3.5.78 \(\int \frac {\sqrt {x}}{(-a+b x)^2} \, dx\) [478]

Optimal. Leaf size=47 \[ \frac {\sqrt {x}}{b (a-b x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {43, 65, 214} \begin {gather*} \frac {\sqrt {x}}{b (a-b x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]/(-a + b*x)^2,x]

[Out]

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

Rule 43

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, n
}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, -1] &&  !IntegerQ[n] && GtQ[n, 0]

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 214

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

Rubi steps

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

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Mathematica [A]
time = 0.06, size = 49, normalized size = 1.04 \begin {gather*} -\frac {\sqrt {x}}{b (-a+b x)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]/(-a + b*x)^2,x]

[Out]

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

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Maple [A]
time = 0.12, size = 38, normalized size = 0.81

method	result	size
derivativedivides	\(\frac {\sqrt {x}}{b \left (-b x +a \right )}-\frac {\arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\)	\(38\)
default	\(\frac {\sqrt {x}}{b \left (-b x +a \right )}-\frac {\arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{b \sqrt {a b}}\)	\(38\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^(1/2)/b/(-b*x+a)-1/b/(a*b)^(1/2)*arctanh(b*x^(1/2)/(a*b)^(1/2))

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Maxima [A]
time = 0.49, size = 56, normalized size = 1.19 \begin {gather*} -\frac {\sqrt {x}}{b^{2} x - a b} + \frac {\log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{2 \, \sqrt {a b} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-sqrt(x)/(b^2*x - a*b) + 1/2*log((b*sqrt(x) - sqrt(a*b))/(b*sqrt(x) + sqrt(a*b)))/(sqrt(a*b)*b)

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Fricas [A]
time = 0.42, size = 123, normalized size = 2.62 \begin {gather*} \left [-\frac {2 \, a b \sqrt {x} - \sqrt {a b} {\left (b x - a\right )} \log \left (\frac {b x + a - 2 \, \sqrt {a b} \sqrt {x}}{b x - a}\right )}{2 \, {\left (a b^{3} x - a^{2} b^{2}\right )}}, -\frac {a b \sqrt {x} - \sqrt {-a b} {\left (b x - a\right )} \arctan \left (\frac {\sqrt {-a b}}{b \sqrt {x}}\right )}{a b^{3} x - a^{2} b^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (37) = 74\).
time = 1.63, size = 243, normalized size = 5.17 \begin {gather*} \begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{b^{2} \sqrt {x}} & \text {for}\: a = 0 \\\frac {2 x^{\frac {3}{2}}}{3 a^{2}} & \text {for}\: b = 0 \\- \frac {a \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{2} \sqrt {\frac {a}{b}} + 2 b^{3} x \sqrt {\frac {a}{b}}} + \frac {a \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{2} \sqrt {\frac {a}{b}} + 2 b^{3} x \sqrt {\frac {a}{b}}} - \frac {2 b \sqrt {x} \sqrt {\frac {a}{b}}}{- 2 a b^{2} \sqrt {\frac {a}{b}} + 2 b^{3} x \sqrt {\frac {a}{b}}} + \frac {b x \log {\left (\sqrt {x} - \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{2} \sqrt {\frac {a}{b}} + 2 b^{3} x \sqrt {\frac {a}{b}}} - \frac {b x \log {\left (\sqrt {x} + \sqrt {\frac {a}{b}} \right )}}{- 2 a b^{2} \sqrt {\frac {a}{b}} + 2 b^{3} x \sqrt {\frac {a}{b}}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.94, size = 40, normalized size = 0.85 \begin {gather*} \frac {\arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b} b} - \frac {\sqrt {x}}{{\left (b x - a\right )} b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

arctan(b*sqrt(x)/sqrt(-a*b))/(sqrt(-a*b)*b) - sqrt(x)/((b*x - a)*b)

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Mupad [B]
time = 0.11, size = 35, normalized size = 0.74 \begin {gather*} \frac {\sqrt {x}}{b\,\left (a-b\,x\right )}-\frac {\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a}\,b^{3/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(1/2)/(a - b*x)^2,x)

[Out]

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

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