\(\int \frac {\sqrt {a-b x}}{x^{3/2}} \, dx\) [501]

Optimal result

Integrand size = 16, antiderivative size = 47 \[ \int \frac {\sqrt {a-b x}}{x^{3/2}} \, dx=-\frac {2 \sqrt {a-b x}}{\sqrt {x}}-2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right ) \]

[Out]

-2*arctan(b^(1/2)*x^(1/2)/(-b*x+a)^(1/2))*b^(1/2)-2*(-b*x+a)^(1/2)/x^(1/2)

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {49, 65, 223, 209} \[ \int \frac {\sqrt {a-b x}}{x^{3/2}} \, dx=-2 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right )-\frac {2 \sqrt {a-b x}}{\sqrt {x}} \]

[In]

Int[Sqrt[a - b*x]/x^(3/2),x]

[Out]

(-2*Sqrt[a - b*x])/Sqrt[x] - 2*Sqrt[b]*ArcTan[(Sqrt[b]*Sqrt[x])/Sqrt[a - b*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 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 209

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

Rule 223

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

Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {a-b x}}{\sqrt {x}}-b \int \frac {1}{\sqrt {x} \sqrt {a-b x}} \, dx \\ & = -\frac {2 \sqrt {a-b x}}{\sqrt {x}}-(2 b) \text {Subst}\left (\int \frac {1}{\sqrt {a-b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = -\frac {2 \sqrt {a-b x}}{\sqrt {x}}-(2 b) \text {Subst}\left (\int \frac {1}{1+b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a-b x}}\right ) \\ & = -\frac {2 \sqrt {a-b x}}{\sqrt {x}}-2 \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a-b x}}\right ) \\ \end{align*}

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.21 \[ \int \frac {\sqrt {a-b x}}{x^{3/2}} \, dx=-\frac {2 \sqrt {a-b x}}{\sqrt {x}}-4 \sqrt {b} \arctan \left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a-b x}}\right ) \]

[In]

Integrate[Sqrt[a - b*x]/x^(3/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 66, normalized size of antiderivative = 1.40

method	result	size
risch	\(-\frac {2 \sqrt {-b x +a}}{\sqrt {x}}-\frac {\sqrt {b}\, \arctan \left (\frac {\sqrt {b}\, \left (x -\frac {a}{2 b}\right )}{\sqrt {-b \,x^{2}+a x}}\right ) \sqrt {x \left (-b x +a \right )}}{\sqrt {x}\, \sqrt {-b x +a}}\)	\(66\)

[In]

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

[Out]

-2*(-b*x+a)^(1/2)/x^(1/2)-b^(1/2)*arctan(b^(1/2)*(x-1/2*a/b)/(-b*x^2+a*x)^(1/2))*(x*(-b*x+a))^(1/2)/x^(1/2)/(-
b*x+a)^(1/2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 91, normalized size of antiderivative = 1.94 \[ \int \frac {\sqrt {a-b x}}{x^{3/2}} \, dx=\left [\frac {\sqrt {-b} x \log \left (-2 \, b x + 2 \, \sqrt {-b x + a} \sqrt {-b} \sqrt {x} + a\right ) - 2 \, \sqrt {-b x + a} \sqrt {x}}{x}, \frac {2 \, {\left (\sqrt {b} x \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - \sqrt {-b x + a} \sqrt {x}\right )}}{x}\right ] \]

[In]

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

[Out]

[(sqrt(-b)*x*log(-2*b*x + 2*sqrt(-b*x + a)*sqrt(-b)*sqrt(x) + a) - 2*sqrt(-b*x + a)*sqrt(x))/x, 2*(sqrt(b)*x*a
rctan(sqrt(-b*x + a)/(sqrt(b)*sqrt(x))) - sqrt(-b*x + a)*sqrt(x))/x]

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 1.83 (sec) , antiderivative size = 148, normalized size of antiderivative = 3.15 \[ \int \frac {\sqrt {a-b x}}{x^{3/2}} \, dx=\begin {cases} \frac {2 i \sqrt {a}}{\sqrt {x} \sqrt {-1 + \frac {b x}{a}}} + 2 i \sqrt {b} \operatorname {acosh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - \frac {2 i b \sqrt {x}}{\sqrt {a} \sqrt {-1 + \frac {b x}{a}}} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {2 \sqrt {a}}{\sqrt {x} \sqrt {1 - \frac {b x}{a}}} - 2 \sqrt {b} \operatorname {asin}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} + \frac {2 b \sqrt {x}}{\sqrt {a} \sqrt {1 - \frac {b x}{a}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((2*I*sqrt(a)/(sqrt(x)*sqrt(-1 + b*x/a)) + 2*I*sqrt(b)*acosh(sqrt(b)*sqrt(x)/sqrt(a)) - 2*I*b*sqrt(x)
/(sqrt(a)*sqrt(-1 + b*x/a)), Abs(b*x/a) > 1), (-2*sqrt(a)/(sqrt(x)*sqrt(1 - b*x/a)) - 2*sqrt(b)*asin(sqrt(b)*s
qrt(x)/sqrt(a)) + 2*b*sqrt(x)/(sqrt(a)*sqrt(1 - b*x/a)), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.74 \[ \int \frac {\sqrt {a-b x}}{x^{3/2}} \, dx=2 \, \sqrt {b} \arctan \left (\frac {\sqrt {-b x + a}}{\sqrt {b} \sqrt {x}}\right ) - \frac {2 \, \sqrt {-b x + a}}{\sqrt {x}} \]

[In]

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

[Out]

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (35) = 70\).

Time = 77.24 (sec) , antiderivative size = 73, normalized size of antiderivative = 1.55 \[ \int \frac {\sqrt {a-b x}}{x^{3/2}} \, dx=-\frac {2 \, b^{2} {\left (\frac {\log \left ({\left | -\sqrt {-b x + a} \sqrt {-b} + \sqrt {{\left (b x - a\right )} b + a b} \right |}\right )}{\sqrt {-b}} + \frac {\sqrt {-b x + a}}{\sqrt {{\left (b x - a\right )} b + a b}}\right )}}{{\left | b \right |}} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

int((a - b*x)^(1/2)/x^(3/2),x)

[Out]

int((a - b*x)^(1/2)/x^(3/2), x)