∫ log(x)__√ b + ax dx [175]

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

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 = 0.02, 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 = {2356, 52, 65, 214} \begin {gather*} -\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} \end {gather*}

(-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

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]

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,

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},

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

\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) \text {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 \begin {gather*} \frac {4 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b+a x}}{\sqrt {b}}\right )+2 \sqrt {b+a x} (-2+\log (x))}{a} \end {gather*}

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

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Mathics [C] Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
time = 14.13, size = 1019, normalized size = 17.88

Piecewise[{{2 (a (-2 + Log[-a x / (a x + b)] - Log[b / (a x + b)]) Sqrt[(a x + b) / a] + 2 Sqrt[a] Sqrt[b] Arc

Coth[Sqrt[b] / (Sqrt[a] Sqrt[(a x + b) / a])]) / a ^ (3 / 2), Abs[b / a + x] < 1 && Abs[a / (a x + b)] < 1 &&

Abs[b / (a x + b)] > 1}, {2 (a (-2 - I Pi + Log[a x / (a x + b)] - Log[b / (a x + b)]) Sqrt[(a x + b) / a] + 2

Sqrt[a] Sqrt[b] ArcTanh[Sqrt[b] / (Sqrt[a] Sqrt[(a x + b) / a])]) / a ^ (3 / 2), Abs[b / a + x] < 1 && Abs[a

/ (a x + b)] < 1}, {2 (a (-2 + I Pi + Log[-a x / (a x + b)] + Log[b / a] - Log[b / (a x + b)]) Sqrt[(a x + b)

/ a] + 2 Sqrt[a] Sqrt[b] ArcCoth[Sqrt[b] / (Sqrt[a] Sqrt[(a x + b) / a])]) / a ^ (3 / 2), Abs[b / a + x] < 1 &

& Abs[b / (a x + b)] > 1}, {2 (a (-2 + Log[a x / (a x + b)] + Log[b / a] - Log[b / (a x + b)]) Sqrt[(a x + b)

/ a] + 2 Sqrt[a] Sqrt[b] ArcTanh[Sqrt[b] / (Sqrt[a] Sqrt[(a x + b) / a])]) / a ^ (3 / 2), Abs[b / a + x] < 1},

{2 (a (-2 + I Pi + Log[-a x / (a x + b)] + Log[b / a] - Log[b / (a x + b)]) Sqrt[(a x + b) / a] + 2 Sqrt[a] S

qrt[b] ArcCoth[Sqrt[b] / (Sqrt[a] Sqrt[(a x + b) / a])]) / a ^ (3 / 2), Abs[a / (a x + b)] < 1 && Abs[b / (a x

+ b)] > 1}, {2 (a (-2 + Log[a x / (a x + b)] + Log[b / a] - Log[b / (a x + b)]) Sqrt[(a x + b) / a] + 2 Sqrt[

a] Sqrt[b] ArcTanh[Sqrt[b] / (Sqrt[a] Sqrt[(a x + b) / a])]) / a ^ (3 / 2), Abs[a / (a x + b)] < 1}, {(a (I Pi

meijerg[{{1}, {3 / 2}}, {{1 / 2}, {0}}, b / a + x] + I Pi meijerg[{{3 / 2, 1}, {}}, {{}, {1 / 2, 0}}, b / a +

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

) / a] + Log[b / a] meijerg[{{1}, {3 / 2}}, {{1 / 2}, {0}}, b / a + x] + Log[b / a] meijerg[{{3 / 2, 1}, {}},

{{}, {1 / 2, 0}}, b / a + x]) + 4 Sqrt[a] Sqrt[b] ArcCoth[Sqrt[b] / (Sqrt[a] Sqrt[(a x + b) / a])]) / a ^ (3 /

2), Abs[b / (a x + b)] > 1}}, -4 Sqrt[b / a + x] / Sqrt[a] - 2 Log[b / (a (b / a + x))] Sqrt[b / a + x] / Sqr

t[a] + 2 Log[1 - b / (a (b / a + x))] Sqrt[b / a + x] / Sqrt[a] + Log[b / a] meijerg[{{1}, {3 / 2}}, {{1 / 2},

{0}}, b / a + x] / Sqrt[a] + Log[b / a] meijerg[{{3 / 2, 1}, {}}, {{}, {1 / 2, 0}}, b / a + x] / Sqrt[a] - 2

I Pi Sqrt[b / a + x] / Sqrt[a] + I Pi meijerg[{{1}, {3 / 2}}, {{1 / 2}, {0}}, b / a + x] / Sqrt[a] + I Pi meij

erg[{{3 / 2, 1}, {}}, {{}, {1 / 2, 0}}, b / a + x] / Sqrt[a] + 4 Sqrt[b] ArcTanh[Sqrt[b] / (Sqrt[a] Sqrt[b / a

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method	result	size

derivativedivides	\(\frac {2 \sqrt {a x +b}\, \ln \left (x \right )-4 \sqrt {a x +b}+4 \sqrt {b}\, \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )}{a}\)	\(43\)
default	\(\frac {2 \sqrt {a x +b}\, \ln \left (x \right )-4 \sqrt {a x +b}+4 \sqrt {b}\, \arctanh \left (\frac {\sqrt {a x +b}}{\sqrt {b}}\right )}{a}\)	\(43\)

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

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Maxima [A]
time = 0.33, size = 58, normalized size = 1.02 \begin {gather*} \frac {2 \, {\left (\sqrt {a x + b} \log \left (x\right ) - \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} \end {gather*}

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))/

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Fricas [A]
time = 0.34, size = 89, normalized size = 1.56 \begin {gather*} \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 {gather*}

[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

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Piecewise((4*sqrt(b)*acoth(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), (Abs(x + b/a) < 1) & (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*(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) & (1/Abs(x + b/a) < 1)), (4*sqrt(b)*acoth(sqrt(b)/(sqrt(a)*sqr

t(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)*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/(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)/sqr

t(a), 1/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*(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)/sqrt(a) + I*pi*meijerg(((1,), (3/2,)), ((1/2,), (0,)), x + b/a)/sq

rt(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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Giac [A]
time = 0.00, size = 70, normalized size = 1.23 \begin {gather*} \frac {2 \left (\sqrt {a x+b} \ln \left (\frac {a x+b-b}{a}\right )+2 \left (-\sqrt {a x+b}-\frac {2 b \arctan \left (\frac {\sqrt {a x+b}}{\sqrt {-b}}\right )}{2 \sqrt {-b}}\right )\right )}{a} \end {gather*}

-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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Mupad [B]
time = 0.09, size = 49, normalized size = 0.86 \begin {gather*} \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 \left (x\right )-2\right )\,\sqrt {b+a\,x}}{a} \end {gather*}

(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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3.2.75 \(\int \frac {\log (x)}{\sqrt {b+a x}} \, dx\) [175]