3.2819 \(\int \frac {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx\)

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

[Out]

2/21*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)-2/7*(1-2*x)^(1/2)*(3
+5*x)^(1/2)/(2+3*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 = {99, 21, 114, 113} \[ \frac {2 \sqrt {\frac {5}{7}} \sqrt {-5 x-3} E\left (\sin ^{-1}\left (\sqrt {5} \sqrt {3 x+2}\right )|\frac {2}{35}\right )}{3 \sqrt {5 x+3}}-\frac {2 \sqrt {1-2 x} \sqrt {5 x+3}}{7 \sqrt {3 x+2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[3 + 5*x])/(7*Sqrt[2 + 3*x]) + (2*Sqrt[5/7]*Sqrt[-3 - 5*x]*EllipticE[ArcSin[Sqrt[5]*Sqrt
[2 + 3*x]], 2/35])/(3*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 99

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 + 1))/((m + 1)*(b*e - a*f)), x] - Dist[1/((m + 1)*(b*e - a*f)), Int[(a +
b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p*Simp[d*e*n + c*f*(m + p + 2) + d*f*(m + n + p + 2)*x, x], x], x] /;
FreeQ[{a, b, c, d, e, f, p}, x] && LtQ[m, -1] && GtQ[n, 0] && (IntegersQ[2*m, 2*n, 2*p] || IntegersQ[m, n + p
] || IntegersQ[p, m + n])

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 {\sqrt {3+5 x}}{\sqrt {1-2 x} (2+3 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 x}}+\frac {2}{7} \int \frac {\frac {5}{2}-5 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 x}}+\frac {5}{7} \int \frac {\sqrt {1-2 x}}{\sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 x}}+\frac {\left (5 \sqrt {-3-5 x}\right ) \int \frac {\sqrt {\frac {3}{7}-\frac {6 x}{7}}}{\sqrt {-9-15 x} \sqrt {2+3 x}} \, dx}{\sqrt {7} \sqrt {3+5 x}}\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {3+5 x}}{7 \sqrt {2+3 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 )}{3 \sqrt {3+5 x}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.11, size = 70, normalized size = 0.86 \[ \frac {-6 \sqrt {1-2 x} \sqrt {3 x+2} \sqrt {5 x+3}-2 i \sqrt {33} (3 x+2) E\left (i \sinh ^{-1}\left (\sqrt {15 x+9}\right )|-\frac {2}{33}\right )}{63 x+42} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-6*Sqrt[1 - 2*x]*Sqrt[2 + 3*x]*Sqrt[3 + 5*x] - (2*I)*Sqrt[33]*(2 + 3*x)*EllipticE[I*ArcSinh[Sqrt[9 + 15*x]],
-2/33])/(42 + 63*x)

________________________________________________________________________________________

fricas [F]  time = 0.76, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {5 \, x + 3} \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{18 \, x^{3} + 15 \, x^{2} - 4 \, x - 4}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((3+5*x)^(1/2)/(2+3*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)/(18*x^3 + 15*x^2 - 4*x - 4), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [C]  time = 0.02, size = 135, normalized size = 1.67 \[ -\frac {\sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \left (60 x^{2}+6 x -2 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticE \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )+35 \sqrt {2}\, \sqrt {5 x +3}\, \sqrt {3 x +2}\, \sqrt {-2 x +1}\, \EllipticF \left (\frac {\sqrt {110 x +66}}{11}, \frac {i \sqrt {66}}{2}\right )-18\right )}{21 \left (30 x^{3}+23 x^{2}-7 x -6\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/21*(5*x+3)^(1/2)*(3*x+2)^(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))-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+6*x-18)/(30*x^3+23*x^2-7*x-6)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________