3.2806 \(\int \frac{(1-2 x)^{5/2} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=253 \[ -\frac{595387 \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{4921875}-\frac{524}{225} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{7/2}-\frac{442 (1-2 x)^{3/2} (3 x+2)^{7/2}}{75 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}+\frac{59662 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{7875}+\frac{373022 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{196875}+\frac{500501 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{984375}-\frac{1065118 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4921875} \]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(15*(3 + 5*x)^(3/2)) - (442*(1 - 2*x)^(3/2)*(2 + 3*x)^(7/2))/(75*Sqrt[3 +
5*x]) + (500501*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/984375 + (373022*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqr
t[3 + 5*x])/196875 + (59662*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/7875 - (524*Sqrt[1 - 2*x]*(2 + 3*x)^(
7/2)*Sqrt[3 + 5*x])/225 - (1065118*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/4921875 - (59
5387*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/4921875

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Rubi [A]  time = 0.099954, antiderivative size = 253, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214, Rules used = {97, 150, 154, 158, 113, 119} \[ -\frac{524}{225} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{7/2}-\frac{442 (1-2 x)^{3/2} (3 x+2)^{7/2}}{75 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{5/2} (3 x+2)^{7/2}}{15 (5 x+3)^{3/2}}+\frac{59662 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{5/2}}{7875}+\frac{373022 \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}}{196875}+\frac{500501 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{984375}-\frac{595387 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4921875}-\frac{1065118 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4921875} \]

Antiderivative was successfully verified.

[In]

Int[((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(3 + 5*x)^(5/2),x]

[Out]

(-2*(1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(15*(3 + 5*x)^(3/2)) - (442*(1 - 2*x)^(3/2)*(2 + 3*x)^(7/2))/(75*Sqrt[3 +
5*x]) + (500501*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/984375 + (373022*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqr
t[3 + 5*x])/196875 + (59662*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x])/7875 - (524*Sqrt[1 - 2*x]*(2 + 3*x)^(
7/2)*Sqrt[3 + 5*x])/225 - (1065118*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/4921875 - (59
5387*Sqrt[11/3]*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/4921875

Rule 97

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)/(b*(m + 1)), x] - Dist[1/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n
- 1)*(e + f*x)^(p - 1)*Simp[d*e*n + c*f*p + d*f*(n + p)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && LtQ[m
, -1] && GtQ[n, 0] && GtQ[p, 0] && (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 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} (2+3 x)^{7/2}}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{\left (\frac{1}{2}-36 x\right ) (1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac{442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt{3+5 x}}+\frac{4}{75} \int \frac{\left (\frac{627}{2}-\frac{5895 x}{2}\right ) \sqrt{1-2 x} (2+3 x)^{5/2}}{\sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac{442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt{3+5 x}}-\frac{524}{225} \sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}+\frac{8 \int \frac{\left (111060-\frac{1342395 x}{4}\right ) (2+3 x)^{5/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{10125}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac{442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt{3+5 x}}+\frac{59662 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{7875}-\frac{524}{225} \sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}-\frac{8 \int \frac{(2+3 x)^{3/2} \left (-\frac{4470615}{8}+\frac{8392995 x}{4}\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{354375}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac{442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt{3+5 x}}+\frac{373022 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{196875}+\frac{59662 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{7875}-\frac{524}{225} \sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}+\frac{8 \int \frac{\left (\frac{13705875}{8}-\frac{67567635 x}{8}\right ) \sqrt{2+3 x}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{8859375}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac{442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt{3+5 x}}+\frac{500501 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{984375}+\frac{373022 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{196875}+\frac{59662 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{7875}-\frac{524}{225} \sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}-\frac{8 \int \frac{-\frac{349379055}{16}-\frac{71895465 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{132890625}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac{442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt{3+5 x}}+\frac{500501 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{984375}+\frac{373022 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{196875}+\frac{59662 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{7875}-\frac{524}{225} \sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}+\frac{1065118 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{4921875}+\frac{6549257 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{9843750}\\ &=-\frac{2 (1-2 x)^{5/2} (2+3 x)^{7/2}}{15 (3+5 x)^{3/2}}-\frac{442 (1-2 x)^{3/2} (2+3 x)^{7/2}}{75 \sqrt{3+5 x}}+\frac{500501 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{984375}+\frac{373022 \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}}{196875}+\frac{59662 \sqrt{1-2 x} (2+3 x)^{5/2} \sqrt{3+5 x}}{7875}-\frac{524}{225} \sqrt{1-2 x} (2+3 x)^{7/2} \sqrt{3+5 x}-\frac{1065118 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4921875}-\frac{595387 \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{4921875}\\ \end{align*}

Mathematica [A]  time = 0.319787, size = 117, normalized size = 0.46 \[ \frac{17517535 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{30 \sqrt{1-2 x} \sqrt{3 x+2} \left (4725000 x^5+1327500 x^4-5654250 x^3+470675 x^2+4026600 x+1215489\right )}{(5 x+3)^{3/2}}+2130236 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{29531250} \]

Antiderivative was successfully verified.

[In]

Integrate[((1 - 2*x)^(5/2)*(2 + 3*x)^(7/2))/(3 + 5*x)^(5/2),x]

[Out]

((30*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(1215489 + 4026600*x + 470675*x^2 - 5654250*x^3 + 1327500*x^4 + 4725000*x^5))
/(3 + 5*x)^(3/2) + 2130236*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 17517535*Sqrt[2]*Ellip
ticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])/29531250

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Maple [C]  time = 0.021, size = 239, normalized size = 0.9 \begin{align*} -{\frac{1}{177187500\,{x}^{2}[phone]\,x-59062500} \left ( -850500000\,{x}^{7}[phone]\,\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}[phone]\,\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}-380700000\,{x}^{6}[phone]\,\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 ) +6390708\,\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}^{5}[phone]\,{x}^{4}-1078163250\,{x}^{3}-311345520\,{x}^{2}[phone]\,x+72929340 \right ) \sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1-2*x)^(5/2)*(2+3*x)^(7/2)/(3+5*x)^(5/2),x)

[Out]

-1/29531250*(-850500000*x^7+87587675*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)[phone]*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)-380700000*x^6+52552605*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))+6390708*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^5+164556000*x^4-1078163250*x^3-311345520*x^2+205131330*x+7292934
0)*(2+3*x)^(1/2)*(1-2*x)^(1/2)/(6*x^2+x-2)/(3+5*x)^(3/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (3 \, x + 2\right )}^{\frac{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(7/2)*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (108 \, x^{5} + 108 \, x^{4} - 45 \, x^{3} - 58 \, x^{2} + 4 \, x + 8\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{125 \, x^{3} + 225 \, x^{2} + 135 \, x + 27}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

integral((108*x^5 + 108*x^4 - 45*x^3 - 58*x^2 + 4*x + 8)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(125*x^3 +
225*x^2 + 135*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((1-2*x)**(5/2)*(2+3*x)**(7/2)/(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{7}{2}}{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((1-2*x)^(5/2)*(2+3*x)^(7/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

integrate((3*x + 2)^(7/2)*(-2*x + 1)^(5/2)/(5*x + 3)^(5/2), x)