3.2897 \(\int \frac{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=158 \[ -\frac{212 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{343 \sqrt{33}}+\frac{38 \sqrt{1-2 x} \sqrt{5 x+3}}{343 \sqrt{3 x+2}}-\frac{8 \sqrt{1-2 x} \sqrt{5 x+3}}{49 (3 x+2)^{3/2}}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{38}{343} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) - (8*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(49*(2 + 3*x)^(3/2)) + (
38*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(343*Sqrt[2 + 3*x]) - (38*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/343 - (212*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(343*Sqrt[33])

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Rubi [A]  time = 0.0510116, antiderivative size = 158, 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 = {99, 152, 158, 113, 119} \[ \frac{38 \sqrt{1-2 x} \sqrt{5 x+3}}{343 \sqrt{3 x+2}}-\frac{8 \sqrt{1-2 x} \sqrt{5 x+3}}{49 (3 x+2)^{3/2}}+\frac{2 \sqrt{5 x+3}}{7 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{212 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}-\frac{38}{343} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3 + 5*x])/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) - (8*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(49*(2 + 3*x)^(3/2)) + (
38*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(343*Sqrt[2 + 3*x]) - (38*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/343 - (212*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(343*Sqrt[33])

Rule 99

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

Rule 152

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 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{\sqrt{3+5 x}}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{2}{7} \int \frac{-13-\frac{45 x}{2}}{\sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{8 \sqrt{1-2 x} \sqrt{3+5 x}}{49 (2+3 x)^{3/2}}-\frac{4}{147} \int \frac{-\frac{99}{4}-30 x}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{8 \sqrt{1-2 x} \sqrt{3+5 x}}{49 (2+3 x)^{3/2}}+\frac{38 \sqrt{1-2 x} \sqrt{3+5 x}}{343 \sqrt{2+3 x}}-\frac{8 \int \frac{-\frac{165}{2}-\frac{285 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{1029}\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{8 \sqrt{1-2 x} \sqrt{3+5 x}}{49 (2+3 x)^{3/2}}+\frac{38 \sqrt{1-2 x} \sqrt{3+5 x}}{343 \sqrt{2+3 x}}+\frac{38}{343} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx+\frac{106}{343} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{2 \sqrt{3+5 x}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{8 \sqrt{1-2 x} \sqrt{3+5 x}}{49 (2+3 x)^{3/2}}+\frac{38 \sqrt{1-2 x} \sqrt{3+5 x}}{343 \sqrt{2+3 x}}-\frac{38}{343} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )-\frac{212 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{343 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.122717, size = 99, normalized size = 0.63 \[ \frac{2 \left (\sqrt{2} \left (140 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+19 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )-\frac{3 \sqrt{5 x+3} \left (114 x^2-37 x-59\right )}{\sqrt{1-2 x} (3 x+2)^{3/2}}\right )}{1029} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-3*Sqrt[3 + 5*x]*(-59 - 37*x + 114*x^2))/(Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) + Sqrt[2]*(19*EllipticE[ArcSin[S
qrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 140*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/1029

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Maple [C]  time = 0.024, size = 219, normalized size = 1.4 \begin{align*} -{\frac{2}{10290\,{x}^{2}+1029\,x-3087}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 420\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+57\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ) x\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+280\,\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 ) +38\,\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 ) -1710\,{x}^{3}-471\,{x}^{2}+1218\,x+531 \right ) \left ( 2+3\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1029*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(420*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x*(3+5*x)^(1/
2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)+57*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x*(3+5*x)^(1/2)*(2+3
*x)^(1/2)*(1-2*x)^(1/2)+280*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))+38*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))-1710*x^3-471*x^2+1218*x+531)/(2+3*x)^(3/2)/(10*x^2+x-3)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{5 \, x + 3}}{{\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((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(5/2),x, algorithm="maxima")

[Out]

integrate(sqrt(5*x + 3)/((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 (\frac{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8), 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((3+5*x)**(1/2)/(1-2*x)**(3/2)/(2+3*x)**(5/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{5 \, x + 3}}{{\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((3+5*x)^(1/2)/(1-2*x)^(3/2)/(2+3*x)^(5/2),x, algorithm="giac")

[Out]

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