3.2992 \(\int \frac{(2+3 x)^{9/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=187 \[ -\frac{76163 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{166375 \sqrt{33}}+\frac{7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{140 (3 x+2)^{5/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{2063 \sqrt{1-2 x} (3 x+2)^{3/2}}{19965 (5 x+3)^{3/2}}+\frac{70226 \sqrt{1-2 x} \sqrt{3 x+2}}{1098075 \sqrt{5 x+3}}-\frac{4971289 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{332750 \sqrt{33}} \]

[Out]

(2063*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(19965*(3 + 5*x)^(3/2)) - (140*(2 + 3*x)^(5/2))/(121*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2)) + (7*(2 + 3*x)^(7/2))/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (70226*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1
098075*Sqrt[3 + 5*x]) - (4971289*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(332750*Sqrt[33]) - (76163
*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(166375*Sqrt[33])

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Rubi [A]  time = 0.0641548, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ \frac{7 (3 x+2)^{7/2}}{33 (1-2 x)^{3/2} (5 x+3)^{3/2}}-\frac{140 (3 x+2)^{5/2}}{121 \sqrt{1-2 x} (5 x+3)^{3/2}}+\frac{2063 \sqrt{1-2 x} (3 x+2)^{3/2}}{19965 (5 x+3)^{3/2}}+\frac{70226 \sqrt{1-2 x} \sqrt{3 x+2}}{1098075 \sqrt{5 x+3}}-\frac{76163 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{166375 \sqrt{33}}-\frac{4971289 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{332750 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^(9/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

(2063*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2))/(19965*(3 + 5*x)^(3/2)) - (140*(2 + 3*x)^(5/2))/(121*Sqrt[1 - 2*x]*(3 + 5
*x)^(3/2)) + (7*(2 + 3*x)^(7/2))/(33*(1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + (70226*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(1
098075*Sqrt[3 + 5*x]) - (4971289*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(332750*Sqrt[33]) - (76163
*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(166375*Sqrt[33])

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{(2+3 x)^{9/2}}{(1-2 x)^{5/2} (3+5 x)^{5/2}} \, dx &=\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{33} \int \frac{(2+3 x)^{5/2} \left (\frac{219}{2}+201 x\right )}{(1-2 x)^{3/2} (3+5 x)^{5/2}} \, dx\\ &=-\frac{140 (2+3 x)^{5/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{1}{363} \int \frac{\left (-3261-\frac{12933 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{2063 \sqrt{1-2 x} (2+3 x)^{3/2}}{19965 (3+5 x)^{3/2}}-\frac{140 (2+3 x)^{5/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}-\frac{2 \int \frac{\left (-\frac{748365}{4}-\frac{1317501 x}{4}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} (3+5 x)^{3/2}} \, dx}{59895}\\ &=\frac{2063 \sqrt{1-2 x} (2+3 x)^{3/2}}{19965 (3+5 x)^{3/2}}-\frac{140 (2+3 x)^{5/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{70226 \sqrt{1-2 x} \sqrt{2+3 x}}{1098075 \sqrt{3+5 x}}-\frac{4 \int \frac{-\frac{7088247}{2}-\frac{44741601 x}{8}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{3294225}\\ &=\frac{2063 \sqrt{1-2 x} (2+3 x)^{3/2}}{19965 (3+5 x)^{3/2}}-\frac{140 (2+3 x)^{5/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{70226 \sqrt{1-2 x} \sqrt{2+3 x}}{1098075 \sqrt{3+5 x}}+\frac{76163 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{332750}+\frac{4971289 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{3660250}\\ &=\frac{2063 \sqrt{1-2 x} (2+3 x)^{3/2}}{19965 (3+5 x)^{3/2}}-\frac{140 (2+3 x)^{5/2}}{121 \sqrt{1-2 x} (3+5 x)^{3/2}}+\frac{7 (2+3 x)^{7/2}}{33 (1-2 x)^{3/2} (3+5 x)^{3/2}}+\frac{70226 \sqrt{1-2 x} \sqrt{2+3 x}}{1098075 \sqrt{3+5 x}}-\frac{4971289 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{332750 \sqrt{33}}-\frac{76163 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{166375 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.252306, size = 107, normalized size = 0.57 \[ \frac{-2457910 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{10 \sqrt{3 x+2} \left (31924075 x^3+30619782 x^2+2244393 x-2780992\right )}{(1-2 x)^{3/2} (5 x+3)^{3/2}}+4971289 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{10980750} \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^(9/2)/((1 - 2*x)^(5/2)*(3 + 5*x)^(5/2)),x]

[Out]

((10*Sqrt[2 + 3*x]*(-2780992 + 2244393*x + 30619782*x^2 + 31924075*x^3))/((1 - 2*x)^(3/2)*(3 + 5*x)^(3/2)) + 4
971289*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2457910*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11
]*Sqrt[3 + 5*x]], -33/2])/10980750

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Maple [C]  time = 0.023, size = 311, normalized size = 1.7 \begin{align*}{\frac{1}{10980750\, \left ( 2\,x-1 \right ) ^{2}}\sqrt{1-2\,x} \left ( 24579100\,\sqrt{2}{\it EllipticF} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}-49712890\,\sqrt{2}{\it EllipticE} \left ( 1/11\,\sqrt{66+110\,x},i/2\sqrt{66} \right ){x}^{2}\sqrt{3+5\,x}\sqrt{2+3\,x}\sqrt{1-2\,x}+2457910\,\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}-4971289\,\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}-7373730\,\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 ) [phone]\,\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 ) [phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}-38541900\,x-55619840 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{2+3\,x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x)

[Out]

1/10980750*(1-2*x)^(1/2)*(24579100*2^(1/2)*EllipticF(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^2*(3+5*x)^(1/2)*(
2+3*x)^(1/2)*(1-2*x)^(1/2)-49712890*2^(1/2)*EllipticE(1/11*(66+110*x)^(1/2),1/2*I*66^(1/2))*x^2*(3+5*x)^(1/2)*
(2+3*x)^(1/2)*(1-2*x)^(1/2)+2457910*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)-4971289*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)-7373730*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))[phone]*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))[phone]*x^4+1557074960*x^3+679727430*x^2-38541900*x-55619840)/(3+5*x)^(3/2)/(2*x-1)^2/(2+3*
x)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{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((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (81 \, x^{4} + 216 \, x^{3} + 216 \, x^{2} + 96 \, x + 16\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{1000 \, x^{6} + 300 \, x^{5} - 870 \, x^{4} - 179 \, x^{3} + 261 \, x^{2} + 27 \, x - 27}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

integral(-(81*x^4 + 216*x^3 + 216*x^2 + 96*x + 16)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(1000*x^6 + 300*
x^5 - 870*x^4 - 179*x^3 + 261*x^2 + 27*x - 27), 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((2+3*x)**(9/2)/(1-2*x)**(5/2)/(3+5*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 (3 \, x + 2\right )}^{\frac{9}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{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((2+3*x)^(9/2)/(1-2*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(9/2)/((5*x + 3)^(5/2)*(-2*x + 1)^(5/2)), x)