3.26.23 \(\int \frac {\sqrt {3+5 x}}{(1-2 x)^{3/2}} \, dx\) [2523]

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

[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 = {49, 56, 222} \begin {gather*} \frac {\sqrt {5 x+3}}{\sqrt {1-2 x}}-\sqrt {\frac {5}{2}} \text {ArcSin}\left (\sqrt {\frac {2}{11}} \sqrt {5 x+3}\right ) \end {gather*}

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 49

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 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 {\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} \text {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 = 0.06, size = 50, normalized size = 1.06 \begin {gather*} \frac {\sqrt {3+5 x}}{\sqrt {1-2 x}}+\sqrt {\frac {5}{2}} \tan ^{-1}\left (\frac {\sqrt {\frac {5}{2}-5 x}}{\sqrt {3+5 x}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]
time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {\sqrt {3+5 x}}{\left (1-2 x \right )^{\frac {3}{2}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.53, size = 36, normalized size = 0.77 \begin {gather*} -\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} \end {gather*}

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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (34) = 68\).
time = 0.44, size = 76, normalized size = 1.62 \begin {gather*} \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 )}} \end {gather*}

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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Sympy [C] Result contains complex when optimal does not.
time = 0.77, size = 94, normalized size = 2.00 \begin {gather*} \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}\: \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 )}}{2} + \frac {5 \sqrt {x + \frac {3}{5}}}{\sqrt {5 - 10 x}} & \text {otherwise} \end {cases} \end {gather*}

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, Abs(x + 3/5) >
11/10), (-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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Giac [A]
time = 0.67, size = 45, normalized size = 0.96 \begin {gather*} -\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 )}} \end {gather*}

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

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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