3.2783 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{13235368 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{5250987 \sqrt{33}}+\frac{36980 \sqrt{1-2 x} (5 x+3)^{5/2}}{18711 (3 x+2)^{7/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{891 (3 x+2)^{9/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{55772 \sqrt{1-2 x} (5 x+3)^{3/2}}{43659 (3 x+2)^{5/2}}+\frac{584888452 \sqrt{1-2 x} \sqrt{5 x+3}}{57760857 \sqrt{3 x+2}}-\frac{17089252 \sqrt{1-2 x} \sqrt{5 x+3}}{8251551 (3 x+2)^{3/2}}-\frac{584888452 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250987 \sqrt{33}} \]

[Out]

(-17089252*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8251551*(2 + 3*x)^(3/2)) + (584888452*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5
7760857*Sqrt[2 + 3*x]) - (55772*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(43659*(2 + 3*x)^(5/2)) - (2*(1 - 2*x)^(5/2)*(3
+ 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(891*(2 + 3*x)^(9/2)) + (36980*Sq
rt[1 - 2*x]*(3 + 5*x)^(5/2))/(18711*(2 + 3*x)^(7/2)) - (584888452*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3
5/33])/(5250987*Sqrt[33]) - (13235368*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5250987*Sqrt[33])

________________________________________________________________________________________

Rubi [A]  time = 0.0999762, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{36980 \sqrt{1-2 x} (5 x+3)^{5/2}}{18711 (3 x+2)^{7/2}}+\frac{370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{891 (3 x+2)^{9/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{55772 \sqrt{1-2 x} (5 x+3)^{3/2}}{43659 (3 x+2)^{5/2}}+\frac{584888452 \sqrt{1-2 x} \sqrt{5 x+3}}{57760857 \sqrt{3 x+2}}-\frac{17089252 \sqrt{1-2 x} \sqrt{5 x+3}}{8251551 (3 x+2)^{3/2}}-\frac{13235368 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250987 \sqrt{33}}-\frac{584888452 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250987 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(13/2),x]

[Out]

(-17089252*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8251551*(2 + 3*x)^(3/2)) + (584888452*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5
7760857*Sqrt[2 + 3*x]) - (55772*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(43659*(2 + 3*x)^(5/2)) - (2*(1 - 2*x)^(5/2)*(3
+ 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(891*(2 + 3*x)^(9/2)) + (36980*Sq
rt[1 - 2*x]*(3 + 5*x)^(5/2))/(18711*(2 + 3*x)^(7/2)) - (584888452*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3
5/33])/(5250987*Sqrt[33]) - (13235368*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5250987*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)^{13/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{2}{33} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac{4}{891} \int \frac{\sqrt{1-2 x} (3+5 x)^{3/2} \left (-\frac{3065}{2}+\frac{25 x}{2}\right )}{(2+3 x)^{9/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac{36980 \sqrt{1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac{8 \int \frac{\left (\frac{147745}{4}-23025 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx}{18711}\\ &=-\frac{55772 \sqrt{1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac{36980 \sqrt{1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac{16 \int \frac{\left (\frac{14799465}{8}-\frac{4921575 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx}{1964655}\\ &=-\frac{17089252 \sqrt{1-2 x} \sqrt{3+5 x}}{8251551 (2+3 x)^{3/2}}-\frac{55772 \sqrt{1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac{36980 \sqrt{1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac{32 \int \frac{\frac{469321365}{16}-\frac{49085475 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{123773265}\\ &=-\frac{17089252 \sqrt{1-2 x} \sqrt{3+5 x}}{8251551 (2+3 x)^{3/2}}+\frac{584888452 \sqrt{1-2 x} \sqrt{3+5 x}}{57760857 \sqrt{2+3 x}}-\frac{55772 \sqrt{1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac{36980 \sqrt{1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac{64 \int \frac{\frac{3426487275}{8}+\frac{10966658475 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{866412855}\\ &=-\frac{17089252 \sqrt{1-2 x} \sqrt{3+5 x}}{8251551 (2+3 x)^{3/2}}+\frac{584888452 \sqrt{1-2 x} \sqrt{3+5 x}}{57760857 \sqrt{2+3 x}}-\frac{55772 \sqrt{1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac{36980 \sqrt{1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac{6617684 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{5250987}+\frac{584888452 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{57760857}\\ &=-\frac{17089252 \sqrt{1-2 x} \sqrt{3+5 x}}{8251551 (2+3 x)^{3/2}}+\frac{584888452 \sqrt{1-2 x} \sqrt{3+5 x}}{57760857 \sqrt{2+3 x}}-\frac{55772 \sqrt{1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac{36980 \sqrt{1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}-\frac{584888452 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250987 \sqrt{33}}-\frac{13235368 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{5250987 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.305907, size = 112, normalized size = 0.45 \[ \frac{-5864078080 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (71063946918 x^5+237923150688 x^4+320012032635 x^3+215597947743 x^2+72620507583 x+9770732477\right )}{(3 x+2)^{11/2}}[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1386260568 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(13/2),x]

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(9770732477 + 72620507583*x + 215597947743*x^2 + 320012032635*x^3 + 237923150
688*x^4 + 71063946918*x^5))/(2 + 3*x)^(11/2) + 9358215232*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] -
5864078080*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(1386260568*Sqrt[2])

________________________________________________________________________________________

Maple [C]  time = 0.023, size = 599, normalized size = 2.4 \begin{align*}{\frac{2}{1732825710\,{x}^{2}[phone]\,x-519847713} \left ( 44530342920\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{5}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-71063946918\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{5}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}[phone]\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}-236879823060\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{4}\sqrt{2+3\,x}\sqrt{1-2\,x}\sqrt{3+5\,x}[phone]\,\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}-315839764080\,\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}[phone]\,\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}-210559842720\,\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}[phone]\,{x}^{7}[phone]\,\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}-70186614240\,\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}[phone]\,{x}^{6}[phone]\,\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 ) -9358215232\,\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 ) [phone]\,{x}^{5}[phone]\,{x}^{4}-54699222996\,{x}^{3}-1429398032628\,{x}^{2}-624272370816\,x-87936592293 \right ) \sqrt{3+5\,x}\sqrt{1-2\,x} \left ( 2+3\,x \right ) ^{-{\frac{11}{2}}}} \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)^(13/2),x)

[Out]

2/173282571*(44530342920*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)-71063946918*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)-236879823060*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)-315839764080*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)[phone]*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)-210559842720*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)[phone]*x^7+43980585600*2^(1/2)*EllipticF(1/11*(66+11
0*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)-70186614240*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)[phone]*x^6+5864078080*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))-9358215232*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))[phone]*x^5+528
6666174003*x^4-54699222996*x^3-1429398032628*x^2-624272370816*x-87936592293)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x
^2+x-3)/(2+3*x)^(11/2)

________________________________________________________________________________________

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{13}{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)^(13/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2), x)

________________________________________________________________________________________

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}}{2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128}, 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)^(13/2),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)/(2187*x^7 + 10206*x^
6 + 20412*x^5 + 22680*x^4 + 15120*x^3 + 6048*x^2 + 1344*x + 128), x)

________________________________________________________________________________________

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)**(13/2),x)

[Out]

Timed out

________________________________________________________________________________________

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{13}{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)^(13/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(13/2), x)