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

Optimal. Leaf size=25 \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{\sqrt {a}} \]

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Rubi [A]  time = 0.01, antiderivative size = 25, 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, 205} \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/(x*Sqrt[-a + b*x]),x]

[Out]

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

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

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Mathematica [A]  time = 0.01, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/(x*Sqrt[-a + b*x]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 25, normalized size = 1.00 \begin {gather*} \frac {2 \tan ^{-1}\left (\frac {\sqrt {b x-a}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[1/(x*Sqrt[-a + b*x]),x]

[Out]

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

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fricas [A]  time = 1.40, size = 58, normalized size = 2.32 \begin {gather*} \left [-\frac {\sqrt {-a} \log \left (\frac {b x - 2 \, \sqrt {b x - a} \sqrt {-a} - 2 \, a}{x}\right )}{a}, \frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.99, size = 19, normalized size = 0.76 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 20, normalized size = 0.80 \begin {gather*} \frac {2 \arctan \left (\frac {\sqrt {b x -a}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 2.99, size = 19, normalized size = 0.76 \begin {gather*} \frac {2 \, \arctan \left (\frac {\sqrt {b x - a}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.05, size = 19, normalized size = 0.76 \begin {gather*} \frac {2\,\mathrm {atan}\left (\frac {\sqrt {b\,x-a}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(b*x - a)^(1/2)),x)

[Out]

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

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sympy [A]  time = 1.23, size = 54, normalized size = 2.16 \begin {gather*} \begin {cases} \frac {2 i \operatorname {acosh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} & \text {for}\: \left |{\frac {a}{b x}}\right | > 1 \\- \frac {2 \operatorname {asin}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{\sqrt {a}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((2*I*acosh(sqrt(a)/(sqrt(b)*sqrt(x)))/sqrt(a), Abs(a/(b*x)) > 1), (-2*asin(sqrt(a)/(sqrt(b)*sqrt(x))
)/sqrt(a), True))

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