3.28.94 \(\int \frac {(1-2 x)^{5/2}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx\) [2794]

Optimal. Leaf size=222 \[ \frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {512 \sqrt {1-2 x} \sqrt {3+5 x}}{81 (2+3 x)^{7/2}}+\frac {20420 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}+\frac {950584 \sqrt {1-2 x} \sqrt {3+5 x}}{3969 (2+3 x)^{3/2}}+\frac {66055016 \sqrt {1-2 x} \sqrt {3+5 x}}{27783 \sqrt {2+3 x}}-\frac {66055016 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{27783}-\frac {1986944 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{27783} \]

[Out]

-66055016/83349*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)-1986944/83349*EllipticF(1/7*21^
(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+14/27*(1-2*x)^(3/2)*(3+5*x)^(1/2)/(2+3*x)^(9/2)+512/81*(1-2*x)^(
1/2)*(3+5*x)^(1/2)/(2+3*x)^(7/2)+20420/567*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(5/2)+950584/3969*(1-2*x)^(1/2)
*(3+5*x)^(1/2)/(2+3*x)^(3/2)[phone]/27783*(1-2*x)^(1/2)*(3+5*x)^(1/2)/(2+3*x)^(1/2)

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Rubi [A]
time = 0.06, 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 = {100, 155, 157, 164, 114, 120} \begin {gather*} -\frac {1986944 \sqrt {\frac {11}{3}} F\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{27783}-\frac {66055016 \sqrt {\frac {11}{3}} E\left (\text {ArcSin}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{27783}+\frac {14 \sqrt {5 x+3} (1-2 x)^{3/2}}{27 (3 x+2)^{9/2}}+\frac {66055016 \sqrt {5 x+3} \sqrt {1-2 x}}{27783 \sqrt {3 x+2}}+\frac {950584 \sqrt {5 x+3} \sqrt {1-2 x}}{3969 (3 x+2)^{3/2}}+\frac {20420 \sqrt {5 x+3} \sqrt {1-2 x}}{567 (3 x+2)^{5/2}}+\frac {512 \sqrt {5 x+3} \sqrt {1-2 x}}{81 (3 x+2)^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(14*(1 - 2*x)^(3/2)*Sqrt[3 + 5*x])/(27*(2 + 3*x)^(9/2)) + (512*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(81*(2 + 3*x)^(7/2
)) + (20420*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(567*(2 + 3*x)^(5/2)) + (950584*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3969*(2
+ 3*x)^(3/2)) + (66055016*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(27783*Sqrt[2 + 3*x]) - (66055016*Sqrt[11/3]*EllipticE
[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/27783 - (1986944*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]
], 35/33])/27783

Rule 100

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*c -
a*d)*(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*((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 - 2)*(e + f*x)^p*Simp[a*d*(d*e*(n - 1) + c*f*(p + 1)) + b*c*(d
*e*(m - n + 2) - c*f*(m + p + 2)) + d*(a*d*f*(n + p) + b*(d*e*(m + 1) - c*f*(m + n + p + 1)))*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 1] && (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)^{5/2}}{(2+3 x)^{11/2} \sqrt {3+5 x}} \, dx &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {2}{27} \int \frac {(194-157 x) \sqrt {1-2 x}}{(2+3 x)^{9/2} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {512 \sqrt {1-2 x} \sqrt {3+5 x}}{81 (2+3 x)^{7/2}}-\frac {4}{567} \int \frac {-\frac {31157}{2}+21301 x}{\sqrt {1-2 x} (2+3 x)^{7/2} \sqrt {3+5 x}} \, dx\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {512 \sqrt {1-2 x} \sqrt {3+5 x}}{81 (2+3 x)^{7/2}}+\frac {20420 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}-\frac {8 \int \frac {-\frac {2372055}{2}+\frac {2680125 x}{2}}{\sqrt {1-2 x} (2+3 x)^{5/2} \sqrt {3+5 x}} \, dx}{19845}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {512 \sqrt {1-2 x} \sqrt {3+5 x}}{81 (2+3 x)^{7/2}}+\frac {20420 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}+\frac {950584 \sqrt {1-2 x} \sqrt {3+5 x}}{3969 (2+3 x)^{3/2}}-\frac {16 \int \frac {-\frac {205814595}{4}+\frac {62382075 x}{2}}{\sqrt {1-2 x} (2+3 x)^{3/2} \sqrt {3+5 x}} \, dx}{416745}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {512 \sqrt {1-2 x} \sqrt {3+5 x}}{81 (2+3 x)^{7/2}}+\frac {20420 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}+\frac {950584 \sqrt {1-2 x} \sqrt {3+5 x}}{3969 (2+3 x)^{3/2}}+\frac {66055016 \sqrt {1-2 x} \sqrt {3+5 x}}{27783 \sqrt {2+3 x}}-\frac {32 \int \frac {-\frac {2744348775}{4}-\frac {4334860425 x}{4}}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{2917215}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {512 \sqrt {1-2 x} \sqrt {3+5 x}}{81 (2+3 x)^{7/2}}+\frac {20420 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}+\frac {950584 \sqrt {1-2 x} \sqrt {3+5 x}}{3969 (2+3 x)^{3/2}}+\frac {66055016 \sqrt {1-2 x} \sqrt {3+5 x}}{27783 \sqrt {2+3 x}}+\frac {10928192 \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx}{27783}+\frac {66055016 \int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} \sqrt {2+3 x}} \, dx}{27783}\\ &=\frac {14 (1-2 x)^{3/2} \sqrt {3+5 x}}{27 (2+3 x)^{9/2}}+\frac {512 \sqrt {1-2 x} \sqrt {3+5 x}}{81 (2+3 x)^{7/2}}+\frac {20420 \sqrt {1-2 x} \sqrt {3+5 x}}{567 (2+3 x)^{5/2}}+\frac {950584 \sqrt {1-2 x} \sqrt {3+5 x}}{3969 (2+3 x)^{3/2}}+\frac {66055016 \sqrt {1-2 x} \sqrt {3+5 x}}{27783 \sqrt {2+3 x}}-\frac {66055016 \sqrt {\frac {11}{3}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{27783}-\frac {1986944 \sqrt {\frac {11}{3}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )}{27783}\\ \end {align*}

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Mathematica [A]
time = 8.34, size = 111, normalized size = 0.50 \begin {gather*} \frac {8 \left (\frac {3 \sqrt {1-2 x} \sqrt {3+5 x} \left (557240459+3296666850 x+7318104714 x^2+7223771916 x^3+2675228148 x^4\right )}{4 (2+3 x)^{9/2}}+\sqrt {2} \left (8256877 E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )-4158805 F\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right )\right )\right )}{83349} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(8*((3*Sqrt[1 - 2*x]*Sqrt[3 + 5*x]*(557240459 + 3296666850*x + 7318104714*x^2 + 7223771916*x^3 + 2675228148*x^
4))/(4*(2 + 3*x)^(9/2)) + Sqrt[2]*(8256877*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 4158805*Ellipt
icF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/83349

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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	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {98 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{19683 \left (\frac {2}{3}+x \right )^{5}}+\frac {20420 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{15309 \left (\frac {2}{3}+x \right )^{3}}+\frac {484 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{6561 \left (\frac {2}{3}+x \right )^{4}}+\frac {-\frac {660550160}{27783} x^{2}-\frac {66055016}{27783} x +\frac {66055016}{9261}}{\sqrt {\left (\frac {2}{3}+x \right ) \left (-30 x^{2}-3 x +9\right )}}+\frac {950584 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}{35721 \left (\frac {2}{3}+x \right )^{2}}+\frac {209093240 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{583443 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}+\frac {330275080 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{583443 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(297\)
default	\(-\frac {2 \left (1327775328 \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-2675228148 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{4} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-7133941728 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{3} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-7133941728 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x^{2} \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}[phone] \sqrt {2}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-3170640768 \sqrt {2}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right ) x \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}-80256844440 x^{6}[phone] \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-528440128 \sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-224738841924 x^{5}-217137403836 x^{4}-55840372398 x^{3}[phone] x^{2}[phone] x [phone]\right ) \sqrt {3+5 x}\, \sqrt {1-2 x}}{83349 \left (10 x^{2}+x -3\right ) \left (2+3 x \right )^{\frac {9}{2}}}\)	\(494\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/83349*(1327775328*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)-2675228148*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)-7133941728*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)-7133941728*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)-3170640768*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)-80256844440*x^6+262276608*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))-528440128*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))-224738841924*x^5-217137403836*x^4-55840372398*x^3+39255728106*x^2+27998280273*x+5015164131)*(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)^(5/2)/(2+3*x)^(11/2)/(3+5*x)^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 0.25, size = 70, normalized size = 0.32 \begin {gather*} \frac {2 \, {\left (2675228148 \, x^{4} + 7223771916 \, x^{3} + 7318104714 \, x^{2} + 3296666850 \, x + 557240459\right )} \sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{27783 \, {\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)^(5/2)/(2+3*x)^(11/2)/(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

2/27783*(2675228148*x^4 + 7223771916*x^3 + 7318104714*x^2 + 3296666850*x + 557240459)*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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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