3.2727 \(\int \frac{(1-2 x)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{103970992 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{61261515 \sqrt{33}}+\frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{891 (3 x+2)^{9/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{13292 \sqrt{1-2 x} (5 x+3)^{3/2}}{43659 (3 x+2)^{7/2}}+\frac{3316711588 \sqrt{1-2 x} \sqrt{5 x+3}}{673876665 \sqrt{3 x+2}}+\frac{45748292 \sqrt{1-2 x} \sqrt{5 x+3}}{96268095 (3 x+2)^{3/2}}-\frac{1366496 \sqrt{1-2 x} \sqrt{5 x+3}}{4584195 (3 x+2)^{5/2}}-\frac{3316711588 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{61261515 \sqrt{33}} \]

[Out]

(-1366496*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4584195*(2 + 3*x)^(5/2)) + (45748292*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(962
68095*(2 + 3*x)^(3/2)) + (3316711588*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(673876665*Sqrt[2 + 3*x]) - (13292*Sqrt[1 -
2*x]*(3 + 5*x)^(3/2))/(43659*(2 + 3*x)^(7/2)) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) + (3
62*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(891*(2 + 3*x)^(9/2)) - (3316711588*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/(61261515*Sqrt[33]) - (103970992*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(61261515*Sqrt[
33])

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Rubi [A]  time = 0.0967433, 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} \[ \frac{362 \sqrt{1-2 x} (5 x+3)^{5/2}}{891 (3 x+2)^{9/2}}-\frac{2 (1-2 x)^{3/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{13292 \sqrt{1-2 x} (5 x+3)^{3/2}}{43659 (3 x+2)^{7/2}}+\frac{3316711588 \sqrt{1-2 x} \sqrt{5 x+3}}{673876665 \sqrt{3 x+2}}+\frac{45748292 \sqrt{1-2 x} \sqrt{5 x+3}}{96268095 (3 x+2)^{3/2}}-\frac{1366496 \sqrt{1-2 x} \sqrt{5 x+3}}{4584195 (3 x+2)^{5/2}}-\frac{103970992 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{61261515 \sqrt{33}}-\frac{3316711588 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{61261515 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-1366496*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(4584195*(2 + 3*x)^(5/2)) + (45748292*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(962
68095*(2 + 3*x)^(3/2)) + (3316711588*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(673876665*Sqrt[2 + 3*x]) - (13292*Sqrt[1 -
2*x]*(3 + 5*x)^(3/2))/(43659*(2 + 3*x)^(7/2)) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) + (3
62*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(891*(2 + 3*x)^(9/2)) - (3316711588*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/(61261515*Sqrt[33]) - (103970992*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(61261515*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)^{3/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{2}{33} \int \frac{\left (\frac{7}{2}-40 x\right ) \sqrt{1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac{4}{891} \int \frac{(3+5 x)^{3/2} \left (-\frac{1993}{2}+\frac{1995 x}{2}\right )}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx\\ &=-\frac{13292 \sqrt{1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac{8 \int \frac{\sqrt{3+5 x} \left (-\frac{107577}{2}+\frac{189705 x}{4}\right )}{\sqrt{1-2 x} (2+3 x)^{7/2}} \, dx}{130977}\\ &=-\frac{1366496 \sqrt{1-2 x} \sqrt{3+5 x}}{4584195 (2+3 x)^{5/2}}-\frac{13292 \sqrt{1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac{16 \int \frac{-\frac{9802263}{8}+\frac{2452215 x}{8}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{13752585}\\ &=-\frac{1366496 \sqrt{1-2 x} \sqrt{3+5 x}}{4584195 (2+3 x)^{5/2}}+\frac{45748292 \sqrt{1-2 x} \sqrt{3+5 x}}{96268095 (2+3 x)^{3/2}}-\frac{13292 \sqrt{1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac{32 \int \frac{-\frac{600436437}{16}+\frac{171556095 x}{8}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{288804285}\\ &=-\frac{1366496 \sqrt{1-2 x} \sqrt{3+5 x}}{4584195 (2+3 x)^{5/2}}+\frac{45748292 \sqrt{1-2 x} \sqrt{3+5 x}}{96268095 (2+3 x)^{3/2}}+\frac{3316711588 \sqrt{1-2 x} \sqrt{3+5 x}}{673876665 \sqrt{2+3 x}}-\frac{13292 \sqrt{1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac{64 \int \frac{-\frac{7891481415}{16}-\frac{12437668455 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2021629995}\\ &=-\frac{1366496 \sqrt{1-2 x} \sqrt{3+5 x}}{4584195 (2+3 x)^{5/2}}+\frac{45748292 \sqrt{1-2 x} \sqrt{3+5 x}}{96268095 (2+3 x)^{3/2}}+\frac{3316711588 \sqrt{1-2 x} \sqrt{3+5 x}}{673876665 \sqrt{2+3 x}}-\frac{13292 \sqrt{1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac{51985496 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{61261515}+\frac{3316711588 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{673876665}\\ &=-\frac{1366496 \sqrt{1-2 x} \sqrt{3+5 x}}{4584195 (2+3 x)^{5/2}}+\frac{45748292 \sqrt{1-2 x} \sqrt{3+5 x}}{96268095 (2+3 x)^{3/2}}+\frac{3316711588 \sqrt{1-2 x} \sqrt{3+5 x}}{673876665 \sqrt{2+3 x}}-\frac{13292 \sqrt{1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{7/2}}-\frac{2 (1-2 x)^{3/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{362 \sqrt{1-2 x} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac{3316711588 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{61261515 \sqrt{33}}-\frac{103970992 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{61261515 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.295558, size = 112, normalized size = 0.45 \[ \frac{-25619043520 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (402980457942 x^5+1356237833922 x^4+1829570010885 x^3+1234133449713 x^2+415681177941 x+55875107717\right )}{(3 x+2)^{11/2}}[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{16173039960 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(55875107717 + 415681177941*x + 1234133449713*x^2 + 1829570010885*x^3 + 13562
37833922*x^4 + 402980457942*x^5))/(2 + 3*x)^(11/2) + 53067385408*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -
33/2] - 25619043520*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(16173039960*Sqrt[2])

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Maple [C]  time = 0.022, size = 599, normalized size = 2.4 \begin{align*} -{\frac{2}{20216299950\,{x}^{2}[phone]\,x-6064889985} \left ( 402980457942\,\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}-194544611730\,\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}[phone]\,\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}-648482039100\,\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}-864642718800\,\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}-576428479200\,\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}-12089413738260\,{x}^{7}[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}-192142826400\,\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}-41896076391486\,{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 ) -25619043520\,\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 ) -55328989706838\,{x}^{5}-30306573018747\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}[phone]\,x+502875969453 \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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2),x)

[Out]

-2/2021629995*(402980457942*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)-194544611730*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)[phone]*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)-648482039100*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)-864642718800*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)-576428479200*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)-12089413738260*x^7+398005390560*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)-192142826400*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)-41896076391486*x^6+53067385
408*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))-25619043
520*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))-55328989
706838*x^5-30306573018747*x^4+293294410596*x^3+8183904282084*x^2+3573505278318*x+502875969453)*(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{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(3/2)/(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 (50 \, x^{3} + 35 \, x^{2} - 12 \, 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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2),x, algorithm="fricas")

[Out]

integral(-(50*x^3 + 35*x^2 - 12*x - 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2187*x^7 + 10206*x^6 + 2041
2*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)**(3/2)*(3+5*x)**(5/2)/(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{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{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)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2),x, algorithm="giac")

[Out]

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