3.2785 \(\int \frac{(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx\)

Optimal. Leaf size=311 \[ -\frac{380220959152 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{16724393595 \sqrt{33}}+\frac{16636 \sqrt{1-2 x} (5 x+3)^{5/2}}{11583 (3 x+2)^{11/2}}+\frac{74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{351 (3 x+2)^{13/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}-\frac{1085156 \sqrt{1-2 x} (5 x+3)^{3/2}}{729729 (3 x+2)^{9/2}}+\frac{12641611554328 \sqrt{1-2 x} \sqrt{5 x+3}}{183968329545 \sqrt{3 x+2}}+\frac{181941877952 \sqrt{1-2 x} \sqrt{5 x+3}}{26281189935 (3 x+2)^{3/2}}+\frac{3914701972 \sqrt{1-2 x} \sqrt{5 x+3}}{3754455705 (3 x+2)^{5/2}}-\frac{112817764 \sqrt{1-2 x} \sqrt{5 x+3}}{107270163 (3 x+2)^{7/2}}-\frac{12641611554328 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16724393595 \sqrt{33}} \]

[Out]

(-112817764*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(107270163*(2 + 3*x)^(7/2)) + (3914701972*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(3754455705*(2 + 3*x)^(5/2)) + (181941877952*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(26281189935*(2 + 3*x)^(3/2)) + (1
2641611554328*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(183968329545*Sqrt[2 + 3*x]) - (1085156*Sqrt[1 - 2*x]*(3 + 5*x)^(3/
2))/(729729*(2 + 3*x)^(9/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(45*(2 + 3*x)^(15/2)) + (74*(1 - 2*x)^(3/2)
*(3 + 5*x)^(5/2))/(351*(2 + 3*x)^(13/2)) + (16636*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(11583*(2 + 3*x)^(11/2)) - (1
2641611554328*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(16724393595*Sqrt[33]) - (380220959152*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(16724393595*Sqrt[33])

________________________________________________________________________________________

Rubi [A]  time = 0.14136, antiderivative size = 311, normalized size of antiderivative = 1., number of steps used = 11, 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} \[ \frac{16636 \sqrt{1-2 x} (5 x+3)^{5/2}}{11583 (3 x+2)^{11/2}}+\frac{74 (1-2 x)^{3/2} (5 x+3)^{5/2}}{351 (3 x+2)^{13/2}}-\frac{2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{45 (3 x+2)^{15/2}}-\frac{1085156 \sqrt{1-2 x} (5 x+3)^{3/2}}{729729 (3 x+2)^{9/2}}+\frac{12641611554328 \sqrt{1-2 x} \sqrt{5 x+3}}{183968329545 \sqrt{3 x+2}}+\frac{181941877952 \sqrt{1-2 x} \sqrt{5 x+3}}{26281189935 (3 x+2)^{3/2}}+\frac{3914701972 \sqrt{1-2 x} \sqrt{5 x+3}}{3754455705 (3 x+2)^{5/2}}-\frac{112817764 \sqrt{1-2 x} \sqrt{5 x+3}}{107270163 (3 x+2)^{7/2}}-\frac{380220959152 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16724393595 \sqrt{33}}-\frac{12641611554328 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16724393595 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(17/2),x]

[Out]

(-112817764*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(107270163*(2 + 3*x)^(7/2)) + (3914701972*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]
)/(3754455705*(2 + 3*x)^(5/2)) + (181941877952*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(26281189935*(2 + 3*x)^(3/2)) + (1
2641611554328*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(183968329545*Sqrt[2 + 3*x]) - (1085156*Sqrt[1 - 2*x]*(3 + 5*x)^(3/
2))/(729729*(2 + 3*x)^(9/2)) - (2*(1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(45*(2 + 3*x)^(15/2)) + (74*(1 - 2*x)^(3/2)
*(3 + 5*x)^(5/2))/(351*(2 + 3*x)^(13/2)) + (16636*Sqrt[1 - 2*x]*(3 + 5*x)^(5/2))/(11583*(2 + 3*x)^(11/2)) - (1
2641611554328*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(16724393595*Sqrt[33]) - (380220959152*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(16724393595*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)^{5/2}}{(2+3 x)^{17/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac{2}{45} \int \frac{\left (-\frac{5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{15/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}-\frac{4 \int \frac{\sqrt{1-2 x} (3+5 x)^{3/2} \left (-\frac{4715}{2}+\frac{3325 x}{2}\right )}{(2+3 x)^{13/2}} \, dx}{1755}\\ &=-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac{16636 \sqrt{1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac{8 \int \frac{\left (\frac{712045}{4}-241650 x\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} (2+3 x)^{11/2}} \, dx}{57915}\\ &=-\frac{1085156 \sqrt{1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac{16636 \sqrt{1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac{16 \int \frac{\left (\frac{73680705}{8}-\frac{50506125 x}{4}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} (2+3 x)^{9/2}} \, dx}{10945935}\\ &=-\frac{112817764 \sqrt{1-2 x} \sqrt{3+5 x}}{107270163 (2+3 x)^{7/2}}-\frac{1085156 \sqrt{1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac{16636 \sqrt{1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac{32 \int \frac{\frac{2496930465}{16}-\frac{898667625 x}{4}}{\sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}} \, dx}{1609052445}\\ &=-\frac{112817764 \sqrt{1-2 x} \sqrt{3+5 x}}{107270163 (2+3 x)^{7/2}}+\frac{3914701972 \sqrt{1-2 x} \sqrt{3+5 x}}{3754455705 (2+3 x)^{5/2}}-\frac{1085156 \sqrt{1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac{16636 \sqrt{1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac{64 \int \frac{\frac{97169848605}{8}-\frac{220201985925 x}{16}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx}{56316835575}\\ &=-\frac{112817764 \sqrt{1-2 x} \sqrt{3+5 x}}{107270163 (2+3 x)^{7/2}}+\frac{3914701972 \sqrt{1-2 x} \sqrt{3+5 x}}{3754455705 (2+3 x)^{5/2}}+\frac{181941877952 \sqrt{1-2 x} \sqrt{3+5 x}}{26281189935 (2+3 x)^{3/2}}-\frac{1085156 \sqrt{1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac{16636 \sqrt{1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac{128 \int \frac{\frac{16880201241165}{32}-\frac{639639414675 x}{2}}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx}{1182653547075}\\ &=-\frac{112817764 \sqrt{1-2 x} \sqrt{3+5 x}}{107270163 (2+3 x)^{7/2}}+\frac{3914701972 \sqrt{1-2 x} \sqrt{3+5 x}}{3754455705 (2+3 x)^{5/2}}+\frac{181941877952 \sqrt{1-2 x} \sqrt{3+5 x}}{26281189935 (2+3 x)^{3/2}}+\frac{12641611554328 \sqrt{1-2 x} \sqrt{3+5 x}}{183968329545 \sqrt{2+3 x}}-\frac{1085156 \sqrt{1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac{16636 \sqrt{1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac{256 \int \frac{\frac{112545140451525}{16}+\frac{355545324965475 x}{32}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{8278574829525}\\ &=-\frac{112817764 \sqrt{1-2 x} \sqrt{3+5 x}}{107270163 (2+3 x)^{7/2}}+\frac{3914701972 \sqrt{1-2 x} \sqrt{3+5 x}}{3754455705 (2+3 x)^{5/2}}+\frac{181941877952 \sqrt{1-2 x} \sqrt{3+5 x}}{26281189935 (2+3 x)^{3/2}}+\frac{12641611554328 \sqrt{1-2 x} \sqrt{3+5 x}}{183968329545 \sqrt{2+3 x}}-\frac{1085156 \sqrt{1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac{16636 \sqrt{1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}+\frac{190110479576 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{16724393595}+\frac{12641611554328 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{183968329545}\\ &=-\frac{112817764 \sqrt{1-2 x} \sqrt{3+5 x}}{107270163 (2+3 x)^{7/2}}+\frac{3914701972 \sqrt{1-2 x} \sqrt{3+5 x}}{3754455705 (2+3 x)^{5/2}}+\frac{181941877952 \sqrt{1-2 x} \sqrt{3+5 x}}{26281189935 (2+3 x)^{3/2}}+\frac{12641611554328 \sqrt{1-2 x} \sqrt{3+5 x}}{183968329545 \sqrt{2+3 x}}-\frac{1085156 \sqrt{1-2 x} (3+5 x)^{3/2}}{729729 (2+3 x)^{9/2}}-\frac{2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{45 (2+3 x)^{15/2}}+\frac{74 (1-2 x)^{3/2} (3+5 x)^{5/2}}{351 (2+3 x)^{13/2}}+\frac{16636 \sqrt{1-2 x} (3+5 x)^{5/2}}{11583 (2+3 x)^{11/2}}-\frac{12641611554328 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16724393595 \sqrt{33}}-\frac{380220959152 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{16724393595 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.320808, size = 122, normalized size = 0.39 \[ \frac{-203774903306240 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{96 \sqrt{2-4 x} \sqrt{5 x+3} \left (13823602234657668 x^7+64974368463330312 x^6+130900492508039982 x^5+146528498784887100 x^4+98427465692862075 x^3+39676146370896231 x^2+8886579657279639 x+853124799464729\right )}{(3 x+2)^{15/2}}[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{8830479818160 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2))/(2 + 3*x)^(17/2),x]

[Out]

((96*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(853124799464729 + 8886579657279639*x + 39676146370896231*x^2 + 9842746569286
2075*x^3 + 146528498784887100*x^4 + 130900492508039982*x^5 + 64974368463330312*x^6 + 13823602234657668*x^7))/(
2 + 3*x)^(15/2) + 404531569738496*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 203774903306240*Ellipti
cF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(8830479818160*Sqrt[2])

________________________________________________________________________________________

Maple [C]  time = 0.042, size = 789, normalized size = 2.5 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(17/2),x)

[Out]

2/551904988635*(-7678123195182561-77419842517122564*x+32495729111616960*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2
),1/2*I*66^(1/2))*x^6*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+64991458223233920*2^(1/2)*EllipticF(1/11*(66+1
10*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)+72212731359148800*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)-304831834382285292*x^2+221
4305034568163712*x^5+166810299141489255*x^4-500221362404680812*x^3+6963370523917920*2^(1/2)*EllipticF(1/11*(66
+110*x)^(1/2),1/2*I*66^(1/2))*x^7*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-13823602234657668*2^(1/2)*Elliptic
E(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^7*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+1990701860603882364*x^8+
4203787124900760138*x^6+3997525460519271384*x^7+414708067039730040*x^9+48141820906099200*2^(1/2)*EllipticF(1/1
1*(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)+4279272969431040*2^(1/2)*Elli
pticF(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)+19256728362439680*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)-645101437617
35784*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^6*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)-38
228233340287872*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)-143355875026079520*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)*(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))-809063139476992*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/11*(66+11
0*x)^(1/2),1/2*I*66^(1/2))-8495162964508416*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)-95570583350719680*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)-129020287523471568*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*6
6^(1/2))*x^5*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(15/2
)

________________________________________________________________________________________

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{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{17}{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)^(5/2)/(2+3*x)^(17/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(17/2), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{19683 \, x^{9} + 118098 \, x^{8} + 314928 \, x^{7} + 489888 \, x^{6} + 489888 \, x^{5} + 326592 \, x^{4} + 145152 \, x^{3} + 41472 \, x^{2} + 6912 \, x + 512}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(17/2),x, algorithm="fricas")

[Out]

integral((100*x^4 + 20*x^3 - 59*x^2 - 6*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(19683*x^9 + 118098*
x^8 + 314928*x^7 + 489888*x^6 + 489888*x^5 + 326592*x^4 + 145152*x^3 + 41472*x^2 + 6912*x + 512), x)

________________________________________________________________________________________

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)**(5/2)/(2+3*x)**(17/2),x)

[Out]

Timed out

________________________________________________________________________________________

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{5}{2}}}{{\left (3 \, x + 2\right )}^{\frac{17}{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)^(5/2)/(2+3*x)^(17/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)/(3*x + 2)^(17/2), x)