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

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

________________________________________________________________________________________

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 = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {63, 205} \[ \frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a} \sqrt {b}} \]

Antiderivative was successfully verified.

[In]

Int[1/(Sqrt[x]*(b + a*x)),x]

[Out]

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

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

Integrate[1/(Sqrt[x]*(b + a*x)),x]

[Out]

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

________________________________________________________________________________________

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

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 1.50, size = 68, normalized size = 2.34 \[ \left [-\frac {\sqrt {-a b} \log \left (\frac {a x - b - 2 \, \sqrt {-a b} \sqrt {x}}{a x + b}\right )}{a b}, -\frac {2 \, \sqrt {a b} \arctan \left (\frac {\sqrt {a b}}{a \sqrt {x}}\right )}{a b}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

giac [A]  time = 0.96, size = 18, normalized size = 0.62 \[ \frac {2 \, \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.32, size = 19, normalized size = 0.66

method	result	size
derivativedivides	\(\frac {2 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\)	\(19\)
default	\(\frac {2 \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}}\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [A]  time = 0.97, size = 18, normalized size = 0.62 \[ \frac {2 \, \arctan \left (\frac {a \sqrt {x}}{\sqrt {a b}}\right )}{\sqrt {a b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

mupad [B]  time = 0.20, size = 19, normalized size = 0.66 \[ \frac {2\,\mathrm {atan}\left (\frac {\sqrt {a}\,\sqrt {x}}{\sqrt {b}}\right )}{\sqrt {a}\,\sqrt {b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(x^(1/2)*(b + a*x)),x)

[Out]

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

________________________________________________________________________________________

sympy [A]  time = 1.32, size = 94, normalized size = 3.24 \[ \begin {cases} \tilde {\infty } \sqrt {x} & \text {for}\: a = 0 \wedge b = 0 \\\frac {2 \sqrt {x}}{b} & \text {for}\: a = 0 \\- \frac {2}{a \sqrt {x}} & \text {for}\: b = 0 \\- \frac {i \log {\left (- i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{a \sqrt {b} \sqrt {\frac {1}{a}}} + \frac {i \log {\left (i \sqrt {b} \sqrt {\frac {1}{a}} + \sqrt {x} \right )}}{a \sqrt {b} \sqrt {\frac {1}{a}}} & \text {otherwise} \end {cases} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________