\(\int \frac {1}{x^{3/2} (-a+b x)} \, dx\) [473]

Optimal result

Integrand size = 15, antiderivative size = 40 \[ \int \frac {1}{x^{3/2} (-a+b x)} \, dx=\frac {2}{a \sqrt {x}}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}} \]

[Out]

-2*arctanh(b^(1/2)*x^(1/2)/a^(1/2))*b^(1/2)/a^(3/2)+2/a/x^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {53, 65, 214} \[ \int \frac {1}{x^{3/2} (-a+b x)} \, dx=\frac {2}{a \sqrt {x}}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}} \]

[In]

Int[1/(x^(3/2)*(-a + b*x)),x]

[Out]

2/(a*Sqrt[x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(3/2)

Rule 53

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^(n + 1
)/((b*c - a*d)*(m + 1))), x] - Dist[d*((m + n + 2)/((b*c - a*d)*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^n,
x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] &&  !(LtQ[n, -1] && (EqQ[a, 0] || (NeQ[
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && 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 214

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

Rubi steps \begin{align*} \text {integral}& = \frac {2}{a \sqrt {x}}+\frac {b \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{a} \\ & = \frac {2}{a \sqrt {x}}+\frac {(2 b) \text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{a} \\ & = \frac {2}{a \sqrt {x}}-\frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 40, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x^{3/2} (-a+b x)} \, dx=\frac {2}{a \sqrt {x}}-\frac {2 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}} \]

[In]

Integrate[1/(x^(3/2)*(-a + b*x)),x]

[Out]

2/(a*Sqrt[x]) - (2*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(3/2)

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 32, normalized size of antiderivative = 0.80

method	result	size
derivativedivides	\(-\frac {2 b \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}+\frac {2}{a \sqrt {x}}\)	\(32\)
default	\(-\frac {2 b \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}+\frac {2}{a \sqrt {x}}\)	\(32\)
risch	\(-\frac {2 b \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{a \sqrt {a b}}+\frac {2}{a \sqrt {x}}\)	\(32\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 91, normalized size of antiderivative = 2.28 \[ \int \frac {1}{x^{3/2} (-a+b x)} \, dx=\left [\frac {x \sqrt {\frac {b}{a}} \log \left (\frac {b x - 2 \, a \sqrt {x} \sqrt {\frac {b}{a}} + a}{b x - a}\right ) + 2 \, \sqrt {x}}{a x}, \frac {2 \, {\left (x \sqrt {-\frac {b}{a}} \arctan \left (\frac {a \sqrt {-\frac {b}{a}}}{b \sqrt {x}}\right ) + \sqrt {x}\right )}}{a x}\right ] \]

[In]

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

[Out]

[(x*sqrt(b/a)*log((b*x - 2*a*sqrt(x)*sqrt(b/a) + a)/(b*x - a)) + 2*sqrt(x))/(a*x), 2*(x*sqrt(-b/a)*arctan(a*sq
rt(-b/a)/(b*sqrt(x))) + sqrt(x))/(a*x)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 76 vs. \(2 (36) = 72\).

Time = 1.34 (sec) , antiderivative size = 76, normalized size of antiderivative = 1.90 \[ \int \frac {1}{x^{3/2} (-a+b x)} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {3}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{3 b x^{\frac {3}{2}}} & \text {for}\: a = 0 \\\frac {2}{a \sqrt {x}} & \text {for}\: b = 0 \\\frac {\log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{a \sqrt {\frac {a}{b}}} - \frac {\log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{a \sqrt {\frac {a}{b}}} + \frac {2}{a \sqrt {x}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo/x**(3/2), Eq(a, 0) & Eq(b, 0)), (-2/(3*b*x**(3/2)), Eq(a, 0)), (2/(a*sqrt(x)), Eq(b, 0)), (log(
sqrt(x) - sqrt(a/b))/(a*sqrt(a/b)) - log(sqrt(x) + sqrt(a/b))/(a*sqrt(a/b)) + 2/(a*sqrt(x)), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.18 \[ \int \frac {1}{x^{3/2} (-a+b x)} \, dx=\frac {b \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{\sqrt {a b} a} + \frac {2}{a \sqrt {x}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 33, normalized size of antiderivative = 0.82 \[ \int \frac {1}{x^{3/2} (-a+b x)} \, dx=\frac {2 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} a} + \frac {2}{a \sqrt {x}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.70 \[ \int \frac {1}{x^{3/2} (-a+b x)} \, dx=\frac {2}{a\,\sqrt {x}}-\frac {2\,\sqrt {b}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{3/2}} \]

[In]

int(-1/(x^(3/2)*(a - b*x)),x)

[Out]

2/(a*x^(1/2)) - (2*b^(1/2)*atanh((b^(1/2)*x^(1/2))/a^(1/2)))/a^(3/2)