3.27.74 \(\int \frac {\sqrt {1-2 x}}{\sqrt {-3-5 x} \sqrt {2+3 x}} \, dx\) [2674]

Optimal. Leaf size=31 \[ \frac {2}{3} \sqrt {\frac {7}{5}} E\left (\sin ^{-1}\left (\sqrt {5} \sqrt {2+3 x}\right )|\frac {2}{35}\right ) \]

[Out]

2/15*EllipticE(5^(1/2)*(2+3*x)^(1/2),1/35*70^(1/2))*35^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 31, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {114} \begin {gather*} \frac {2}{3} \sqrt {\frac {7}{5}} E\left (\text {ArcSin}\left (\sqrt {5} \sqrt {3 x+2}\right )|\frac {2}{35}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 114

Int[Sqrt[(e_.) + (f_.)*(x_)]/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[(2/b)*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)))], 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])

Rubi steps

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

________________________________________________________________________________________

Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(109\) vs. \(2(31)=62\).
time = 1.38, size = 109, normalized size = 3.52 \begin {gather*} -\frac {2 \left (\frac {3 \left (-3+x+10 x^2\right )}{\sqrt {2+3 x}}+\sqrt {35} \sqrt {\frac {-1+2 x}{2+3 x}} (2+3 x) \sqrt {\frac {3+5 x}{2+3 x}} E\left (\sin ^{-1}\left (\frac {\sqrt {\frac {7}{2}}}{\sqrt {2+3 x}}\right )|\frac {2}{35}\right )\right )}{15 \sqrt {-3-5 x} \sqrt {1-2 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*((3*(-3 + x + 10*x^2))/Sqrt[2 + 3*x] + Sqrt[35]*Sqrt[(-1 + 2*x)/(2 + 3*x)]*(2 + 3*x)*Sqrt[(3 + 5*x)/(2 + 3
*x)]*EllipticE[ArcSin[Sqrt[7/2]/Sqrt[2 + 3*x]], 2/35]))/(15*Sqrt[-3 - 5*x]*Sqrt[1 - 2*x])

________________________________________________________________________________________

Maple [A]
time = 0.09, size = 41, normalized size = 1.32

method	result	size
default	\(\frac {\left (33 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )+2 \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )\right ) \sqrt {2}}{15}\)	\(41\)
elliptic	\(\frac {\sqrt {\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (\frac {\sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{21 \sqrt {30 x^{3}+23 x^{2}-7 x -6}}-\frac {2 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{21 \sqrt {30 x^{3}+23 x^{2}-7 x -6}}\right )}{\sqrt {1-2 x}\, \sqrt {-3-5 x}\, \sqrt {2+3 x}}\)	\(172\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/15*(33*EllipticF(1/7*(28+42*x)^(1/2),1/2*70^(1/2))+2*EllipticE(1/7*(28+42*x)^(1/2),1/2*70^(1/2)))*2^(1/2)

________________________________________________________________________________________

Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 0.13, size = 26, normalized size = 0.84 \begin {gather*} \frac {68}{675} \, \sqrt {30} {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right ) + \frac {2}{15} \, \sqrt {30} {\rm weierstrassZeta}\left (\frac {1159}{675}, \frac {38998}{91125}, {\rm weierstrassPInverse}\left (\frac {1159}{675}, \frac {38998}{91125}, x + \frac {23}{90}\right )\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

68/675*sqrt(30)*weierstrassPInverse(1159/675, 38998/91125, x + 23/90) + 2/15*sqrt(30)*weierstrassZeta(1159/675
, 38998/91125, weierstrassPInverse(1159/675, 38998/91125, x + 23/90))

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {1 - 2 x}}{\sqrt {- 5 x - 3} \sqrt {3 x + 2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int \frac {\sqrt {1-2\,x}}{\sqrt {3\,x+2}\,\sqrt {-5\,x-3}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________