{1-2 x}}{6553600000}+\frac{932023556783 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6553600000 \sqrt{10}} \]

[Out]

(84729414253*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6553600000 + (7702674023*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1966080000 +
(700243093*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/491520000 - (63658463*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/16384000 - (57
87133*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/3072000 - (526103*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/768000 - ((1 - 2*x)^
(7/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/30 - ((1 - 2*x)^(7/2)*(3 + 5*x)^(7/2)*(245011 + 170940*x))/672000 + (932023
556783*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6553600000*Sqrt[10])

________________________________________________________________________________________

Rubi [A]  time = 0.0758393, antiderivative size = 216, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {100, 147, 50, 54, 216} \[ -\frac{1}{30} (3 x+2)^2 (5 x+3)^{7/2} (1-2 x)^{7/2}-\frac{526103 (5 x+3)^{5/2} (1-2 x)^{7/2}}{768000}-\frac{5787133 (5 x+3)^{3/2} (1-2 x)^{7/2}}{3072000}-\frac{(5 x+3)^{7/2} (170940 x+245011) (1-2 x)^{7/2}}{672000}-\frac{63658463 \sqrt{5 x+3} (1-2 x)^{7/2}}{16384000}+\frac{700243093 \sqrt{5 x+3} (1-2 x)^{5/2}}{491520000}+\frac{7702674023 \sqrt{5 x+3} (1-2 x)^{3/2}}{1966080000}+\frac{84729414253 \sqrt{5 x+3} \sqrt{1-2 x}}{6553600000}+\frac{932023556783 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{6553600000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(84729414253*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/6553600000 + (7702674023*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/1966080000 +
(700243093*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/491520000 - (63658463*(1 - 2*x)^(7/2)*Sqrt[3 + 5*x])/16384000 - (57
87133*(1 - 2*x)^(7/2)*(3 + 5*x)^(3/2))/3072000 - (526103*(1 - 2*x)^(7/2)*(3 + 5*x)^(5/2))/768000 - ((1 - 2*x)^
(7/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/30 - ((1 - 2*x)^(7/2)*(3 + 5*x)^(7/2)*(245011 + 170940*x))/672000 + (932023
556783*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(6553600000*Sqrt[10])

Rule 100

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))/(d*f*(m + n + p + 1)), x] + Dist[1/(d*f*(m + n + p + 1)), I
nt[(a + b*x)^(m - 2)*(c + d*x)^n*(e + f*x)^p*Simp[a^2*d*f*(m + n + p + 1) - b*(b*c*e*(m - 1) + a*(d*e*(n + 1)
+ c*f*(p + 1))) + b*(a*d*f*(2*m + n + p) - b*(d*e*(m + n) + c*f*(m + p)))*x, x], x], x] /; FreeQ[{a, b, c, d,
e, f, n, p}, x] && GtQ[m, 1] && NeQ[m + n + p + 1, 0] && IntegerQ[m]

Rule 147

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_))*((g_.) + (h_.)*(x_)), x_Symbol]
:> -Simp[((a*d*f*h*(n + 2) + b*c*f*h*(m + 2) - b*d*(f*g + e*h)*(m + n + 3) - b*d*f*h*(m + n + 2)*x)*(a + b*x)^
(m + 1)*(c + d*x)^(n + 1))/(b^2*d^2*(m + n + 2)*(m + n + 3)), x] + Dist[(a^2*d^2*f*h*(n + 1)*(n + 2) + a*b*d*(
n + 1)*(2*c*f*h*(m + 1) - d*(f*g + e*h)*(m + n + 3)) + b^2*(c^2*f*h*(m + 1)*(m + 2) - c*d*(f*g + e*h)*(m + 1)*
(m + n + 3) + d^2*e*g*(m + n + 2)*(m + n + 3)))/(b^2*d^2*(m + n + 2)*(m + n + 3)), Int[(a + b*x)^m*(c + d*x)^n
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && NeQ[m + n + 2, 0] && NeQ[m + n + 3, 0]

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 54

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 216

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[(Rt[-b, 2]*x)/Sqrt[a]]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rubi steps

\begin{align*} \int (1-2 x)^{5/2} (2+3 x)^3 (3+5 x)^{5/2} \, dx &=-\frac{1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{1}{90} \int \left (-393-\frac{1221 x}{2}\right ) (1-2 x)^{5/2} (2+3 x) (3+5 x)^{5/2} \, dx\\ &=-\frac{1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac{526103 \int (1-2 x)^{5/2} (3+5 x)^{5/2} \, dx}{64000}\\ &=-\frac{526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac{1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac{5787133 \int (1-2 x)^{5/2} (3+5 x)^{3/2} \, dx}{307200}\\ &=-\frac{5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac{526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac{1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac{63658463 \int (1-2 x)^{5/2} \sqrt{3+5 x} \, dx}{2048000}\\ &=-\frac{63658463 (1-2 x)^{7/2} \sqrt{3+5 x}}{16384000}-\frac{5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac{526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac{1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac{700243093 \int \frac{(1-2 x)^{5/2}}{\sqrt{3+5 x}} \, dx}{32768000}\\ &=\frac{700243093 (1-2 x)^{5/2} \sqrt{3+5 x}}{491520000}-\frac{63658463 (1-2 x)^{7/2} \sqrt{3+5 x}}{16384000}-\frac{5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac{526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac{1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac{7702674023 \int \frac{(1-2 x)^{3/2}}{\sqrt{3+5 x}} \, dx}{196608000}\\ &=\frac{7702674023 (1-2 x)^{3/2} \sqrt{3+5 x}}{1966080000}+\frac{700243093 (1-2 x)^{5/2} \sqrt{3+5 x}}{491520000}-\frac{63658463 (1-2 x)^{7/2} \sqrt{3+5 x}}{16384000}-\frac{5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac{526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac{1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac{84729414253 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{1310720000}\\ &=\frac{84729414253 \sqrt{1-2 x} \sqrt{3+5 x}}{6553600000}+\frac{7702674023 (1-2 x)^{3/2} \sqrt{3+5 x}}{1966080000}+\frac{700243093 (1-2 x)^{5/2} \sqrt{3+5 x}}{491520000}-\frac{63658463 (1-2 x)^{7/2} \sqrt{3+5 x}}{16384000}-\frac{5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac{526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac{1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac{932023556783 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{13107200000}\\ &=\frac{84729414253 \sqrt{1-2 x} \sqrt{3+5 x}}{6553600000}+\frac{7702674023 (1-2 x)^{3/2} \sqrt{3+5 x}}{1966080000}+\frac{700243093 (1-2 x)^{5/2} \sqrt{3+5 x}}{491520000}-\frac{63658463 (1-2 x)^{7/2} \sqrt{3+5 x}}{16384000}-\frac{5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac{526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac{1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac{932023556783 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{6553600000 \sqrt{5}}\\ &=\frac{84729414253 \sqrt{1-2 x} \sqrt{3+5 x}}{6553600000}+\frac{7702674023 (1-2 x)^{3/2} \sqrt{3+5 x}}{1966080000}+\frac{700243093 (1-2 x)^{5/2} \sqrt{3+5 x}}{491520000}-\frac{63658463 (1-2 x)^{7/2} \sqrt{3+5 x}}{16384000}-\frac{5787133 (1-2 x)^{7/2} (3+5 x)^{3/2}}{3072000}-\frac{526103 (1-2 x)^{7/2} (3+5 x)^{5/2}}{768000}-\frac{1}{30} (1-2 x)^{7/2} (2+3 x)^2 (3+5 x)^{7/2}-\frac{(1-2 x)^{7/2} (3+5 x)^{7/2} (245011+170940 x)}{672000}+\frac{932023556783 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{6553600000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.213425, size = 99, normalized size = 0.46 \[ -\frac{10 \sqrt{5 x+3} \left (82575360000000 x^9+163602432000000 x^8+16806297600000 x^7-152280832000000 x^6-74172819968000 x^5+48825346630400 x^4+38603789187520 x^3-3650664293320 x^2-9390934073894 x+1496712721437\right )[phone] \sqrt{10-20 x} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{1376256000000 \sqrt{1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(10*Sqrt[3 + 5*x]*(1496712721437 - 9390934073894*x - 3650664293320*x^2 + 38603789187520*x^3 + 48825346630400*
x^4 - 74172819968000*x^5 - 152280832000000*x^6 + 16806297600000*x^7 + 163602432000000*x^8 + 82575360000000*x^9
) + 19572494692443*Sqrt[10 - 20*x]*ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]])/(1376256000000*Sqrt[1 - 2*x])

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 189, normalized size = 0.9 \begin{align*}{\frac{1}{2752512000000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 825753600000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{8}+2048901120000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{7}+1192513536000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}-926551552000000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}-1205003975680000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}-114248521536000\,{x}^{3}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{x}^{2}\sqrt{-10\,{x}^{2}-x+3}[phone]\,\sqrt{10}\arcsin \left ({\frac{20\,x}{11}}+1/11 \right ) [phone]\,x\sqrt{-10\,{x}^{2}-x+3}-29934254428740\,\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)^(5/2)*(2+3*x)^3*(3+5*x)^(5/2),x)

[Out]

1/2752512000000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(825753600000000*(-10*x^2-x+3)^(1/2)*x^8+2048901120000000*(-10*x^2
-x+3)^(1/2)*x^7+1192513536000000*(-10*x^2-x+3)^(1/2)*x^6-926551552000000*x^5*(-10*x^2-x+3)^(1/2)-1205003975680
000*x^4*(-10*x^2-x+3)^(1/2)-114248521536000*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)+19
572494692443*10^(1/2)*arcsin(20/11*x+1/11)[phone]*x*(-10*x^2-x+3)^(1/2)-29934254428740*(-10*x^2-x+3)^
(1/2))/(-10*x^2-x+3)^(1/2)

________________________________________________________________________________________

Maxima [A]  time = 3.398, size = 196, normalized size = 0.91 \begin{align*} -\frac{3}{10} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x^{2} - \frac{1047}{1600} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}} x - \frac{111537}{224000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{7}{2}} + \frac{526103}{384000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} x + \frac{526103}{7680000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{5}{2}} + \frac{63658463}{12288000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x + \frac{63658463}{245760000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{7702674023}{327680000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{932023556783}{131072000000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{7702674023}{6553600000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/10*(-10*x^2 - x + 3)^(7/2)*x^2 - 1047/1600*(-10*x^2 - x + 3)^(7/2)*x - 111537/224000*(-10*x^2 - x + 3)^(7/2
) + 526103/384000*(-10*x^2 - x + 3)^(5/2)*x + 526103/7680000*(-10*x^2 - x + 3)^(5/2) + 63658463/12288000*(-10*
x^2 - x + 3)^(3/2)*x + 63658463/245760000*(-10*x^2 - x + 3)^(3/2) + 7702674023/327680000*sqrt(-10*x^2 - x + 3)
*x - 932023556783/131072000000*sqrt(10)*arcsin(-20/11*x - 1/11) + 7702674023/6553600000*sqrt(-10*x^2 - x + 3)

________________________________________________________________________________________

Fricas [A]  time = 1.82818, size = 462, normalized size = 2.14 \begin{align*} \frac{1}{137625600000} \,{\left (41287680000000 \, x^{8} + 102445056000000 \, x^{7} + 59625676800000 \, x^{6} - 46327577600000 \, x^{5} - 60250198784000 \, x^{4} - 5712426076800 \, x^{3} + 16445681555360 \, x^{2} + 6397508631020 \, x - 1496712721437\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{932023556783}{131072000000} \, \sqrt{10} \arctan \left (\frac{\sqrt{10}{\left (20 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1}}{20 \,{\left (10 \, x^{2} + x - 3\right )}}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/137625600000*(41287680000000*x^8 + 102445056000000*x^7 + 59625676800000*x^6 - 46327577600000*x^5 - 602501987
84000*x^4 - 5712426076800*x^3 + 16445681555360*x^2 + 6397508631020*x - 1496712721437)*sqrt(5*x + 3)*sqrt(-2*x
+ 1) - 932023556783/131072000000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2
+ x - 3))

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 2.39533, size = 828, normalized size = 3.83 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/2293760000000*sqrt(5)*(2*(4*(8*(4*(16*(4*(8*(28*(160*x - 671)*(5*x + 3) + 224229)*(5*x + 3) - 10649095)*(5*x
+ 3) + 156846013)*(5*x + 3) - 5833974249)*(5*x + 3) + 32860221177)*(5*x + 3) - 191242758273)*(5*x + 3) + 4583
8370655)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 504222077205*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 99/286720
0000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(24*(140*x - 503)*(5*x + 3) + 125723)*(5*x + 3) - 12366397)*(5*x + 3) + 57
5611497)*(5*x + 3) - 3898324857)*(5*x + 3) + 26381882625)*(5*x + 3) - 12293622495)*sqrt(5*x + 3)*sqrt(-10*x +
5) + 135229847445*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 147/51200000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(
120*x - 359)*(5*x + 3) + 63769)*(5*x + 3) - 3968469)*(5*x + 3) + 33617829)*(5*x + 3) - 276044685)*(5*x + 3) +
87356115)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 960917265*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 457/1920000
000*sqrt(5)*(2*(4*(8*(4*(16*(100*x - 239)*(5*x + 3) + 27999)*(5*x + 3) - 318159)*(5*x + 3) + 3237255)*(5*x + 3
) - 2656665)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 29223315*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 409/38400
000*sqrt(5)*(2*(4*(8*(12*(80*x - 143)*(5*x + 3) + 9773)*(5*x + 3) - 136405)*(5*x + 3) + 60555)*sqrt(5*x + 3)*s
qrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) - 101/960000*sqrt(5)*(2*(4*(8*(60*x - 71)
*(5*x + 3) + 2179)*(5*x + 3) - 4125)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 45375*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5
*x + 3))) + 23/2000*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 363*sqrt(2)*arcs
in(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/50*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(2)*arcs
in(1/11*sqrt(22)*sqrt(5*x + 3)))