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

Optimal. Leaf size=26 \[ \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{21}} \sqrt {5 x+3}\right ) \]

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Rubi [A]  time = 0.01, 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 = {54, 216} \begin {gather*} \sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{21}} \sqrt {5 x+3}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

Sqrt[2/5]*ArcSin[Sqrt[2/21]*Sqrt[3 + 5*x]]

Rule 54

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 216

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 {3-2 x} \sqrt {3+5 x}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{\sqrt {21-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )}{\sqrt {5}}\\ &=\sqrt {\frac {2}{5}} \sin ^{-1}\left (\sqrt {\frac {2}{21}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 45, normalized size = 1.73 \begin {gather*} -\frac {\sqrt {\frac {2}{5}} \sqrt {3-2 x} \sinh ^{-1}\left (\sqrt {\frac {5}{21}} \sqrt {2 x-3}\right )}{\sqrt {2 x-3}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[2/5]*Sqrt[3 - 2*x]*ArcSinh[Sqrt[5/21]*Sqrt[-3 + 2*x]])/Sqrt[-3 + 2*x])

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IntegrateAlgebraic [A]  time = 0.05, size = 36, normalized size = 1.38 \begin {gather*} -\sqrt {\frac {2}{5}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}} \sqrt {3-2 x}}{\sqrt {5 x+3}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.77, size = 44, normalized size = 1.69 \begin {gather*} -\frac {1}{5} \, \sqrt {5} \sqrt {2} \arctan \left (\frac {\sqrt {5} \sqrt {2} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 3} - 3 \, \sqrt {5} \sqrt {2}}{10 \, x}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.93, size = 21, normalized size = 0.81 \begin {gather*} \frac {1}{5} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {1}{21} \, \sqrt {42} \sqrt {5 \, x + 3}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/5*sqrt(5)*sqrt(2)*arcsin(1/21*sqrt(42)*sqrt(5*x + 3))

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maple [B]  time = 0.01, size = 39, normalized size = 1.50 \begin {gather*} \frac {\sqrt {\left (-2 x +3\right ) \left (5 x +3\right )}\, \sqrt {10}\, \arcsin \left (\frac {20 x}{21}-\frac {3}{7}\right )}{10 \sqrt {-2 x +3}\, \sqrt {5 x +3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/10*((-2*x+3)*(3+5*x))^(1/2)/(-2*x+3)^(1/2)/(3+5*x)^(1/2)*10^(1/2)*arcsin(20/21*x-3/7)

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maxima [A]  time = 3.00, size = 11, normalized size = 0.42 \begin {gather*} -\frac {1}{10} \, \sqrt {10} \arcsin \left (-\frac {20}{21} \, x + \frac {3}{7}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*sqrt(10)*arcsin(-20/21*x + 3/7)

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mupad [B]  time = 0.08, size = 40, normalized size = 1.54 \begin {gather*} -\frac {2\,\sqrt {10}\,\mathrm {atan}\left (\frac {\sqrt {10}\,\left (\sqrt {3}-\sqrt {3-2\,x}\right )}{2\,\left (\sqrt {3}-\sqrt {5\,x+3}\right )}\right )}{5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.07, size = 58, normalized size = 2.23 \begin {gather*} \begin {cases} - \frac {\sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {210} \sqrt {x + \frac {3}{5}}}{21} \right )}}{5} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{21} > 1 \\\frac {\sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {210} \sqrt {x + \frac {3}{5}}}{21} \right )}}{5} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-sqrt(10)*I*acosh(sqrt(210)*sqrt(x + 3/5)/21)/5, 10*Abs(x + 3/5)/21 > 1), (sqrt(10)*asin(sqrt(210)*
sqrt(x + 3/5)/21)/5, True))

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