3.472 \(\int \frac {1}{\sqrt {x} (-a+b x)} \, dx\)

Optimal. Leaf size=29 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 29, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {63, 208} \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(-a + b*x)),x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 29, normalized size = 1.00 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(-a + b*x)),x]

[Out]

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

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fricas [A]  time = 0.45, size = 67, normalized size = 2.31 \[ \left [\frac {\sqrt {a b} \log \left (\frac {b x + a - 2 \, \sqrt {a b} \sqrt {x}}{b x - a}\right )}{a b}, \frac {2 \, \sqrt {-a b} \arctan \left (\frac {\sqrt {-a b}}{b \sqrt {x}}\right )}{a b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.00, size = 20, normalized size = 0.69 \[ \frac {2 \, \arctan \left (\frac {b \sqrt {x}}{\sqrt {-a b}}\right )}{\sqrt {-a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 19, normalized size = 0.66 \[ -\frac {2 \arctanh \left (\frac {b \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.03, size = 34, normalized size = 1.17 \[ \frac {\log \left (\frac {b \sqrt {x} - \sqrt {a b}}{b \sqrt {x} + \sqrt {a b}}\right )}{\sqrt {a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

log((b*sqrt(x) - sqrt(a*b))/(b*sqrt(x) + sqrt(a*b)))/sqrt(a*b)

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mupad [B]  time = 0.13, size = 19, normalized size = 0.66 \[ -\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {b}\,\sqrt {x}}{\sqrt {a}}\right )}{\sqrt {a}\,\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-1/(x^(1/2)*(a - b*x)),x)

[Out]

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

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sympy [A]  time = 1.25, size = 88, normalized size = 3.03 \[ \begin {cases} \frac {\tilde {\infty }}{\sqrt {x}} & \text {for}\: a = 0 \wedge b = 0 \\- \frac {2}{b \sqrt {x}} & \text {for}\: a = 0 \\- \frac {2 \sqrt {x}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (- \sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{\sqrt {a} b \sqrt {\frac {1}{b}}} - \frac {\log {\left (\sqrt {a} \sqrt {\frac {1}{b}} + \sqrt {x} \right )}}{\sqrt {a} b \sqrt {\frac {1}{b}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((zoo/sqrt(x), Eq(a, 0) & Eq(b, 0)), (-2/(b*sqrt(x)), Eq(a, 0)), (-2*sqrt(x)/a, Eq(b, 0)), (log(-sqrt
(a)*sqrt(1/b) + sqrt(x))/(sqrt(a)*b*sqrt(1/b)) - log(sqrt(a)*sqrt(1/b) + sqrt(x))/(sqrt(a)*b*sqrt(1/b)), True)
)

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