3.2916 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac{220028 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1764735}+\frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{7/2}}-\frac{106558 \sqrt{1-2 x} \sqrt{5 x+3}}{1764735 \sqrt{3 x+2}}-\frac{106772 \sqrt{1-2 x} \sqrt{5 x+3}}{252105 (3 x+2)^{3/2}}-\frac{37117 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 (3 x+2)^{5/2}}+\frac{229 \sqrt{1-2 x} \sqrt{5 x+3}}{1029 (3 x+2)^{7/2}}+\frac{106558 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1764735} \]

[Out]

(229*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1029*(2 + 3*x)^(7/2)) - (37117*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36015*(2 + 3*x
)^(5/2)) - (106772*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(252105*(2 + 3*x)^(3/2)) - (106558*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(1764735*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) + (106558*Sqrt[11/3]*Ellipt
icE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1764735 - (220028*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/1764735

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Rubi [A]  time = 0.0795845, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{7/2}}-\frac{106558 \sqrt{1-2 x} \sqrt{5 x+3}}{1764735 \sqrt{3 x+2}}-\frac{106772 \sqrt{1-2 x} \sqrt{5 x+3}}{252105 (3 x+2)^{3/2}}-\frac{37117 \sqrt{1-2 x} \sqrt{5 x+3}}{36015 (3 x+2)^{5/2}}+\frac{229 \sqrt{1-2 x} \sqrt{5 x+3}}{1029 (3 x+2)^{7/2}}-\frac{220028 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1764735}+\frac{106558 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1764735} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(229*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1029*(2 + 3*x)^(7/2)) - (37117*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(36015*(2 + 3*x
)^(5/2)) - (106772*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(252105*(2 + 3*x)^(3/2)) - (106558*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(1764735*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(7/2)) + (106558*Sqrt[11/3]*Ellipt
icE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/1764735 - (220028*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 -
2*x]], 35/33])/1764735

Rule 98

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 150

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 152

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 158

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]

Rule 113

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2*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))])/b, 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 119

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

Rubi steps

\begin{align*} \int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{9/2}} \, dx &=\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{1}{7} \int \frac{\left (-\frac{357}{2}-325 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx\\ &=\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 (2+3 x)^{7/2}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{2 \int \frac{-\frac{86161}{4}-36950 x}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{1029}\\ &=\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 (2+3 x)^{7/2}}-\frac{37117 \sqrt{1-2 x} \sqrt{3+5 x}}{36015 (2+3 x)^{5/2}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{4 \int \frac{-79446-\frac{556755 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{36015}\\ &=\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 (2+3 x)^{7/2}}-\frac{37117 \sqrt{1-2 x} \sqrt{3+5 x}}{36015 (2+3 x)^{5/2}}-\frac{106772 \sqrt{1-2 x} \sqrt{3+5 x}}{252105 (2+3 x)^{3/2}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{8 \int \frac{-\frac{1014441}{8}-\frac{400395 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{756315}\\ &=\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 (2+3 x)^{7/2}}-\frac{37117 \sqrt{1-2 x} \sqrt{3+5 x}}{36015 (2+3 x)^{5/2}}-\frac{106772 \sqrt{1-2 x} \sqrt{3+5 x}}{252105 (2+3 x)^{3/2}}-\frac{106558 \sqrt{1-2 x} \sqrt{3+5 x}}{1764735 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{16 \int \frac{-166965+\frac{799185 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5294205}\\ &=\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 (2+3 x)^{7/2}}-\frac{37117 \sqrt{1-2 x} \sqrt{3+5 x}}{36015 (2+3 x)^{5/2}}-\frac{106772 \sqrt{1-2 x} \sqrt{3+5 x}}{252105 (2+3 x)^{3/2}}-\frac{106558 \sqrt{1-2 x} \sqrt{3+5 x}}{1764735 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}-\frac{106558 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{1764735}+\frac{1210154 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1764735}\\ &=\frac{229 \sqrt{1-2 x} \sqrt{3+5 x}}{1029 (2+3 x)^{7/2}}-\frac{37117 \sqrt{1-2 x} \sqrt{3+5 x}}{36015 (2+3 x)^{5/2}}-\frac{106772 \sqrt{1-2 x} \sqrt{3+5 x}}{252105 (2+3 x)^{3/2}}-\frac{106558 \sqrt{1-2 x} \sqrt{3+5 x}}{1764735 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{7/2}}+\frac{106558 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1764735}-\frac{220028 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1764735}\\ \end{align*}

Mathematica [A]  time = 0.177004, size = 109, normalized size = 0.49 \[ \frac{2 \left (\sqrt{2} \left (1868510 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )-53279 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )+\frac{3 \sqrt{5 x+3} \left (2877066 x^4+11042235 x^3+12020751 x^2+4889131 x+616327\right )}{\sqrt{1-2 x} (3 x+2)^{7/2}}\right )}{5294205} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((3*Sqrt[3 + 5*x]*(616327 + 4889131*x + 12020751*x^2 + 11042235*x^3 + 2877066*x^4))/(Sqrt[1 - 2*x]*(2 + 3*x
)^(7/2)) + Sqrt[2]*(-53279*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 1868510*EllipticF[ArcSin[Sqrt[
2/11]*Sqrt[3 + 5*x]], -33/2])))/5294205

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Maple [C]  time = 0.025, size = 409, normalized size = 1.8 \begin{align*}{\frac{2}{52942050\,{x}^{2}+5294205\,x-15882615}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 1438533\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-50449770\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{3}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2877066\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-100899540\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+1918044\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-67266360\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+426232\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -14948080\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -43155990\,{x}^{5}-191527119\,{x}^{4}-279691380\,{x}^{3}-181523724\,{x}^{2}-53247084\,x-5546943 \right ) \left ( 2+3\,x \right ) ^{-{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5294205*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(1438533*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^3*(3+
5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-50449770*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^3*(3
+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+2877066*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^2*(3
+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-100899540*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^2*
(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+1918044*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x*(3
+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-67266360*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x*(3+
5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+426232*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/11
*(66+110*x)^(1/2),1/2*I*66^(1/2))-14948080*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*(6
6+110*x)^(1/2),1/2*I*66^(1/2))-43155990*x^5-191527119*x^4-279691380*x^3-181523724*x^2-53247084*x-5546943)/(2+3
*x)^(7/2)/(10*x^2+x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{972 \, x^{7} + 2268 \, x^{6} + 1323 \, x^{5} - 630 \, x^{4} - 840 \, x^{3} - 112 \, x^{2} + 112 \, x + 32}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(972*x^7 + 2268*x^6 + 1323*x^5 - 630*x
^4 - 840*x^3 - 112*x^2 + 112*x + 32), x)

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{9}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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