3.28.85 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{17/2}} \, dx\) [2785]

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

[Out]

-2/45*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(15/2)+74/351*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2)-12641611554
328/551904988635*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-380220959152/551904988635*Elli
pticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1085156/729729*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^
(9/2)+16636/11583*(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2)-112817764/107270163*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(
2+3*x)^(7/2)[phone]/3754455705*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)[phone]/26281189935*(1-2*x)^
(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)[phone]/183968329545*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.09, antiderivative size = 311, normalized size of antiderivative = 1.00, 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 = {99, 155, 157, 164, 114, 120} \begin {gather*} -\frac {380220959152 F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{16724393595 \sqrt {33}}-\frac {12641611554328 E\left (\text {ArcSin}\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}} \end {gather*}

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 99

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 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*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)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-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]))

Rule 155

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 157

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 164

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]

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*}

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Mathematica [A]
time = 9.32, size = 122, normalized size = 0.39 \begin {gather*} \frac {\frac {96 \sqrt {2-4 x} \sqrt {3+5 x} \left (853124799464729+8886579657279639 x+39676146370896231 x^2+98427465692862075 x^3+146528498784887100 x^4+130900492508039982 x^5+64974368463330312 x^6+13823602234657668 x^7\right )}{(2+3 x)^{15/2}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-203774903306240 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{8830479818160 \sqrt {2}} \end {gather*}

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])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(772\) vs. \(2(231)=462\).
time = 0.10, size = 773, normalized size = 2.49

method	result	size
elliptic	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {1813814 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{531972441 \left (\frac {2}{3}+x \right )^{5}}-\frac {641434 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{379980315 \left (\frac {2}{3}+x \right )^{6}}+\frac {16058 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{103630995 \left (\frac {2}{3}+x \right )^{7}}-\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{23914845 \left (\frac {2}{3}+x \right )^{8}}+\frac {3914701972 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{101370304035 \left (\frac {2}{3}+x \right )^{3}}+\frac {1513936 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8688883203 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {25283223108656}{36793665909} x^{2}-\frac {12641611554328}{183968329545} x +\frac {12641611554328}{61322776515}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {181941877952 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{236530709415 \left (\frac {2}{3}+x \right )^{2}}+\frac {8003209987664 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{772666984089 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {12641611554328 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{772666984089 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\)	\(380\)
default	\(\frac {2 \left (-7678123195182561-77419842517122564 x +13823602234657668 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{7} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-6860231710739748 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{7} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+129020287523471568 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-64028829300237648 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{5} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-71143143666930720 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+143355875026079520 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+95570583350719680 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-47428762444620480 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-18971504977848192 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+38228233340287872 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-4215889995077376 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}+8495162964508416 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-500221362404680812 x^{3}+166810299141489255 x^{4}+4203787124900760138 x^{6}+3997525460519271384 x^{7}+2214305034568163712 x^{5}+1990701860603882364 x^{8}+414708067039730040 x^{9}-304831834382285292 x^{2}-401513332864512 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+64510143761735784 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-32014414650118824 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{6} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{551904988635 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {15}{2}}}\)	\(773\)

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,method=_RETURNVERBOSE)

[Out]

2/551904988635*(-7678123195182561-77419842517122564*x+13823602234657668*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),
1/2*70^(1/2))*x^7*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-6860231710739748*2^(1/2)*EllipticF(1/7*(28+42*x)^
(1/2),1/2*70^(1/2))*x^7*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+64510143761735784*2^(1/2)*EllipticE(1/7*(28
+42*x)^(1/2),1/2*70^(1/2))*x^6*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-32014414650118824*2^(1/2)*EllipticF(
1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^6*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-64028829300237648*2^(1/2)*Ell
ipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^5*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+129020287523471568*2^(
1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^5*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-71143143666930
720*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^4*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+1433558
75026079520*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^4*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)
-47428762444620480*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x
)^(1/2)+95570583350719680*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)
*(1-2*x)^(1/2)-18971504977848192*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x
)^(1/2)*(1-2*x)^(1/2)+38228233340287872*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*
(-3-5*x)^(1/2)*(1-2*x)^(1/2)-4215889995077376*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1
/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)+8495162964508416*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x
)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)-500221362404680812*x^3+166810299141489255*x^4+4203787124900760138*x^6+399
7525460519271384*x^7+2214305034568163712*x^5+1990701860603882364*x^8+414708067039730040*x^9-304831834382285292
*x^2-401513332864512*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(
1/2))[phone]*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^
(1/2)))*(3+5*x)^(1/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(15/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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)

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Fricas [A]
time = 0.18, size = 100, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (13823602234657668 \, x^{7} + 64974368463330312 \, x^{6} + 130900492508039982 \, x^{5} + 146528498784887100 \, x^{4} + 98427465692862075 \, x^{3} + 39676146370896231 \, x^{2} + 8886579657279639 \, x + 853124799464729\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{183968329545 \, {\left (6561 \, x^{8} + 34992 \, x^{7} + 81648 \, x^{6} + 108864 \, x^{5} + 90720 \, x^{4} + 48384 \, x^{3} + 16128 \, x^{2} + 3072 \, x + 256\right )}} \end {gather*}

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]

2/183968329545*(13823602234657668*x^7 + 64974368463330312*x^6 + 130900492508039982*x^5 + 146528498784887100*x^
4 + 98427465692862075*x^3 + 39676146370896231*x^2 + 8886579657279639*x + 853124799464729)*sqrt(5*x + 3)*sqrt(3
*x + 2)*sqrt(-2*x + 1)/(6561*x^8 + 34992*x^7 + 81648*x^6 + 108864*x^5 + 90720*x^4 + 48384*x^3 + 16128*x^2 + 30
72*x + 256)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{17/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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