3.9.63 \(\int \frac {\sqrt {1-x} \sqrt {2+3 x}}{\sqrt {1-x^2}} \, dx\) [863]

Optimal. Leaf size=35 \[ \sqrt {1+x} \sqrt {2+3 x}-\frac {\sinh ^{-1}\left (\sqrt {2+3 x}\right )}{\sqrt {3}} \]

[Out]

-1/3*arcsinh((2+3*x)^(1/2))*3^(1/2)+(1+x)^(1/2)*(2+3*x)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 35, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {26, 52, 56, 221} \begin {gather*} \sqrt {x+1} \sqrt {3 x+2}-\frac {\sinh ^{-1}\left (\sqrt {3 x+2}\right )}{\sqrt {3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[1 + x]*Sqrt[2 + 3*x] - ArcSinh[Sqrt[2 + 3*x]]/Sqrt[3]

Rule 26

Int[(u_.)*((a_) + (b_.)*(x_)^(n_.))^(m_.)*((c_) + (d_.)*(x_)^(j_))^(p_.), x_Symbol] :> Dist[(-b^2/d)^m, Int[u/
(a - b*x^n)^m, x], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[j, 2*n] && EqQ[p, -m] && EqQ[b^2*c + a^2*d, 0]
&& GtQ[a, 0] && LtQ[d, 0]

Rule 52

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + n + 1))), x] + Dist[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

Rule 56

Int[1/(Sqrt[(a_.) + (b_.)*(x_)]*Sqrt[(c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[2/Sqrt[b], Subst[Int[1/Sqrt[b*c -
a*d + d*x^2], x], x, Sqrt[a + b*x]], x] /; FreeQ[{a, b, c, d}, x] && GtQ[b*c - a*d, 0] && GtQ[b, 0]

Rule 221

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[Rt[b, 2]*(x/Sqrt[a])]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin {align*} \int \frac {\sqrt {1-x} \sqrt {2+3 x}}{\sqrt {1-x^2}} \, dx &=\int \frac {\sqrt {2+3 x}}{\sqrt {1+x}} \, dx\\ &=\sqrt {1+x} \sqrt {2+3 x}-\frac {1}{2} \int \frac {1}{\sqrt {1+x} \sqrt {2+3 x}} \, dx\\ &=\sqrt {1+x} \sqrt {2+3 x}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {1+x^2}} \, dx,x,\sqrt {2+3 x}\right )}{\sqrt {3}}\\ &=\sqrt {1+x} \sqrt {2+3 x}-\frac {\sinh ^{-1}\left (\sqrt {2+3 x}\right )}{\sqrt {3}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.74, size = 49, normalized size = 1.40 \begin {gather*} \frac {3 \sqrt {1+x} (2+3 x)-\sqrt {6+9 x} \sinh ^{-1}\left (\sqrt {2+3 x}\right )}{3 \sqrt {2+3 x}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(3*Sqrt[1 + x]*(2 + 3*x) - Sqrt[6 + 9*x]*ArcSinh[Sqrt[2 + 3*x]])/(3*Sqrt[2 + 3*x])

________________________________________________________________________________________

Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(85\) vs. \(2(27)=54\).
time = 0.64, size = 86, normalized size = 2.46

method	result	size
default	\(\frac {\sqrt {1-x}\, \sqrt {2+3 x}\, \sqrt {-x^{2}+1}\, \left (\ln \left (\frac {5 \sqrt {3}}{6}+x \sqrt {3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}-6 \sqrt {3 x^{2}+5 x +2}\right )}{6 \left (-1+x \right ) \sqrt {3 x^{2}+5 x +2}}\)	\(86\)
risch	\(-\frac {\left (1+x \right ) \sqrt {2+3 x}\, \sqrt {\frac {\left (1-x \right ) \left (2+3 x \right ) \left (-x^{2}+1\right )}{\left (-1+x \right )^{2}}}\, \left (-1+x \right )}{\sqrt {\left (1+x \right ) \left (2+3 x \right )}\, \sqrt {1-x}\, \sqrt {-x^{2}+1}}+\frac {\ln \left (\frac {\left (\frac {5}{2}+3 x \right ) \sqrt {3}}{3}+\sqrt {3 x^{2}+5 x +2}\right ) \sqrt {3}\, \sqrt {\frac {\left (1-x \right ) \left (2+3 x \right ) \left (-x^{2}+1\right )}{\left (-1+x \right )^{2}}}\, \left (-1+x \right )}{6 \sqrt {1-x}\, \sqrt {2+3 x}\, \sqrt {-x^{2}+1}}\)	\(149\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(1-x)^(1/2)*(2+3*x)^(1/2)*(-x^2+1)^(1/2)*(ln(5/6*3^(1/2)+x*3^(1/2)+(3*x^2+5*x+2)^(1/2))*3^(1/2)-6*(3*x^2+5
*x+2)^(1/2))/(-1+x)/(3*x^2+5*x+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-x)^(1/2)*(2+3*x)^(1/2)/(-x^2+1)^(1/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 96 vs. \(2 (27) = 54\).
time = 0.42, size = 96, normalized size = 2.74 \begin {gather*} \frac {\sqrt {3} {\left (x - 1\right )} \log \left (-\frac {72 \, x^{3} + 4 \, \sqrt {3} \sqrt {-x^{2} + 1} {\left (6 \, x + 5\right )} \sqrt {3 \, x + 2} \sqrt {-x + 1} + 48 \, x^{2} - 71 \, x - 49}{x - 1}\right ) - 12 \, \sqrt {-x^{2} + 1} \sqrt {3 \, x + 2} \sqrt {-x + 1}}{12 \, {\left (x - 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(sqrt(3)*(x - 1)*log(-(72*x^3 + 4*sqrt(3)*sqrt(-x^2 + 1)*(6*x + 5)*sqrt(3*x + 2)*sqrt(-x + 1) + 48*x^2 -
71*x - 49)/(x - 1)) - 12*sqrt(-x^2 + 1)*sqrt(3*x + 2)*sqrt(-x + 1))/(x - 1)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________