3.2881 \(\int \frac{1}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx\)

Optimal. Leaf size=156 \[ -\frac{536 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right ),\frac{35}{33}\right )}{77 \sqrt{33}}+\frac{89020 \sqrt{1-2 x} \sqrt{3 x+2}}{2541 \sqrt{5 x+3}}-\frac{1340 \sqrt{1-2 x} \sqrt{3 x+2}}{231 (5 x+3)^{3/2}}+\frac{6 \sqrt{1-2 x}}{7 \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{17804 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}} \]

[Out]

(6*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (1340*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(231*(3 + 5*x)^(3/2))
+ (89020*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(2541*Sqrt[3 + 5*x]) - (17804*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/(77*Sqrt[33]) - (536*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(77*Sqrt[33])

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Rubi [A]  time = 0.0519411, antiderivative size = 156, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {104, 152, 158, 113, 119} \[ \frac{89020 \sqrt{1-2 x} \sqrt{3 x+2}}{2541 \sqrt{5 x+3}}-\frac{1340 \sqrt{1-2 x} \sqrt{3 x+2}}{231 (5 x+3)^{3/2}}+\frac{6 \sqrt{1-2 x}}{7 \sqrt{3 x+2} (5 x+3)^{3/2}}-\frac{536 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}}-\frac{17804 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) - (1340*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(231*(3 + 5*x)^(3/2))
+ (89020*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(2541*Sqrt[3 + 5*x]) - (17804*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]]
, 35/33])/(77*Sqrt[33]) - (536*EllipticF[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/(77*Sqrt[33])

Rule 104

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(b*(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*(m + 1) - b*(d*e*(m + n + 2) +
c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && LtQ[m, -1] &&
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}{\sqrt{1-2 x} (2+3 x)^{3/2} (3+5 x)^{5/2}} \, dx &=\frac{6 \sqrt{1-2 x}}{7 \sqrt{2+3 x} (3+5 x)^{3/2}}+\frac{2}{7} \int \frac{40-45 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{5/2}} \, dx\\ &=\frac{6 \sqrt{1-2 x}}{7 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{1340 \sqrt{1-2 x} \sqrt{2+3 x}}{231 (3+5 x)^{3/2}}-\frac{4}{231} \int \frac{\frac{3245}{2}-1005 x}{\sqrt{1-2 x} \sqrt{2+3 x} (3+5 x)^{3/2}} \, dx\\ &=\frac{6 \sqrt{1-2 x}}{7 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{1340 \sqrt{1-2 x} \sqrt{2+3 x}}{231 (3+5 x)^{3/2}}+\frac{89020 \sqrt{1-2 x} \sqrt{2+3 x}}{2541 \sqrt{3+5 x}}+\frac{8 \int \frac{21135+\frac{66765 x}{2}}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx}{2541}\\ &=\frac{6 \sqrt{1-2 x}}{7 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{1340 \sqrt{1-2 x} \sqrt{2+3 x}}{231 (3+5 x)^{3/2}}+\frac{89020 \sqrt{1-2 x} \sqrt{2+3 x}}{2541 \sqrt{3+5 x}}+\frac{268}{77} \int \frac{1}{\sqrt{1-2 x} \sqrt{2+3 x} \sqrt{3+5 x}} \, dx+\frac{17804}{847} \int \frac{\sqrt{3+5 x}}{\sqrt{1-2 x} \sqrt{2+3 x}} \, dx\\ &=\frac{6 \sqrt{1-2 x}}{7 \sqrt{2+3 x} (3+5 x)^{3/2}}-\frac{1340 \sqrt{1-2 x} \sqrt{2+3 x}}{231 (3+5 x)^{3/2}}+\frac{89020 \sqrt{1-2 x} \sqrt{2+3 x}}{2541 \sqrt{3+5 x}}-\frac{17804 E\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}}-\frac{536 F\left (\sin ^{-1}\left (\sqrt{\frac{3}{7}} \sqrt{1-2 x}\right )|\frac{35}{33}\right )}{77 \sqrt{33}}\\ \end{align*}

Mathematica [A]  time = 0.122425, size = 99, normalized size = 0.63 \[ \frac{2 \left (2 \sqrt{2} \left (4451 E\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right )|-\frac{33}{2}\right )-2240 \text{EllipticF}\left (\sin ^{-1}\left (\sqrt{\frac{2}{11}} \sqrt{5 x+3}\right ),-\frac{33}{2}\right )\right )+\frac{\sqrt{1-2 x} \left (667650 x^2+823580 x+253409\right )}{\sqrt{3 x+2} (5 x+3)^{3/2}}\right )}{2541} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((Sqrt[1 - 2*x]*(253409 + 823580*x + 667650*x^2))/(Sqrt[2 + 3*x]*(3 + 5*x)^(3/2)) + 2*Sqrt[2]*(4451*Ellipti
cE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 2240*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2])))/2541

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Maple [C]  time = 0.027, size = 219, normalized size = 1.4 \begin{align*}{\frac{2}{15246\,{x}^{2}+2541\,x-5082}\sqrt{2+3\,x}\sqrt{1-2\,x} \left ( 22400\,\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}-44510\,\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}+13440\,\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 ) -26706\,\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 ) +1335300\,{x}^{3}+979510\,{x}^{2}-316762\,x-253409 \right ) \left ( 3+5\,x \right ) ^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/2541*(2+3*x)^(1/2)*(1-2*x)^(1/2)*(22400*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)-44510*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)+13440*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))-26706*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))+1335300*x^3+979510*x^2-316762*x-253409)/(3+5*x)^(3/2)/(6*x^2+x-2)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{\sqrt{5 \, x + 3} \sqrt{3 \, x + 2} \sqrt{-2 \, x + 1}}{2250 \, x^{6} + 5925 \, x^{5} + 5305 \, x^{4} + 1111 \, x^{3} - 1035 \, x^{2} - 648 \, x - 108}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(2250*x^6 + 5925*x^5 + 5305*x^4 + 1111*x^3 - 1035*x^2 - 6
48*x - 108), 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+3*x)**(3/2)/(3+5*x)**(5/2)/(1-2*x)**(1/2),x)

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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