\(\int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\) [2736]

Optimal result

Integrand size = 28, antiderivative size = 191 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {388 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}+\frac {18068 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^{3/2}}+\frac {1255552 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 \sqrt {2+3 x}}-\frac {1255552 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5145}-\frac {37768 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5145} \]

[Out]

-1255552/15435*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-37768/15435*EllipticF(1/7*21^(1/
2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+2/3*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+388/105*(1-2*x)^(1/2)
*(3+5*x)^(1/2)/(2+3*x)^(5/2)+18068/735*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+1255552/5145*(1-2*x)^(1/2)*(3
+5*x)^(1/2)/(2+3*x)^(1/2)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {100, 157, 164, 114, 120} \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=-\frac {37768 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5145}-\frac {1255552 \sqrt {\frac {11}{3}} E\left (\arcsin \left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5145}+\frac {1255552 \sqrt {1-2 x} \sqrt {5 x+3}}{5145 \sqrt {3 x+2}}+\frac {18068 \sqrt {1-2 x} \sqrt {5 x+3}}{735 (3 x+2)^{3/2}}+\frac {388 \sqrt {1-2 x} \sqrt {5 x+3}}{105 (3 x+2)^{5/2}}+\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{3 (3 x+2)^{7/2}} \]

[In]

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

[Out]

(2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3*(2 + 3*x)^(7/2)) + (388*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(105*(2 + 3*x)^(5/2))
+ (18068*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(735*(2 + 3*x)^(3/2)) + (1255552*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5145*Sqrt
[2 + 3*x]) - (1255552*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5145 - (37768*Sqrt[11/3]*E
llipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/5145

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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 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*} \text {integral}& = \frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {2}{21} \int \frac {119-161 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx \\ & = \frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {388 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}+\frac {4}{735} \int \frac {\frac {18039}{2}-10185 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx \\ & = \frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {388 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}+\frac {18068 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^{3/2}}+\frac {8 \int \frac {391209-\frac {474285 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{15435} \\ & = \frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {388 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}+\frac {18068 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^{3/2}}+\frac {1255552 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 \sqrt {2+3 x}}+\frac {16 \int \frac {\frac {20865495}{4}+8239560 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{108045} \\ & = \frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {388 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}+\frac {18068 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^{3/2}}+\frac {1255552 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 \sqrt {2+3 x}}+\frac {207724 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5145}+\frac {1255552 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{5145} \\ & = \frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{3 (2+3 x)^{7/2}}+\frac {388 \sqrt {1-2 x} \sqrt {3+5 x}}{105 (2+3 x)^{5/2}}+\frac {18068 \sqrt {1-2 x} \sqrt {3+5 x}}{735 (2+3 x)^{3/2}}+\frac {1255552 \sqrt {1-2 x} \sqrt {3+5 x}}{5145 \sqrt {2+3 x}}-\frac {1255552 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5145}-\frac {37768 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5145} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 7.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.53 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (5295887+23387310 x+34469046 x^2+16949952 x^3\right )}{4 (2+3 x)^{7/2}}+i \sqrt {33} \left (156944 E\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right )|-\frac {2}{33}\right )-161665 \operatorname {EllipticF}\left (i \text {arcsinh}\left (\sqrt {9+15 x}\right ),-\frac {2}{33}\right )\right )\right )}{15435} \]

[In]

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

[Out]

(8*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(5295887 + 23387310*x + 34469046*x^2 + 16949952*x^3))/(4*(2 + 3*x)^(7/2)) +
I*Sqrt[33]*(156944*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33] - 161665*EllipticF[I*ArcSinh[Sqrt[9 + 15*x]],
-2/33])))/15435

Maple [A] (verified)

Time = 1.29 (sec) , antiderivative size = 267, normalized size of antiderivative = 1.40

method	result	size
elliptic	\(\frac {\sqrt {-\left (-1+2 x \right ) \left (3+5 x \right ) \left (2+3 x \right )}\, \left (\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{243 \left (\frac {2}{3}+x \right )^{4}}+\frac {388 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{2835 \left (\frac {2}{3}+x \right )^{3}}+\frac {18068 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6615 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {2511104}{1029} x^{2}-\frac {1255552}{5145} x +\frac {1255552}{1715}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {1589752 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{108045 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {2511104 \sqrt {10+15 x}\, \sqrt {21-42 x}\, \sqrt {-15 x -9}\, \left (-\frac {7 E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{6}+\frac {F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )}{2}\right )}{108045 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(267\)
default	\(-\frac {2 \left (16462116 \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-16949952 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-33899904 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}[phone] \sqrt {5}\, \sqrt {7}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}-22599936 \sqrt {5}\, \sqrt {7}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right ) x \sqrt {2+3 x}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}+4877664 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, F\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-5022208 \sqrt {5}\, \sqrt {2+3 x}\, \sqrt {7}\, \sqrt {1-2 x}\, \sqrt {-3-5 x}\, E\left (\sqrt {10+15 x}, \frac {\sqrt {70}}{35}\right )-508498560 x^{5}-1084921236 x^{4}-652476870 x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{15435 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {7}{2}}}\)	\(409\)

[In]

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

[Out]

(-(-1+2*x)*(3+5*x)*(2+3*x))^(1/2)/(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)*(2/243*(-30*x^3-23*x^2+7*x+6)^(1/2
)/(2/3+x)^4+388/2835*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^3+18068/6615*(-30*x^3-23*x^2+7*x+6)^(1/2)/(2/3+x)^2+
1255552/15435*(-30*x^2-3*x+9)/((2/3+x)*(-30*x^2-3*x+9))^(1/2)+1589752/108045*(10+15*x)^(1/2)*(21-42*x)^(1/2)*(
-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))+2511104/108045*(10+15*x)^
(1/2)*(21-42*x)^(1/2)*(-15*x-9)^(1/2)/(-30*x^3-23*x^2+7*x+6)^(1/2)*(-7/6*EllipticE((10+15*x)^(1/2),1/35*70^(1/
2))+1/2*EllipticF((10+15*x)^(1/2),1/35*70^(1/2))))

Fricas [C] (verification not implemented)

Result contains higher order function than in optimal. Order 9 vs. order 4.

Time = 0.07 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.67 \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\frac {2 \, {\left (135 \, {\left (16949952 \, x^{3} + 34469046 \, x^{2} + 23387310 \, x + 5295887\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1} - 10665286 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + 28249920 \, \sqrt {-30} {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right )\right )}}{694575 \, {\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )}} \]

[In]

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

[Out]

2/694575*(135*(16949952*x^3 + 34469046*x^2 + 23387310*x + 5295887)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)
- 10665286*sqrt(-30)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*weierstrassPInverse(1159/675, 38998/91125, x + 2
3/90) + 28249920*sqrt(-30)*(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*weierstrassZeta(1159/675, 38998/91125, wei
erstrassPInverse(1159/675, 38998/91125, x + 23/90)))/(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)

Sympy [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\text {Timed out} \]

[In]

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

[Out]

Timed out

Maxima [F]

\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [F]

\[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\int { \frac {{\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{\sqrt {5 \, x + 3} {\left (3 \, x + 2\right )}^{\frac {9}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(1-2 x)^{3/2}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx=\int \frac {{\left (1-2\,x\right )}^{3/2}}{{\left (3\,x+2\right )}^{9/2}\,\sqrt {5\,x+3}} \,d x \]

[In]

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

[Out]

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