3.28.16 \(\int \frac {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx\) [2716]

Optimal. Leaf size=222 \[ -\frac {8252 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {280904 \sqrt {1-2 x} \sqrt {3+5 x}}{416745 (2+3 x)^{3/2}}+\frac {19885156 \sqrt {1-2 x} \sqrt {3+5 x}}{2917215 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{7/2}}-\frac {19885156 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2917215}-\frac {609304 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2917215} \]

[Out]

-2/27*(1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(9/2)-19885156/8751645*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*115
5^(1/2))*33^(1/2)-609304/8751645*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+74/189*(3+5*x)
^(3/2)*(1-2*x)^(1/2)/(2+3*x)^(7/2)-8252/19845*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+280904/416745*(1-2*x)^
(1/2)*(3+5*x)^(1/2)/(2+3*x)^(3/2)[phone]/2917215*(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 {609304 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2917215}-\frac {19885156 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2917215}+\frac {74 \sqrt {1-2 x} (5 x+3)^{3/2}}{189 (3 x+2)^{7/2}}-\frac {2 (1-2 x)^{3/2} (5 x+3)^{3/2}}{27 (3 x+2)^{9/2}}+\frac {19885156 \sqrt {1-2 x} \sqrt {5 x+3}}{2917215 \sqrt {3 x+2}}+\frac {280904 \sqrt {1-2 x} \sqrt {5 x+3}}{416745 (3 x+2)^{3/2}}-\frac {8252 \sqrt {1-2 x} \sqrt {5 x+3}}{19845 (3 x+2)^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-8252*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(19845*(2 + 3*x)^(5/2)) + (280904*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(416745*(2
+ 3*x)^(3/2)) + (19885156*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(2917215*Sqrt[2 + 3*x]) - (2*(1 - 2*x)^(3/2)*(3 + 5*x)^
(3/2))/(27*(2 + 3*x)^(9/2)) + (74*Sqrt[1 - 2*x]*(3 + 5*x)^(3/2))/(189*(2 + 3*x)^(7/2)) - (19885156*Sqrt[11/3]*
EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/2917215 - (609304*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqr
t[1 - 2*x]], 35/33])/2917215

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 {(1-2 x)^{3/2} (3+5 x)^{3/2}}{(2+3 x)^{11/2}} \, dx &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {2}{27} \int \frac {\left (-\frac {3}{2}-30 x\right ) \sqrt {1-2 x} \sqrt {3+5 x}}{(2+3 x)^{9/2}} \, dx\\ &=-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{7/2}}-\frac {4}{567} \int \frac {\sqrt {3+5 x} \left (-\frac {1233}{2}+\frac {1245 x}{2}\right )}{\sqrt {1-2 x} (2+3 x)^{7/2}} \, dx\\ &=-\frac {8252 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{7/2}}-\frac {8 \int \frac {-12189+\frac {32205 x}{4}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{59535}\\ &=-\frac {8252 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {280904 \sqrt {1-2 x} \sqrt {3+5 x}}{416745 (2+3 x)^{3/2}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{7/2}}-\frac {16 \int \frac {-\frac {3566769}{8}+\frac {526695 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{1250235}\\ &=-\frac {8252 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {280904 \sqrt {1-2 x} \sqrt {3+5 x}}{416745 (2+3 x)^{3/2}}+\frac {19885156 \sqrt {1-2 x} \sqrt {3+5 x}}{2917215 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{7/2}}-\frac {32 \int \frac {-\frac {11813745}{2}-\frac {74569335 x}{8}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{8751645}\\ &=-\frac {8252 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {280904 \sqrt {1-2 x} \sqrt {3+5 x}}{416745 (2+3 x)^{3/2}}+\frac {19885156 \sqrt {1-2 x} \sqrt {3+5 x}}{2917215 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{7/2}}+\frac {3351172 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2917215}+\frac {19885156 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{2917215}\\ &=-\frac {8252 \sqrt {1-2 x} \sqrt {3+5 x}}{19845 (2+3 x)^{5/2}}+\frac {280904 \sqrt {1-2 x} \sqrt {3+5 x}}{416745 (2+3 x)^{3/2}}+\frac {19885156 \sqrt {1-2 x} \sqrt {3+5 x}}{2917215 \sqrt {2+3 x}}-\frac {2 (1-2 x)^{3/2} (3+5 x)^{3/2}}{27 (2+3 x)^{9/2}}+\frac {74 \sqrt {1-2 x} (3+5 x)^{3/2}}{189 (2+3 x)^{7/2}}-\frac {19885156 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2917215}-\frac {609304 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{2917215}\\ \end {align*}

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Mathematica [A]
time = 7.86, size = 111, normalized size = 0.50 \begin {gather*} \frac {4 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (167622907+993561978 x+2204875881 x^2+2174142276 x^3+805348818 x^4\right )}{2 (2+3 x)^{9/2}}+\sqrt {2} \left (4971289 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-2457910 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{8751645} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(167622907 + 993561978*x + 2204875881*x^2 + 2174142276*x^3 + 805348818*x^4)
)/(2*(2 + 3*x)^(9/2)) + Sqrt[2]*(4971289*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2457910*Elliptic
F[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/8751645

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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	\(-\frac {\sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {18994 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{1607445 \left (\frac {2}{3}+x \right )^{3}}+\frac {280904 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{3750705 \left (\frac {2}{3}+x \right )^{2}}+\frac {-\frac {39770312}{583443} x^{2}-\frac {19885156}{2917215} x +\frac {19885156}{972405}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {12601328 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{12252303 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {19885156 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{12252303 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {14 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{59049 \left (\frac {2}{3}+x \right )^{5}}-\frac {740 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{137781 \left (\frac {2}{3}+x \right )^{4}}\right )}{\left (10 x^{2}+x -3\right ) \sqrt {2+3 x}}\)	\(308\)
default	\(-\frac {2 \left (407167398 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-805348818 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2147596848 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2147596848 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-954487488 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-24160464540 x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-159081248 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-67640314734 x^{5}-65420563896 x^{4}-16854206499 x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{8751645 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\)	\(494\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/8751645*(407167398*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)-805348818*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)-2147596848*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)-2147596848*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)-954487488*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)-24160464540*x^6+80428128*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))-159081248*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*7
0^(1/2))-67640314734*x^5-65420563896*x^4-16854206499*x^3+11834509785*x^2+8439189081*x+1508606163)*(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((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(11/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.26, size = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (805348818 \, x^{4} + 2174142276 \, x^{3} + 2204875881 \, x^{2} + 993561978 \, x + 167622907\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{2917215 \, {\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((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(11/2),x, algorithm="fricas")

[Out]

2/2917215*(805348818*x^4 + 2174142276*x^3 + 2204875881*x^2 + 993561978*x + 167622907)*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((1-2*x)**(3/2)*(3+5*x)**(3/2)/(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((1-2*x)^(3/2)*(3+5*x)^(3/2)/(2+3*x)^(11/2),x, algorithm="giac")

[Out]

integrate((5*x + 3)^(3/2)*(-2*x + 1)^(3/2)/(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 {{\left (1-2\,x\right )}^{3/2}\,{\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)^(3/2)*(5*x + 3)^(3/2))/(3*x + 2)^(11/2),x)

[Out]

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

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