3.2788 \(\int \frac{(1-2 x)^{5/2} \sqrt{2+3 x}}{\sqrt{3+5 x}} \, dx\)

Optimal. Leaf size=160 \[ -\frac{132824 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{590625}+\frac{2}{35} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{326 \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}}{2625}+\frac{30922 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{118125}-\frac{408311 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625} \]

[Out]

(30922*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/118125 + (326*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2
625 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/35 - (408311*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[
1 - 2*x]], 35/33])/590625 - (132824*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/590625

________________________________________________________________________________________

Rubi [A]  time = 0.0528908, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, 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} \[ \frac{2}{35} \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{5/2}+\frac{326 \sqrt{3 x+2} \sqrt{5 x+3} (1-2 x)^{3/2}}{2625}+\frac{30922 \sqrt{3 x+2} \sqrt{5 x+3} \sqrt{1-2 x}}{118125}-\frac{132824 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625}-\frac{408311 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(30922*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/118125 + (326*(1 - 2*x)^(3/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/2
625 + (2*(1 - 2*x)^(5/2)*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/35 - (408311*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[
1 - 2*x]], 35/33])/590625 - (132824*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/590625

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 \frac{(1-2 x)^{5/2} \sqrt{2+3 x}}{\sqrt{3+5 x}} \, dx &=\frac{2}{35} (1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{2}{35} \int \frac{\left (-\frac{111}{2}-\frac{163 x}{2}\right ) (1-2 x)^{3/2}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}}{2625}+\frac{2}{35} (1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{4 \int \frac{\left (-2774-\frac{15461 x}{4}\right ) \sqrt{1-2 x}}{\sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2625}\\ &=\frac{30922 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{118125}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}}{2625}+\frac{2}{35} (1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{8 \int \frac{-\frac{391093}{8}-\frac{408311 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{118125}\\ &=\frac{30922 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{118125}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}}{2625}+\frac{2}{35} (1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x}+\frac{408311 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{590625}+\frac{730532 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{590625}\\ &=\frac{30922 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{118125}+\frac{326 (1-2 x)^{3/2} \sqrt{2+3 x} \sqrt{3+5 x}}{2625}+\frac{2}{35} (1-2 x)^{5/2} \sqrt{2+3 x} \sqrt{3+5 x}-\frac{408311 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625}-\frac{132824 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{590625}\\ \end{align*}

Mathematica [A]  time = 0.195055, size = 102, normalized size = 0.64 \[ \frac{1783285 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+30 \sqrt{1-2 x} \sqrt{3 x+2} \sqrt{5 x+3} \left (13500 x^2-28170 x+26171\right )+408311 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{1771875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(26171 - 28170*x + 13500*x^2) + 408311*Sqrt[2]*EllipticE[ArcSin[
Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 1783285*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/177187
5

________________________________________________________________________________________

Maple [C]  time = 0.011, size = 150, normalized size = 0.9 \begin{align*} -{\frac{1}{53156250\,{x}^{3}[phone]\,{x}^{2}-12403125\,x-10631250}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1783285\,\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 ) +408311\,\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 ) -12150000\,{x}^{5}[phone]\,{x}^{4}-1281600\,{x}^{3}-21543690\,{x}^{2}+425310\,x+4710780 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/1771875*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(3+5*x)^(1/2)*(1783285*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))+408311*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ell
ipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))-12150000*x^5+16038000*x^4-1281600*x^3-21543690*x^2+425310*x+47107
80)/(30*x^3+23*x^2-7*x-6)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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