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

Optimal. Leaf size=42 \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{b x-a}} \]

[Out]

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

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Rubi [A]  time = 0.0099389, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {51, 63, 205} \[ -\frac{2 \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{3/2}}-\frac{2}{a \sqrt{b x-a}} \]

Antiderivative was successfully verified.

[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 51

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 63

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 205

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

Rubi steps

\begin{align*} \int \frac{1}{x (-a+b x)^{3/2}} \, dx &=-\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 \operatorname{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 [C]  time = 0.0082974, size = 33, normalized size = 0.79 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};1-\frac{b x}{a}\right )}{a \sqrt{b x-a}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x*(-a + b*x)^(3/2)),x]

[Out]

(-2*Hypergeometric2F1[-1/2, 1, 1/2, 1 - (b*x)/a])/(a*Sqrt[-a + b*x])

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Maple [A]  time = 0.007, size = 35, normalized size = 0.8 \begin{align*} -2\,{\frac{1}{{a}^{3/2}}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) }-2\,{\frac{1}{a\sqrt{bx-a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.44623, size = 266, normalized size = 6.33 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [C]  time = 2.73861, size = 439, normalized size = 10.45 \begin{align*} \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}\: \frac{\left |{b x}\right |}{\left |{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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)/Abs(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))

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Giac [A]  time = 1.17981, size = 46, normalized size = 1.1 \begin{align*} -\frac{2 \, \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{a^{\frac{3}{2}}} - \frac{2}{\sqrt{b x - a} a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)