\(\int \frac {1}{x^{3/2} (-a+b x)^2} \, dx\) [480]

Optimal result

Integrand size = 15, antiderivative size = 57 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a-b x)}+\frac {3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \]

[Out]

3*arctanh(b^(1/2)*x^(1/2)/a^(1/2))*b^(1/2)/a^(5/2)-3/a^2/x^(1/2)+1/a/(-b*x+a)/x^(1/2)

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {44, 53, 65, 214} \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=\frac {3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a-b x)} \]

[In]

Int[1/(x^(3/2)*(-a + b*x)^2),x]

[Out]

-3/(a^2*Sqrt[x]) + 1/(a*Sqrt[x]*(a - b*x)) + (3*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(5/2)

Rule 44

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] && ILtQ[m, -1] &&  !IntegerQ[n] && LtQ[n, 0]

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 {1}{a \sqrt {x} (a-b x)}-\frac {3 \int \frac {1}{x^{3/2} (-a+b x)} \, dx}{2 a} \\ & = -\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a-b x)}-\frac {(3 b) \int \frac {1}{\sqrt {x} (-a+b x)} \, dx}{2 a^2} \\ & = -\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a-b x)}-\frac {(3 b) \text {Subst}\left (\int \frac {1}{-a+b x^2} \, dx,x,\sqrt {x}\right )}{a^2} \\ & = -\frac {3}{a^2 \sqrt {x}}+\frac {1}{a \sqrt {x} (a-b x)}+\frac {3 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=\frac {-2 a+3 b x}{a^2 \sqrt {x} (a-b x)}+\frac {3 \sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}} \]

[In]

Integrate[1/(x^(3/2)*(-a + b*x)^2),x]

[Out]

(-2*a + 3*b*x)/(a^2*Sqrt[x]*(a - b*x)) + (3*Sqrt[b]*ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]])/a^(5/2)

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 48, normalized size of antiderivative = 0.84

method	result	size
derivativedivides	\(\frac {2 b \left (\frac {\sqrt {x}}{-2 b x +2 a}+\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\)	\(48\)
default	\(\frac {2 b \left (\frac {\sqrt {x}}{-2 b x +2 a}+\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{2 \sqrt {a b}}\right )}{a^{2}}-\frac {2}{a^{2} \sqrt {x}}\)	\(48\)
risch	\(-\frac {2}{a^{2} \sqrt {x}}-\frac {b \left (\frac {\sqrt {x}}{b x -a}-\frac {3 \,\operatorname {arctanh}\left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\right )}{a^{2}}\)	\(48\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 151, normalized size of antiderivative = 2.65 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=\left [\frac {3 \, {\left (b x^{2} - a x\right )} \sqrt {\frac {b}{a}} \log \left (\frac {b x + 2 \, a \sqrt {x} \sqrt {\frac {b}{a}} + a}{b x - a}\right ) - 2 \, {\left (3 \, b x - 2 \, a\right )} \sqrt {x}}{2 \, {\left (a^{2} b x^{2} - a^{3} x\right )}}, -\frac {3 \, {\left (b x^{2} - a x\right )} \sqrt {-\frac {b}{a}} \arctan \left (\frac {a \sqrt {-\frac {b}{a}}}{b \sqrt {x}}\right ) + {\left (3 \, b x - 2 \, a\right )} \sqrt {x}}{a^{2} b x^{2} - a^{3} x}\right ] \]

[In]

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

[Out]

[1/2*(3*(b*x^2 - a*x)*sqrt(b/a)*log((b*x + 2*a*sqrt(x)*sqrt(b/a) + a)/(b*x - a)) - 2*(3*b*x - 2*a)*sqrt(x))/(a
^2*b*x^2 - a^3*x), -(3*(b*x^2 - a*x)*sqrt(-b/a)*arctan(a*sqrt(-b/a)/(b*sqrt(x))) + (3*b*x - 2*a)*sqrt(x))/(a^2
*b*x^2 - a^3*x)]

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 354 vs. \(2 (51) = 102\).

Time = 12.42 (sec) , antiderivative size = 354, normalized size of antiderivative = 6.21 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=\begin {cases} \frac {\tilde {\infty }}{x^{\frac {5}{2}}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{a^{2} \sqrt {x}} & \text {for}\: b = 0 \\- \frac {2}{5 b^{2} x^{\frac {5}{2}}} & \text {for}\: a = 0 \\- \frac {3 a \sqrt {x} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} + \frac {3 a \sqrt {x} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} - \frac {4 a \sqrt {\frac {a}{b}}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} + \frac {3 b x^{\frac {3}{2}} \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} - \frac {3 b x^{\frac {3}{2}} \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} + \frac {6 b x \sqrt {\frac {a}{b}}}{2 a^{3} \sqrt {x} \sqrt {\frac {a}{b}} - 2 a^{2} b x^{\frac {3}{2}} \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-2/(a**2*sqrt(x)), Eq(b, 0)), (-2/(5*b**2*x**(5/2)), Eq(a, 0))
, (-3*a*sqrt(x)*log(sqrt(x) - sqrt(a/b))/(2*a**3*sqrt(x)*sqrt(a/b) - 2*a**2*b*x**(3/2)*sqrt(a/b)) + 3*a*sqrt(x
)*log(sqrt(x) + sqrt(a/b))/(2*a**3*sqrt(x)*sqrt(a/b) - 2*a**2*b*x**(3/2)*sqrt(a/b)) - 4*a*sqrt(a/b)/(2*a**3*sq
rt(x)*sqrt(a/b) - 2*a**2*b*x**(3/2)*sqrt(a/b)) + 3*b*x**(3/2)*log(sqrt(x) - sqrt(a/b))/(2*a**3*sqrt(x)*sqrt(a/
b) - 2*a**2*b*x**(3/2)*sqrt(a/b)) - 3*b*x**(3/2)*log(sqrt(x) + sqrt(a/b))/(2*a**3*sqrt(x)*sqrt(a/b) - 2*a**2*b
*x**(3/2)*sqrt(a/b)) + 6*b*x*sqrt(a/b)/(2*a**3*sqrt(x)*sqrt(a/b) - 2*a**2*b*x**(3/2)*sqrt(a/b)), True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.21 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=-\frac {3 \, b x - 2 \, a}{a^{2} b x^{\frac {3}{2}} - a^{3} \sqrt {x}} - \frac {3 \, b \log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{2 \, \sqrt {a b} a^{2}} \]

[In]

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

[Out]

-(3*b*x - 2*a)/(a^2*b*x^(3/2) - a^3*sqrt(x)) - 3/2*b*log((b*sqrt(x) - sqrt(a*b))/(b*sqrt(x) + sqrt(a*b)))/(sqr
t(a*b)*a^2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 52, normalized size of antiderivative = 0.91 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=-\frac {3 \, b \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} a^{2}} - \frac {3 \, b x - 2 \, a}{{\left (b x^{\frac {3}{2}} - a \sqrt {x}\right )} a^{2}} \]

[In]

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

[Out]

-3*b*arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*a^2) - (3*b*x - 2*a)/((b*x^(3/2) - a*sqrt(x))*a^2)

Mupad [B] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.86 \[ \int \frac {1}{x^{3/2} (-a+b x)^2} \, dx=\frac {3\,\sqrt {b}\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{a^{5/2}}-\frac {\frac {2}{a}-\frac {3\,b\,x}{a^2}}{a\,\sqrt {x}-b\,x^{3/2}} \]

[In]

int(1/(x^(3/2)*(a - b*x)^2),x)

[Out]

(3*b^(1/2)*atanh((b^(1/2)*x^(1/2))/a^(1/2)))/a^(5/2) - (2/a - (3*b*x)/a^2)/(a*x^(1/2) - b*x^(3/2))