3.30.34 \(\int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\) [2934]

Optimal. Leaf size=222 \[ \frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {46585232 \sqrt {1-2 x} \sqrt {2+3 x}}{290521 \sqrt {3+5 x}}+\frac {46585232 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055}+\frac {1400888 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055} \]

[Out]

46585232/1452605*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+1400888/1452605*EllipticF(1/7*
21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/77/(2+3*x)^(5/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)+138/2695*(1-2*
x)^(1/2)/(2+3*x)^(5/2)/(3+5*x)^(1/2)+14928/18865*(1-2*x)^(1/2)/(2+3*x)^(3/2)/(3+5*x)^(1/2)+2101332/132055*(1-2
*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)-46585232/290521*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.06, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {106, 157, 164, 114, 120} \begin {gather*} \frac {1400888 \sqrt {\frac {3}{11}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055}+\frac {46585232 \sqrt {\frac {3}{11}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055}-\frac {46585232 \sqrt {1-2 x} \sqrt {3 x+2}}{290521 \sqrt {5 x+3}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {14928 \sqrt {1-2 x}}{18865 (3 x+2)^{3/2} \sqrt {5 x+3}}+\frac {138 \sqrt {1-2 x}}{2695 (3 x+2)^{5/2} \sqrt {5 x+3}}+\frac {4}{77 \sqrt {1-2 x} (3 x+2)^{5/2} \sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)),x]

[Out]

4/(77*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (138*Sqrt[1 - 2*x])/(2695*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])
+ (14928*Sqrt[1 - 2*x])/(18865*(2 + 3*x)^(3/2)*Sqrt[3 + 5*x]) + (2101332*Sqrt[1 - 2*x])/(132055*Sqrt[2 + 3*x]*
Sqrt[3 + 5*x]) - (46585232*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(290521*Sqrt[3 + 5*x]) + (46585232*Sqrt[3/11]*Elliptic
E[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/132055 + (1400888*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*
x]], 35/33])/132055

Rule 106

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)/((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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
IntegersQ[2*m, 2*n, 2*p]

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*} \int \frac {1}{(1-2 x)^{3/2} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {2}{77} \int \frac {-\frac {163}{2}-105 x}{\sqrt {1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}-\frac {4 \int \frac {-1291+\frac {1725 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{2695}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}-\frac {8 \int \frac {-\frac {301413}{4}+83970 x}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{56595}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {16 \int \frac {-\frac {12741465}{4}+\frac {7879995 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} (3+5 x)^{3/2}} \, dx}{396165}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {46585232 \sqrt {1-2 x} \sqrt {2+3 x}}{290521 \sqrt {3+5 x}}+\frac {32 \int \frac {-\frac {331786305}{8}-\frac {131020965 x}{2}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{4357815}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {46585232 \sqrt {1-2 x} \sqrt {2+3 x}}{290521 \sqrt {3+5 x}}-\frac {2101332 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{132055}-\frac {139755696 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{1452605}\\ &=\frac {4}{77 \sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {138 \sqrt {1-2 x}}{2695 (2+3 x)^{5/2} \sqrt {3+5 x}}+\frac {14928 \sqrt {1-2 x}}{18865 (2+3 x)^{3/2} \sqrt {3+5 x}}+\frac {2101332 \sqrt {1-2 x}}{132055 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {46585232 \sqrt {1-2 x} \sqrt {2+3 x}}{290521 \sqrt {3+5 x}}+\frac {46585232 \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055}+\frac {1400888 \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{132055}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 7.69, size = 109, normalized size = 0.49 \begin {gather*} \frac {2 \left (\frac {-884250959-2283681406 x+1919527182 x^2+9225477612 x^3+6289006320 x^4}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}}-2 \sqrt {2} \left (11646308 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-5867645 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{1452605} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((1 - 2*x)^(3/2)*(2 + 3*x)^(7/2)*(3 + 5*x)^(3/2)),x]

[Out]

(2*((-884250959 - 2283681406*x + 1919527182*x^2 + 9225477612*x^3 + 6289006320*x^4)/(Sqrt[1 - 2*x]*(2 + 3*x)^(5
/2)*Sqrt[3 + 5*x]) - 2*Sqrt[2]*(11646308*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5867645*Elliptic
F[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/1452605

________________________________________________________________________________________

Maple [A]
time = 0.12, size = 308, normalized size = 1.39

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {2 \left (-20-30 x \right ) \left (-\frac {7503029}{2905210}+\frac {1500641 x}{290521}\right )}{\sqrt {\left (x^{2}+\frac {1}{10} x -\frac {3}{10}\right ) \left (-20-30 x \right )}}-\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{245 \left (\frac {2}{3}+x \right )^{3}}-\frac {666 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1715 \left (\frac {2}{3}+x \right )^{2}}-\frac {260982 \left (-30 x^{2}-3 x +9\right )}{12005 \sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}-\frac {29492116 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{2033647 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {46585232 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{2033647 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(277\)
default	\(-\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \left (209633544 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-104015934 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-138687912 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-46229304 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )[phone] x^{4}[phone] x^{3}[phone] x^{2}-2283681406 x -884250959\right )}{1452605 \left (2+3 x \right )^{\frac {5}{2}} \left (10 x^{2}+x -3\right )}\)	\(308\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x,method=_RETURNVERBOSE)

[Out]

-2/1452605*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(209633544*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)-104015934*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)[phone]*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)-138687912*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)[phone]*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))-46229304*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))[phone]*x^4+9225477612*x^3+1919527182*x^2-2283681406*x-884250959)/(2+3*x)^(5/2)/(10*x^2
+x-3)

________________________________________________________________________________________

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/(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)), x)

________________________________________________________________________________________

Fricas [A]
time = 0.24, size = 70, normalized size = 0.32 \begin {gather*} -\frac {2 \, {\left (6289006320 \, x^{4} + 9225477612 \, x^{3} + 1919527182 \, x^{2} - 2283681406 \, x - 884250959\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{1452605 \, {\left (270 \, x^{5} + 567 \, x^{4} + 333 \, x^{3} - 46 \, x^{2} - 100 \, x - 24\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

-2/1452605*(6289006320*x^4 + 9225477612*x^3 + 1919527182*x^2 - 2283681406*x - 884250959)*sqrt(5*x + 3)*sqrt(3*
x + 2)*sqrt(-2*x + 1)/(270*x^5 + 567*x^4 + 333*x^3 - 46*x^2 - 100*x - 24)

________________________________________________________________________________________

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/(1-2*x)**(3/2)/(2+3*x)**(7/2)/(3+5*x)**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 5988 deep

________________________________________________________________________________________

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/(1-2*x)^(3/2)/(2+3*x)^(7/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

integrate(1/((5*x + 3)^(3/2)*(3*x + 2)^(7/2)*(-2*x + 1)^(3/2)), x)

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (1-2\,x\right )}^{3/2}\,{\left (3\,x+2\right )}^{7/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((1 - 2*x)^(3/2)*(3*x + 2)^(7/2)*(5*x + 3)^(3/2)),x)

[Out]

int(1/((1 - 2*x)^(3/2)*(3*x + 2)^(7/2)*(5*x + 3)^(3/2)), x)

________________________________________________________________________________________