3.2784 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx\)

Optimal. Leaf size=280 \[ -\frac{3894280616 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{477839817 \sqrt{33}}+\frac{60080 \sqrt{1-2 x} (5 x+3)^{5/2}}{34749 (3 x+2)^{9/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1287 (3 x+2)^{11/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}-\frac{2622980 \sqrt{1-2 x} (5 x+3)^{3/2}}{1702701 (3 x+2)^{7/2}}+\frac{129922578224 \sqrt{1-2 x} \sqrt{5 x+3}}{5256237987 \sqrt{3 x+2}}+\frac{1876198516 \sqrt{1-2 x} \sqrt{5 x+3}}{750891141 (3 x+2)^{3/2}}-\frac{54281308 \sqrt{1-2 x} \sqrt{5 x+3}}{35756721 (3 x+2)^{5/2}}-\frac{129922578224 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{477839817 \sqrt{33}} \]

[Out]

(-54281308*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35756721*(2 + 3*x)^(5/2)) + (1876198516*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/
(750891141*(2 + 3*x)^(3/2)) + (129922578224*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5256237987*Sqrt[2 + 3*x]) - (2622980
*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(1702701*(2 + 3*x)^(7/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(39*(2 + 3*x)^
(13/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(1287*(2 + 3*x)^(11/2)) + (60080*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)
)/(34749*(2 + 3*x)^(9/2)) - (129922578224*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(477839817*Sqrt[3
3]) - (3894280616*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(477839817*Sqrt[33])

________________________________________________________________________________________

Rubi [A]  time = 0.112277, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 152, 158, 113, 119} \[ \frac{60080 \sqrt{1-2 x} (5 x+3)^{5/2}}{34749 (3 x+2)^{9/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{1287 (3 x+2)^{11/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{39 (3 x+2)^{13/2}}-\frac{2622980 \sqrt{1-2 x} (5 x+3)^{3/2}}{1702701 (3 x+2)^{7/2}}+\frac{129922578224 \sqrt{1-2 x} \sqrt{5 x+3}}{5256237987 \sqrt{3 x+2}}+\frac{1876198516 \sqrt{1-2 x} \sqrt{5 x+3}}{750891141 (3 x+2)^{3/2}}-\frac{54281308 \sqrt{1-2 x} \sqrt{5 x+3}}{35756721 (3 x+2)^{5/2}}-\frac{3894280616 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{477839817 \sqrt{33}}-\frac{129922578224 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{477839817 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(15/2),x]

[Out]

(-54281308*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(35756721*(2 + 3*x)^(5/2)) + (1876198516*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/
(750891141*(2 + 3*x)^(3/2)) + (129922578224*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5256237987*Sqrt[2 + 3*x]) - (2622980
*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(1702701*(2 + 3*x)^(7/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(39*(2 + 3*x)^
(13/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(1287*(2 + 3*x)^(11/2)) + (60080*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2)
)/(34749*(2 + 3*x)^(9/2)) - (129922578224*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(477839817*Sqrt[3
3]) - (3894280616*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(477839817*Sqrt[33])

Rule 97

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 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{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{15/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac{2}{39} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}-\frac{4 \int \frac{\sqrt{1-2 x} (3+5 x)^{3/2} \left (-1945+\frac{1675 x}{2}\right )}{(2+3 x)^{11/2}} \, dx}{1287}\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac{60080 \sqrt{1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac{8 \int \frac{\left (\frac{375445}{4}-\frac{210225 x}{2}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx}{34749}\\ &=-\frac{2622980 \sqrt{1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac{60080 \sqrt{1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac{16 \int \frac{\left (\frac{38522835}{8}-5499150 x\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx}{5108103}\\ &=-\frac{54281308 \sqrt{1-2 x} \sqrt{3+5 x}}{35756721 (2+3 x)^{5/2}}-\frac{2622980 \sqrt{1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac{60080 \sqrt{1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac{32 \int \frac{\frac{1283806245}{16}-\frac{796081425 x}{8}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{536350815}\\ &=-\frac{54281308 \sqrt{1-2 x} \sqrt{3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac{1876198516 \sqrt{1-2 x} \sqrt{3+5 x}}{750891141 (2+3 x)^{3/2}}-\frac{2622980 \sqrt{1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac{60080 \sqrt{1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac{64 \int \frac{\frac{14437282485}{4}-\frac{35178722175 x}{16}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{11263367115}\\ &=-\frac{54281308 \sqrt{1-2 x} \sqrt{3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac{1876198516 \sqrt{1-2 x} \sqrt{3+5 x}}{750891141 (2+3 x)^{3/2}}+\frac{129922578224 \sqrt{1-2 x} \sqrt{3+5 x}}{5256237987 \sqrt{2+3 x}}-\frac{2622980 \sqrt{1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac{60080 \sqrt{1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac{128 \int \frac{\frac{1541948542725}{32}+\frac{609012085425 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{78843569805}\\ &=-\frac{54281308 \sqrt{1-2 x} \sqrt{3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac{1876198516 \sqrt{1-2 x} \sqrt{3+5 x}}{750891141 (2+3 x)^{3/2}}+\frac{129922578224 \sqrt{1-2 x} \sqrt{3+5 x}}{5256237987 \sqrt{2+3 x}}-\frac{2622980 \sqrt{1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac{60080 \sqrt{1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}+\frac{1947140308 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{477839817}+\frac{129922578224 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{5256237987}\\ &=-\frac{54281308 \sqrt{1-2 x} \sqrt{3+5 x}}{35756721 (2+3 x)^{5/2}}+\frac{1876198516 \sqrt{1-2 x} \sqrt{3+5 x}}{750891141 (2+3 x)^{3/2}}+\frac{129922578224 \sqrt{1-2 x} \sqrt{3+5 x}}{5256237987 \sqrt{2+3 x}}-\frac{2622980 \sqrt{1-2 x} (3+5 x)^{3/2}}{1702701 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{39 (2+3 x)^{13/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{1287 (2+3 x)^{11/2}}+\frac{60080 \sqrt{1-2 x} (3+5 x)^{5/2}}{34749 (2+3 x)^{9/2}}-\frac{129922578224 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{477839817 \sqrt{33}}-\frac{3894280616 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{477839817 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.308405, size = 117, normalized size = 0.42 \[ \frac{-1050671168960 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (47356779762648 x^6+191022825888450 x^5+321056742490902 x^4+287874442427697 x^3+145238558453649 x^2+39086872650957 x+4382625184685\right )}{(3 x+2)^{13/2}}[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{126149711688 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(15/2),x]

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(4382625184685 + 39086872650957*x + 145238558453649*x^2 + 287874442427697*x^3
+ 321056742490902*x^4 + 191022825888450*x^5 + 47356779762648*x^6))/(2 + 3*x)^(13/2) + 2078761251584*EllipticE
[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 1050671168960*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/
(126149711688*Sqrt[2])

________________________________________________________________________________________

Maple [C]  time = 0.025, size = 694, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x)

[Out]

2/15768713961*(23935602567870*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^6*(3+5*x)^(1/2)*(2+3*x
)^(1/2)*(1-2*x)^(1/2)-47356779762648*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^6*(3+5*x)^(1/2)
*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^5*(3+5*x
)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-189427119050592*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^
5*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(
1/2))*x^4*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)-315711865084320*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/
2*I*66^(1/2))*x^4*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(3+5*x)^(1/2)[phone]*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)-280632768963840*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)+1420703392879440*x^8+7092030390480
0*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)-140316
384481920*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
)+5872755115941444*x^7+18912081041280*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)-37417702528512*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)+9778559734528578*x^6+2101342337920*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x
)^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))-4157522503168*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))+7880198067307566*x^5+2331269398474443*x^4-9825481269
59616*x^3-1058407652589420*x^2-338633978304558*x-39443626662165)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3
*x)^(13/2)

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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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{15}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x, algorithm="fricas")

[Out]

integral((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(6561*x^8 + 34992*x^
7 + 81648*x^6 + 108864*x^5 + 90720*x^4 + 48384*x^3 + 16128*x^2 + 3072*x + 256), 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((1-2*x)**(5/2)*(3+5*x)**(5/2)/(2+3*x)**(15/2),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 (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{15}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(15/2), x)