3.2661 \(\int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx\)

Optimal. Leaf size=222 \[ -\frac {673072 \sqrt {\frac {11}{3}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )}{6806835}-\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}+\frac {22738708 \sqrt {1-2 x} \sqrt {5 x+3}}{6806835 \sqrt {3 x+2}}+\frac {332372 \sqrt {1-2 x} \sqrt {5 x+3}}{972405 (3 x+2)^{3/2}}+\frac {8842 \sqrt {1-2 x} \sqrt {5 x+3}}{138915 (3 x+2)^{5/2}}-\frac {214 \sqrt {1-2 x} \sqrt {5 x+3}}{3969 (3 x+2)^{7/2}}-\frac {22738708 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835} \]

[Out]

-22738708/20420505*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-673072/20420505*EllipticF(1/
7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-2/27*(3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(9/2)-214/3969*(1-
2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+8842/138915*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+332372/972405*(1-
2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)[phone]/6806835*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]  time = 0.08, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 8, 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} \[ -\frac {2 \sqrt {1-2 x} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}+\frac {22738708 \sqrt {1-2 x} \sqrt {5 x+3}}{6806835 \sqrt {3 x+2}}+\frac {332372 \sqrt {1-2 x} \sqrt {5 x+3}}{972405 (3 x+2)^{3/2}}+\frac {8842 \sqrt {1-2 x} \sqrt {5 x+3}}{138915 (3 x+2)^{5/2}}-\frac {214 \sqrt {1-2 x} \sqrt {5 x+3}}{3969 (3 x+2)^{7/2}}-\frac {673072 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835}-\frac {22738708 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835} \]

Antiderivative was successfully verified.

[In]

Int[(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^(11/2),x]

[Out]

(-214*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3969*(2 + 3*x)^(7/2)) + (8842*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(138915*(2 + 3*
x)^(5/2)) + (332372*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(972405*(2 + 3*x)^(3/2)) + (22738708*Sqrt[1 - 2*x]*Sqrt[3 + 5
*x])/(6806835*Sqrt[2 + 3*x]) - (2*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(27*(2 + 3*x)^(9/2)) - (22738708*Sqrt[11/3]*E
llipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/6806835 - (673072*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt
[1 - 2*x]], 35/33])/6806835

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 {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx &=-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {2}{27} \int \frac {\left (\frac {9}{2}-20 x\right ) \sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{9/2}} \, dx\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{3969 (2+3 x)^{7/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {4 \int \frac {-\frac {1493}{4}-\frac {2225 x}{2}}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx}{3969}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{3969 (2+3 x)^{7/2}}+\frac {8842 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{5/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {8 \int \frac {\frac {38883}{4}-\frac {66315 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{138915}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{3969 (2+3 x)^{7/2}}+\frac {8842 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{5/2}}+\frac {332372 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 (2+3 x)^{3/2}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {16 \int \frac {\frac {4022817}{8}-\frac {1246395 x}{4}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{2917215}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{3969 (2+3 x)^{7/2}}+\frac {8842 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{5/2}}+\frac {332372 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 (2+3 x)^{3/2}}+\frac {22738708 \sqrt {1-2 x} \sqrt {3+5 x}}{6806835 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {32 \int \frac {\frac {53938515}{8}+\frac {85270155 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{20420505}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{3969 (2+3 x)^{7/2}}+\frac {8842 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{5/2}}+\frac {332372 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 (2+3 x)^{3/2}}+\frac {22738708 \sqrt {1-2 x} \sqrt {3+5 x}}{6806835 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {3701896 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{6806835}+\frac {22738708 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{6806835}\\ &=-\frac {214 \sqrt {1-2 x} \sqrt {3+5 x}}{3969 (2+3 x)^{7/2}}+\frac {8842 \sqrt {1-2 x} \sqrt {3+5 x}}{138915 (2+3 x)^{5/2}}+\frac {332372 \sqrt {1-2 x} \sqrt {3+5 x}}{972405 (2+3 x)^{3/2}}+\frac {22738708 \sqrt {1-2 x} \sqrt {3+5 x}}{6806835 \sqrt {2+3 x}}-\frac {2 \sqrt {1-2 x} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}-\frac {22738708 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835}-\frac {673072 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 107, normalized size = 0.48 \[ \frac {-93064160 \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+\frac {24 \sqrt {2-4 x} \sqrt {5 x+3} \left (920917674 x^4+2487189618 x^3+2520548433 x^2+1134125364 x+190959271\right )}{(3 x+2)^{9/2}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )}{81682020 \sqrt {2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(2 + 3*x)^(11/2),x]

[Out]

((24*Sqrt[2 - 4*x]*Sqrt[3 + 5*x]*(190959271 + 1134125364*x + 2520548433*x^2 + 2487189618*x^3 + 920917674*x^4))
/(2 + 3*x)^(9/2) + 181909664*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 93064160*EllipticF[ArcSin[Sq
rt[2/11]*Sqrt[3 + 5*x]], -33/2])/(81682020*Sqrt[2])

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fricas [F]  time = 1.26, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{729 \, x^{6} + 2916 \, x^{5} + 4860 \, x^{4} + 4320 \, x^{3} + 2160 \, x^{2} + 576 \, x + 64}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2),x, algorithm="fricas")

[Out]

integral((5*x + 3)^(3/2)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(729*x^6 + 2916*x^5 + 4860*x^4 + 4320*x^3 + 2160*x^2 + 5
76*x + 64), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^(11/2), x)

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maple [C]  time = 0.04, size = 504, normalized size = 2.27 \[ \frac {2 \left (27627530220 x^{6}[phone] x^{5}-920917674 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{4} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x^{4}-2455780464 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{3} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] x^{3}-2455780464 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x^{2} \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-13553781675 x^{2}-1091457984 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, x \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-9634250463 x -181909664 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )[phone] \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-1718633439\right ) \sqrt {-2 x +1}\, \sqrt {5 x +3}}{20420505 \left (10 x^{2}+x -3\right ) \left (3 x +2\right )^{\frac {9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((5*x+3)^(3/2)*(-2*x+1)^(1/2)/(3*x+2)^(11/2),x)

[Out]

2/20420505*(471137310*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)-920917674*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)-2455780464*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)-2455780464*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]*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)-1091457984*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+93064160*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))-181909664*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]*x^5+74789762778*x^4+19200699657*x^3-13553781675*
x^2-9634250463*x-1718633439)*(-2*x+1)^(1/2)*(5*x+3)^(1/2)/(10*x^2+x-3)/(3*x+2)^(9/2)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (5 \, x + 3\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}{{\left (3 \, x + 2\right )}^{\frac {11}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(11/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(3/2)*sqrt(-2*x + 1)/(3*x + 2)^(11/2), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {1-2\,x}\,{\left (5\,x+3\right )}^{3/2}}{{\left (3\,x+2\right )}^{11/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((1 - 2*x)^(1/2)*(5*x + 3)^(3/2))/(3*x + 2)^(11/2),x)

[Out]

int(((1 - 2*x)^(1/2)*(5*x + 3)^(3/2))/(3*x + 2)^(11/2), 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((3+5*x)**(3/2)*(1-2*x)**(1/2)/(2+3*x)**(11/2),x)

[Out]

Timed out

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