\(\int \frac {\sqrt {2-b x}}{x^{3/2}} \, dx\) [517]

Optimal result

Integrand size = 16, antiderivative size = 42 \[ \int \frac {\sqrt {2-b x}}{x^{3/2}} \, dx=-\frac {2 \sqrt {2-b x}}{\sqrt {x}}-2 \sqrt {b} \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \]

[Out]

-2*arcsin(1/2*b^(1/2)*x^(1/2)*2^(1/2))*b^(1/2)-2*(-b*x+2)^(1/2)/x^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {49, 56, 222} \[ \int \frac {\sqrt {2-b x}}{x^{3/2}} \, dx=-2 \sqrt {b} \arcsin \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right )-\frac {2 \sqrt {2-b x}}{\sqrt {x}} \]

[In]

Int[Sqrt[2 - b*x]/x^(3/2),x]

[Out]

(-2*Sqrt[2 - b*x])/Sqrt[x] - 2*Sqrt[b]*ArcSin[(Sqrt[b]*Sqrt[x])/Sqrt[2]]

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*} \text {integral}& = -\frac {2 \sqrt {2-b x}}{\sqrt {x}}-b \int \frac {1}{\sqrt {x} \sqrt {2-b x}} \, dx \\ & = -\frac {2 \sqrt {2-b x}}{\sqrt {x}}-(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {2-b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 \sqrt {2-b x}}{\sqrt {x}}-2 \sqrt {b} \sin ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.36 \[ \int \frac {\sqrt {2-b x}}{x^{3/2}} \, dx=-\frac {2 \sqrt {2-b x}}{\sqrt {x}}+4 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {2}-\sqrt {2-b x}}\right ) \]

[In]

Integrate[Sqrt[2 - b*x]/x^(3/2),x]

[Out]

(-2*Sqrt[2 - b*x])/Sqrt[x] + 4*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/(Sqrt[2] - Sqrt[2 - b*x])]

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(63\) vs. \(2(31)=62\).

Time = 0.08 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.52

method	result	size
meijerg	\(\frac {\left (-b \right )^{\frac {3}{2}} \left (\frac {4 \sqrt {\pi }\, \sqrt {2}\, \sqrt {-\frac {b x}{2}+1}}{\sqrt {x}\, \sqrt {-b}}+\frac {4 \sqrt {\pi }\, \sqrt {b}\, \arcsin \left (\frac {\sqrt {b}\, \sqrt {x}\, \sqrt {2}}{2}\right )}{\sqrt {-b}}\right )}{2 \sqrt {\pi }\, b}\)	\(64\)
risch	\(\frac {2 \left (b x -2\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {-x \left (b x -2\right )}\, \sqrt {x}\, \sqrt {-b x +2}}-\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {1}{b}\right )}{\sqrt {-b \,x^{2}+2 x}}\right ) \sqrt {\left (-b x +2\right ) x}}{\sqrt {x}\, \sqrt {-b x +2}}\)	\(90\)

[In]

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

[Out]

1/2*(-b)^(3/2)/Pi^(1/2)/b*(4*Pi^(1/2)/x^(1/2)*2^(1/2)/(-b)^(1/2)*(-1/2*b*x+1)^(1/2)+4*Pi^(1/2)/(-b)^(1/2)*b^(1
/2)*arcsin(1/2*b^(1/2)*x^(1/2)*2^(1/2)))

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 90, normalized size of antiderivative = 2.14 \[ \int \frac {\sqrt {2-b x}}{x^{3/2}} \, dx=\left [\frac {\sqrt {-b} x \log \left (-b x + \sqrt {-b x + 2} \sqrt {-b} \sqrt {x} + 1\right ) - 2 \, \sqrt {-b x + 2} \sqrt {x}}{x}, \frac {2 \, {\left (\sqrt {b} x \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - \sqrt {-b x + 2} \sqrt {x}\right )}}{x}\right ] \]

[In]

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

[Out]

[(sqrt(-b)*x*log(-b*x + sqrt(-b*x + 2)*sqrt(-b)*sqrt(x) + 1) - 2*sqrt(-b*x + 2)*sqrt(x))/x, 2*(sqrt(b)*x*arcta
n(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))) - sqrt(-b*x + 2)*sqrt(x))/x]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.01 (sec) , antiderivative size = 122, normalized size of antiderivative = 2.90 \[ \int \frac {\sqrt {2-b x}}{x^{3/2}} \, dx=\begin {cases} 2 i \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} - \frac {2 i b \sqrt {x}}{\sqrt {b x - 2}} + \frac {4 i}{\sqrt {x} \sqrt {b x - 2}} & \text {for}\: \left |{b x}\right | > 2 \\- 2 \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {2} \sqrt {b} \sqrt {x}}{2} \right )} + \frac {2 b \sqrt {x}}{\sqrt {- b x + 2}} - \frac {4}{\sqrt {x} \sqrt {- b x + 2}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*I*sqrt(b)*acosh(sqrt(2)*sqrt(b)*sqrt(x)/2) - 2*I*b*sqrt(x)/sqrt(b*x - 2) + 4*I/(sqrt(x)*sqrt(b*x
- 2)), Abs(b*x) > 2), (-2*sqrt(b)*asin(sqrt(2)*sqrt(b)*sqrt(x)/2) + 2*b*sqrt(x)/sqrt(-b*x + 2) - 4/(sqrt(x)*sq
rt(-b*x + 2)), True))

Maxima [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83 \[ \int \frac {\sqrt {2-b x}}{x^{3/2}} \, dx=2 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + 2}}{\sqrt {b} \sqrt {x}}\right ) - \frac {2 \, \sqrt {-b x + 2}}{\sqrt {x}} \]

[In]

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

[Out]

2*sqrt(b)*arctan(sqrt(-b*x + 2)/(sqrt(b)*sqrt(x))) - 2*sqrt(-b*x + 2)/sqrt(x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 69 vs. \(2 (31) = 62\).

Time = 5.81 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.64 \[ \int \frac {\sqrt {2-b x}}{x^{3/2}} \, dx=-\frac {2 \, b^{2} {\left (\frac {\log \left ({\left | -\sqrt {-b x + 2} \sqrt {-b} + \sqrt {{\left (b x - 2\right )} b + 2 \, b} \right |}\right )}{\sqrt {-b}} + \frac {\sqrt {-b x + 2}}{\sqrt {{\left (b x - 2\right )} b + 2 \, b}}\right )}}{{\left | b \right |}} \]

[In]

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

[Out]

-2*b^2*(log(abs(-sqrt(-b*x + 2)*sqrt(-b) + sqrt((b*x - 2)*b + 2*b)))/sqrt(-b) + sqrt(-b*x + 2)/sqrt((b*x - 2)*
b + 2*b))/abs(b)

Mupad [F(-1)]

Timed out. \[ \int \frac {\sqrt {2-b x}}{x^{3/2}} \, dx=\int \frac {\sqrt {2-b\,x}}{x^{3/2}} \,d x \]

[In]

int((2 - b*x)^(1/2)/x^(3/2),x)

[Out]

int((2 - b*x)^(1/2)/x^(3/2), x)