\(\int \frac {1}{x (-a+b x)^{3/2}} \, dx\) [362]

Optimal result

Integrand size = 15, antiderivative size = 42 \[ \int \frac {1}{x (-a+b x)^{3/2}} \, dx=-\frac {2}{a \sqrt {-a+b x}}-\frac {2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{3/2}} \]

[Out]

-2*arctan((b*x-a)^(1/2)/a^(1/2))/a^(3/2)-2/a/(b*x-a)^(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, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {53, 65, 211} \[ \int \frac {1}{x (-a+b x)^{3/2}} \, dx=-\frac {2 \arctan \left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a \sqrt {b x-a}} \]

[In]

Int[1/(x*(-a + b*x)^(3/2)),x]

[Out]

-2/(a*Sqrt[-a + b*x]) - (2*ArcTan[Sqrt[-a + b*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 211

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x (-a+b x)^{3/2}} \, dx=-\frac {2}{a \sqrt {-a+b x}}-\frac {2 \arctan \left (\frac {\sqrt {-a+b x}}{\sqrt {a}}\right )}{a^{3/2}} \]

[In]

Integrate[1/(x*(-a + b*x)^(3/2)),x]

[Out]

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

Maple [A] (verified)

Time = 0.08 (sec) , antiderivative size = 35, normalized size of antiderivative = 0.83

method	result	size
derivativedivides	\(-\frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}-\frac {2}{a \sqrt {b x -a}}\)	\(35\)
default	\(-\frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}-\frac {2}{a \sqrt {b x -a}}\)	\(35\)
pseudoelliptic	\(-\frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}}-\frac {2}{a \sqrt {b x -a}}\)	\(35\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 124, normalized size of antiderivative = 2.95 \[ \int \frac {1}{x (-a+b x)^{3/2}} \, dx=\left [-\frac {{\left (b x - a\right )} \sqrt {-a} \log \left (\frac {b x + 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right ) + 2 \, \sqrt {b x - a} a}{a^{2} b x - a^{3}}, -\frac {2 \, {\left ({\left (b x - a\right )} \sqrt {a} \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right ) + \sqrt {b x - a} a\right )}}{a^{2} b x - a^{3}}\right ] \]

[In]

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

[Out]

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

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 2.14 (sec) , antiderivative size = 437, normalized size of antiderivative = 10.40 \[ \int \frac {1}{x (-a+b x)^{3/2}} \, dx=\begin {cases} - \frac {2 a^{3} \sqrt {-1 + \frac {b x}{a}}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {i a^{3} \log {\left (\frac {b x}{a} \right )}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {2 i a^{3} \log {\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {2 a^{3} \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {i a^{2} b x \log {\left (\frac {b x}{a} \right )}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {2 i a^{2} b x \log {\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {2 a^{2} b x \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} & \text {for}\: \left |{\frac {b x}{a}}\right | > 1 \\- \frac {2 i a^{3} \sqrt {1 - \frac {b x}{a}}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {i a^{3} \log {\left (\frac {b x}{a} \right )}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {2 i a^{3} \log {\left (\sqrt {1 - \frac {b x}{a}} + 1 \right )}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {\pi a^{3}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {i a^{2} b x \log {\left (\frac {b x}{a} \right )}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} - \frac {2 i a^{2} b x \log {\left (\sqrt {1 - \frac {b x}{a}} + 1 \right )}}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} + \frac {\pi a^{2} b x}{- a^{\frac {9}{2}} + a^{\frac {7}{2}} b x} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*a**3*sqrt(-1 + b*x/a)/(-a**(9/2) + a**(7/2)*b*x) - I*a**3*log(b*x/a)/(-a**(9/2) + a**(7/2)*b*x)
+ 2*I*a**3*log(sqrt(b)*sqrt(x)/sqrt(a))/(-a**(9/2) + a**(7/2)*b*x) - 2*a**3*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-
a**(9/2) + a**(7/2)*b*x) + I*a**2*b*x*log(b*x/a)/(-a**(9/2) + a**(7/2)*b*x) - 2*I*a**2*b*x*log(sqrt(b)*sqrt(x)
/sqrt(a))/(-a**(9/2) + a**(7/2)*b*x) + 2*a**2*b*x*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/(-a**(9/2) + a**(7/2)*b*x),
Abs(b*x/a) > 1), (-2*I*a**3*sqrt(1 - b*x/a)/(-a**(9/2) + a**(7/2)*b*x) - I*a**3*log(b*x/a)/(-a**(9/2) + a**(7/
2)*b*x) + 2*I*a**3*log(sqrt(1 - b*x/a) + 1)/(-a**(9/2) + a**(7/2)*b*x) - pi*a**3/(-a**(9/2) + a**(7/2)*b*x) +
I*a**2*b*x*log(b*x/a)/(-a**(9/2) + a**(7/2)*b*x) - 2*I*a**2*b*x*log(sqrt(1 - b*x/a) + 1)/(-a**(9/2) + a**(7/2)
*b*x) + pi*a**2*b*x/(-a**(9/2) + a**(7/2)*b*x), True))

Maxima [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x (-a+b x)^{3/2}} \, dx=-\frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {2}{\sqrt {b x - a} a} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x (-a+b x)^{3/2}} \, dx=-\frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{a^{\frac {3}{2}}} - \frac {2}{\sqrt {b x - a} a} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.81 \[ \int \frac {1}{x (-a+b x)^{3/2}} \, dx=-\frac {2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{a^{3/2}}-\frac {2}{a\,\sqrt {b\,x-a}} \]

[In]

int(1/(x*(b*x - a)^(3/2)),x)

[Out]

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