3.28.61 \(\int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx\) [2761]

Optimal. Leaf size=191 \[ -\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{7 (2+3 x)^{5/2}}+\frac {524 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {36052 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 \sqrt {2+3 x}}-\frac {36052 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}-\frac {1048 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323} \]

[Out]

-36052/3969*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1048/3969*EllipticF(1/7*21^(1/2)*(1
-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/21*(1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+2/7*(1-2*x)^(3/2)*(3+5*x)
^(1/2)/(2+3*x)^(5/2)+524/189*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+36052/1323*(1-2*x)^(1/2)*(3+5*x)^(1/2)/
(2+3*x)^(1/2)

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Rubi [A]
time = 0.04, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {99, 155, 157, 164, 114, 120} \begin {gather*} -\frac {1048 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}-\frac {36052 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}-\frac {2 \sqrt {5 x+3} (1-2 x)^{5/2}}{21 (3 x+2)^{7/2}}+\frac {2 \sqrt {5 x+3} (1-2 x)^{3/2}}{7 (3 x+2)^{5/2}}+\frac {36052 \sqrt {5 x+3} \sqrt {1-2 x}}{1323 \sqrt {3 x+2}}+\frac {524 \sqrt {5 x+3} \sqrt {1-2 x}}{189 (3 x+2)^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^(9/2),x]

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(21*(2 + 3*x)^(7/2)) + (2*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(7*(2 + 3*x)^(5/2)
) + (524*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(189*(2 + 3*x)^(3/2)) + (36052*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1323*Sqrt[2
+ 3*x]) - (36052*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1323 - (1048*Sqrt[11/3]*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1323

Rule 99

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 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*Rt[-(b
*e - a*f)/d, 2]*EllipticE[ArcSin[Sqrt[a + b*x]/Rt[-(b*c - a*d)/d, 2]], f*((b*c - a*d)/(d*(b*e - a*f)))], x] /;
FreeQ[{a, b, c, d, e, f}, x] && GtQ[b/(b*c - a*d), 0] && GtQ[b/(b*e - a*f), 0] &&  !LtQ[-(b*c - a*d)/d, 0] &&
!(SimplerQ[c + d*x, a + b*x] && GtQ[-d/(b*c - a*d), 0] && GtQ[d/(d*e - c*f), 0] &&  !LtQ[(b*c - a*d)/b, 0])

Rule 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*EllipticF[ArcSin[Sqrt[a + b*x]/(Rt[-b/d, 2]*Sqrt[(b*c - a*d)/b])], f*((b*c - a*d)
/(d*(b*e - a*f)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-b/d] &&  !(SimplerQ[c + d*x, a + b*x] && GtQ[(d*e - c*f)/d, 0] && GtQ[-d/b, 0]) &&  !(SimplerQ[c + d*x, a
+ b*x] && GtQ[((-b)*e + a*f)/f, 0] && GtQ[-f/b, 0]) &&  !(SimplerQ[e + f*x, a + b*x] && GtQ[((-d)*e + c*f)/f,
0] && GtQ[((-b)*e + a*f)/f, 0] && (PosQ[-f/d] || PosQ[-f/b]))

Rule 155

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*((e + f*x)^(p + 1)/(b*(b*e - a*f)*(m + 1))), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 0] && IntegersQ[2*m, 2*n, 2*p]

Rule 157

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 164

Int[((g_.) + (h_.)*(x_))/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol]
:> Dist[h/f, Int[Sqrt[e + f*x]/(Sqrt[a + b*x]*Sqrt[c + d*x]), x], x] + Dist[(f*g - e*h)/f, Int[1/(Sqrt[a + b*
x]*Sqrt[c + d*x]*Sqrt[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && SimplerQ[a + b*x, e + f*x] &&
SimplerQ[c + d*x, e + f*x]

Rubi steps

\begin {align*} \int \frac {(1-2 x)^{5/2} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {\left (-\frac {25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{7 (2+3 x)^{5/2}}-\frac {4}{315} \int \frac {\left (-\frac {705}{2}-\frac {75 x}{2}\right ) \sqrt {1-2 x}}{(2+3 x)^{5/2} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{7 (2+3 x)^{5/2}}+\frac {524 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {8 \int \frac {\frac {31665}{4}-5025 x}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{2835}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{7 (2+3 x)^{5/2}}+\frac {524 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {36052 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 \sqrt {2+3 x}}+\frac {16 \int \frac {106800+\frac {675975 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{19845}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{7 (2+3 x)^{5/2}}+\frac {524 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {36052 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 \sqrt {2+3 x}}+\frac {5764 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1323}+\frac {36052 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1323}\\ &=-\frac {2 (1-2 x)^{5/2} \sqrt {3+5 x}}{21 (2+3 x)^{7/2}}+\frac {2 (1-2 x)^{3/2} \sqrt {3+5 x}}{7 (2+3 x)^{5/2}}+\frac {524 \sqrt {1-2 x} \sqrt {3+5 x}}{189 (2+3 x)^{3/2}}+\frac {36052 \sqrt {1-2 x} \sqrt {3+5 x}}{1323 \sqrt {2+3 x}}-\frac {36052 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}-\frac {1048 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1323}\\ \end {align*}

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Mathematica [A]
time = 8.16, size = 106, normalized size = 0.55 \begin {gather*} \frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (151859+671007 x+988524 x^2+486702 x^3\right )}{2 (2+3 x)^{7/2}}+\sqrt {2} \left (9013 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-4690 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{3969} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(2 + 3*x)^(9/2),x]

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(151859 + 671007*x + 988524*x^2 + 486702*x^3))/(2*(2 + 3*x)^(7/2)) + Sqrt[2
]*(9013*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 4690*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]],
-33/2])))/3969

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs. \(2(139)=278\).
time = 0.15, size = 401, normalized size = 2.10

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {26 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{729 \left (\frac {2}{3}+x \right )^{3}}+\frac {160 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{567 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {360520}{1323} x^{2}-\frac {36052}{1323} x +\frac {36052}{441}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {113920 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{27783 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {180260 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{27783 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2187 \left (\frac {2}{3}+x \right )^{4}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(273\)
default	\(-\frac {2 \left (233442 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-486702 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+466884 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-973404 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+311256 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-648936 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+69168 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-144208 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-14601060 x^{5}-31115826 x^{4}-18715464 x^{3}+2327925 x^{2}+5583486 x +1366731\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{3969 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\)	\(401\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2),x,method=_RETURNVERBOSE)

[Out]

-2/3969*(233442*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(
1/2)-486702*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)
+466884*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-973
404*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+311256*
2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-648936*2^(1/2
)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+69168*2^(1/2)*(2+3*
x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-144208*2^(1/2)*(2+3*x)^(1/2)
*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))-14601060*x^5-31115826*x^4-18715464*x
^3+2327925*x^2+5583486*x+1366731)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(7/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2),x, algorithm="maxima")

[Out]

integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2), x)

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Fricas [A]
time = 0.37, size = 60, normalized size = 0.31 \begin {gather*} \frac {2 \, {\left (486702 \, x^{3} + 988524 \, x^{2} + 671007 \, x + 151859\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1323 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2),x, algorithm="fricas")

[Out]

2/1323*(486702*x^3 + 988524*x^2 + 671007*x + 151859)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(81*x^4 + 216*
x^3 + 216*x^2 + 96*x + 16)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)**(5/2)*(3+5*x)**(1/2)/(2+3*x)**(9/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3877 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2),x, algorithm="giac")

[Out]

integrate(sqrt(5*x + 3)*(-2*x + 1)^(5/2)/(3*x + 2)^(9/2), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,\sqrt {5\,x+3}}{{\left (3\,x+2\right )}^{9/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(1/2))/(3*x + 2)^(9/2),x)

[Out]

int(((1 - 2*x)^(5/2)*(5*x + 3)^(1/2))/(3*x + 2)^(9/2), x)

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