3.2763 \(\int \frac{(1-2 x)^{5/2} \sqrt{3+5 x}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{7442032 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{64827 \sqrt{33}}-\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac{10 \sqrt{5 x+3} (1-2 x)^{3/2}}{99 (3 x+2)^{9/2}}+\frac{247408648 \sqrt{5 x+3} \sqrt{1-2 x}}{713097 \sqrt{3 x+2}}+\frac{3560432 \sqrt{5 x+3} \sqrt{1-2 x}}{101871 (3 x+2)^{3/2}}+\frac{76492 \sqrt{5 x+3} \sqrt{1-2 x}}{14553 (3 x+2)^{5/2}}+\frac{1900 \sqrt{5 x+3} \sqrt{1-2 x}}{2079 (3 x+2)^{7/2}}-\frac{247408648 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{64827 \sqrt{33}} \]

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(33*(2 + 3*x)^(11/2)) + (10*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(99*(2 + 3*x)^(9
/2)) + (1900*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2079*(2 + 3*x)^(7/2)) + (76492*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14553*
(2 + 3*x)^(5/2)) + (3560432*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(101871*(2 + 3*x)^(3/2)) + (247408648*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(713097*Sqrt[2 + 3*x]) - (247408648*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(64827*Sq
rt[33]) - (7442032*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(64827*Sqrt[33])

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Rubi [A]  time = 0.0981, 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} \[ -\frac{2 \sqrt{5 x+3} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac{10 \sqrt{5 x+3} (1-2 x)^{3/2}}{99 (3 x+2)^{9/2}}+\frac{247408648 \sqrt{5 x+3} \sqrt{1-2 x}}{713097 \sqrt{3 x+2}}+\frac{3560432 \sqrt{5 x+3} \sqrt{1-2 x}}{101871 (3 x+2)^{3/2}}+\frac{76492 \sqrt{5 x+3} \sqrt{1-2 x}}{14553 (3 x+2)^{5/2}}+\frac{1900 \sqrt{5 x+3} \sqrt{1-2 x}}{2079 (3 x+2)^{7/2}}-\frac{7442032 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{64827 \sqrt{33}}-\frac{247408648 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{64827 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(5/2)*Sqrt[3 + 5*x])/(33*(2 + 3*x)^(11/2)) + (10*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(99*(2 + 3*x)^(9
/2)) + (1900*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2079*(2 + 3*x)^(7/2)) + (76492*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(14553*
(2 + 3*x)^(5/2)) + (3560432*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(101871*(2 + 3*x)^(3/2)) + (247408648*Sqrt[1 - 2*x]*S
qrt[3 + 5*x])/(713097*Sqrt[2 + 3*x]) - (247408648*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(64827*Sq
rt[33]) - (7442032*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(64827*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} \sqrt{3+5 x}}{(2+3 x)^{13/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{33 (2+3 x)^{11/2}}+\frac{2}{33} \int \frac{\left (-\frac{25}{2}-30 x\right ) (1-2 x)^{3/2}}{(2+3 x)^{11/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{33 (2+3 x)^{11/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{99 (2+3 x)^{9/2}}-\frac{4}{891} \int \frac{\sqrt{1-2 x} \left (-\frac{1035}{2}+\frac{585 x}{2}\right )}{(2+3 x)^{9/2} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{33 (2+3 x)^{11/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{99 (2+3 x)^{9/2}}+\frac{1900 \sqrt{1-2 x} \sqrt{3+5 x}}{2079 (2+3 x)^{7/2}}+\frac{8 \int \frac{\frac{149805}{4}-51390 x}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{18711}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{33 (2+3 x)^{11/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{99 (2+3 x)^{9/2}}+\frac{1900 \sqrt{1-2 x} \sqrt{3+5 x}}{2079 (2+3 x)^{7/2}}+\frac{76492 \sqrt{1-2 x} \sqrt{3+5 x}}{14553 (2+3 x)^{5/2}}+\frac{16 \int \frac{2855520-\frac{12908025 x}{4}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{654885}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{33 (2+3 x)^{11/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{99 (2+3 x)^{9/2}}+\frac{1900 \sqrt{1-2 x} \sqrt{3+5 x}}{2079 (2+3 x)^{7/2}}+\frac{76492 \sqrt{1-2 x} \sqrt{3+5 x}}{14553 (2+3 x)^{5/2}}+\frac{3560432 \sqrt{1-2 x} \sqrt{3+5 x}}{101871 (2+3 x)^{3/2}}+\frac{32 \int \frac{\frac{991125045}{8}-\frac{150205725 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{13752585}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{33 (2+3 x)^{11/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{99 (2+3 x)^{9/2}}+\frac{1900 \sqrt{1-2 x} \sqrt{3+5 x}}{2079 (2+3 x)^{7/2}}+\frac{76492 \sqrt{1-2 x} \sqrt{3+5 x}}{14553 (2+3 x)^{5/2}}+\frac{3560432 \sqrt{1-2 x} \sqrt{3+5 x}}{101871 (2+3 x)^{3/2}}+\frac{247408648 \sqrt{1-2 x} \sqrt{3+5 x}}{713097 \sqrt{2+3 x}}+\frac{64 \int \frac{1651972050+\frac{20875104675 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{96268095}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{33 (2+3 x)^{11/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{99 (2+3 x)^{9/2}}+\frac{1900 \sqrt{1-2 x} \sqrt{3+5 x}}{2079 (2+3 x)^{7/2}}+\frac{76492 \sqrt{1-2 x} \sqrt{3+5 x}}{14553 (2+3 x)^{5/2}}+\frac{3560432 \sqrt{1-2 x} \sqrt{3+5 x}}{101871 (2+3 x)^{3/2}}+\frac{247408648 \sqrt{1-2 x} \sqrt{3+5 x}}{713097 \sqrt{2+3 x}}+\frac{3721016 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{64827}+\frac{247408648 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{713097}\\ &=-\frac{2 (1-2 x)^{5/2} \sqrt{3+5 x}}{33 (2+3 x)^{11/2}}+\frac{10 (1-2 x)^{3/2} \sqrt{3+5 x}}{99 (2+3 x)^{9/2}}+\frac{1900 \sqrt{1-2 x} \sqrt{3+5 x}}{2079 (2+3 x)^{7/2}}+\frac{76492 \sqrt{1-2 x} \sqrt{3+5 x}}{14553 (2+3 x)^{5/2}}+\frac{3560432 \sqrt{1-2 x} \sqrt{3+5 x}}{101871 (2+3 x)^{3/2}}+\frac{247408648 \sqrt{1-2 x} \sqrt{3+5 x}}{713097 \sqrt{2+3 x}}-\frac{247408648 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{64827 \sqrt{33}}-\frac{7442032 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{64827 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.288768, size = 115, normalized size = 0.46 \[ \frac{32 \sqrt{2} \left (30926081 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-15576890 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{24 \sqrt{1-2 x} \sqrt{5 x+3} \left (30060150732 x^5+101209884912 x^4+136342955970 x^3+91862628912 x^2+30956769477 x+4174268813\right )}{(3 x+2)^{11/2}}}{8557164} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((24*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(4174268813 + 30956769477*x + 91862628912*x^2 + 136342955970*x^3 + 1012098849
12*x^4 + 30060150732*x^5))/(2 + 3*x)^(11/2) + 32*Sqrt[2]*(30926081*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]],
-33/2] - 15576890*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/8557164

________________________________________________________________________________________

Maple [C]  time = 0.021, size = 599, normalized size = 2.4 \begin{align*}{\frac{2}{21392910\,{x}^{2}+2139291\,x-6417873} \left ( 15140737080\,\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}-30060150732\,\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}-100200502440\,\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}-133600669920\,\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}-89067113280\,\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}-29689037760\,\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 ) -3958538368\,\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}-22795632684\,{x}^{3}-608665287387\,{x}^{2}-266088118854\,x-37568419317 \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)^(1/2)/(2+3*x)^(13/2),x)

[Out]

2/2139291*(15140737080*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)-30060150732*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)-100200502440*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)-133600669920*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)-89067113280*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+14953814400*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)-29689037760*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+1993841920*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))-3958538368*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+2254018771
062*x^4-22795632684*x^3-608665287387*x^2-266088118854*x-37568419317)*(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{\sqrt{5 \, x + 3}{\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)^(1/2)/(2+3*x)^(13/2),x, algorithm="maxima")

[Out]

integrate(sqrt(5*x + 3)*(-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 (4 \, x^{2} - 4 \, x + 1\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)^(1/2)/(2+3*x)^(13/2),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 2268
0*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)**(1/2)/(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{\sqrt{5 \, x + 3}{\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)^(1/2)/(2+3*x)^(13/2),x, algorithm="giac")

[Out]

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