3.2773 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac {23441272 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{1750329 \sqrt {33}}-\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{891 (3 x+2)^{9/2}}+\frac {12280 (5 x+3)^{3/2} \sqrt {1-2 x}}{6237 (3 x+2)^{7/2}}+\frac {780320008 \sqrt {5 x+3} \sqrt {1-2 x}}{19253619 \sqrt {3 x+2}}+\frac {11243972 \sqrt {5 x+3} \sqrt {1-2 x}}{2750517 (3 x+2)^{3/2}}-\frac {325796 \sqrt {5 x+3} \sqrt {1-2 x}}{130977 (3 x+2)^{5/2}}-\frac {780320008 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}} \]

[Out]

-2/33*(1-2*x)^(5/2)*(3+5*x)^(3/2)/(2+3*x)^(11/2)+230/891*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(9/2)-780320008/5
7760857*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-23441272/57760857*EllipticF(1/7*21^(1/2
)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+12280/6237*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(7/2)-325796/130977*(
1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)[phone]/2750517*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)[phone]/
19253619*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]  time = 0.10, antiderivative size = 249, normalized size of antiderivative = 1.00, 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} \[ -\frac {2 (5 x+3)^{3/2} (1-2 x)^{5/2}}{33 (3 x+2)^{11/2}}+\frac {230 (5 x+3)^{3/2} (1-2 x)^{3/2}}{891 (3 x+2)^{9/2}}+\frac {12280 (5 x+3)^{3/2} \sqrt {1-2 x}}{6237 (3 x+2)^{7/2}}+\frac {780320008 \sqrt {5 x+3} \sqrt {1-2 x}}{19253619 \sqrt {3 x+2}}+\frac {11243972 \sqrt {5 x+3} \sqrt {1-2 x}}{2750517 (3 x+2)^{3/2}}-\frac {325796 \sqrt {5 x+3} \sqrt {1-2 x}}{130977 (3 x+2)^{5/2}}-\frac {23441272 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}}-\frac {780320008 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-325796*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(130977*(2 + 3*x)^(5/2)) + (11243972*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(27505
17*(2 + 3*x)^(3/2)) + (780320008*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(19253619*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(5/2)*(3
+ 5*x)^(3/2))/(33*(2 + 3*x)^(11/2)) + (230*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(891*(2 + 3*x)^(9/2)) + (12280*Sq
rt[1 - 2*x]*(3 + 5*x)^(3/2))/(6237*(2 + 3*x)^(7/2)) - (780320008*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/(1750329*Sqrt[33]) - (23441272*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(1750329*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)^{3/2}}{(2+3 x)^{13/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (-\frac {15}{2}-40 x\right ) (1-2 x)^{3/2} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {\sqrt {1-2 x} \sqrt {3+5 x} \left (-1200+\frac {1005 x}{2}\right )}{(2+3 x)^{9/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {8 \int \frac {\left (\frac {232425}{4}-\frac {131115 x}{2}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx}{18711}\\ &=-\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {16 \int \frac {\frac {7896165}{8}-1154775 x}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{1964655}\\ &=-\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {32 \int \frac {\frac {347150355}{8}-\frac {210824475 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{41257755}\\ &=-\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac {780320008 \sqrt {1-2 x} \sqrt {3+5 x}}{19253619 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {64 \int \frac {\frac {9262076325}{16}+\frac {7315500075 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{288804285}\\ &=-\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac {780320008 \sqrt {1-2 x} \sqrt {3+5 x}}{19253619 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}+\frac {11720636 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{1750329}+\frac {780320008 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{19253619}\\ &=-\frac {325796 \sqrt {1-2 x} \sqrt {3+5 x}}{130977 (2+3 x)^{5/2}}+\frac {11243972 \sqrt {1-2 x} \sqrt {3+5 x}}{2750517 (2+3 x)^{3/2}}+\frac {780320008 \sqrt {1-2 x} \sqrt {3+5 x}}{19253619 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {230 (1-2 x)^{3/2} (3+5 x)^{3/2}}{891 (2+3 x)^{9/2}}+\frac {12280 \sqrt {1-2 x} (3+5 x)^{3/2}}{6237 (2+3 x)^{7/2}}-\frac {780320008 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}}-\frac {23441272 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{1750329 \sqrt {33}}\\ \end {align*}

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Mathematica [A]  time = 0.40, size = 115, normalized size = 0.46 \[ \frac {16 \sqrt {2} \left (195080002 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )-98384755 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )\right )+\frac {24 \sqrt {1-2 x} \sqrt {5 x+3} \left (94808880972 x^5+319217269302 x^4+429993423180 x^3+289719086787 x^2+97637232762 x+13163824553\right )}{(3 x+2)^{11/2}}}{231043428} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((24*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(13163824553 + 97637232762*x + 289719086787*x^2 + 429993423180*x^3 + 31921726
9302*x^4 + 94808880972*x^5))/(2 + 3*x)^(11/2) + 16*Sqrt[2]*(195080002*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x
]], -33/2] - 98384755*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/231043428

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fricas [F]  time = 1.24, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (20 \, x^{3} - 8 \, x^{2} - 7 \, x + 3\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((20*x^3 - 8*x^2 - 7*x + 3)*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)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.03, size = 599, normalized size = 2.41 \[ \frac {2 \left (2844266429160 x^{7}[phone] x^{6}-94808880972 \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 )[phone] \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 )[phone] x^{5}-316029603240 \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 )[phone] \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 )[phone] x^{4}-421372804320 \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 )[phone] \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 )-71666565399 x^{3}-280915202880 \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 )[phone] \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 )-1919645346207 x^{2}-93638400960 \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 )[phone] \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 )-839243621199 x -12485120128 \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 )[phone] \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 )-118474420977\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{57760857 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/57760857*(47814990930*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)-94808880972*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)[phone]*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)-316029603240*2^(1/2)*EllipticE(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)[phone]*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)-421372804320*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)[phone]*2^(1/2)*EllipticF(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)-280915202880*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)[phone]*x^7+47224682400*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)-93638400960*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)[phone]*x^6+6296624320
*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))-1248512012
8*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))[phone]
74558*x^5+7108597449432*x^4-71666565399*x^3-1919645346207*x^2-839243621199*x-118474420977)*(5*x+3)^(1/2)*(-2*x
+1)^(1/2)/(10*x^2+x-3)/(3*x+2)^(11/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{13/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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