3.2671 \(\int \frac{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac{37904696 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{47647845 \sqrt{33}}-\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{2079 (3 x+2)^{9/2}}+\frac{1305025844 \sqrt{1-2 x} \sqrt{5 x+3}}{524126295 \sqrt{3 x+2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{5 x+3}}{74875185 (3 x+2)^{3/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{5 x+3}}{10696455 (3 x+2)^{5/2}}-\frac{13022 \sqrt{1-2 x} \sqrt{5 x+3}}{305613 (3 x+2)^{7/2}}-\frac{1305025844 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}} \]

[Out]

(-13022*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(305613*(2 + 3*x)^(7/2)) + (627806*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10696455
*(2 + 3*x)^(5/2)) + (19417096*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(74875185*(2 + 3*x)^(3/2)) + (1305025844*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x])/(524126295*Sqrt[2 + 3*x]) - (118*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2079*(2 + 3*x)^(9/2)) - (2
*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) - (1305025844*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/(47647845*Sqrt[33]) - (37904696*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(47647845*Sqrt[33]
)

________________________________________________________________________________________

Rubi [A]  time = 0.0971437, 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} \[ -\frac{2 \sqrt{1-2 x} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac{118 \sqrt{1-2 x} (5 x+3)^{3/2}}{2079 (3 x+2)^{9/2}}+\frac{1305025844 \sqrt{1-2 x} \sqrt{5 x+3}}{524126295 \sqrt{3 x+2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{5 x+3}}{74875185 (3 x+2)^{3/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{5 x+3}}{10696455 (3 x+2)^{5/2}}-\frac{13022 \sqrt{1-2 x} \sqrt{5 x+3}}{305613 (3 x+2)^{7/2}}-\frac{37904696 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}}-\frac{1305025844 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-13022*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(305613*(2 + 3*x)^(7/2)) + (627806*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10696455
*(2 + 3*x)^(5/2)) + (19417096*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(74875185*(2 + 3*x)^(3/2)) + (1305025844*Sqrt[1 - 2
*x]*Sqrt[3 + 5*x])/(524126295*Sqrt[2 + 3*x]) - (118*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2079*(2 + 3*x)^(9/2)) - (2
*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) - (1305025844*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/(47647845*Sqrt[33]) - (37904696*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(47647845*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{\sqrt{1-2 x} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx &=-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{2}{33} \int \frac{\left (\frac{19}{2}-30 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{11/2}} \, dx\\ &=-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{4 \int \frac{\left (-\frac{189}{4}-\frac{5025 x}{2}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx}{6237}\\ &=-\frac{13022 \sqrt{1-2 x} \sqrt{3+5 x}}{305613 (2+3 x)^{7/2}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{8 \int \frac{-\frac{676497}{8}-185700 x}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{916839}\\ &=-\frac{13022 \sqrt{1-2 x} \sqrt{3+5 x}}{305613 (2+3 x)^{7/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{5/2}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{16 \int \frac{\frac{1286433}{2}-\frac{14125635 x}{8}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{32089365}\\ &=-\frac{13022 \sqrt{1-2 x} \sqrt{3+5 x}}{305613 (2+3 x)^{7/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{3+5 x}}{74875185 (2+3 x)^{3/2}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{32 \int \frac{\frac{687512943}{16}-\frac{109221165 x}{4}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{673876665}\\ &=-\frac{13022 \sqrt{1-2 x} \sqrt{3+5 x}}{305613 (2+3 x)^{7/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac{1305025844 \sqrt{1-2 x} \sqrt{3+5 x}}{524126295 \sqrt{2+3 x}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{64 \int \frac{\frac{2319498765}{4}+\frac{14681540745 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{4717136655}\\ &=-\frac{13022 \sqrt{1-2 x} \sqrt{3+5 x}}{305613 (2+3 x)^{7/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac{1305025844 \sqrt{1-2 x} \sqrt{3+5 x}}{524126295 \sqrt{2+3 x}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac{18952348 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{47647845}+\frac{1305025844 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{524126295}\\ &=-\frac{13022 \sqrt{1-2 x} \sqrt{3+5 x}}{305613 (2+3 x)^{7/2}}+\frac{627806 \sqrt{1-2 x} \sqrt{3+5 x}}{10696455 (2+3 x)^{5/2}}+\frac{19417096 \sqrt{1-2 x} \sqrt{3+5 x}}{74875185 (2+3 x)^{3/2}}+\frac{1305025844 \sqrt{1-2 x} \sqrt{3+5 x}}{524126295 \sqrt{2+3 x}}-\frac{118 \sqrt{1-2 x} (3+5 x)^{3/2}}{2079 (2+3 x)^{9/2}}-\frac{2 \sqrt{1-2 x} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}-\frac{1305025844 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}}-\frac{37904696 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{47647845 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.300951, size = 112, normalized size = 0.45 \[ \frac{-10873573760 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{48 \sqrt{2-4 x} \sqrt{5 x+3} \left (158560640046 x^5+534040213536 x^4+719808574005 x^3+484598540169 x^2+162787885893 x+21813966691\right )}{(3 x+2)^{11/2}}[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{12579031080 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(21813966691 + 162787885893*x + 484598540169*x^2 + 719808574005*x^3 + 5340402
13536*x^4 + 158560640046*x^5))/(2 + 3*x)^(11/2) + 20880413504*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/
2] - 10873573760*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(12579031080*Sqrt[2])

________________________________________________________________________________________

Maple [C]  time = 0.036, size = 599, normalized size = 2.4 \begin{align*}{\frac{2}{15723788850\,{x}^{2}[phone]\,x-4717136655} \left ( 82571200740\,\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}-158560640046\,\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}-528535466820\,\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}-704713955760\,\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}-469809303840\,\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}-156603101280\,\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 ) -20880413504\,\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}-140844968748\,{x}^{3}-3218604203112\,{x}^{2}-1399649072964\,x-196325700219 \right ) \sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 2+3\,x \right ) ^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1572378885*(82571200740*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)-158560640046*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)-528535466820*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)-704713955760*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)-469809303840*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+81551803200*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)-156603101280*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+10873573760*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))-20880413504*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+11891020005261*x^4-140844968748*x^3-3218604203112*x^2-1399649072964*x-196325700219)*(1-2*x)^(1/2)*(3+5*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}} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(13/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*sqrt(-2*x + 1)/(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 (25 \, x^{2} + 30 \, 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((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(13/2),x, algorithm="fricas")

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 22
680*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((3+5*x)**(5/2)*(1-2*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{{\left (5 \, x + 3\right )}^{\frac{5}{2}} \sqrt{-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac{13}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(13/2),x, algorithm="giac")

[Out]

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