3.2286 \(\int \sqrt{1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=201 \[ -\frac{3}{80} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{7/2}-\frac{1419 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}}{11200}-\frac{3 (1-2 x)^{3/2} (522420 x+899099) (5 x+3)^{7/2}}{1280000}-\frac{135817609 (1-2 x)^{3/2} (5 x+3)^{5/2}}{20480000}-\frac{1493993699 (1-2 x)^{3/2} (5 x+3)^{3/2}}{49152000}-\frac{16433930689 (1-2 x)^{3/2} \sqrt{5 x+3}}{131072000}+\frac{180773237579 \sqrt{1-2 x} \sqrt{5 x+3}}{1310720000}+\frac{1988505613369 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1310720000 \sqrt{10}} \]

[Out]

(180773237579*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1310720000 - (16433930689*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/131072000
- (1493993699*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/49152000 - (135817609*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/20480000
- (1419*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/11200 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^(7/2))/
80 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2)*(899099 + 522420*x))/1280000 + (1988505613369*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/(1310720000*Sqrt[10])

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Rubi [A]  time = 0.0663079, antiderivative size = 201, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231, Rules used = {100, 153, 147, 50, 54, 216} \[ -\frac{3}{80} (1-2 x)^{3/2} (3 x+2)^3 (5 x+3)^{7/2}-\frac{1419 (1-2 x)^{3/2} (3 x+2)^2 (5 x+3)^{7/2}}{11200}-\frac{3 (1-2 x)^{3/2} (522420 x+899099) (5 x+3)^{7/2}}{1280000}-\frac{135817609 (1-2 x)^{3/2} (5 x+3)^{5/2}}{20480000}-\frac{1493993699 (1-2 x)^{3/2} (5 x+3)^{3/2}}{49152000}-\frac{16433930689 (1-2 x)^{3/2} \sqrt{5 x+3}}{131072000}+\frac{180773237579 \sqrt{1-2 x} \sqrt{5 x+3}}{1310720000}+\frac{1988505613369 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )}{1310720000 \sqrt{10}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(180773237579*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/1310720000 - (16433930689*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/131072000
- (1493993699*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/49152000 - (135817609*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/20480000
- (1419*(1 - 2*x)^(3/2)*(2 + 3*x)^2*(3 + 5*x)^(7/2))/11200 - (3*(1 - 2*x)^(3/2)*(2 + 3*x)^3*(3 + 5*x)^(7/2))/
80 - (3*(1 - 2*x)^(3/2)*(3 + 5*x)^(7/2)*(899099 + 522420*x))/1280000 + (1988505613369*ArcSin[Sqrt[2/11]*Sqrt[3
+ 5*x]])/(1310720000*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 153

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 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 \sqrt{1-2 x} (2+3 x)^4 (3+5 x)^{5/2} \, dx &=-\frac{3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac{1}{80} \int \left (-452-\frac{1419 x}{2}\right ) \sqrt{1-2 x} (2+3 x)^2 (3+5 x)^{5/2} \, dx\\ &=-\frac{1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}+\frac{\int \sqrt{1-2 x} (2+3 x) (3+5 x)^{5/2} \left (\frac{176225}{2}+\frac{548541 x}{4}\right ) \, dx}{5600}\\ &=-\frac{1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac{135817609 \int \sqrt{1-2 x} (3+5 x)^{5/2} \, dx}{2560000}\\ &=-\frac{135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac{1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac{1493993699 \int \sqrt{1-2 x} (3+5 x)^{3/2} \, dx}{8192000}\\ &=-\frac{1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac{135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac{1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac{16433930689 \int \sqrt{1-2 x} \sqrt{3+5 x} \, dx}{32768000}\\ &=-\frac{16433930689 (1-2 x)^{3/2} \sqrt{3+5 x}}{131072000}-\frac{1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac{135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac{1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac{180773237579 \int \frac{\sqrt{1-2 x}}{\sqrt{3+5 x}} \, dx}{262144000}\\ &=\frac{180773237579 \sqrt{1-2 x} \sqrt{3+5 x}}{1310720000}-\frac{16433930689 (1-2 x)^{3/2} \sqrt{3+5 x}}{131072000}-\frac{1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac{135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac{1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac{1988505613369 \int \frac{1}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{2621440000}\\ &=\frac{180773237579 \sqrt{1-2 x} \sqrt{3+5 x}}{1310720000}-\frac{16433930689 (1-2 x)^{3/2} \sqrt{3+5 x}}{131072000}-\frac{1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac{135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac{1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac{1988505613369 \operatorname{Subst}\left (\int \frac{1}{\sqrt{11-2 x^2}} \, dx,x,\sqrt{3+5 x}\right )}{1310720000 \sqrt{5}}\\ &=\frac{180773237579 \sqrt{1-2 x} \sqrt{3+5 x}}{1310720000}-\frac{16433930689 (1-2 x)^{3/2} \sqrt{3+5 x}}{131072000}-\frac{1493993699 (1-2 x)^{3/2} (3+5 x)^{3/2}}{49152000}-\frac{135817609 (1-2 x)^{3/2} (3+5 x)^{5/2}}{20480000}-\frac{1419 (1-2 x)^{3/2} (2+3 x)^2 (3+5 x)^{7/2}}{11200}-\frac{3}{80} (1-2 x)^{3/2} (2+3 x)^3 (3+5 x)^{7/2}-\frac{3 (1-2 x)^{3/2} (3+5 x)^{7/2} (899099+522420 x)}{1280000}+\frac{1988505613369 \sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{3+5 x}\right )}{1310720000 \sqrt{10}}\\ \end{align*}

Mathematica [A]  time = 0.0757373, size = 85, normalized size = 0.42 \[ \frac{10 \sqrt{1-2 x} \sqrt{5 x+3} \left (6967296000000 x^7+30838579200000 x^6+57746856960000 x^5+58346097408000 x^4+32457421737600 x^3+6882844528480 x^2-3991703112140 x-5973304472091\right )-41758617880749 \sqrt{10} \sin ^{-1}\left (\sqrt{\frac{5}{11}} \sqrt{1-2 x}\right )}{275251200000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(10*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(-5973304472091 - 3991703112140*x + 6882844528480*x^2 + 32457421737600*x^3 + 5
8346097408000*x^4 + 57746856960000*x^5 + 30838579200000*x^6 + 6967296000000*x^7) - 41758617880749*Sqrt[10]*Arc
Sin[Sqrt[5/11]*Sqrt[1 - 2*x]])/275251200000

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Maple [A]  time = 0.012, size = 172, normalized size = 0.9 \begin{align*}{\frac{1}{550502400000}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 139345920000000\,\sqrt{-10\,{x}^{2}-x+3}{x}^{7}[phone]\,\sqrt{-10\,{x}^{2}-x+3}{x}^{6}+1154937139200000\,{x}^{5}\sqrt{-10\,{x}^{2}-x+3}+1166921948160000\,{x}^{4}\sqrt{-10\,{x}^{2}-x+3}[phone]\,{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 ) -79834062242800\,x\sqrt{-10\,{x}^{2}-x+3}-119466089441820\,\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((2+3*x)^4*(3+5*x)^(5/2)*(1-2*x)^(1/2),x)

[Out]

1/550502400000*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(139345920000000*(-10*x^2-x+3)^(1/2)*x^7+616771584000000*(-10*x^2-x
+3)^(1/2)*x^6+1154937139200000*x^5*(-10*x^2-x+3)^(1/2)+1166921948160000*x^4*(-10*x^2-x+3)^(1/2)[phone]
0*x^3*(-10*x^2-x+3)^(1/2)[phone]*x^2*(-10*x^2-x+3)^(1/2)[phone]*10^(1/2)*arcsin(20/11*x+1/11)
-79834062242800*x*(-10*x^2-x+3)^(1/2)-119466089441820*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^(1/2)

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Maxima [A]  time = 2.50797, size = 186, normalized size = 0.93 \begin{align*} -\frac{405}{16} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{5} - \frac{49059}{448} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{4} - \frac{739881}{3584} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{3} - \frac{80346831}{358400} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x^{2} - \frac{4513921183}{28672000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} x - \frac{26326737569}{344064000} \,{\left (-10 \, x^{2} - x + 3\right )}^{\frac{3}{2}} + \frac{16433930689}{65536000} \, \sqrt{-10 \, x^{2} - x + 3} x - \frac{1988505613369}{26214400000} \, \sqrt{10} \arcsin \left (-\frac{20}{11} \, x - \frac{1}{11}\right ) + \frac{16433930689}{1310720000} \, \sqrt{-10 \, x^{2} - x + 3} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-405/16*(-10*x^2 - x + 3)^(3/2)*x^5 - 49059/448*(-10*x^2 - x + 3)^(3/2)*x^4 - 739881/3584*(-10*x^2 - x + 3)^(3
/2)*x^3 - 80346831/358400*(-10*x^2 - x + 3)^(3/2)*x^2 - 4513921183/28672000*(-10*x^2 - x + 3)^(3/2)*x - 263267
37569/344064000*(-10*x^2 - x + 3)^(3/2) + 16433930689/65536000*sqrt(-10*x^2 - x + 3)*x - 1988505613369/2621440
0000*sqrt(10)*arcsin(-20/11*x - 1/11) + 16433930689/1310720000*sqrt(-10*x^2 - x + 3)

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Fricas [A]  time = 1.78357, size = 429, normalized size = 2.13 \begin{align*} \frac{1}{27525120000} \,{\left (6967296000000 \, x^{7} + 30838579200000 \, x^{6} + 57746856960000 \, x^{5} + 58346097408000 \, x^{4} + 32457421737600 \, x^{3} + 6882844528480 \, x^{2} - 3991703112140 \, x - 5973304472091\right )} \sqrt{5 \, x + 3} \sqrt{-2 \, x + 1} - \frac{1988505613369}{26214400000} \, \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((2+3*x)^4*(3+5*x)^(5/2)*(1-2*x)^(1/2),x, algorithm="fricas")

[Out]

1/27525120000*(6967296000000*x^7 + 30838579200000*x^6 + 57746856960000*x^5 + 58346097408000*x^4 + 324574217376
00*x^3 + 6882844528480*x^2 - 3991703112140*x - 5973304472091)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 1988505613369/262
14400000*sqrt(10)*arctan(1/20*sqrt(10)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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

[Out]

Timed out

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Giac [B]  time = 2.29659, size = 682, normalized size = 3.39 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

27/2293760000000*sqrt(5)*(2*(4*(8*(4*(16*(20*(24*(140*x - 503)*(5*x + 3) + 125723)*(5*x + 3) - 12366397)*(5*x
+ 3) + 575611497)*(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))) + 261/35840000000*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))) + 42
03/2560000000*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))) +
451/8000000*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)*sqrt(-10*x + 5) - 666105*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 653/240000*sqrt(5)*(2*(4*(8*(6
0*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(2
2)*sqrt(5*x + 3))) + 7/125*sqrt(5)*(2*(4*(40*x - 23)*(5*x + 3) + 33)*sqrt(5*x + 3)*sqrt(-10*x + 5) - 363*sqrt(
2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))) + 9/25*sqrt(5)*(2*(20*x + 1)*sqrt(5*x + 3)*sqrt(-10*x + 5) + 121*sqrt(
2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))