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

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

[Out]

-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 = 0.14, 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 = {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 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)]))

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]

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 = 0.44, size = 122, normalized size = 0.39 \[ \frac {-203774903306240 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {96 \sqrt {2-4 x} \sqrt {5 x+3} \left (13823602234657668 x^7+64974368463330312 x^6+130900492508039982 x^5+146528498784887100 x^4+98427465692862075 x^3+39676146370896231 x^2+8886579657279639 x+853124799464729\right )}{(3 x+2)^{15/2}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{8830479818160 \sqrt {2}} \]

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])

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fricas [F]  time = 0.79, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{19683 \, x^{9} + 118098 \, x^{8} + 314928 \, x^{7} + 489888 \, x^{6} + 489888 \, x^{5} + 326592 \, x^{4} + 145152 \, x^{3} + 41472 \, x^{2} + 6912 \, x + 512}, x\right ) \]

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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {17}{2}}}\,{d x} \]

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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maple [C]  time = 0.05, size = 789, normalized size = 2.54 \[ -\frac {2 \left (77419842517122564 x -3997525460519271384 x^{7}+500221362404680812 x^{3}-166810299141489255 x^{4}+304831834382285292 x^{2}-2214305034568163712 x^{5}-4203787124900760138 x^{6}+8495162964508416 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-4279272969431040 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1990701860603882364 x^{8}-414708067039730040 x^{9}[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-407549806612480 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+38228233340287872 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-19256728362439680 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+143355875026079520 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-72212731359148800 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+95570583350719680 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-48141820906099200 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+129020287523471568 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{5} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-64991458223233920 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{5} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+13823602234657668 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{7} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-6963370523917920 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{7} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+64510143761735784 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{6} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-32495729111616960 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{6} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+7678123195182561\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{551904988635 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {15}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/551904988635*(77419842517122564*x+64510143761735784*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))
*x^6*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-72212731359148800*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*
I*66^(1/2))*x^4*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-64991458223233920*2^(1/2)*EllipticF(1/11*(110*x+66)
^(1/2),1/2*I*66^(1/2))*x^5*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-48141820906099200*2^(1/2)*EllipticF(1/11
*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^3*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-19256728362439680*2^(1/2)*Ell
ipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^2*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-4279272969431040*2
^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-3997525460
519271384*x^7+500221362404680812*x^3-166810299141489255*x^4+304831834382285292*x^2-2214305034568163712*x^5-420
3787124900760138*x^6+8495162964508416*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x*(5*x+3)^(1/2)*
(3*x+2)^(1/2)*(-2*x+1)^(1/2)-1990701860603882364*x^8+95570583350719680*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2)
,1/2*I*66^(1/2))*x^3*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)+38228233340287872*2^(1/2)*EllipticE(1/11*(110*
x+66)^(1/2),1/2*I*66^(1/2))*x^2*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)+143355875026079520*2^(1/2)*Elliptic
E(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^4*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-414708067039730040*x^9+
129020287523471568*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^5*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2
*x+1)^(1/2)-407549806612480*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticF(1/11*(110*x+66)^(1/2)
,1/2*I*66^(1/2))[phone]*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticE(1/11*(110*x+66)^
(1/2),1/2*I*66^(1/2))-32495729111616960*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^6*(5*x+3)^(1
/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)+13823602234657668*2^(1/2)*EllipticE(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))*x^7
*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)-6963370523917920*2^(1/2)*EllipticF(1/11*(110*x+66)^(1/2),1/2*I*66^
(1/2))*x^7*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)+7678123195182561)*(5*x+3)^(1/2)*(-2*x+1)^(1/2)/(10*x^2+x
-3)/(3*x+2)^(15/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {17}{2}}}\,{d x} \]

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

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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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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