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

Optimal. Leaf size=113 \[ -\frac {3}{40} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3-\frac {259}{800} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (77820 x+187559)}{128000}+\frac {10866247 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{128000 \sqrt {10}} \]

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Rubi [A]  time = 0.03, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {100, 153, 147, 54, 216} \begin {gather*} -\frac {3}{40} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^3-\frac {259}{800} \sqrt {1-2 x} \sqrt {5 x+3} (3 x+2)^2-\frac {7 \sqrt {1-2 x} \sqrt {5 x+3} (77820 x+187559)}{128000}+\frac {10866247 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )}{128000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-259*Sqrt[1 - 2*x]*(2 + 3*x)^2*Sqrt[3 + 5*x])/800 - (3*Sqrt[1 - 2*x]*(2 + 3*x)^3*Sqrt[3 + 5*x])/40 - (7*Sqrt[
1 - 2*x]*Sqrt[3 + 5*x]*(187559 + 77820*x))/128000 + (10866247*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]])/(128000*Sqrt[1
0])

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 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 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 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 \frac {(2+3 x)^4}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx &=-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {1}{40} \int \frac {\left (-238-\frac {777 x}{2}\right ) (2+3 x)^2}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}+\frac {\int \frac {(2+3 x) \left (\frac {41769}{2}+\frac {136185 x}{4}\right )}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{1200}\\ &=-\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (187559+77820 x)}{128000}+\frac {10866247 \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx}{256000}\\ &=-\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (187559+77820 x)}{128000}+\frac {10866247 \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{128000 \sqrt {5}}\\ &=-\frac {259}{800} \sqrt {1-2 x} (2+3 x)^2 \sqrt {3+5 x}-\frac {3}{40} \sqrt {1-2 x} (2+3 x)^3 \sqrt {3+5 x}-\frac {7 \sqrt {1-2 x} \sqrt {3+5 x} (187559+77820 x)}{128000}+\frac {10866247 \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )}{128000 \sqrt {10}}\\ \end {align*}

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Mathematica [A]  time = 0.07, size = 83, normalized size = 0.73 \begin {gather*} -\frac {\sqrt {1-2 x} \left (30 \sqrt {2 x-1} \sqrt {5 x+3} \left (86400 x^3+297120 x^2+462540 x+518491\right )[phone] \sqrt {10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )\right )}{1280000 \sqrt {2 x-1}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/1280000*(Sqrt[1 - 2*x]*(30*Sqrt[-1 + 2*x]*Sqrt[3 + 5*x]*(518491 + 462540*x + 297120*x^2 + 86400*x^3) + 1086
6247*Sqrt[10]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]]))/Sqrt[-1 + 2*x]

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IntegrateAlgebraic [A]  time = 0.20, size = 125, normalized size = 1.11 \begin {gather*} -\frac {33 \sqrt {1-2 x} \left (\frac {41158475 (1-2 x)^3}{(5 x+3)^3}+\frac {60352110 (1-2 x)^2}{(5 x+3)^2}+\frac {31952340 (1-2 x)}{5 x+3}+6678728\right )}{128000 \sqrt {5 x+3} \left (\frac {5 (1-2 x)}{5 x+3}+2\right )^4}-\frac {10866247 \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {1-2 x}}{\sqrt {5 x+3}}\right )}{128000 \sqrt {10}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-33*Sqrt[1 - 2*x]*(6678728 + (41158475*(1 - 2*x)^3)/(3 + 5*x)^3 + (60352110*(1 - 2*x)^2)/(3 + 5*x)^2 + (31952
340*(1 - 2*x))/(3 + 5*x)))/(128000*Sqrt[3 + 5*x]*(2 + (5*(1 - 2*x))/(3 + 5*x))^4) - (10866247*ArcTan[(Sqrt[5/2
]*Sqrt[1 - 2*x])/Sqrt[3 + 5*x]])/(128000*Sqrt[10])

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fricas [A]  time = 1.03, size = 72, normalized size = 0.64 \begin {gather*} -\frac {3}{128000} \, {\left (86400 \, x^{3} + 297120 \, x^{2} + 462540 \, x + 518491\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1} - \frac {10866247}{2560000} \, \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3/128000*(86400*x^3 + 297120*x^2 + 462540*x + 518491)*sqrt(5*x + 3)*sqrt(-2*x + 1) - 10866247/2560000*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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giac [A]  time = 1.01, size = 63, normalized size = 0.56 \begin {gather*} -\frac {1}{6400000} \, \sqrt {5} {\left (6 \, {\left (12 \, {\left (8 \, {\left (180 \, x + 403\right )} {\left (5 \, x + 3\right )} + 16609\right )} {\left (5 \, x + 3\right )} + 1646339\right )} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5} - 54331235 \, \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right )\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/6400000*sqrt(5)*(6*(12*(8*(180*x + 403)*(5*x + 3) + 16609)*(5*x + 3) + 1646339)*sqrt(5*x + 3)*sqrt(-10*x +
5) - 54331235*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3)))

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maple [A]  time = 0.02, size = 104, normalized size = 0.92 \begin {gather*} \frac {\sqrt {-2 x +1}\, \sqrt {5 x +3}\, \left (-5184000 \sqrt {-10 x^{2}-x +3}\, x^{3}-17827200 \sqrt {-10 x^{2}-x +3}\, x^{2}-27752400 \sqrt {-10 x^{2}-x +3}\, x [phone] \sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right )-31109460 \sqrt {-10 x^{2}-x +3}\right )}{2560000 \sqrt {-10 x^{2}-x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2560000*(-2*x+1)^(1/2)*(5*x+3)^(1/2)*(-5184000*(-10*x^2-x+3)^(1/2)*x^3-17827200*(-10*x^2-x+3)^(1/2)*x^2+1086
6247*10^(1/2)*arcsin(20/11*x+1/11)-27752400*(-10*x^2-x+3)^(1/2)*x-31109460*(-10*x^2-x+3)^(1/2))/(-10*x^2-x+3)^
(1/2)

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maxima [A]  time = 1.23, size = 75, normalized size = 0.66 \begin {gather*} -\frac {81}{40} \, \sqrt {-10 \, x^{2} - x + 3} x^{3} - \frac {5571}{800} \, \sqrt {-10 \, x^{2} - x + 3} x^{2} - \frac {69381}{6400} \, \sqrt {-10 \, x^{2} - x + 3} x - \frac {10866247}{2560000} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) - \frac {1555473}{128000} \, \sqrt {-10 \, x^{2} - x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-81/40*sqrt(-10*x^2 - x + 3)*x^3 - 5571/800*sqrt(-10*x^2 - x + 3)*x^2 - 69381/6400*sqrt(-10*x^2 - x + 3)*x - 1
0866247/2560000*sqrt(10)*arcsin(-20/11*x - 1/11) - 1555473/128000*sqrt(-10*x^2 - x + 3)

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mupad [B]  time = 11.93, size = 708, normalized size = 6.27 \begin {gather*} \frac {10866247\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {1-2\,x}-1\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{640000}-\frac {\frac {6770247\,\left (\sqrt {1-2\,x}-1\right )}{195312500\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}-\frac {33291573\,{\left (\sqrt {1-2\,x}-1\right )}^3}{78125000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^3}-\frac {883182573\,{\left (\sqrt {1-2\,x}-1\right )}^5}{156250000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^5}+\frac {451883391\,{\left (\sqrt {1-2\,x}-1\right )}^7}{62500000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^7}-\frac {451883391\,{\left (\sqrt {1-2\,x}-1\right )}^9}{25000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^9}+\frac {883182573\,{\left (\sqrt {1-2\,x}-1\right )}^{11}}{10000000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{11}}+\frac {33291573\,{\left (\sqrt {1-2\,x}-1\right )}^{13}}{800000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{13}}-\frac {6770247\,{\left (\sqrt {1-2\,x}-1\right )}^{15}}{320000\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{15}}+\frac {49152\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^2}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {258048\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^4}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1032192\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^6}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {16147968\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^8}{390625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {258048\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {16128\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {768\,\sqrt {3}\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}}{\frac {1024\,{\left (\sqrt {1-2\,x}-1\right )}^2}{78125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^2}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^4}{15625\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^4}+\frac {1792\,{\left (\sqrt {1-2\,x}-1\right )}^6}{3125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^6}+\frac {224\,{\left (\sqrt {1-2\,x}-1\right )}^8}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^8}+\frac {448\,{\left (\sqrt {1-2\,x}-1\right )}^{10}}{125\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{10}}+\frac {112\,{\left (\sqrt {1-2\,x}-1\right )}^{12}}{25\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{12}}+\frac {16\,{\left (\sqrt {1-2\,x}-1\right )}^{14}}{5\,{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{14}}+\frac {{\left (\sqrt {1-2\,x}-1\right )}^{16}}{{\left (\sqrt {3}-\sqrt {5\,x+3}\right )}^{16}}+\frac {256}{390625}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(10866247*10^(1/2)*atan((10^(1/2)*((1 - 2*x)^(1/2) - 1))/(2*(3^(1/2) - (5*x + 3)^(1/2)))))/640000 - ((6770247*
((1 - 2*x)^(1/2) - 1))/(195312500*(3^(1/2) - (5*x + 3)^(1/2))) - (33291573*((1 - 2*x)^(1/2) - 1)^3)/(78125000*
(3^(1/2) - (5*x + 3)^(1/2))^3) - (883182573*((1 - 2*x)^(1/2) - 1)^5)/(156250000*(3^(1/2) - (5*x + 3)^(1/2))^5)
+ (451883391*((1 - 2*x)^(1/2) - 1)^7)/(62500000*(3^(1/2) - (5*x + 3)^(1/2))^7) - (451883391*((1 - 2*x)^(1/2)
- 1)^9)/(25000000*(3^(1/2) - (5*x + 3)^(1/2))^9) + (883182573*((1 - 2*x)^(1/2) - 1)^11)/(10000000*(3^(1/2) - (
5*x + 3)^(1/2))^11) + (33291573*((1 - 2*x)^(1/2) - 1)^13)/(800000*(3^(1/2) - (5*x + 3)^(1/2))^13) - (6770247*(
(1 - 2*x)^(1/2) - 1)^15)/(320000*(3^(1/2) - (5*x + 3)^(1/2))^15) + (49152*3^(1/2)*((1 - 2*x)^(1/2) - 1)^2)/(39
0625*(3^(1/2) - (5*x + 3)^(1/2))^2) + (258048*3^(1/2)*((1 - 2*x)^(1/2) - 1)^4)/(78125*(3^(1/2) - (5*x + 3)^(1/
2))^4) + (1032192*3^(1/2)*((1 - 2*x)^(1/2) - 1)^6)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^6) + (16147968*3^(1/2)*(
(1 - 2*x)^(1/2) - 1)^8)/(390625*(3^(1/2) - (5*x + 3)^(1/2))^8) + (258048*3^(1/2)*((1 - 2*x)^(1/2) - 1)^10)/(31
25*(3^(1/2) - (5*x + 3)^(1/2))^10) + (16128*3^(1/2)*((1 - 2*x)^(1/2) - 1)^12)/(125*(3^(1/2) - (5*x + 3)^(1/2))
^12) + (768*3^(1/2)*((1 - 2*x)^(1/2) - 1)^14)/(25*(3^(1/2) - (5*x + 3)^(1/2))^14))/((1024*((1 - 2*x)^(1/2) - 1
)^2)/(78125*(3^(1/2) - (5*x + 3)^(1/2))^2) + (1792*((1 - 2*x)^(1/2) - 1)^4)/(15625*(3^(1/2) - (5*x + 3)^(1/2))
^4) + (1792*((1 - 2*x)^(1/2) - 1)^6)/(3125*(3^(1/2) - (5*x + 3)^(1/2))^6) + (224*((1 - 2*x)^(1/2) - 1)^8)/(125
*(3^(1/2) - (5*x + 3)^(1/2))^8) + (448*((1 - 2*x)^(1/2) - 1)^10)/(125*(3^(1/2) - (5*x + 3)^(1/2))^10) + (112*(
(1 - 2*x)^(1/2) - 1)^12)/(25*(3^(1/2) - (5*x + 3)^(1/2))^12) + (16*((1 - 2*x)^(1/2) - 1)^14)/(5*(3^(1/2) - (5*
x + 3)^(1/2))^14) + ((1 - 2*x)^(1/2) - 1)^16/(3^(1/2) - (5*x + 3)^(1/2))^16 + 256/390625)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 x + 2\right )^{4}}{\sqrt {1 - 2 x} \sqrt {5 x + 3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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