3.4.70 \(\int \frac {x^{\frac {1-n}{2}+\frac {1}{2} (-3+n)}}{\sqrt {a+b x}} \, dx\)

Optimal. Leaf size=23 \[ -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \]

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Rubi [A]  time = 0.01, antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {7, 63, 208} \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^((1 - n)/2 + (-3 + n)/2)/Sqrt[a + b*x],x]

[Out]

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

Rule 7

Int[(u_.)*(Px_)^(p_), x_Symbol] :> Int[u*Px^Simplify[p], x] /; PolyQ[Px, x] &&  !RationalQ[p] && FreeQ[p, x] &
& RationalQ[Simplify[p]]

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

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Mathematica [A]  time = 0.00, size = 23, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^((1 - n)/2 + (-3 + n)/2)/Sqrt[a + b*x],x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 23, normalized size = 1.00 \begin {gather*} -\frac {2 \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^((1 - n)/2 + (-3 + n)/2)/Sqrt[a + b*x],x]

[Out]

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

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fricas [A]  time = 1.11, size = 56, normalized size = 2.43 \begin {gather*} \left [\frac {\log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right )}{\sqrt {a}}, \frac {2 \, \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right )}{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]

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

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giac [A]  time = 0.85, size = 21, normalized size = 0.91 \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 = 18, normalized size = 0.78 \begin {gather*} -\frac {2 \arctanh \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*arctanh((b*x+a)^(1/2)/a^(1/2))/a^(1/2)

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maxima [A]  time = 2.99, size = 32, normalized size = 1.39 \begin {gather*} \frac {\log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\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]

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

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mupad [B]  time = 0.00, size = 17, normalized size = 0.74 \begin {gather*} -\frac {2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )}{\sqrt {a}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x*(a + b*x)^(1/2)),x)

[Out]

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

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sympy [A]  time = 1.09, size = 24, normalized size = 1.04 \begin {gather*} - \frac {2 \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \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)

[Out]

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

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