3.2922 \(\int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\)

Optimal. Leaf size=81 \[ \frac {4 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}}+\frac {2 \sqrt {\frac {5}{7}} \sqrt {-5 x-3} E\left (\sin ^{-1}\left (\sqrt {5} \sqrt {3 x+2}\right )|\frac {2}{35}\right )}{11 \sqrt {5 x+3}} \]

[Out]

2/77*EllipticE(5^(1/2)*(2+3*x)^(1/2),1/35*70^(1/2))*35^(1/2)*(-3-5*x)^(1/2)/(3+5*x)^(1/2)+4/77*(2+3*x)^(1/2)*(
3+5*x)^(1/2)/(1-2*x)^(1/2)

________________________________________________________________________________________

Rubi [A]  time = 0.02, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {104, 21, 114, 113} \[ \frac {4 \sqrt {3 x+2} \sqrt {5 x+3}}{77 \sqrt {1-2 x}}+\frac {2 \sqrt {\frac {5}{7}} \sqrt {-5 x-3} E\left (\sin ^{-1}\left (\sqrt {5} \sqrt {3 x+2}\right )|\frac {2}{35}\right )}{11 \sqrt {5 x+3}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/(77*Sqrt[1 - 2*x]) + (2*Sqrt[5/7]*Sqrt[-3 - 5*x]*EllipticE[ArcSin[Sqrt[5]*Sqrt
[2 + 3*x]], 2/35])/(11*Sqrt[3 + 5*x])

Rule 21

Int[(u_.)*((a_) + (b_.)*(v_))^(m_.)*((c_) + (d_.)*(v_))^(n_.), x_Symbol] :> Dist[(b/d)^m, Int[u*(c + d*v)^(m +
n), x], x] /; FreeQ[{a, b, c, d, n}, x] && EqQ[b*c - a*d, 0] && IntegerQ[m] && ( !IntegerQ[n] || SimplerQ[c +
d*x, a + b*x])

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 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[(Sqrt[e + f*
x]*Sqrt[(b*(c + d*x))/(b*c - a*d)])/(Sqrt[c + d*x]*Sqrt[(b*(e + f*x))/(b*e - a*f)]), Int[Sqrt[(b*e)/(b*e - a*f
) + (b*f*x)/(b*e - a*f)]/(Sqrt[a + b*x]*Sqrt[(b*c)/(b*c - a*d) + (b*d*x)/(b*c - a*d)]), x], 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]

Rubi steps

\begin {align*} \int \frac {1}{(1-2 x)^{3/2} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx &=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{77 \sqrt {1-2 x}}-\frac {2}{77} \int \frac {-\frac {15}{2}+15 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{77 \sqrt {1-2 x}}+\frac {15}{77} \int \frac {\sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{77 \sqrt {1-2 x}}+\frac {\left (15 \sqrt {-3-5 x}\right ) \int \frac {\sqrt {\frac {3}{7}-\frac {6 x}{7}}}{\sqrt {-9-15 x} \sqrt {2+3 x}} \, dx}{11 \sqrt {7} \sqrt {3+5 x}}\\ &=\frac {4 \sqrt {2+3 x} \sqrt {3+5 x}}{77 \sqrt {1-2 x}}+\frac {2 \sqrt {\frac {5}{7}} \sqrt {-3-5 x} E\left (\sin ^{-1}\left (\sqrt {5} \sqrt {2+3 x}\right )|\frac {2}{35}\right )}{11 \sqrt {3+5 x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.09, size = 61, normalized size = 0.75 \[ \frac {2}{77} \left (\frac {2 \sqrt {3 x+2} \sqrt {5 x+3}}{\sqrt {1-2 x}}-i \sqrt {33} E\left (i \sinh ^{-1}\left (\sqrt {15 x+9}\right )|-\frac {2}{33}\right )\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*((2*Sqrt[2 + 3*x]*Sqrt[3 + 5*x])/Sqrt[1 - 2*x] - I*Sqrt[33]*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]], -2/33]))/7
7

________________________________________________________________________________________

fricas [F]  time = 0.56, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{60 \, x^{4} + 16 \, x^{3} - 37 \, x^{2} - 5 \, x + 6}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(sqrt(5*x + 3)*sqrt(3*x + 2)*sqrt(-2*x + 1)/(60*x^4 + 16*x^3 - 37*x^2 - 5*x + 6), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.02, size = 135, normalized size = 1.67 \[ -\frac {\sqrt {-2 x +1}\, \sqrt {3 x +2}\, \sqrt {5 x +3}\, \left (60 x^{2}+76 x -2 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+35 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+24\right )}{77 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/77*(-2*x+1)^(1/2)*(3*x+2)^(1/2)*(5*x+3)^(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))-2*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))+60*x^2+76*x+24)/(30*x^3+23*x^2-7*x-6)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________