3.2773 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{23441272 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{1750329 \sqrt{33}}-\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{891 (3 x+2)^{9/2}}+\frac{12280 (5 x+3)^{3/2} \sqrt{1-2 x}}{6237 (3 x+2)^{7/2}}+\frac{780320008 \sqrt{5 x+3} \sqrt{1-2 x}}{19253619 \sqrt{3 x+2}}+\frac{11243972 \sqrt{5 x+3} \sqrt{1-2 x}}{2750517 (3 x+2)^{3/2}}-\frac{325796 \sqrt{5 x+3} \sqrt{1-2 x}}{130977 (3 x+2)^{5/2}}-\frac{780320008 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}} \]

[Out]

(-325796*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(130977*(2 + 3*x)^(5/2)) + (11243972*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(27505
17*(2 + 3*x)^(3/2)) + (780320008*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(19253619*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(5/2)*(3
+ 5*x)^(3/2))/(33*(2 + 3*x)^(11/2)) + (230*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(891*(2 + 3*x)^(9/2)) + (12280*Sq
rt[1 - 2*x]*(3 + 5*x)^(3/2))/(6237*(2 + 3*x)^(7/2)) - (780320008*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/(1750329*Sqrt[33]) - (23441272*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1750329*Sqrt[33])

________________________________________________________________________________________

Rubi [A]  time = 0.0971486, 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} \[ -\frac{2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac{230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{891 (3 x+2)^{9/2}}+\frac{12280 (5 x+3)^{3/2} \sqrt{1-2 x}}{6237 (3 x+2)^{7/2}}+\frac{780320008 \sqrt{5 x+3} \sqrt{1-2 x}}{19253619 \sqrt{3 x+2}}+\frac{11243972 \sqrt{5 x+3} \sqrt{1-2 x}}{2750517 (3 x+2)^{3/2}}-\frac{325796 \sqrt{5 x+3} \sqrt{1-2 x}}{130977 (3 x+2)^{5/2}}-\frac{23441272 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}}-\frac{780320008 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-325796*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(130977*(2 + 3*x)^(5/2)) + (11243972*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(27505
17*(2 + 3*x)^(3/2)) + (780320008*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(19253619*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(5/2)*(3
+ 5*x)^(3/2))/(33*(2 + 3*x)^(11/2)) + (230*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(891*(2 + 3*x)^(9/2)) + (12280*Sq
rt[1 - 2*x]*(3 + 5*x)^(3/2))/(6237*(2 + 3*x)^(7/2)) - (780320008*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/(1750329*Sqrt[33]) - (23441272*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1750329*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)^{3/2}}{(2+3 x)^{13/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{2}{33} \int \frac{\left (-\frac{15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt{3+5 x}}{(2+3 x)^{11/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}-\frac{4}{891} \int \frac{\sqrt{1-2 x} \sqrt{3+5 x} \left (-1200+\frac{1005 x}{2}\right )}{(2+3 x)^{9/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{12280 \sqrt{1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac{8 \int \frac{\left (\frac{232425}{4}-\frac{131115 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx}{18711}\\ &=-\frac{325796 \sqrt{1-2 x} \sqrt{3+5 x}}{130977 (2+3 x)^{5/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{12280 \sqrt{1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac{16 \int \frac{\frac{7896165}{8}-1154775 x}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{1964655}\\ &=-\frac{325796 \sqrt{1-2 x} \sqrt{3+5 x}}{130977 (2+3 x)^{5/2}}+\frac{11243972 \sqrt{1-2 x} \sqrt{3+5 x}}{2750517 (2+3 x)^{3/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{12280 \sqrt{1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac{32 \int \frac{\frac{347150355}{8}-\frac{210824475 x}{8}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{41257755}\\ &=-\frac{325796 \sqrt{1-2 x} \sqrt{3+5 x}}{130977 (2+3 x)^{5/2}}+\frac{11243972 \sqrt{1-2 x} \sqrt{3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac{780320008 \sqrt{1-2 x} \sqrt{3+5 x}}{19253619 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{12280 \sqrt{1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac{64 \int \frac{\frac{9262076325}{16}+\frac{7315500075 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{288804285}\\ &=-\frac{325796 \sqrt{1-2 x} \sqrt{3+5 x}}{130977 (2+3 x)^{5/2}}+\frac{11243972 \sqrt{1-2 x} \sqrt{3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac{780320008 \sqrt{1-2 x} \sqrt{3+5 x}}{19253619 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{12280 \sqrt{1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac{11720636 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1750329}+\frac{780320008 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{19253619}\\ &=-\frac{325796 \sqrt{1-2 x} \sqrt{3+5 x}}{130977 (2+3 x)^{5/2}}+\frac{11243972 \sqrt{1-2 x} \sqrt{3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac{780320008 \sqrt{1-2 x} \sqrt{3+5 x}}{19253619 \sqrt{2+3 x}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac{230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac{12280 \sqrt{1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}-\frac{780320008 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}}-\frac{23441272 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{1750329 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.300426, size = 115, normalized size = 0.46 \[ \frac{16 \sqrt{2} \left (195080002 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-98384755 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{24 \sqrt{1-2 x} \sqrt{5 x+3} \left (94808880972 x^5+319217269302 x^4+429993423180 x^3+289719086787 x^2+97637232762 x+13163824553\right )}{(3 x+2)^{11/2}}}{231043428} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((24*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(13163824553 + 97637232762*x + 289719086787*x^2 + 429993423180*x^3 + 31921726
9302*x^4 + 94808880972*x^5))/(2 + 3*x)^(11/2) + 16*Sqrt[2]*(195080002*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]], -33/2] - 98384755*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/231043428

________________________________________________________________________________________

Maple [C]  time = 0.022, size = 599, normalized size = 2.4 \begin{align*}{\frac{2}{577608570\,{x}^{2}[phone]\,x-173282571} \left ( 47814990930\,\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}-94808880972\,\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}-316029603240\,\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}-421372804320\,\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}-280915202880\,\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}-93638400960\,\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 ) -12485120128\,\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}-71666565399\,{x}^{3}-1919645346207\,{x}^{2}-839243621199\,x-118474420977 \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)^(3/2)/(2+3*x)^(13/2),x)

[Out]

2/57760857*(47814990930*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)-94808880972*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)-316029603240*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)-421372804320*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)-280915202880*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+47224682400*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)-93638400960*2^(1/2)*EllipticE(1/11*(66+1
10*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+6296624320*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))-12485120128*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+71
08597449432*x^4-71666565399*x^3-1919645346207*x^2-839243621199*x-118474420977)*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10
*x^2+x-3)/(2+3*x)^(11/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (5 \, x + 3\right )}^{\frac{3}{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)^(3/2)/(2+3*x)^(13/2),x, algorithm="maxima")

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\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)^(3/2)/(2+3*x)^(13/2),x, algorithm="fricas")

[Out]

integral((20*x^3 - 8*x^2 - 7*x + 3)*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)

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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)**(3/2)/(2+3*x)**(13/2),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{3}{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)^(3/2)/(2+3*x)^(13/2),x, algorithm="giac")

[Out]

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