3.2747 \(\int \frac{(1-2 x)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=191 \[ -\frac{8366 \sqrt{\frac{3}{11}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{15625}-\frac{106 \sqrt{1-2 x} (3 x+2)^{5/2}}{25 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}+\frac{1558}{625} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{2264 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{3125}+\frac{1973 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625} \]

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2))/(15*(3 + 5*x)^(3/2)) - (106*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(25*Sqrt[3 + 5
*x]) + (2264*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3125 + (1558*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*
x])/625 + (1973*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625 - (8366*Sqrt[3/11]*Ellipti
cF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625

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Rubi [A]  time = 0.0673948, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, 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} \[ -\frac{106 \sqrt{1-2 x} (3 x+2)^{5/2}}{25 \sqrt{5 x+3}}-\frac{2 (1-2 x)^{3/2} (3 x+2)^{5/2}}{15 (5 x+3)^{3/2}}+\frac{1558}{625} \sqrt{1-2 x} \sqrt{5 x+3} (3 x+2)^{3/2}+\frac{2264 \sqrt{1-2 x} \sqrt{5 x+3} \sqrt{3 x+2}}{3125}-\frac{8366 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}+\frac{1973 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(1 - 2*x)^(3/2)*(2 + 3*x)^(5/2))/(15*(3 + 5*x)^(3/2)) - (106*Sqrt[1 - 2*x]*(2 + 3*x)^(5/2))/(25*Sqrt[3 + 5
*x]) + (2264*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/3125 + (1558*Sqrt[1 - 2*x]*(2 + 3*x)^(3/2)*Sqrt[3 + 5*
x])/625 + (1973*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625 - (8366*Sqrt[3/11]*Ellipti
cF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/15625

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)^{3/2} (2+3 x)^{5/2}}{(3+5 x)^{5/2}} \, dx &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}+\frac{2}{15} \int \frac{\left (\frac{3}{2}-24 x\right ) \sqrt{1-2 x} (2+3 x)^{3/2}}{(3+5 x)^{3/2}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac{106 \sqrt{1-2 x} (2+3 x)^{5/2}}{25 \sqrt{3+5 x}}+\frac{4}{75} \int \frac{\left (306-\frac{2337 x}{2}\right ) (2+3 x)^{3/2}}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac{106 \sqrt{1-2 x} (2+3 x)^{5/2}}{25 \sqrt{3+5 x}}+\frac{1558}{625} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{4 \int \frac{\sqrt{2+3 x} \left (-\frac{2775}{4}+5094 x\right )}{\sqrt{1-2 x} \sqrt{3+5 x}} \, dx}{1875}\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac{106 \sqrt{1-2 x} (2+3 x)^{5/2}}{25 \sqrt{3+5 x}}+\frac{2264 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3125}+\frac{1558}{625} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{4 \int \frac{\frac{5967}{2}-\frac{17757 x}{4}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{28125}\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac{106 \sqrt{1-2 x} (2+3 x)^{5/2}}{25 \sqrt{3+5 x}}+\frac{2264 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3125}+\frac{1558}{625} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}-\frac{1973 \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx}{15625}+\frac{12549 \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{15625}\\ &=-\frac{2 (1-2 x)^{3/2} (2+3 x)^{5/2}}{15 (3+5 x)^{3/2}}-\frac{106 \sqrt{1-2 x} (2+3 x)^{5/2}}{25 \sqrt{3+5 x}}+\frac{2264 \sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}}{3125}+\frac{1558}{625} \sqrt{1-2 x} (2+3 x)^{3/2} \sqrt{3+5 x}+\frac{1973 \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}-\frac{8366 \sqrt{\frac{3}{11}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{15625}\\ \end{align*}

Mathematica [A]  time = 0.314651, size = 107, normalized size = 0.56 \[ \frac{39620 \sqrt{2} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )+\frac{10 \sqrt{1-2 x} \sqrt{3 x+2} \left (-6750 x^3+1050 x^2+2975 x-106\right )}{(5 x+3)^{3/2}}-1973 \sqrt{2} E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )}{46875} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((10*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*(-106 + 2975*x + 1050*x^2 - 6750*x^3))/(3 + 5*x)^(3/2) - 1973*Sqrt[2]*Ellipti
cE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] + 39620*Sqrt[2]*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]
)/46875

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Maple [C]  time = 0.019, size = 229, normalized size = 1.2 \begin{align*} -{\frac{1}{281250\,{x}^{2}+46875\,x-93750} \left ( 198100\,\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}-9865\,\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}+118860\,\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 ) -5919\,\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 ) +405000\,{x}^{5}+4500\,{x}^{4}-324000\,{x}^{3}-2390\,{x}^{2}+60560\,x-2120 \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)^(3/2)*(2+3*x)^(5/2)/(3+5*x)^(5/2),x)

[Out]

-1/46875*(198100*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)-9865*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
)+118860*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))-591
9*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))+405000*x^5
+4500*x^4-324000*x^3-2390*x^2+60560*x-2120)*(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{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{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)^(3/2)*(2+3*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((3*x + 2)^(5/2)*(-2*x + 1)^(3/2)/(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 (18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4\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)^(3/2)*(2+3*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="fricas")

[Out]

integral(-(18*x^3 + 15*x^2 - 4*x - 4)*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)**(3/2)*(2+3*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{5}{2}}{\left (-2 \, x + 1\right )}^{\frac{3}{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)^(3/2)*(2+3*x)^(5/2)/(3+5*x)^(5/2),x, algorithm="giac")

[Out]

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