3.2719 \(\int (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{5/2} \, dx\)

Optimal. Leaf size=249 \[ -\frac{70536439 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{13820625 \sqrt{33}}+\frac{2}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac{67 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{160875}-\frac{160084 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{3378375}-\frac{2133359 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{6756750}-\frac{70536439 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{30405375}-\frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55282500 \sqrt{33}} \]

[Out]

(-70536439*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/30405375 - (2133359*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x
)^(3/2))/6756750 - (160084*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/3378375 - (67*Sqrt[1 - 2*x]*Sqrt[2 + 3
*x]*(3 + 5*x)^(7/2))/160875 + (62*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/3575 + (2*(1 - 2*x)^(3/2)*(2
+ 3*x)^(3/2)*(3 + 5*x)^(7/2))/65 - (9380126059*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(55282500*Sq
rt[33]) - (70536439*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(13820625*Sqrt[33])

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Rubi [A]  time = 0.102512, antiderivative size = 249, normalized size of antiderivative = 1., number of steps used = 9, 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} \[ \frac{2}{65} (1-2 x)^{3/2} (3 x+2)^{3/2} (5 x+3)^{7/2}+\frac{62 \sqrt{1-2 x} (3 x+2)^{3/2} (5 x+3)^{7/2}}{3575}-\frac{67 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{7/2}}{160875}-\frac{160084 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{5/2}}{3378375}-\frac{2133359 \sqrt{1-2 x} \sqrt{3 x+2} (5 x+3)^{3/2}}{6756750}-\frac{70536439 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3}}{30405375}-\frac{70536439 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13820625 \sqrt{33}}-\frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55282500 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-70536439*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/30405375 - (2133359*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x
)^(3/2))/6756750 - (160084*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(5/2))/3378375 - (67*Sqrt[1 - 2*x]*Sqrt[2 + 3
*x]*(3 + 5*x)^(7/2))/160875 + (62*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(7/2))/3575 + (2*(1 - 2*x)^(3/2)*(2
+ 3*x)^(3/2)*(3 + 5*x)^(7/2))/65 - (9380126059*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(55282500*Sq
rt[33]) - (70536439*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(13820625*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)^{3/2} (3+5 x)^{5/2} \, dx &=\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{2}{65} \int \left (-\frac{69}{2}-\frac{93 x}{2}\right ) \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2} \, dx\\ &=\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{4 \int \frac{\left (-870-\frac{201 x}{4}\right ) \sqrt{2+3 x} (3+5 x)^{5/2}}{\sqrt{1-2 x}} \, dx}{10725}\\ &=-\frac{67 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{160875}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac{4 \int \frac{(3+5 x)^{5/2} \left (\frac{639867}{8}+120063 x\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{482625}\\ &=-\frac{160084 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{3378375}-\frac{67 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{160875}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{4 \int \frac{\left (-\frac{62883465}{8}-\frac{96001155 x}{8}\right ) (3+5 x)^{3/2}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{10135125}\\ &=-\frac{2133359 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{6756750}-\frac{160084 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{3378375}-\frac{67 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{160875}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac{4 \int \frac{\sqrt{3+5 x} \left (\frac{8251543035}{16}+\frac{3174139755 x}{4}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{152026875}\\ &=-\frac{70536439 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{30405375}-\frac{2133359 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{6756750}-\frac{160084 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{3378375}-\frac{67 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{160875}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{4 \int \frac{-\frac{267229618515}{16}-\frac{422105672655 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1368241875}\\ &=-\frac{70536439 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{30405375}-\frac{2133359 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{6756750}-\frac{160084 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{3378375}-\frac{67 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{160875}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}+\frac{70536439 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{27641250}+\frac{9380126059 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{608107500}\\ &=-\frac{70536439 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{30405375}-\frac{2133359 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{6756750}-\frac{160084 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}}{3378375}-\frac{67 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{7/2}}{160875}+\frac{62 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{7/2}}{3575}+\frac{2}{65} (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{7/2}-\frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{55282500 \sqrt{33}}-\frac{70536439 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{13820625 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.292554, size = 115, normalized size = 0.46 \[ \frac{9380126059 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-5 \left (944944217 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+3 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (1403325000 x^5+2364390000 x^4+496455750 x^3-1110242250 x^2-638983395 x+67302101\right )\right )}{912161250 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(9380126059*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 5*(3*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x
]*(67302101 - 638983395*x - 1110242250*x^2 + 496455750*x^3 + 2364390000*x^4 + 1403325000*x^5) + 944944217*Elli
pticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/(912161250*Sqrt[2])

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Maple [C]  time = 0.011, size = 165, normalized size = 0.7 \begin{align*}{\frac{1}{54729675000\,{x}^{3}+41959417500\,{x}^{2}-12770257500\,x-10945935000}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( -1262992500000\,{x}^{8}-3096245250000\,{x}^{7}-1783541025000\,{x}^{6}+4724721085\,\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 ) -9380126059\,\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 ) +1405783957500\,{x}^{5}+1870998115500\,{x}^{4}+236537814150\,{x}^{3}-380468567640\,{x}^{2}-100883569890\,x+12114378180 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/1824322500*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(-1262992500000*x^8-3096245250000*x^7-1783541025000*x^6
+4724721085*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))-
9380126059*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))+1
405783957500*x^5+1870998115500*x^4+236537814150*x^3-380468567640*x^2-100883569890*x+12114378180)/(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{3}{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)^(3/2)*(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)*(3*x + 2)^(3/2)*(-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 (150 \, x^{4} + 205 \, x^{3} + 34 \, x^{2} - 51 \, x - 18\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)^(3/2)*(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

integral(-(150*x^4 + 205*x^3 + 34*x^2 - 51*x - 18)*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)**(3/2)*(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{3}{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)^(3/2)*(3+5*x)^(5/2),x, algorithm="giac")

[Out]

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