3.27.86 \(\int \frac {\sqrt {1-2 x}}{\sqrt {2+3 x} (3+5 x)^{3/2}} \, dx\) [2686]

Optimal. Leaf size=61 \[ -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}}+2 \sqrt {\frac {7}{5}} E\left (\sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right ) \]

[Out]

2/5*EllipticE(1/11*55^(1/2)*(1-2*x)^(1/2),1/35*1155^(1/2))*35^(1/2)-2*(1-2*x)^(1/2)*(2+3*x)^(1/2)/(3+5*x)^(1/2
)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {101, 21, 114} \begin {gather*} 2 \sqrt {\frac {7}{5}} E\left (\text {ArcSin}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )-\frac {2 \sqrt {1-2 x} \sqrt {3 x+2}}{\sqrt {5 x+3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/Sqrt[3 + 5*x] + 2*Sqrt[7/5]*EllipticE[ArcSin[Sqrt[5/11]*Sqrt[1 - 2*x]], 33/35
]

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 101

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 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 {2+3 x} (3+5 x)^{3/2}} \, dx &=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}}+2 \int \frac {-2-3 x}{\sqrt {1-2 x} \sqrt {2+3 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}}-2 \int \frac {\sqrt {2+3 x}}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=-\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}}+2 \sqrt {\frac {7}{5}} E\left (\sin ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {1-2 x}\right )|\frac {33}{35}\right )\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 2.68, size = 61, normalized size = 1.00 \begin {gather*} -\frac {2 \sqrt {1-2 x} \sqrt {2+3 x}}{\sqrt {3+5 x}}-\frac {2}{5} \sqrt {2} E\left (\sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )|-\frac {33}{2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[1 - 2*x]*Sqrt[2 + 3*x])/Sqrt[3 + 5*x] - (2*Sqrt[2]*EllipticE[ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]], -33/2]
)/5

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(131\) vs. \(2(47)=94\).
time = 0.09, size = 132, normalized size = 2.16

method	result	size
default	\(\frac {2 \sqrt {1-2 x}\, \sqrt {3+5 x}\, \sqrt {2+3 x}\, \left (\sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-\sqrt {2}\, \sqrt {2+3 x}\, \sqrt {-3-5 x}\, \sqrt {1-2 x}\, \EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )-30 x^{2}-5 x +10\right )}{5 \left (30 x^{3}+23 x^{2}-7 x -6\right )}\)	\(132\)
elliptic	\(\frac {\sqrt {-\left (3+5 x \right ) \left (-1+2 x \right ) \left (2+3 x \right )}\, \left (-\frac {4 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{21 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \sqrt {28+42 x}\, \sqrt {-15 x -9}\, \sqrt {21-42 x}\, \left (-\frac {\EllipticE \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{15}-\frac {3 \EllipticF \left (\frac {\sqrt {28+42 x}}{7}, \frac {\sqrt {70}}{2}\right )}{5}\right )}{7 \sqrt {-30 x^{3}-23 x^{2}+7 x +6}}-\frac {2 \left (-30 x^{2}-5 x +10\right )}{5 \sqrt {\left (x +\frac {3}{5}\right ) \left (-30 x^{2}-5 x +10\right )}}\right )}{\sqrt {1-2 x}\, \sqrt {2+3 x}\, \sqrt {3+5 x}}\)	\(201\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/5*(1-2*x)^(1/2)*(3+5*x)^(1/2)*(2+3*x)^(1/2)*(2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticF(1/
7*(28+42*x)^(1/2),1/2*70^(1/2))-2^(1/2)*(2+3*x)^(1/2)*(-3-5*x)^(1/2)*(1-2*x)^(1/2)*EllipticE(1/7*(28+42*x)^(1/
2),1/2*70^(1/2))-30*x^2-5*x+10)/(30*x^3+23*x^2-7*x-6)

________________________________________________________________________________________

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.11, size = 23, normalized size = 0.38 \begin {gather*} -\frac {2 \, \sqrt {3 \, x + 2} \sqrt {-2 \, x + 1}}{\sqrt {5 \, x + 3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

________________________________________________________________________________________

Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {1-2\,x}}{\sqrt {3\,x+2}\,{\left (5\,x+3\right )}^{3/2}} \,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)^(3/2)),x)

[Out]

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

________________________________________________________________________________________