3.2803 \(\int \frac{(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=191 \[ \frac{3176}{45} \sqrt{\frac{11}{3}} \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )+\frac{14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} \sqrt{5 x+3}}-\frac{105584 \sqrt{3 x+2} \sqrt{1-2 x}}{27 \sqrt{5 x+3}}+\frac{17468 \sqrt{1-2 x}}{45 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2716 \sqrt{1-2 x}}{135 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{105584}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

[Out]

(14*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (2716*Sqrt[1 - 2*x])/(135*(2 + 3*x)^(3/2)*Sqrt[3 + 5
*x]) + (17468*Sqrt[1 - 2*x])/(45*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (105584*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27*Sqrt[
3 + 5*x]) + (105584*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45 + (3176*Sqrt[11/3]*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45

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Rubi [A]  time = 0.0636568, 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 = {98, 150, 152, 158, 113, 119} \[ \frac{14 (1-2 x)^{3/2}}{15 (3 x+2)^{5/2} \sqrt{5 x+3}}-\frac{105584 \sqrt{3 x+2} \sqrt{1-2 x}}{27 \sqrt{5 x+3}}+\frac{17468 \sqrt{1-2 x}}{45 \sqrt{3 x+2} \sqrt{5 x+3}}+\frac{2716 \sqrt{1-2 x}}{135 (3 x+2)^{3/2} \sqrt{5 x+3}}+\frac{3176}{45} \sqrt{\frac{11}{3}} F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{105584}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(14*(1 - 2*x)^(3/2))/(15*(2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) + (2716*Sqrt[1 - 2*x])/(135*(2 + 3*x)^(3/2)*Sqrt[3 + 5
*x]) + (17468*Sqrt[1 - 2*x])/(45*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (105584*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(27*Sqrt[
3 + 5*x]) + (105584*Sqrt[11/3]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45 + (3176*Sqrt[11/3]*Ellipt
icF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/45

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{(1-2 x)^{5/2}}{(2+3 x)^{7/2} (3+5 x)^{3/2}} \, dx &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2}{15} \int \frac{(163-95 x) \sqrt{1-2 x}}{(2+3 x)^{5/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2716 \sqrt{1-2 x}}{135 (2+3 x)^{3/2} \sqrt{3+5 x}}-\frac{4}{135} \int \frac{-\frac{17369}{2}+9900 x}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2716 \sqrt{1-2 x}}{135 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{17468 \sqrt{1-2 x}}{45 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{8}{945} \int \frac{-\frac{741125}{2}+\frac{458535 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2716 \sqrt{1-2 x}}{135 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{17468 \sqrt{1-2 x}}{45 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{105584 \sqrt{1-2 x} \sqrt{2+3 x}}{27 \sqrt{3+5 x}}+\frac{16 \int \frac{-\frac{19301205}{4}-7621845 x}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{10395}\\ &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2716 \sqrt{1-2 x}}{135 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{17468 \sqrt{1-2 x}}{45 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{105584 \sqrt{1-2 x} \sqrt{2+3 x}}{27 \sqrt{3+5 x}}-\frac{17468}{45} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx-\frac{105584}{45} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{14 (1-2 x)^{3/2}}{15 (2+3 x)^{5/2} \sqrt{3+5 x}}+\frac{2716 \sqrt{1-2 x}}{135 (2+3 x)^{3/2} \sqrt{3+5 x}}+\frac{17468 \sqrt{1-2 x}}{45 \sqrt{2+3 x} \sqrt{3+5 x}}-\frac{105584 \sqrt{1-2 x} \sqrt{2+3 x}}{27 \sqrt{3+5 x}}+\frac{105584}{45} \sqrt{\frac{11}{3}} E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )+\frac{3176}{45} \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 = 0.233169, size = 105, normalized size = 0.55 \[ \frac{2}{135} \left (-2 \sqrt{2} \left (26396 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-13295 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )-\frac{3 \sqrt{1-2 x} \left (2375640 x^3+4672674 x^2+3061396 x+668031\right )}{(3 x+2)^{5/2} \sqrt{5 x+3}}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((-3*Sqrt[1 - 2*x]*(668031 + 3061396*x + 4672674*x^2 + 2375640*x^3))/((2 + 3*x)^(5/2)*Sqrt[3 + 5*x]) - 2*Sq
rt[2]*(26396*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 13295*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5
*x]], -33/2])))/135

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Maple [C]  time = 0.025, size = 314, normalized size = 1.6 \begin{align*}{\frac{2}{1350\,{x}^{2}+135\,x-405}\sqrt{1-2\,x}\sqrt{3+5\,x} \left ( 475128\,\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}-239310\,\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}+633504\,\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}-319080\,\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}+211168\,\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 ) -106360\,\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 ) -14253840\,{x}^{4}-20909124\,{x}^{3}-4350354\,{x}^{2}+5176002\,x+2004093 \right ) \left ( 2+3\,x \right ) ^{-{\frac{5}{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)^(3/2),x)

[Out]

2/135*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(475128*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)-239310*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)+633504*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)-319080*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)+211168*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))-106360*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))-14253840*x^4-20909124*x^3-4350354*x^2+5176002*x+2004093)/(2+3*x)^(5/2)/(10*x^2+x-3)

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (4 \, x^{2} - 4 \, x + 1\right )} \sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{2025 \, x^{6} + 7830 \, x^{5} + 12609 \, x^{4} + 10824 \, x^{3} + 5224 \, x^{2} + 1344 \, x + 144}, 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)^(3/2),x, algorithm="fricas")

[Out]

integral((4*x^2 - 4*x + 1)*sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2025*x^6 + 7830*x^5 + 12609*x^4 + 10824
*x^3 + 5224*x^2 + 1344*x + 144), 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)**(3/2),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-2 \, x + 1\right )}^{\frac{5}{2}}}{{\left (5 \, x + 3\right )}^{\frac{3}{2}}{\left (3 \, x + 2\right )}^{\frac{7}{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)^(3/2),x, algorithm="giac")

[Out]

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