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

Optimal. Leaf size=57 \[ \frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a-b x)} \]

[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)

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Rubi [A]  time = 0.0179159, antiderivative size = 57, normalized size of antiderivative = 1., 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, 208} \[ \frac{3 \sqrt{b} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{a^{5/2}}-\frac{3}{a^2 \sqrt{x}}+\frac{1}{a \sqrt{x} (a-b x)} \]

Antiderivative was successfully verified.

[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 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 208

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

Rubi steps

\begin{align*} \int \frac{1}{x^{3/2} (-a+b x)^2} \, dx &=\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) \operatorname{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 [C]  time = 0.0058549, size = 24, normalized size = 0.42 \[ -\frac{2 \, _2F_1\left (-\frac{1}{2},2;\frac{1}{2};\frac{b x}{a}\right )}{a^2 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(x^(3/2)*(-a + b*x)^2),x]

[Out]

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

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Maple [A]  time = 0.01, size = 49, normalized size = 0.9 \begin{align*} -2\,{\frac{1}{{a}^{2}\sqrt{x}}}-2\,{\frac{b}{{a}^{2}} \left ( 1/2\,{\frac{\sqrt{x}}{bx-a}}-3/2\,{\frac{1}{\sqrt{ab}}{\it Artanh} \left ({\frac{b\sqrt{x}}{\sqrt{ab}}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/a^2/x^(1/2)-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)))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.62697, size = 324, normalized size = 5.68 \begin{align*} \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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)]

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Sympy [A]  time = 42.7898, size = 403, normalized size = 7.07 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{5}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{5 b^{2} x^{\frac{5}{2}}} & \text{for}\: a = 0 \\- \frac{2}{a^{2} \sqrt{x}} & \text{for}\: b = 0 \\- \frac{4 a^{\frac{3}{2}} \sqrt{\frac{1}{b}}}{2 a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} - 2 a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} + \frac{6 \sqrt{a} b x \sqrt{\frac{1}{b}}}{2 a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} - 2 a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} - \frac{3 a \sqrt{x} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} - 2 a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} + \frac{3 a \sqrt{x} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} - 2 a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} + \frac{3 b x^{\frac{3}{2}} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} - 2 a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} - \frac{3 b x^{\frac{3}{2}} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{2 a^{\frac{7}{2}} \sqrt{x} \sqrt{\frac{1}{b}} - 2 a^{\frac{5}{2}} b x^{\frac{3}{2}} \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(5/2), Eq(a, 0) & Eq(b, 0)), (-2/(5*b**2*x**(5/2)), Eq(a, 0)), (-2/(a**2*sqrt(x)), Eq(b, 0))
, (-4*a**(3/2)*sqrt(1/b)/(2*a**(7/2)*sqrt(x)*sqrt(1/b) - 2*a**(5/2)*b*x**(3/2)*sqrt(1/b)) + 6*sqrt(a)*b*x*sqrt
(1/b)/(2*a**(7/2)*sqrt(x)*sqrt(1/b) - 2*a**(5/2)*b*x**(3/2)*sqrt(1/b)) - 3*a*sqrt(x)*log(-sqrt(a)*sqrt(1/b) +
sqrt(x))/(2*a**(7/2)*sqrt(x)*sqrt(1/b) - 2*a**(5/2)*b*x**(3/2)*sqrt(1/b)) + 3*a*sqrt(x)*log(sqrt(a)*sqrt(1/b)
+ sqrt(x))/(2*a**(7/2)*sqrt(x)*sqrt(1/b) - 2*a**(5/2)*b*x**(3/2)*sqrt(1/b)) + 3*b*x**(3/2)*log(-sqrt(a)*sqrt(1
/b) + sqrt(x))/(2*a**(7/2)*sqrt(x)*sqrt(1/b) - 2*a**(5/2)*b*x**(3/2)*sqrt(1/b)) - 3*b*x**(3/2)*log(sqrt(a)*sqr
t(1/b) + sqrt(x))/(2*a**(7/2)*sqrt(x)*sqrt(1/b) - 2*a**(5/2)*b*x**(3/2)*sqrt(1/b)), True))

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Giac [A]  time = 1.20185, size = 70, normalized size = 1.23 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)