3.2872 \(\int \frac {1}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ \frac {4}{7} \sqrt {\frac {3}{11}} \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right ),\frac {35}{33}\right )-\frac {680 \sqrt {1-2 x} \sqrt {3 x+2}}{77 \sqrt {5 x+3}}+\frac {6 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {136}{7} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

[Out]

136/77*EllipticE(1/7*21^(1/2)*(1-2*x)^(1/2),1/33*1155^(1/2))*33^(1/2)+4/77*EllipticF(1/7*21^(1/2)*(1-2*x)^(1/2
),1/33*1155^(1/2))*33^(1/2)+6/7*(1-2*x)^(1/2)/(2+3*x)^(1/2)/(3+5*x)^(1/2)-680/77*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(
3+5*x)^(1/2)

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Rubi [A]  time = 0.04, antiderivative size = 129, normalized size of antiderivative = 1.00, number of steps used = 5, 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} \[ -\frac {680 \sqrt {1-2 x} \sqrt {3 x+2}}{77 \sqrt {5 x+3}}+\frac {6 \sqrt {1-2 x}}{7 \sqrt {3 x+2} \sqrt {5 x+3}}+\frac {4}{7} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {136}{7} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(6*Sqrt[1 - 2*x])/(7*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]) - (680*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/(77*Sqrt[3 + 5*x]) + (13
6*Sqrt[3/11]*EllipticE[ArcSin[Sqrt[3/7]*Sqrt[1 - 2*x]], 35/33])/7 + (4*Sqrt[3/11]*EllipticF[ArcSin[Sqrt[3/7]*S
qrt[1 - 2*x]], 35/33])/7

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 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)]))

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]

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-2 x} (2+3 x)^{3/2} (3+5 x)^{3/2}} \, dx &=\frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} \sqrt {3+5 x}}+\frac {2}{7} \int \frac {25-15 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} \sqrt {3+5 x}}-\frac {680 \sqrt {1-2 x} \sqrt {2+3 x}}{77 \sqrt {3+5 x}}-\frac {4}{77} \int \frac {\frac {645}{2}+510 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {6 \sqrt {1-2 x}}{7 \sqrt {2+3 x} \sqrt {3+5 x}}-\frac {680 \sqrt {1-2 x} \sqrt {2+3 x}}{77 \sqrt {3+5 x}}-\frac {6}{7} \int \frac {1}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx-\frac {408}{77} \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} \sqrt {3+5 x}}-\frac {680 \sqrt {1-2 x} \sqrt {2+3 x}}{77 \sqrt {3+5 x}}+\frac {136}{7} \sqrt {\frac {3}{11}} E\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )+\frac {4}{7} \sqrt {\frac {3}{11}} F\left (\sin ^{-1}\left (\sqrt {\frac {3}{7}} \sqrt {1-2 x}\right )|\frac {35}{33}\right )\\ \end {align*}

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Mathematica [A]  time = 0.15, size = 130, normalized size = 1.01 \[ -\frac {2 \left (-35 \sqrt {2} \left (15 x^2+19 x+6\right ) \operatorname {EllipticF}\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ),-\frac {33}{2}\right )+68 \sqrt {2} \left (15 x^2+19 x+6\right ) E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right )|-\frac {33}{2}\right )+\sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3} (1020 x+647)\right )}{77 (3 x+2) (5 x+3)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x]*(647 + 1020*x) + 68*Sqrt[2]*(6 + 19*x + 15*x^2)*EllipticE[ArcSi
n[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2] - 35*Sqrt[2]*(6 + 19*x + 15*x^2)*EllipticF[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]
], -33/2]))/(77*(2 + 3*x)*(3 + 5*x))

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fricas [F]  time = 0.84, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{450 \, x^{5} + 915 \, x^{4} + 512 \, x^{3} - 85 \, x^{2} - 156 \, x - 36}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(450*x^5 + 915*x^4 + 512*x^3 - 85*x^2 - 156*x - 36), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [C]  time = 0.02, size = 135, normalized size = 1.05 \[ -\frac {2 \sqrt {3 x +2}\, \sqrt {5 x +3}\, \sqrt {-2 x +1}\, \left (2040 x^{2}+274 x -68 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+35 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-647\right )}{77 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/77*(3*x+2)^(1/2)*(5*x+3)^(1/2)*(-2*x+1)^(1/2)*(35*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*Ellipt
icF(1/11*(110*x+66)^(1/2),1/2*I*66^(1/2))-68*2^(1/2)*(5*x+3)^(1/2)*(3*x+2)^(1/2)*(-2*x+1)^(1/2)*EllipticE(1/11
*(110*x+66)^(1/2),1/2*I*66^(1/2))+2040*x^2+274*x-647)/(30*x^3+23*x^2-7*x-6)

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (5 \, x + 3\right )}^{\frac {3}{2}} {\left (3 \, x + 2\right )}^{\frac {3}{2}} \sqrt {-2 \, x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\sqrt {1-2\,x}\,{\left (3\,x+2\right )}^{3/2}\,{\left (5\,x+3\right )}^{3/2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\sqrt {1 - 2 x} \left (3 x + 2\right )^{\frac {3}{2}} \left (5 x + 3\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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