3.2717 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx\)

Optimal. Leaf size=249 \[ -\frac {43537016 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{6806835 \sqrt {33}}+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{297 (3 x+2)^{9/2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}+\frac {1446357824 \sqrt {1-2 x} \sqrt {5 x+3}}{74875185 \sqrt {3 x+2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {5 x+3}}{10696455 (3 x+2)^{3/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {5 x+3}}{1528065 (3 x+2)^{5/2}}-\frac {12872 \sqrt {1-2 x} \sqrt {5 x+3}}{43659 (3 x+2)^{7/2}}-\frac {1446357824 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835 \sqrt {33}} \]

[Out]

-2/33*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(11/2)-1446357824/224625555*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/3
3*1155^(1/2))*33^(1/2)-43537016/224625555*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+74/29
7*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2)-12872/43659*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+442076/15280
65*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)[phone]/10696455*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)+14463
57824/74875185*(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} \[ \frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{297 (3 x+2)^{9/2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{33 (3 x+2)^{11/2}}+\frac {1446357824 \sqrt {1-2 x} \sqrt {5 x+3}}{74875185 \sqrt {3 x+2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {5 x+3}}{10696455 (3 x+2)^{3/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {5 x+3}}{1528065 (3 x+2)^{5/2}}-\frac {12872 \sqrt {1-2 x} \sqrt {5 x+3}}{43659 (3 x+2)^{7/2}}-\frac {43537016 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835 \sqrt {33}}-\frac {1446357824 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835 \sqrt {33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-12872*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(43659*(2 + 3*x)^(7/2)) + (442076*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(1528065*(
2 + 3*x)^(5/2)) + (20799916*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(10696455*(2 + 3*x)^(3/2)) + (1446357824*Sqrt[1 - 2*x
]*Sqrt[3 + 5*x])/(74875185*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2))/(33*(2 + 3*x)^(11/2)) + (74*Sq
rt[1 - 2*x]*(3 + 5*x)^(3/2))/(297*(2 + 3*x)^(9/2)) - (1446357824*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35
/33])/(6806835*Sqrt[33]) - (43537016*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(6806835*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)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{13/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {2}{33} \int \frac {\left (-\frac {3}{2}-30 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{11/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {4}{891} \int \frac {\sqrt {3+5 x} \left (-864+\frac {2235 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx\\ &=-\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {8 \int \frac {-\frac {67269}{4}+\frac {64875 x}{4}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{130977}\\ &=-\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {3+5 x}}{1528065 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {16 \int \frac {-\frac {8968797}{8}+\frac {4973355 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{4584195}\\ &=-\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {32 \int \frac {-\frac {193192407}{4}+\frac {233999055 x}{8}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{96268095}\\ &=-\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{3/2}}+\frac {1446357824 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {64 \int \frac {-\frac {10301685885}{16}-1016970345 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{673876665}\\ &=-\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{3/2}}+\frac {1446357824 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}+\frac {21768508 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6806835}+\frac {1446357824 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{74875185}\\ &=-\frac {12872 \sqrt {1-2 x} \sqrt {3+5 x}}{43659 (2+3 x)^{7/2}}+\frac {442076 \sqrt {1-2 x} \sqrt {3+5 x}}{1528065 (2+3 x)^{5/2}}+\frac {20799916 \sqrt {1-2 x} \sqrt {3+5 x}}{10696455 (2+3 x)^{3/2}}+\frac {1446357824 \sqrt {1-2 x} \sqrt {3+5 x}}{74875185 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{33 (2+3 x)^{11/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{297 (2+3 x)^{9/2}}-\frac {1446357824 E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835 \sqrt {33}}-\frac {43537016 F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835 \sqrt {33}}\\ \end {align*}

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Mathematica [A]  time = 0.40, size = 112, normalized size = 0.45 \[ \frac {-5823976480 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {24 \sqrt {2-4 x} \sqrt {5 x+3} \left (175732475616 x^5+591671694906 x^4+797050394730 x^3+537061687749 x^2+180988667568 x+24398176891\right )}{(3 x+2)^{11/2}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{898502220 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((24*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(24398176891 + 180988667568*x + 537061687749*x^2 + 797050394730*x^3 + 5916716
94906*x^4 + 175732475616*x^5))/(2 + 3*x)^(11/2) + 11570862592*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/
2] - 5823976480*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(898502220*Sqrt[2])

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fricas [F]  time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (10 \, x^{2} + x - 3\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2187 \, x^{7} + 10206 \, x^{6} + 20412 \, x^{5} + 22680 \, x^{4} + 15120 \, x^{3} + 6048 \, x^{2} + 1344 \, x + 128}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(10*x^2 + x - 3)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2187*x^7 + 10206*x^6 + 20412*x^5 + 2268
0*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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.03, size = 599, normalized size = 2.41 \[ -\frac {2 \left (-5271974268480 x^{7}-18277348274028 x^{6}[phone] \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 )-88451642790 \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 )-24104934646074 x^{5}[phone] \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 )-294838809300 \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 )-13177956562506 x^{4}[phone] \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 )-393118412400 \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 )[phone] x^{3}[phone] \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 )-262078941600 \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 )[phone] x^{2}[phone] \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 )-87359647200 \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 )[phone] x [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 )-11647952960 \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 )[phone]\right ) \sqrt {5 x +3}\, \sqrt {-2 x +1}}{224625555 \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)^(3/2)*(5*x+3)^(3/2)/(3*x+2)^(13/2),x)

[Out]

-2/224625555*(175732475616*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)-88451642790*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)[phone]*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)-294838809300*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)[phone]*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)-393118412400*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)[phone]*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)-262078941600*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)-5271974268480*x^7+173562938880*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)-87359647200*2^(1/2)*Ellipti
cF(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)-18277348274028*x^6+23141
725184*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))-1164
7952960*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))-241
04934646074*x^5-13177956562506*x^4+132608462283*x^3+3558643880307*x^2+1555703477439*x+219583592019)*(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 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (-2 \, x + 1\right )}^{\frac {3}{2}}}{{\left (3 \, x + 2\right )}^{\frac {13}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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