3.2913 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ \frac{31}{147} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{31 \sqrt{1-2 x} \sqrt{5 x+3}}{147 \sqrt{3 x+2}}+\frac{1159}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(31*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(147*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) +
(1159*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/147 + (31*Sqrt[11/3]*EllipticF[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/147

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Rubi [A]  time = 0.0373532, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {98, 150, 158, 113, 119} \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} \sqrt{3 x+2}}+\frac{31 \sqrt{1-2 x} \sqrt{5 x+3}}{147 \sqrt{3 x+2}}+\frac{31}{147} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{1159}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(31*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(147*Sqrt[2 + 3*x]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]) +
(1159*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/147 + (31*Sqrt[11/3]*EllipticF[ArcSin[Sqrt
[3/7]*Sqrt[1 - 2*x]], 35/33])/147

Rule 98

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

Rule 150

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*(e + f*x)^(p + 1))/(b*(b*e - a*f)*(m + 1)), x] - Dist[1
/(b*(b*e - a*f)*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[b*c*(f*g - e*h)*(m + 1) + (
b*g - a*h)*(d*e*n + c*f*(p + 1)) + d*(b*(f*g - e*h)*(m + 1) + f*(b*g - a*h)*(n + p + 1))*x, x], x], x] /; Free
Q[{a, b, c, d, e, f, g, h, p}, x] && LtQ[m, -1] && GtQ[n, 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{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{3/2}} \, dx &=\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{1}{7} \int \frac{\sqrt{3+5 x} \left (\frac{237}{2}+170 x\right )}{\sqrt{1-2 x} (2+3 x)^{3/2}} \, dx\\ &=\frac{31 \sqrt{1-2 x} \sqrt{3+5 x}}{147 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{2}{147} \int \frac{\frac{7295}{4}+\frac{5795 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx\\ &=\frac{31 \sqrt{1-2 x} \sqrt{3+5 x}}{147 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} \sqrt{2+3 x}}-\frac{341}{294} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{1159}{147} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{31 \sqrt{1-2 x} \sqrt{3+5 x}}{147 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} \sqrt{2+3 x}}+\frac{1159}{147} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{31}{147} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )\\ \end{align*}

Mathematica [A]  time = 0.120873, size = 122, normalized size = 0.95 \[ \frac{1295 \sqrt{2-4 x} (3 x+2) \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+6 \sqrt{3 x+2} \sqrt{5 x+3} (1093 x+724)-2318 \sqrt{2-4 x} (3 x+2) E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{882 \sqrt{1-2 x} (3 x+2)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(724 + 1093*x) - 2318*Sqrt[2 - 4*x]*(2 + 3*x)*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[
3 + 5*x]], -33/2] + 1295*Sqrt[2 - 4*x]*(2 + 3*x)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/(882*Sqrt
[1 - 2*x]*(2 + 3*x))

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Maple [C]  time = 0.018, size = 135, normalized size = 1.1 \begin{align*} -{\frac{1}{26460\,{x}^{3}+20286\,{x}^{2}-6174\,x-5292}\sqrt{1-2\,x}\sqrt{2+3\,x}\sqrt{3+5\,x} \left ( 1295\,\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 ) -2318\,\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 ) +32790\,{x}^{2}+41394\,x+13032 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/882*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(2+3*x)^(1/2)*(1295*2^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(1-2*x)^(1/2)*Ellip
ticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))-2318*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))+32790*x^2+41394*x+13032)/(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 \frac{{\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((3+5*x)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(3/2),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 (\frac{{\left (25 \, x^{2} + 30 \, x + 9\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{36 \, x^{4} + 12 \, x^{3} - 23 \, x^{2} - 4 \, x + 4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((25*x^2 + 30*x + 9)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(36*x^4 + 12*x^3 - 23*x^2 - 4*x + 4),
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)**(5/2)/(1-2*x)**(3/2)/(2+3*x)**(3/2),x)

[Out]

Timed out

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

[Out]

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