3.25.93 \(\int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\) [2493]

Optimal. Leaf size=26 \[ \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right ) \]

[Out]

1/5*arcsin(1/11*22^(1/2)*(3+5*x)^(1/2))*10^(1/2)

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Rubi [A]
time = 0.00, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {56, 222} \begin {gather*} \sqrt {\frac {2}{5}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[2/5]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*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 222

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

Rubi steps

\begin {align*} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx &=\frac {2 \text {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{\sqrt {5}}\\ &=\sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [A]
time = 0.04, size = 31, normalized size = 1.19 \begin {gather*} -\sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[2/5]*ArcTan[Sqrt[5/2 - 5*x]/Sqrt[3 + 5*x]])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(38\) vs. \(2(18)=36\).
time = 0.07, size = 39, normalized size = 1.50

method	result	size
default	\(\frac {\sqrt {10}\, \arcsin \left (\frac {20 x}{11}+\frac {1}{11}\right ) \sqrt {\left (1-2 x \right ) \left (3+5 x \right )}}{10 \sqrt {1-2 x}\, \sqrt {3+5 x}}\)	\(39\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*10^(1/2)*arcsin(20/11*x+1/11)*((1-2*x)*(3+5*x))^(1/2)/(1-2*x)^(1/2)/(3+5*x)^(1/2)

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Maxima [A]
time = 0.52, size = 11, normalized size = 0.42 \begin {gather*} -\frac {1}{10} \, \sqrt {10} \arcsin \left (-\frac {20}{11} \, x - \frac {1}{11}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*sqrt(10)*arcsin(-20/11*x - 1/11)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 46 vs. \(2 (18) = 36\).
time = 1.20, size = 46, normalized size = 1.77 \begin {gather*} -\frac {1}{10} \, \sqrt {5} \sqrt {2} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*sqrt(5)*sqrt(2)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x - 3))

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Sympy [C] Result contains complex when optimal does not.
time = 0.55, size = 56, normalized size = 2.15 \begin {gather*} \begin {cases} - \frac {\sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{5} & \text {for}\: \left |{x + \frac {3}{5}}\right | > \frac {11}{10} \\\frac {\sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{5} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/5, Abs(x + 3/5) > 11/10), (sqrt(10)*asin(sqrt(110)*sq
rt(x + 3/5)/11)/5, True))

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Giac [A]
time = 0.72, size = 21, normalized size = 0.81 \begin {gather*} \frac {1}{5} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(5)*sqrt(2)*arcsin(1/11*sqrt(22)*sqrt(5*x + 3))

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Mupad [B]
time = 0.04, size = 36, normalized size = 1.38 \begin {gather*} -\frac {2\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}{5\,\left (\sqrt {1-2\,x}-1\right )}\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(2*10^(1/2)*atan((10^(1/2)*(3^(1/2) - (5*x + 3)^(1/2)))/(5*((1 - 2*x)^(1/2) - 1))))/5

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