3.2805 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{11/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=253 \[ \frac{20549264 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{9261}+\frac{14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt{5 x+3}}-\frac{3415750480 \sqrt{3 x+2} \sqrt{1-2 x}}{27783 \sqrt{5 x+3}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{813208 \sqrt{1-2 x}}{1323 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{11660 \sqrt{1-2 x}}{189 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{652 \sqrt{1-2 x}}{81 (3 x+2)^{7/2} \sqrt{5 x+3}}+\frac{683150096 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261} \]

[Out]

(14*(1 - 2*x)^(3/2))/(27*(2 + 3*x)^(9/2)*Sqrt[3 + 5*x]) + (652*Sqrt[1 - 2*x])/(81*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x
]) + (11660*Sqrt[1 - 2*x])/(189*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (813208*Sqrt[1 - 2*x])/(1323*(2 + 3*x)^(3/2)*
Sqrt[3 + 5*x]) + (113020952*Sqrt[1 - 2*x])/(9261*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (3415750480*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x])/(27783*Sqrt[3 + 5*x]) + (683150096*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/92
61 + (20549264*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/9261

________________________________________________________________________________________

Rubi [A]  time = 0.0963376, antiderivative size = 253, 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 = {98, 150, 152, 158, 113, 119} \[ \frac{14 (1-2 x)^{3/2}}{27 (3 x+2)^{9/2} \sqrt{5 x+3}}-\frac{3415750480 \sqrt{3 x+2} \sqrt{1-2 x}}{27783 \sqrt{5 x+3}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{813208 \sqrt{1-2 x}}{1323 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{11660 \sqrt{1-2 x}}{189 (3 x+2)^{5/2} \sqrt{5 x+3}}+\frac{652 \sqrt{1-2 x}}{81 (3 x+2)^{7/2} \sqrt{5 x+3}}+\frac{20549264 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261}+\frac{683150096 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261} \]

Antiderivative was successfully verified.

[In]

Int[(1 - 2*x)^(5/2)/((2 + 3*x)^(11/2)*(3 + 5*x)^(3/2)),x]

[Out]

(14*(1 - 2*x)^(3/2))/(27*(2 + 3*x)^(9/2)*Sqrt[3 + 5*x]) + (652*Sqrt[1 - 2*x])/(81*(2 + 3*x)^(7/2)*Sqrt[3 + 5*x
]) + (11660*Sqrt[1 - 2*x])/(189*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (813208*Sqrt[1 - 2*x])/(1323*(2 + 3*x)^(3/2)*
Sqrt[3 + 5*x]) + (113020952*Sqrt[1 - 2*x])/(9261*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (3415750480*Sqrt[1 - 2*x]*Sqrt
[2 + 3*x])/(27783*Sqrt[3 + 5*x]) + (683150096*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/92
61 + (20549264*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/9261

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{(1-2 x)^{5/2}}{(2+3 x)^{11/2} (3+5 x)^{3/2}} \, dx &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{2}{27} \int \frac{(229-227 x) \sqrt{1-2 x}}{(2+3 x)^{9/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}-\frac{4}{567} \int \frac{-\frac{50897}{2}+38346 x}{\sqrt{1-2 x} (2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{11660 \sqrt{1-2 x}}{189 (2+3 x)^{5/2} \sqrt{3+5 x}}-\frac{8 \int \frac{-\frac{5572105}{2}+\frac{7651875 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx}{19845}\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{11660 \sqrt{1-2 x}}{189 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{813208 \sqrt{1-2 x}}{1323 (2+3 x)^{3/2} \sqrt{3+5 x}}-\frac{16 \int \frac{-\frac{842998695}{4}+\frac{480300975 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx}{416745}\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{11660 \sqrt{1-2 x}}{189 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{813208 \sqrt{1-2 x}}{1323 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{32 \int \frac{-8991074400+\frac{22250999925 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx}{2917215}\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{11660 \sqrt{1-2 x}}{189 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{813208 \sqrt{1-2 x}}{1323 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{3415750480 \sqrt{1-2 x} \sqrt{2+3 x}}{27783 \sqrt{3+5 x}}+\frac{64 \int \frac{-\frac{936620355825}{8}-\frac{739723463325 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{32089365}\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{11660 \sqrt{1-2 x}}{189 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{813208 \sqrt{1-2 x}}{1323 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{3415750480 \sqrt{1-2 x} \sqrt{2+3 x}}{27783 \sqrt{3+5 x}}-\frac{113020952 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{9261}-\frac{683150096 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{9261}\\ &=\frac{14 (1-2 x)^{3/2}}{27 (2+3 x)^{9/2} \sqrt{3+5 x}}+\frac{652 \sqrt{1-2 x}}{81 (2+3 x)^{7/2} \sqrt{3+5 x}}+\frac{11660 \sqrt{1-2 x}}{189 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{813208 \sqrt{1-2 x}}{1323 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{113020952 \sqrt{1-2 x}}{9261 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{3415750480 \sqrt{1-2 x} \sqrt{2+3 x}}{27783 \sqrt{3+5 x}}+\frac{683150096 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261}+\frac{20549264 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{9261}\\ \end{align*}

Mathematica [A]  time = 0.275378, size = 115, normalized size = 0.45 \[ \frac{2 \left (-4 \sqrt{2} \left (85393762 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-43010905 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-\frac{3 \sqrt{1-2 x} \left (138337894440 x^5+456548966244 x^4+602551975428 x^3+397527527442 x^2+131099014240 x+17289178827\right )}{(3 x+2)^{9/2} \sqrt{5 x+3}}\right )}{27783} \]

Antiderivative was successfully verified.

[In]

Integrate[(1 - 2*x)^(5/2)/((2 + 3*x)^(11/2)*(3 + 5*x)^(3/2)),x]

[Out]

(2*((-3*Sqrt[1 - 2*x]*(17289178827 + 131099014240*x + 397527527442*x^2 + 602551975428*x^3 + 456548966244*x^4 +
138337894440*x^5))/((2 + 3*x)^(9/2)*Sqrt[3 + 5*x]) - 4*Sqrt[2]*(85393762*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 +
5*x]], -33/2] - 43010905*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/27783

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Maple [C]  time = 0.027, size = 504, normalized size = 2. \begin{align*}{\frac{2}{277830\,{x}^{2}+27783\,x-83349}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 27667578888\,\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}-13935533220\,\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}[phone]\,\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}-37161421920\,\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}[phone]\,\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}-37161421920\,\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}[phone]\,\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}-16516187520\,\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}-830027366640\,{x}^{6}[phone]\,\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 ) -2752697920\,\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 ) -2324280114144\,{x}^{5}-2245664953836\,{x}^{4}-577509238368\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+51867536481 \right ) \left ( 2+3\,x \right ) ^{-{\frac{9}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)/(2+3*x)^(11/2)/(3+5*x)^(3/2),x)

[Out]

2/27783*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(27667578888*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)-13935533220*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)[phone]*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)-37161421920*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)[phone]*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)-37161421920*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)[phone]*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)-16516187520*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)-830027366640*x^6+5465200768*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))-2752697920*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))-2324280114144*x^5-224566495383
6*x^4-577509238368*x^3+405988496886*x^2+289561969758*x+51867536481)/(2+3*x)^(9/2)/(10*x^2+x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^(11/2)/(3+5*x)^(3/2),x, algorithm="maxima")

[Out]

integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(11/2)), 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}}{18225 \, x^{8} + 94770 \, x^{7} + 215541 \, x^{6} + 280044 \, x^{5} + 227340 \, x^{4} + 118080 \, x^{3} + 38320 \, x^{2} + 7104 \, x + 576}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^(11/2)/(3+5*x)^(3/2),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(18225*x^8 + 94770*x^7 + 215541*x^6 + 28
0044*x^5 + 227340*x^4 + 118080*x^3 + 38320*x^2 + 7104*x + 576), 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)/(2+3*x)**(11/2)/(3+5*x)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)/(2+3*x)^(11/2)/(3+5*x)^(3/2),x, algorithm="giac")

[Out]

integrate((-2*x + 1)^(5/2)/((5*x + 3)^(3/2)*(3*x + 2)^(11/2)), x)