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

Optimal. Leaf size=62 \[ -\frac{3 b}{a^2 \sqrt{b x-a}}-\frac{3 b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{5/2}}+\frac{1}{a x \sqrt{b x-a}} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0157637, antiderivative size = 65, normalized size of antiderivative = 1.05, number of steps used = 4, number of rules used = 3, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {51, 63, 205} \[ -\frac{3 \sqrt{b x-a}}{a^2 x}-\frac{3 b \tan ^{-1}\left (\frac{\sqrt{b x-a}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{2}{a x \sqrt{b x-a}} \]

Antiderivative was successfully verified.

[In]

Int[1/(x^2*(-a + b*x)^(3/2)),x]

[Out]

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

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

Mathematica [C]  time = 0.0087758, size = 34, normalized size = 0.55 \[ -\frac{2 b \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};1-\frac{b x}{a}\right )}{a^2 \sqrt{b x-a}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^2*(-a + b*x)^(3/2)),x]

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 0.011, size = 54, normalized size = 0.9 \begin{align*} -2\,{\frac{b}{{a}^{2}\sqrt{bx-a}}}-{\frac{1}{{a}^{2}x}\sqrt{bx-a}}-3\,{\frac{b}{{a}^{5/2}}\arctan \left ({\frac{\sqrt{bx-a}}{\sqrt{a}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.62799, size = 344, normalized size = 5.55 \begin{align*} \left [-\frac{3 \,{\left (b^{2} x^{2} - a b x\right )} \sqrt{-a} \log \left (\frac{b x + 2 \, \sqrt{b x - a} \sqrt{-a} - 2 \, a}{x}\right ) + 2 \,{\left (3 \, a b x - a^{2}\right )} \sqrt{b x - a}}{2 \,{\left (a^{3} b x^{2} - a^{4} x\right )}}, -\frac{3 \,{\left (b^{2} x^{2} - a b x\right )} \sqrt{a} \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right ) +{\left (3 \, a b x - a^{2}\right )} \sqrt{b x - a}}{a^{3} b x^{2} - a^{4} x}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/2*(3*(b^2*x^2 - a*b*x)*sqrt(-a)*log((b*x + 2*sqrt(b*x - a)*sqrt(-a) - 2*a)/x) + 2*(3*a*b*x - a^2)*sqrt(b*x
- a))/(a^3*b*x^2 - a^4*x), -(3*(b^2*x^2 - a*b*x)*sqrt(a)*arctan(sqrt(b*x - a)/sqrt(a)) + (3*a*b*x - a^2)*sqrt
(b*x - a))/(a^3*b*x^2 - a^4*x)]

________________________________________________________________________________________

Sympy [A]  time = 4.73075, size = 160, normalized size = 2.58 \begin{align*} \begin{cases} - \frac{i}{a \sqrt{b} x^{\frac{3}{2}} \sqrt{\frac{a}{b x} - 1}} + \frac{3 i \sqrt{b}}{a^{2} \sqrt{x} \sqrt{\frac{a}{b x} - 1}} - \frac{3 i b \operatorname{acosh}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{5}{2}}} & \text{for}\: \frac{\left |{a}\right |}{\left |{b}\right | \left |{x}\right |} > 1 \\\frac{1}{a \sqrt{b} x^{\frac{3}{2}} \sqrt{- \frac{a}{b x} + 1}} - \frac{3 \sqrt{b}}{a^{2} \sqrt{x} \sqrt{- \frac{a}{b x} + 1}} + \frac{3 b \operatorname{asin}{\left (\frac{\sqrt{a}}{\sqrt{b} \sqrt{x}} \right )}}{a^{\frac{5}{2}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-I/(a*sqrt(b)*x**(3/2)*sqrt(a/(b*x) - 1)) + 3*I*sqrt(b)/(a**2*sqrt(x)*sqrt(a/(b*x) - 1)) - 3*I*b*ac
osh(sqrt(a)/(sqrt(b)*sqrt(x)))/a**(5/2), Abs(a)/(Abs(b)*Abs(x)) > 1), (1/(a*sqrt(b)*x**(3/2)*sqrt(-a/(b*x) + 1
)) - 3*sqrt(b)/(a**2*sqrt(x)*sqrt(-a/(b*x) + 1)) + 3*b*asin(sqrt(a)/(sqrt(b)*sqrt(x)))/a**(5/2), True))

________________________________________________________________________________________

Giac [A]  time = 1.18359, size = 86, normalized size = 1.39 \begin{align*} -\frac{3 \, b \arctan \left (\frac{\sqrt{b x - a}}{\sqrt{a}}\right )}{a^{\frac{5}{2}}} - \frac{3 \,{\left (b x - a\right )} b + 2 \, a b}{{\left ({\left (b x - a\right )}^{\frac{3}{2}} + \sqrt{b x - a} a\right )} a^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*b*arctan(sqrt(b*x - a)/sqrt(a))/a^(5/2) - (3*(b*x - a)*b + 2*a*b)/(((b*x - a)^(3/2) + sqrt(b*x - a)*a)*a^2)