3.27.61 \(\int \frac {\sqrt {1-2 x} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx\) [2661]

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

[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 = 0.05, 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 = {99, 155, 157, 164, 114, 120} \begin {gather*} -\frac {673072 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{6806835}-\frac {22738708 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\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}} \end {gather*}

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 99

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 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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 120

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[2*(Rt[-b/d,
2]/(b*Sqrt[(b*e - a*f)/b]))*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)))], x] /; FreeQ[{a, b, c, d, e, f}, x] && GtQ[(b*c - a*d)/b, 0] && GtQ[(b*e - a*f)/b, 0] && Po
sQ[-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 155

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 157

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 164

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 = 4.90, size = 107, normalized size = 0.48 \begin {gather*} \frac {\frac {24 \sqrt {2-4 x} \sqrt {3+5 x} \left (190959271+1134125364 x+2520548433 x^2+2487189618 x^3+920917674 x^4\right )}{(2+3 x)^{9/2}}[phone] E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-93064160 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )}{81682020 \sqrt {2}} \end {gather*}

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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(493\) vs. \(2(162)=324\).
time = 0.10, size = 494, normalized size = 2.23

method	result	size
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {2 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19683 \left (\frac {2}{3}+x \right )^{5}}-\frac {704 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{321489 \left (\frac {2}{3}+x \right )^{4}}+\frac {8842 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3750705 \left (\frac {2}{3}+x \right )^{3}}+\frac {332372 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{8751645 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {45477416}{1361367} x^{2}-\frac {22738708}{6806835} x +\frac {22738708}{2268945}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {14383604 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{28588707 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {22738708 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{28588707 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(297\)
default	\(-\frac {2 \left (449780364 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-920917674 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2455780464 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2455780464 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-1091457984 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-27627530220 x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-181909664 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-77378441562 x^{5}-74789762778 x^{4}-19200699657 x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{20420505 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\)	\(494\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/20420505*(449780364*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^4*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1
-2*x)^(1/2)-920917674*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^4*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-
2*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-
2*x)^(1/2)-2455780464*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^3*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-
2*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-
2*x)^(1/2)-2455780464*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x^2*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-
2*x)^(1/2)[phone]*2^(1/2)*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x
)^(1/2)-1091457984*2^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2))*x*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^
(1/2)-27627530220*x^6+88845504*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/7*(28+42*x)^(1/2
),1/2*70^(1/2))-181909664*2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/2),1/2
*70^(1/2))-77378441562*x^5-74789762778*x^4-19200699657*x^3+13553781675*x^2+9634250463*x+1718633439)*(3+5*x)^(1
/2)*(1-2*x)^(1/2)/(10*x^2+x-3)/(2+3*x)^(9/2)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [A]
time = 0.12, size = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (920917674 \, x^{4} + 2487189618 \, x^{3} + 2520548433 \, x^{2} + 1134125364 \, x + 190959271\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{6806835 \, {\left (243 \, x^{5} + 810 \, x^{4} + 1080 \, x^{3} + 720 \, x^{2} + 240 \, x + 32\right )}} \end {gather*}

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]

2/6806835*(920917674*x^4 + 2487189618*x^3 + 2520548433*x^2 + 1134125364*x + 190959271)*sqrt(5*x + 3)*sqrt(3*x
+ 2)*sqrt(-2*x + 1)/(243*x^5 + 810*x^4 + 1080*x^3 + 720*x^2 + 240*x + 32)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 8856 deep

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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

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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