3.2755 \(\int (1-2 x)^{5/2} (2+3 x)^{3/2} \sqrt{3+5 x} \, dx\)

Optimal. Leaf size=218 \[ -\frac{4738087 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{8859375 \sqrt{33}}+\frac{2}{55} (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{106 (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2475}+\frac{2866 (3 x+2)^{3/2} (5 x+3)^{3/2} \sqrt{1-2 x}}{86625}+\frac{38729 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{2165625}-\frac{4738087 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{19490625}-\frac{326256461 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}} \]

[Out]

(-4738087*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/19490625 + (38729*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(
3/2))/2165625 + (2866*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/86625 + (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3
/2)*(3 + 5*x)^(3/2))/2475 + (2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/55 - (326256461*EllipticE[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(17718750*Sqrt[33]) - (4738087*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/(8859375*Sqrt[33])

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Rubi [A]  time = 0.0826217, antiderivative size = 218, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{2}{55} (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{5/2}+\frac{106 (3 x+2)^{3/2} (5 x+3)^{3/2} (1-2 x)^{3/2}}{2475}+\frac{2866 (3 x+2)^{3/2} (5 x+3)^{3/2} \sqrt{1-2 x}}{86625}+\frac{38729 \sqrt{3 x+2} (5 x+3)^{3/2} \sqrt{1-2 x}}{2165625}-\frac{4738087 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{19490625}-\frac{4738087 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375 \sqrt{33}}-\frac{326256461 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4738087*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/19490625 + (38729*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(3 + 5*x)^(
3/2))/2165625 + (2866*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/86625 + (106*(1 - 2*x)^(3/2)*(2 + 3*x)^(3
/2)*(3 + 5*x)^(3/2))/2475 + (2*(1 - 2*x)^(5/2)*(2 + 3*x)^(3/2)*(3 + 5*x)^(3/2))/55 - (326256461*EllipticE[ArcS
in[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(17718750*Sqrt[33]) - (4738087*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]],
35/33])/(8859375*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)^{5/2} (2+3 x)^{3/2} \sqrt{3+5 x} \, dx &=\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{2}{55} \int \left (-\frac{113}{2}-\frac{159 x}{2}\right ) (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x} \, dx\\ &=\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{4 \int \left (-2979-\frac{12897 x}{4}\right ) \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x} \, dx}{7425}\\ &=\frac{2866 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{8 \int \frac{\sqrt{2+3 x} \sqrt{3+5 x} \left (-\frac{670815}{8}+\frac{348561 x}{8}\right )}{\sqrt{1-2 x}} \, dx}{779625}\\ &=\frac{38729 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2165625}+\frac{2866 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{8 \int \frac{\sqrt{3+5 x} \left (\frac{57670353}{16}+\frac{42642783 x}{8}\right )}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{19490625}\\ &=-\frac{4738087 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{19490625}+\frac{38729 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2165625}+\frac{2866 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{8 \int \frac{-\frac{463899753}{4}-\frac{2936308149 x}{16}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{175415625}\\ &=-\frac{4738087 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{19490625}+\frac{38729 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2165625}+\frac{2866 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}+\frac{4738087 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{17718750}+\frac{326256461 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{194906250}\\ &=-\frac{4738087 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{19490625}+\frac{38729 \sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}}{2165625}+\frac{2866 \sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}}{86625}+\frac{106 (1-2 x)^{3/2} (2+3 x)^{3/2} (3+5 x)^{3/2}}{2475}+\frac{2}{55} (1-2 x)^{5/2} (2+3 x)^{3/2} (3+5 x)^{3/2}-\frac{326256461 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{17718750 \sqrt{33}}-\frac{4738087 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{8859375 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.249302, size = 107, normalized size = 0.49 \[ \frac{-169899590 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+15 \sqrt{2-4 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (42525000 x^4-13702500 x^3-35750250 x^2+16294455 x+9437696\right )[phone] E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{292359375 \sqrt{2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(15*Sqrt[2 - 4*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(9437696 + 16294455*x - 35750250*x^2 - 13702500*x^3 + 42525000*x
^4) + 326256461*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 169899590*EllipticF[ArcSin[Sqrt[2/11]*Sqr
t[3 + 5*x]], -33/2])/(292359375*Sqrt[2])

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Maple [C]  time = 0.01, size = 160, normalized size = 0.7 \begin{align*}{\frac{1}{17541562500\,{x}^{3}[phone]\,{x}^{2}-4093031250\,x-3508312500}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 38272500000\,{x}^{7}[phone]\,{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 ) -326256461\,\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 ) -50560200000\,{x}^{5}-14779638000\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-4914918060\,x-1698785280 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/584718750*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(38272500000*x^7+17010000000*x^6+169899590*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))-326256461*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))-50560200000*x^5-14779638000
*x^4+29711102850*x^3+9525219690*x^2-4914918060*x-1698785280)/(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 \sqrt{5 \, x + 3}{\left (3 \, x + 2\right )}^{\frac{3}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (12 \, x^{3} - 4 \, x^{2} - 5 \, x + 2\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)^(5/2)*(2+3*x)^(3/2)*(3+5*x)^(1/2),x, algorithm="fricas")

[Out]

integral((12*x^3 - 4*x^2 - 5*x + 2)*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)**(5/2)*(2+3*x)**(3/2)*(3+5*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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