3.2783 \(\int \frac {(1-2 x)^{5/2} (3+5 x)^{5/2}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac {13235368 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{5250987 \sqrt {33}}+\frac {36980 \sqrt {1-2 x} (5 x+3)^{5/2}}{18711 (3 x+2)^{7/2}}+\frac {370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{891 (3 x+2)^{9/2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac {55772 \sqrt {1-2 x} (5 x+3)^{3/2}}{43659 (3 x+2)^{5/2}}+\frac {584888452 \sqrt {1-2 x} \sqrt {5 x+3}}{57760857 \sqrt {3 x+2}}-\frac {17089252 \sqrt {1-2 x} \sqrt {5 x+3}}{8251551 (3 x+2)^{3/2}}-\frac {584888452 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250987 \sqrt {33}} \]

[Out]

-2/33*(1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(11/2)+370/891*(1-2*x)^(3/2)*(3+5*x)^(5/2)/(2+3*x)^(9/2)-584888452/1
73282571*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-13235368/173282571*EllipticF(1/7*21^(1
/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-55772/43659*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(5/2)+36980/18711*
(3+5*x)^(5/2)*(1-2*x)^(1/2)/(2+3*x)^(7/2)-17089252/8251551*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)[phone]
/57760857*(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} \[ \frac {36980 \sqrt {1-2 x} (5 x+3)^{5/2}}{18711 (3 x+2)^{7/2}}+\frac {370 (1-2 x)^{3/2} (5 x+3)^{5/2}}{891 (3 x+2)^{9/2}}-\frac {2 (1-2 x)^{5/2} (5 x+3)^{5/2}}{33 (3 x+2)^{11/2}}-\frac {55772 \sqrt {1-2 x} (5 x+3)^{3/2}}{43659 (3 x+2)^{5/2}}+\frac {584888452 \sqrt {1-2 x} \sqrt {5 x+3}}{57760857 \sqrt {3 x+2}}-\frac {17089252 \sqrt {1-2 x} \sqrt {5 x+3}}{8251551 (3 x+2)^{3/2}}-\frac {13235368 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250987 \sqrt {33}}-\frac {584888452 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250987 \sqrt {33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-17089252*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(8251551*(2 + 3*x)^(3/2)) + (584888452*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(5
7760857*Sqrt[2 + 3*x]) - (55772*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(43659*(2 + 3*x)^(5/2)) - (2*(1 - 2*x)^(5/2)*(3
+ 5*x)^(5/2))/(33*(2 + 3*x)^(11/2)) + (370*(1 - 2*x)^(3/2)*(3 + 5*x)^(5/2))/(891*(2 + 3*x)^(9/2)) + (36980*Sq
rt[1 - 2*x]*(3 + 5*x)^(5/2))/(18711*(2 + 3*x)^(7/2)) - (584888452*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 3
5/33])/(5250987*Sqrt[33]) - (13235368*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(5250987*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)^{13/2}} \, dx &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (-\frac {5}{2}-50 x\right ) (1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {\sqrt {1-2 x} (3+5 x)^{3/2} \left (-\frac {3065}{2}+\frac {25 x}{2}\right )}{(2+3 x)^{9/2}} \, dx\\ &=-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac {8 \int \frac {\left (\frac {147745}{4}-23025 x\right ) (3+5 x)^{3/2}}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx}{18711}\\ &=-\frac {55772 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac {16 \int \frac {\left (\frac {14799465}{8}-\frac {4921575 x}{4}\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{5/2}} \, dx}{1964655}\\ &=-\frac {17089252 \sqrt {1-2 x} \sqrt {3+5 x}}{8251551 (2+3 x)^{3/2}}-\frac {55772 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac {32 \int \frac {\frac {469321365}{16}-\frac {49085475 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{123773265}\\ &=-\frac {17089252 \sqrt {1-2 x} \sqrt {3+5 x}}{8251551 (2+3 x)^{3/2}}+\frac {584888452 \sqrt {1-2 x} \sqrt {3+5 x}}{57760857 \sqrt {2+3 x}}-\frac {55772 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac {64 \int \frac {\frac {3426487275}{8}+\frac {10966658475 x}{16}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{866412855}\\ &=-\frac {17089252 \sqrt {1-2 x} \sqrt {3+5 x}}{8251551 (2+3 x)^{3/2}}+\frac {584888452 \sqrt {1-2 x} \sqrt {3+5 x}}{57760857 \sqrt {2+3 x}}-\frac {55772 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}+\frac {6617684 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{5250987}+\frac {584888452 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{57760857}\\ &=-\frac {17089252 \sqrt {1-2 x} \sqrt {3+5 x}}{8251551 (2+3 x)^{3/2}}+\frac {584888452 \sqrt {1-2 x} \sqrt {3+5 x}}{57760857 \sqrt {2+3 x}}-\frac {55772 \sqrt {1-2 x} (3+5 x)^{3/2}}{43659 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{5/2} (3+5 x)^{5/2}}{33 (2+3 x)^{11/2}}+\frac {370 (1-2 x)^{3/2} (3+5 x)^{5/2}}{891 (2+3 x)^{9/2}}+\frac {36980 \sqrt {1-2 x} (3+5 x)^{5/2}}{18711 (2+3 x)^{7/2}}-\frac {584888452 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250987 \sqrt {33}}-\frac {13235368 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{5250987 \sqrt {33}}\\ \end {align*}

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Mathematica [A]  time = 0.43, size = 112, normalized size = 0.45 \[ \frac {-5864078080 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {48 \sqrt {2-4 x} \sqrt {5 x+3} \left (71063946918 x^5+237923150688 x^4+320012032635 x^3+215597947743 x^2+72620507583 x+9770732477\right )}{(3 x+2)^{11/2}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{1386260568 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((48*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(9770732477 + 72620507583*x + 215597947743*x^2 + 320012032635*x^3 + 237923150
688*x^4 + 71063946918*x^5))/(2 + 3*x)^(11/2) + 9358215232*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] -
5864078080*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(1386260568*Sqrt[2])

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fricas [F]  time = 0.77, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (100 \, x^{4} + 20 \, x^{3} - 59 \, x^{2} - 6 \, x + 9\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(3+5*x)^(5/2)/(2+3*x)^(13/2),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)/(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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((5*x + 3)^(5/2)*(-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 (2131918407540 x^{7}[phone] x^{6}-71063946918 \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}-236879823060 \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}-315839764080 \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 )-54699222996 x^{3}-210559842720 \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 )-1429398032628 x^{2}-70186614240 \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 )-624272370816 x -9358215232 \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 )-87936592293\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{173282571 \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)^(5/2)/(3*x+2)^(13/2),x)

[Out]

2/173282571*(44530342920*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)-71063946918*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)-236879823060*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)-315839764080*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)-210559842720*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+43980585600*2^(1/2)*EllipticF(1/1
1*(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)-70186614240*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+586407808
0*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))-935821523
2*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]
8852*x^5+5286666174003*x^4-54699222996*x^3-1429398032628*x^2-624272370816*x-87936592293)*(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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {5}{2}} {\left (-2 \, x + 1\right )}^{\frac {5}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((5*x + 3)^(5/2)*(-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 \[ \int \frac {{\left (1-2\,x\right )}^{5/2}\,{\left (5\,x+3\right )}^{5/2}}{{\left (3\,x+2\right )}^{13/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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