3.2718 \(\int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=280 \[ -\frac{50299451003 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{4146187500 \sqrt{33}}+\frac{2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}+\frac{178 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}}{14625}+\frac{2503 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}-\frac{199721 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{12065625}-\frac{57509209 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{506756250}-\frac{380132617 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{506756250}-\frac{50299451003 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{9121612500}-\frac{836091184171 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2073093750 \sqrt{33}} \]

[Out]

(-50299451003*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9121612500 - (380132617*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(
3 + 5*x)^(3/2))/506756250 - (57509209*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/506756250 - (199721*Sqrt[1
- 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/12065625 + (2503*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/804375 +
(178*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2))/14625 + (2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2
))/75 - (836091184171*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(2073093750*Sqrt[33]) - (50299451003*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(4146187500*Sqrt[33])

________________________________________________________________________________________

Rubi [A]  time = 0.117976, antiderivative size = 280, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {101, 154, 158, 113, 119} \[ \frac{2}{75} (1-2 x)^{3/2} (3 x+2)^{5/2} (5 x+3)^{7/2}+\frac{178 \sqrt{1-2 x} (3 x+2)^{5/2} (5 x+3)^{7/2}}{14625}+\frac{2503 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{804375}-\frac{199721 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{12065625}-\frac{57509209 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{506756250}-\frac{380132617 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{506756250}-\frac{50299451003 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{9121612500}-\frac{50299451003 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4146187500 \sqrt{33}}-\frac{836091184171 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2073093750 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-50299451003*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/9121612500 - (380132617*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(
3 + 5*x)^(3/2))/506756250 - (57509209*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/506756250 - (199721*Sqrt[1
- 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(7/2))/12065625 + (2503*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/804375 +
(178*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2))/14625 + (2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2)*(3 + 5*x)^(7/2
))/75 - (836091184171*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(2073093750*Sqrt[33]) - (50299451003*
EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(4146187500*Sqrt[33])

Rule 101

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[((a +
b*x)^m*(c + d*x)^n*(e + f*x)^(p + 1))/(f*(m + n + p + 1)), x] - Dist[1/(f*(m + n + p + 1)), Int[(a + b*x)^(m -
1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a*f) + b*n*(d*e - c*f))
*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (Integ
ersQ[2*m, 2*n, 2*p] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && 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]

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)]))

Rubi steps

\begin{align*} \int (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{5/2} \, dx &=\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{2}{75} \int \left (-\frac{71}{2}-\frac{89 x}{2}\right ) \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx\\ &=\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{4 \int \frac{(2+3 x)^{3/2} (3+5 x)^{5/2} \left (-\frac{2339}{2}+\frac{2503 x}{4}\right )}{\sqrt{1-2 x}} \, dx}{14625}\\ &=\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac{4 \int \frac{\sqrt{2+3 x} (3+5 x)^{5/2} \left (\frac{816405}{8}+\frac{599163 x}{4}\right )}{\sqrt{1-2 x}} \, dx}{804375}\\ &=-\frac{199721 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{12065625}+\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{4 \int \frac{\left (-\frac{113620371}{8}-\frac{172527627 x}{8}\right ) (3+5 x)^{5/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{36196875}\\ &=-\frac{57509209 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{506756250}-\frac{199721 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{12065625}+\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac{4 \int \frac{(3+5 x)^{3/2} \left (\frac{22424965215}{16}+\frac{17105967765 x}{8}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{760134375}\\ &=-\frac{380132617 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{506756250}-\frac{57509209 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{506756250}-\frac{199721 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{12065625}+\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{4 \int \frac{\left (-\frac{735492282165}{8}-\frac{2263475295135 x}{16}\right ) \sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{11402015625}\\ &=-\frac{50299451003 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{9121612500}-\frac{380132617 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{506756250}-\frac{57509209 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{506756250}-\frac{199721 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{12065625}+\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac{4 \int \frac{\frac{95277493539765}{32}+\frac{37624103287695 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{102618140625}\\ &=-\frac{50299451003 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{9121612500}-\frac{380132617 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{506756250}-\frac{57509209 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{506756250}-\frac{199721 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{12065625}+\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}+\frac{50299451003 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{8292375000}+\frac{836091184171 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{22804031250}\\ &=-\frac{50299451003 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{9121612500}-\frac{380132617 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{506756250}-\frac{57509209 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{506756250}-\frac{199721 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{12065625}+\frac{2503 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{804375}+\frac{178 \sqrt{1-2 x} (2+3 x)^{5/2} (3+5 x)^{7/2}}{14625}+\frac{2}{75} (1-2 x)^{3/2} (2+3 x)^{5/2} (3+5 x)^{7/2}-\frac{836091184171 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{2073093750 \sqrt{33}}-\frac{50299451003 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4146187500 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.265368, size = 119, normalized size = 0.42 \[ \frac{\sqrt{2} \left (3344364736684 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-1684482853585 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (547296750000 x^6+1316318850000 x^5+888419542500 x^4-227285730000 x^3-522917547750 x^2-177853891770 x+44426819351\right )}{273648375000} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(44426819351 - 177853891770*x - 522917547750*x^2 - 227285730000
*x^3 + 888419542500*x^4 + 1316318850000*x^5 + 547296750000*x^6) + Sqrt[2]*(3344364736684*EllipticE[ArcSin[Sqrt
[2/11]*Sqrt[3 + 5*x]], -33/2] - 1684482853585*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/27364837500
0

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Maple [C]  time = 0.02, size = 170, normalized size = 0.6 \begin{align*}{\frac{1}{8209451250000\,{x}^{3}[phone]\,{x}^{2}-1915538625000\,x-1641890250000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -492567075000000\,{x}^{9}-1562321722500000\,{x}^{8}-1592905277250000\,{x}^{7}-33511953825000\,{x}^{6}[phone]\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) -3344364736684\,\sqrt{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) +1050958443600000\,{x}^{5}[phone]\,{x}^{4}-67989068522100\,{x}^{3}-162128981218890\,{x}^{2}-22684068454890\,x+7996827483180 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/273648375000*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-492567075000000*x^9-1562321722500000*x^8-1592905277
250000*x^7-33511953825000*x^6+1684482853585*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/11*(
66+110*x)^(1/2),1/2*I*66^(1/2))-3344364736684*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/11
*(66+110*x)^(1/2),1/2*I*66^(1/2))+1050958443600000*x^5+633067124890500*x^4-67989068522100*x^3-162128981218890*
x^2-22684068454890*x+7996827483180)/(30*x^3+23*x^2-7*x-6)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-{\left (450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (5 \, x + 3\right )}^{\frac{5}{2}}{\left (3 \, x + 2\right )}^{\frac{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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