3.2914 \(\int \frac{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx\)

Optimal. Leaf size=160 \[ \frac{598 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{3087}+\frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{2797 \sqrt{1-2 x} \sqrt{5 x+3}}{3087 \sqrt{3 x+2}}+\frac{97 \sqrt{1-2 x} \sqrt{5 x+3}}{441 (3 x+2)^{3/2}}+\frac{2797 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3087} \]

[Out]

(97*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(441*(2 + 3*x)^(3/2)) - (2797*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3087*Sqrt[2 + 3*x
]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) + (2797*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt
[1 - 2*x]], 35/33])/3087 + (598*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3087

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Rubi [A]  time = 0.0517047, antiderivative size = 160, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {98, 150, 152, 158, 113, 119} \[ \frac{11 (5 x+3)^{3/2}}{7 \sqrt{1-2 x} (3 x+2)^{3/2}}-\frac{2797 \sqrt{1-2 x} \sqrt{5 x+3}}{3087 \sqrt{3 x+2}}+\frac{97 \sqrt{1-2 x} \sqrt{5 x+3}}{441 (3 x+2)^{3/2}}+\frac{598 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3087}+\frac{2797 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3087} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(97*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(441*(2 + 3*x)^(3/2)) - (2797*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(3087*Sqrt[2 + 3*x
]) + (11*(3 + 5*x)^(3/2))/(7*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) + (2797*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt
[1 - 2*x]], 35/33])/3087 + (598*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/3087

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 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{(3+5 x)^{5/2}}{(1-2 x)^{3/2} (2+3 x)^{5/2}} \, dx &=\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{1}{7} \int \frac{\sqrt{3+5 x} \left (\frac{39}{2}+5 x\right )}{\sqrt{1-2 x} (2+3 x)^{5/2}} \, dx\\ &=\frac{97 \sqrt{1-2 x} \sqrt{3+5 x}}{441 (2+3 x)^{3/2}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{2}{441} \int \frac{\frac{2279}{4}+505 x}{\sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}} \, dx\\ &=\frac{97 \sqrt{1-2 x} \sqrt{3+5 x}}{441 (2+3 x)^{3/2}}-\frac{2797 \sqrt{1-2 x} \sqrt{3+5 x}}{3087 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{4 \int \frac{2920+\frac{13985 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3087}\\ &=\frac{97 \sqrt{1-2 x} \sqrt{3+5 x}}{441 (2+3 x)^{3/2}}-\frac{2797 \sqrt{1-2 x} \sqrt{3+5 x}}{3087 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}-\frac{2797 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3087}-\frac{3289 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3087}\\ &=\frac{97 \sqrt{1-2 x} \sqrt{3+5 x}}{441 (2+3 x)^{3/2}}-\frac{2797 \sqrt{1-2 x} \sqrt{3+5 x}}{3087 \sqrt{2+3 x}}+\frac{11 (3+5 x)^{3/2}}{7 \sqrt{1-2 x} (2+3 x)^{3/2}}+\frac{2797 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3087}+\frac{598 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{3087}\\ \end{align*}

Mathematica [A]  time = 0.125958, size = 100, normalized size = 0.62 \[ \frac{\frac{6 \sqrt{5 x+3} \left (8391 x^2+12847 x+4819\right )}{\sqrt{1-2 x} (3 x+2)^{3/2}}-\sqrt{2} \left (7070 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+2797 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )\right )}{9261} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*Sqrt[3 + 5*x]*(4819 + 12847*x + 8391*x^2))/(Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)) - Sqrt[2]*(2797*EllipticE[ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 7070*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]))/9261

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Maple [C]  time = 0.02, size = 219, normalized size = 1.4 \begin{align*}{\frac{1}{92610\,{x}^{2}+9261\,x-27783}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 21210\,\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}+8391\,\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}+14140\,\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 ) +5594\,\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 ) -251730\,{x}^{3}-536448\,{x}^{2}-375816\,x-86742 \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)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(5/2),x)

[Out]

1/9261*(3+5*x)^(1/2)*(1-2*x)^(1/2)*(21210*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)+8391*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)+14140*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))+5594*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))-251730*x^3-536448*x^2-375816*x-86742)/(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{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\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)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((5*x + 3)^(5/2)/((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{{\left (25 \, x^{2} + 30 \, x + 9\right )} \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)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(5/2),x, algorithm="fricas")

[Out]

integral((25*x^2 + 30*x + 9)*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)**(5/2)/(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{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}{{\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)^(5/2)/(1-2*x)^(3/2)/(2+3*x)^(5/2),x, algorithm="giac")

[Out]

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