3.485 \(\int \frac{\sqrt{x}}{(-a+b x)^3} \, dx\)

Optimal. Leaf size=75 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{\sqrt{x}}{4 a b (a-b x)}-\frac{\sqrt{x}}{2 b (a-b x)^2} \]

[Out]

-Sqrt[x]/(2*b*(a - b*x)^2) + Sqrt[x]/(4*a*b*(a - b*x)) + ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]]/(4*a^(3/2)*b^(3/2)
)

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Rubi [A]  time = 0.0199319, antiderivative size = 75, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {47, 51, 63, 208} \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}+\frac{\sqrt{x}}{4 a b (a-b x)}-\frac{\sqrt{x}}{2 b (a-b x)^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(-a + b*x)^3,x]

[Out]

-Sqrt[x]/(2*b*(a - b*x)^2) + Sqrt[x]/(4*a*b*(a - b*x)) + ArcTanh[(Sqrt[b]*Sqrt[x])/Sqrt[a]]/(4*a^(3/2)*b^(3/2)
)

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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{\sqrt{x}}{(-a+b x)^3} \, dx &=-\frac{\sqrt{x}}{2 b (a-b x)^2}+\frac{\int \frac{1}{\sqrt{x} (-a+b x)^2} \, dx}{4 b}\\ &=-\frac{\sqrt{x}}{2 b (a-b x)^2}+\frac{\sqrt{x}}{4 a b (a-b x)}-\frac{\int \frac{1}{\sqrt{x} (-a+b x)} \, dx}{8 a b}\\ &=-\frac{\sqrt{x}}{2 b (a-b x)^2}+\frac{\sqrt{x}}{4 a b (a-b x)}-\frac{\operatorname{Subst}\left (\int \frac{1}{-a+b x^2} \, dx,x,\sqrt{x}\right )}{4 a b}\\ &=-\frac{\sqrt{x}}{2 b (a-b x)^2}+\frac{\sqrt{x}}{4 a b (a-b x)}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{x}}{\sqrt{a}}\right )}{4 a^{3/2} b^{3/2}}\\ \end{align*}

Mathematica [C]  time = 0.0050309, size = 26, normalized size = 0.35 \[ -\frac{2 x^{3/2} \, _2F_1\left (\frac{3}{2},3;\frac{5}{2};\frac{b x}{a}\right )}{3 a^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(-a + b*x)^3,x]

[Out]

(-2*x^(3/2)*Hypergeometric2F1[3/2, 3, 5/2, (b*x)/a])/(3*a^3)

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Maple [A]  time = 0.009, size = 54, normalized size = 0.7 \begin{align*} 2\,{\frac{1}{ \left ( bx-a \right ) ^{2}} \left ( -1/8\,{\frac{{x}^{3/2}}{a}}-1/8\,{\frac{\sqrt{x}}{b}} \right ) }+{\frac{1}{4\,ab}{\it Artanh} \left ({b\sqrt{x}{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{ab}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*(-1/8/a*x^(3/2)-1/8*x^(1/2)/b)/(b*x-a)^2+1/4/b/a/(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(x^(1/2)/(b*x-a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.72903, size = 410, normalized size = 5.47 \begin{align*} \left [\frac{{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{a b} \log \left (\frac{b x + a + 2 \, \sqrt{a b} \sqrt{x}}{b x - a}\right ) - 2 \,{\left (a b^{2} x + a^{2} b\right )} \sqrt{x}}{8 \,{\left (a^{2} b^{4} x^{2} - 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}, -\frac{{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \sqrt{-a b} \arctan \left (\frac{\sqrt{-a b}}{b \sqrt{x}}\right ) +{\left (a b^{2} x + a^{2} b\right )} \sqrt{x}}{4 \,{\left (a^{2} b^{4} x^{2} - 2 \, a^{3} b^{3} x + a^{4} b^{2}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*((b^2*x^2 - 2*a*b*x + a^2)*sqrt(a*b)*log((b*x + a + 2*sqrt(a*b)*sqrt(x))/(b*x - a)) - 2*(a*b^2*x + a^2*b)
*sqrt(x))/(a^2*b^4*x^2 - 2*a^3*b^3*x + a^4*b^2), -1/4*((b^2*x^2 - 2*a*b*x + a^2)*sqrt(-a*b)*arctan(sqrt(-a*b)/
(b*sqrt(x))) + (a*b^2*x + a^2*b)*sqrt(x))/(a^2*b^4*x^2 - 2*a^3*b^3*x + a^4*b^2)]

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Sympy [A]  time = 33.3675, size = 668, normalized size = 8.91 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{3}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{3 b^{3} x^{\frac{3}{2}}} & \text{for}\: a = 0 \\- \frac{2 x^{\frac{3}{2}}}{3 a^{3}} & \text{for}\: b = 0 \\- \frac{2 a^{\frac{3}{2}} b \sqrt{x} \sqrt{\frac{1}{b}}}{8 a^{\frac{7}{2}} b^{2} \sqrt{\frac{1}{b}} - 16 a^{\frac{5}{2}} b^{3} x \sqrt{\frac{1}{b}} + 8 a^{\frac{3}{2}} b^{4} x^{2} \sqrt{\frac{1}{b}}} - \frac{2 \sqrt{a} b^{2} x^{\frac{3}{2}} \sqrt{\frac{1}{b}}}{8 a^{\frac{7}{2}} b^{2} \sqrt{\frac{1}{b}} - 16 a^{\frac{5}{2}} b^{3} x \sqrt{\frac{1}{b}} + 8 a^{\frac{3}{2}} b^{4} x^{2} \sqrt{\frac{1}{b}}} - \frac{a^{2} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 a^{\frac{7}{2}} b^{2} \sqrt{\frac{1}{b}} - 16 a^{\frac{5}{2}} b^{3} x \sqrt{\frac{1}{b}} + 8 a^{\frac{3}{2}} b^{4} x^{2} \sqrt{\frac{1}{b}}} + \frac{a^{2} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 a^{\frac{7}{2}} b^{2} \sqrt{\frac{1}{b}} - 16 a^{\frac{5}{2}} b^{3} x \sqrt{\frac{1}{b}} + 8 a^{\frac{3}{2}} b^{4} x^{2} \sqrt{\frac{1}{b}}} + \frac{2 a b x \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 a^{\frac{7}{2}} b^{2} \sqrt{\frac{1}{b}} - 16 a^{\frac{5}{2}} b^{3} x \sqrt{\frac{1}{b}} + 8 a^{\frac{3}{2}} b^{4} x^{2} \sqrt{\frac{1}{b}}} - \frac{2 a b x \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 a^{\frac{7}{2}} b^{2} \sqrt{\frac{1}{b}} - 16 a^{\frac{5}{2}} b^{3} x \sqrt{\frac{1}{b}} + 8 a^{\frac{3}{2}} b^{4} x^{2} \sqrt{\frac{1}{b}}} - \frac{b^{2} x^{2} \log{\left (- \sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 a^{\frac{7}{2}} b^{2} \sqrt{\frac{1}{b}} - 16 a^{\frac{5}{2}} b^{3} x \sqrt{\frac{1}{b}} + 8 a^{\frac{3}{2}} b^{4} x^{2} \sqrt{\frac{1}{b}}} + \frac{b^{2} x^{2} \log{\left (\sqrt{a} \sqrt{\frac{1}{b}} + \sqrt{x} \right )}}{8 a^{\frac{7}{2}} b^{2} \sqrt{\frac{1}{b}} - 16 a^{\frac{5}{2}} b^{3} x \sqrt{\frac{1}{b}} + 8 a^{\frac{3}{2}} b^{4} x^{2} \sqrt{\frac{1}{b}}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/x**(3/2), Eq(a, 0) & Eq(b, 0)), (-2/(3*b**3*x**(3/2)), Eq(a, 0)), (-2*x**(3/2)/(3*a**3), Eq(b,
0)), (-2*a**(3/2)*b*sqrt(x)*sqrt(1/b)/(8*a**(7/2)*b**2*sqrt(1/b) - 16*a**(5/2)*b**3*x*sqrt(1/b) + 8*a**(3/2)*b
**4*x**2*sqrt(1/b)) - 2*sqrt(a)*b**2*x**(3/2)*sqrt(1/b)/(8*a**(7/2)*b**2*sqrt(1/b) - 16*a**(5/2)*b**3*x*sqrt(1
/b) + 8*a**(3/2)*b**4*x**2*sqrt(1/b)) - a**2*log(-sqrt(a)*sqrt(1/b) + sqrt(x))/(8*a**(7/2)*b**2*sqrt(1/b) - 16
*a**(5/2)*b**3*x*sqrt(1/b) + 8*a**(3/2)*b**4*x**2*sqrt(1/b)) + a**2*log(sqrt(a)*sqrt(1/b) + sqrt(x))/(8*a**(7/
2)*b**2*sqrt(1/b) - 16*a**(5/2)*b**3*x*sqrt(1/b) + 8*a**(3/2)*b**4*x**2*sqrt(1/b)) + 2*a*b*x*log(-sqrt(a)*sqrt
(1/b) + sqrt(x))/(8*a**(7/2)*b**2*sqrt(1/b) - 16*a**(5/2)*b**3*x*sqrt(1/b) + 8*a**(3/2)*b**4*x**2*sqrt(1/b)) -
2*a*b*x*log(sqrt(a)*sqrt(1/b) + sqrt(x))/(8*a**(7/2)*b**2*sqrt(1/b) - 16*a**(5/2)*b**3*x*sqrt(1/b) + 8*a**(3/
2)*b**4*x**2*sqrt(1/b)) - b**2*x**2*log(-sqrt(a)*sqrt(1/b) + sqrt(x))/(8*a**(7/2)*b**2*sqrt(1/b) - 16*a**(5/2)
*b**3*x*sqrt(1/b) + 8*a**(3/2)*b**4*x**2*sqrt(1/b)) + b**2*x**2*log(sqrt(a)*sqrt(1/b) + sqrt(x))/(8*a**(7/2)*b
**2*sqrt(1/b) - 16*a**(5/2)*b**3*x*sqrt(1/b) + 8*a**(3/2)*b**4*x**2*sqrt(1/b)), True))

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Giac [A]  time = 1.20825, size = 74, normalized size = 0.99 \begin{align*} -\frac{\arctan \left (\frac{b \sqrt{x}}{\sqrt{-a b}}\right )}{4 \, \sqrt{-a b} a b} - \frac{b x^{\frac{3}{2}} + a \sqrt{x}}{4 \,{\left (b x - a\right )}^{2} a b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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