3.2523 \(\int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx\)

Optimal. Leaf size=47 \[ \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}-\sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {47, 54, 216} \[ \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}-\sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[3 + 5*x]/(1 - 2*x)^(3/2),x]

[Out]

Sqrt[3 + 5*x]/Sqrt[1 - 2*x] - Sqrt[5/2]*ArcSin[Sqrt[2/11]*Sqrt[3 + 5*x]]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, 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 {\sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx &=\frac {\sqrt {3+5 x}}{\sqrt {1-2 x}}-\frac {5}{2} \int \frac {1}{\sqrt {1-2 x} \sqrt {3+5 x}} \, dx\\ &=\frac {\sqrt {3+5 x}}{\sqrt {1-2 x}}-\sqrt {5} \operatorname {Subst}\left (\int \frac {1}{\sqrt {11-2 x^2}} \, dx,x,\sqrt {3+5 x}\right )\\ &=\frac {\sqrt {3+5 x}}{\sqrt {1-2 x}}-\sqrt {\frac {5}{2}} \sin ^{-1}\left (\sqrt {\frac {2}{11}} \sqrt {3+5 x}\right )\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 54, normalized size = 1.15 \[ \frac {2 \sqrt {5 x+3}-\sqrt {20 x-10} \sinh ^{-1}\left (\sqrt {\frac {5}{11}} \sqrt {2 x-1}\right )}{2 \sqrt {1-2 x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[3 + 5*x] - Sqrt[-10 + 20*x]*ArcSinh[Sqrt[5/11]*Sqrt[-1 + 2*x]])/(2*Sqrt[1 - 2*x])

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fricas [B]  time = 0.72, size = 76, normalized size = 1.62 \[ \frac {\sqrt {5} \sqrt {2} {\left (2 \, x - 1\right )} \arctan \left (\frac {\sqrt {5} \sqrt {2} {\left (20 \, x + 1\right )} \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{20 \, {\left (10 \, x^{2} + x - 3\right )}}\right ) - 4 \, \sqrt {5 \, x + 3} \sqrt {-2 \, x + 1}}{4 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*(sqrt(5)*sqrt(2)*(2*x - 1)*arctan(1/20*sqrt(5)*sqrt(2)*(20*x + 1)*sqrt(5*x + 3)*sqrt(-2*x + 1)/(10*x^2 + x
- 3)) - 4*sqrt(5*x + 3)*sqrt(-2*x + 1))/(2*x - 1)

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giac [A]  time = 1.03, size = 45, normalized size = 0.96 \[ -\frac {1}{2} \, \sqrt {10} \arcsin \left (\frac {1}{11} \, \sqrt {22} \sqrt {5 \, x + 3}\right ) - \frac {\sqrt {5} \sqrt {5 \, x + 3} \sqrt {-10 \, x + 5}}{5 \, {\left (2 \, x - 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.23, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {5 x +3}}{\left (-2 x +1\right )^{\frac {3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 1.34, size = 36, normalized size = 0.77 \[ -\frac {1}{4} \, \sqrt {5} \sqrt {2} \arcsin \left (\frac {20}{11} \, x + \frac {1}{11}\right ) - \frac {\sqrt {-10 \, x^{2} - x + 3}}{2 \, x - 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*sqrt(5)*sqrt(2)*arcsin(20/11*x + 1/11) - sqrt(-10*x^2 - x + 3)/(2*x - 1)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 1.95, size = 95, normalized size = 2.02 \[ \begin {cases} - \frac {5 i \sqrt {x + \frac {3}{5}}}{\sqrt {10 x - 5}} + \frac {\sqrt {10} i \operatorname {acosh}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{2} & \text {for}\: \frac {10 \left |{x + \frac {3}{5}}\right |}{11} > 1 \\- \frac {\sqrt {10} \operatorname {asin}{\left (\frac {\sqrt {110} \sqrt {x + \frac {3}{5}}}{11} \right )}}{2} + \frac {5 \sqrt {x + \frac {3}{5}}}{\sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-5*I*sqrt(x + 3/5)/sqrt(10*x - 5) + sqrt(10)*I*acosh(sqrt(110)*sqrt(x + 3/5)/11)/2, 10*Abs(x + 3/5)
/11 > 1), (-sqrt(10)*asin(sqrt(110)*sqrt(x + 3/5)/11)/2 + 5*sqrt(x + 3/5)/sqrt(5 - 10*x), True))

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