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

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Rubi [A]  time = 0.0296218, antiderivative size = 57, normalized size of antiderivative = 1., 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 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]))

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]

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*}

Mathematica [A]  time = 0.0226035, 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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Maple [A]  time = 0.009, size = 48, normalized size = 0.8 \begin{align*} 4\,{\frac{\sqrt{b}}{a}{\it Artanh} \left ({\frac{\sqrt{ax+b}}{\sqrt{b}}} \right ) }-4\,{\frac{\sqrt{ax+b}}{a}}+2\,{\frac{\ln \left ( x \right ) \sqrt{ax+b}}{a}} \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.88307, size = 234, normalized size = 4.11 \begin{align*} \left [\frac{2 \,{\left (\sqrt{a x + b}{\left (\log \left (x\right ) - 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 \left (x\right ) - 2\right )} - 2 \, \sqrt{-b} \arctan \left (\frac{\sqrt{a x + b} \sqrt{-b}}{b}\right )\right )}}{a}\right ] \end{align*}

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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Sympy [B]  time = 4.27925, size = 930, normalized size = 16.32 \begin{align*} \text{result too large to display} \end{align*}

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)/(Abs(a)*Abs(x + b/a)) > 1)), (4*sqrt(b)*atanh(s
qrt(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)))/sq
rt(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)*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), (1/Abs(x + b/a) < 1) & (Abs(b)/(Abs(a)*Abs(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(sqr
t(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) + meijerg(((1,), (3/2,)), ((1/2,), (0,)), x + b/a)*log(b/a)/sqr
t(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), Abs(b)/(Abs(a)*Abs
(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)))/sq
rt(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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Giac [A]  time = 1.10269, size = 65, normalized size = 1.14 \begin{align*} -\frac{2 \,{\left (\frac{2 \, b \arctan \left (\frac{\sqrt{a x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \sqrt{a x + b} \log \left (x\right ) + 2 \, \sqrt{a x + b}\right )}}{a} \end{align*}

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