3.175 \(\int \frac {\log (x)}{\sqrt {b+a x}} \, dx\)

Optimal. Leaf size=57 \[ -\frac {4 \sqrt {a x+b}}{a}+\frac {2 \log (x) \sqrt {a x+b}}{a}+\frac {4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a x+b}}{\sqrt {b}}\right )}{a} \]

[Out]

4*arctanh((a*x+b)^(1/2)/b^(1/2))*b^(1/2)/a-4*(a*x+b)^(1/2)/a+2*ln(x)*(a*x+b)^(1/2)/a

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Rubi [A]  time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {2319, 50, 63, 208} \[ -\frac {4 \sqrt {a x+b}}{a}+\frac {2 \log (x) \sqrt {a x+b}}{a}+\frac {4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a x+b}}{\sqrt {b}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Int[Log[x]/Sqrt[b + a*x],x]

[Out]

(-4*Sqrt[b + a*x])/a + (4*Sqrt[b]*ArcTanh[Sqrt[b + a*x]/Sqrt[b]])/a + (2*Sqrt[b + a*x]*Log[x])/a

Rule 50

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + n + 1)), x] + Dist[(n*(b*c - a*d))/(b*(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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]

Rule 2319

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((d_) + (e_.)*(x_))^(q_.), x_Symbol] :> Simp[((d + e*x)^(q + 1
)*(a + b*Log[c*x^n])^p)/(e*(q + 1)), x] - Dist[(b*n*p)/(e*(q + 1)), Int[((d + e*x)^(q + 1)*(a + b*Log[c*x^n])^
(p - 1))/x, x], x] /; FreeQ[{a, b, c, d, e, n, p, q}, x] && GtQ[p, 0] && NeQ[q, -1] && (EqQ[p, 1] || (Integers
Q[2*p, 2*q] &&  !IGtQ[q, 0]) || (EqQ[p, 2] && NeQ[q, 1]))

Rubi steps

\begin {align*} \int \frac {\log (x)}{\sqrt {b+a x}} \, dx &=\frac {2 \sqrt {b+a x} \log (x)}{a}-\frac {2 \int \frac {\sqrt {b+a x}}{x} \, dx}{a}\\ &=-\frac {4 \sqrt {b+a x}}{a}+\frac {2 \sqrt {b+a x} \log (x)}{a}-\frac {(2 b) \int \frac {1}{x \sqrt {b+a x}} \, dx}{a}\\ &=-\frac {4 \sqrt {b+a x}}{a}+\frac {2 \sqrt {b+a x} \log (x)}{a}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {1}{-\frac {b}{a}+\frac {x^2}{a}} \, dx,x,\sqrt {b+a x}\right )}{a^2}\\ &=-\frac {4 \sqrt {b+a x}}{a}+\frac {4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )}{a}+\frac {2 \sqrt {b+a x} \log (x)}{a}\\ \end {align*}

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Mathematica [A]  time = 0.02, size = 43, normalized size = 0.75 \[ \frac {2 (\log (x)-2) \sqrt {a x+b}+4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {a x+b}}{\sqrt {b}}\right )}{a} \]

Antiderivative was successfully verified.

[In]

Integrate[Log[x]/Sqrt[b + a*x],x]

[Out]

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

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fricas [A]  time = 0.45, size = 89, normalized size = 1.56 \[ \left [\frac {2 \, {\left (\sqrt {a x + b} {\left (\log \relax (x) - 2\right )} + \sqrt {b} \log \left (\frac {a x + 2 \, \sqrt {a x + b} \sqrt {b} + 2 \, b}{x}\right )\right )}}{a}, \frac {2 \, {\left (\sqrt {a x + b} {\left (\log \relax (x) - 2\right )} - 2 \, \sqrt {-b} \arctan \left (\frac {\sqrt {a x + b} \sqrt {-b}}{b}\right )\right )}}{a}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.17, size = 48, normalized size = 0.84 \[ -\frac {2 \, {\left (\frac {2 \, b \arctan \left (\frac {\sqrt {a x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}} - \sqrt {a x + b} \log \relax (x) + 2 \, \sqrt {a x + b}\right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 48, normalized size = 0.84 \[ \frac {4 \sqrt {b}\, \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )}{a}+\frac {2 \sqrt {a x +b}\, \ln \relax (x )}{a}-\frac {4 \sqrt {a x +b}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*arctanh((a*x+b)^(1/2)/b^(1/2))*b^(1/2)/a-4*(a*x+b)^(1/2)/a+2*ln(x)*(a*x+b)^(1/2)/a

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maxima [A]  time = 0.96, size = 58, normalized size = 1.02 \[ \frac {2 \, {\left (\sqrt {a x + b} \log \relax (x) - \sqrt {b} \log \left (\frac {\sqrt {a x + b} - \sqrt {b}}{\sqrt {a x + b} + \sqrt {b}}\right ) - 2 \, \sqrt {a x + b}\right )}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.09, size = 49, normalized size = 0.86 \[ \frac {2\,\sqrt {b}\,\ln \left (\frac {2\,b+a\,x+2\,\sqrt {b}\,\sqrt {b+a\,x}}{x}\right )}{a}+\frac {2\,\left (\ln \relax (x)-2\right )\,\sqrt {b+a\,x}}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(x)/(b + a*x)^(1/2),x)

[Out]

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

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sympy [B]  time = 4.15, size = 920, normalized size = 16.14 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((4*sqrt(b)*acoth(sqrt(b)/(sqrt(a)*sqrt(x + b/a)))/a + 2*sqrt(x + b/a)*log(b/a)/sqrt(a) - 2*sqrt(x +
b/a)*log(b/(a*(x + b/a)))/sqrt(a) + 2*sqrt(x + b/a)*log(-1 + b/(a*(x + b/a)))/sqrt(a) - 4*sqrt(x + b/a)/sqrt(a
) + 2*I*pi*sqrt(x + b/a)/sqrt(a), (Abs(x + b/a) < 1) & (Abs(b/(a*(x + b/a))) > 1)), (4*sqrt(b)*atanh(sqrt(b)/(
sqrt(a)*sqrt(x + b/a)))/a + 2*sqrt(x + b/a)*log(b/a)/sqrt(a) - 2*sqrt(x + b/a)*log(b/(a*(x + b/a)))/sqrt(a) +
2*sqrt(x + b/a)*log(1 - b/(a*(x + b/a)))/sqrt(a) - 4*sqrt(x + b/a)/sqrt(a), Abs(x + b/a) < 1), (4*sqrt(b)*acot
h(sqrt(b)/(sqrt(a)*sqrt(x + b/a)))/a + 2*sqrt(x + b/a)*log(b/a)/sqrt(a) - 2*sqrt(x + b/a)*log(b/(a*(x + b/a)))
/sqrt(a) + 2*sqrt(x + b/a)*log(-1 + b/(a*(x + b/a)))/sqrt(a) - 4*sqrt(x + b/a)/sqrt(a) + 2*I*pi*sqrt(x + b/a)/
sqrt(a), (1/Abs(x + b/a) < 1) & (Abs(b/(a*(x + b/a))) > 1)), (4*sqrt(b)*atanh(sqrt(b)/(sqrt(a)*sqrt(x + b/a)))
/a + 2*sqrt(x + b/a)*log(b/a)/sqrt(a) - 2*sqrt(x + b/a)*log(b/(a*(x + b/a)))/sqrt(a) + 2*sqrt(x + b/a)*log(1 -
b/(a*(x + b/a)))/sqrt(a) - 4*sqrt(x + b/a)/sqrt(a), 1/Abs(x + b/a) < 1), (4*sqrt(b)*acoth(sqrt(b)/(sqrt(a)*sq
rt(x + b/a)))/a - 2*sqrt(x + b/a)*log(b/(a*(x + b/a)))/sqrt(a) + 2*sqrt(x + b/a)*log(-1 + b/(a*(x + b/a)))/sqr
t(a) - 4*sqrt(x + b/a)/sqrt(a) + meijerg(((1,), (3/2,)), ((1/2,), (0,)), x + b/a)*log(b/a)/sqrt(a) + I*pi*meij
erg(((1,), (3/2,)), ((1/2,), (0,)), x + b/a)/sqrt(a) + meijerg(((3/2, 1), ()), ((), (1/2, 0)), x + b/a)*log(b/
a)/sqrt(a) + I*pi*meijerg(((3/2, 1), ()), ((), (1/2, 0)), x + b/a)/sqrt(a), Abs(b/(a*(x + b/a))) > 1), (4*sqrt
(b)*atanh(sqrt(b)/(sqrt(a)*sqrt(x + b/a)))/a - 2*sqrt(x + b/a)*log(b/(a*(x + b/a)))/sqrt(a) + 2*sqrt(x + b/a)*
log(1 - b/(a*(x + b/a)))/sqrt(a) - 4*sqrt(x + b/a)/sqrt(a) - 2*I*pi*sqrt(x + b/a)/sqrt(a) + meijerg(((1,), (3/
2,)), ((1/2,), (0,)), x + b/a)*log(b/a)/sqrt(a) + I*pi*meijerg(((1,), (3/2,)), ((1/2,), (0,)), x + b/a)/sqrt(a
) + meijerg(((3/2, 1), ()), ((), (1/2, 0)), x + b/a)*log(b/a)/sqrt(a) + I*pi*meijerg(((3/2, 1), ()), ((), (1/2
, 0)), x + b/a)/sqrt(a), True))

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