\(\int \frac {a+b \log (c x^n)}{\sqrt {d+e x}} \, dx\) [147]

Optimal result

Integrand size = 20, antiderivative size = 69 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}} \, dx=-\frac {4 b n \sqrt {d+e x}}{e}+\frac {4 b \sqrt {d} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e} \]

[Out]

4*b*n*arctanh((e*x+d)^(1/2)/d^(1/2))*d^(1/2)/e-4*b*n*(e*x+d)^(1/2)/e+2*(a+b*ln(c*x^n))*(e*x+d)^(1/2)/e

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2356, 52, 65, 214} \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}+\frac {4 b \sqrt {d} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e}-\frac {4 b n \sqrt {d+e x}}{e} \]

[In]

Int[(a + b*Log[c*x^n])/Sqrt[d + e*x],x]

[Out]

(-4*b*n*Sqrt[d + e*x])/e + (4*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/e + (2*Sqrt[d + e*x]*(a + b*Log[c*x^
n]))/e

Rule 52

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 65

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 214

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

Rule 2356

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*} \text {integral}& = \frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {(2 b n) \int \frac {\sqrt {d+e x}}{x} \, dx}{e} \\ & = -\frac {4 b n \sqrt {d+e x}}{e}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {(2 b d n) \int \frac {1}{x \sqrt {d+e x}} \, dx}{e} \\ & = -\frac {4 b n \sqrt {d+e x}}{e}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e}-\frac {(4 b d n) \text {Subst}\left (\int \frac {1}{-\frac {d}{e}+\frac {x^2}{e}} \, dx,x,\sqrt {d+e x}\right )}{e^2} \\ & = -\frac {4 b n \sqrt {d+e x}}{e}+\frac {4 b \sqrt {d} n \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )}{e}+\frac {2 \sqrt {d+e x} \left (a+b \log \left (c x^n\right )\right )}{e} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.80 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}} \, dx=\frac {4 b \sqrt {d} n \text {arctanh}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right )+2 \sqrt {d+e x} \left (a-2 b n+b \log \left (c x^n\right )\right )}{e} \]

[In]

Integrate[(a + b*Log[c*x^n])/Sqrt[d + e*x],x]

[Out]

(4*b*Sqrt[d]*n*ArcTanh[Sqrt[d + e*x]/Sqrt[d]] + 2*Sqrt[d + e*x]*(a - 2*b*n + b*Log[c*x^n]))/e

Maple [A] (verified)

Time = 0.40 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.90

method	result	size
derivativedivides	\(\frac {2 \sqrt {e x +d}\, a +2 b \left (\ln \left (c \,x^{n}\right ) \sqrt {e x +d}+2 n \left (-\sqrt {e x +d}+\sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )\right )\right )}{e}\)	\(62\)
default	\(\frac {2 \sqrt {e x +d}\, a +2 b \left (\ln \left (c \,x^{n}\right ) \sqrt {e x +d}+2 n \left (-\sqrt {e x +d}+\sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )\right )\right )}{e}\)	\(62\)
parts	\(\frac {2 a \sqrt {e x +d}}{e}+\frac {2 b \left (\ln \left (c \,x^{n}\right ) \sqrt {e x +d}-2 n \left (\sqrt {e x +d}-\sqrt {d}\, \operatorname {arctanh}\left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )\right )\right )}{e}\)	\(64\)

[In]

int((a+b*ln(c*x^n))/(e*x+d)^(1/2),x,method=_RETURNVERBOSE)

[Out]

2/e*((e*x+d)^(1/2)*a+b*(ln(c*x^n)*(e*x+d)^(1/2)+2*n*(-(e*x+d)^(1/2)+d^(1/2)*arctanh((e*x+d)^(1/2)/d^(1/2)))))

Fricas [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.68 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}} \, dx=\left [\frac {2 \, {\left (b \sqrt {d} n \log \left (\frac {e x + 2 \, \sqrt {e x + d} \sqrt {d} + 2 \, d}{x}\right ) + {\left (b n \log \left (x\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt {e x + d}\right )}}{e}, -\frac {2 \, {\left (2 \, b \sqrt {-d} n \arctan \left (\frac {\sqrt {e x + d} \sqrt {-d}}{d}\right ) - {\left (b n \log \left (x\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt {e x + d}\right )}}{e}\right ] \]

[In]

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

[Out]

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

Sympy [A] (verification not implemented)

Time = 2.30 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.90 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}} \, dx=a \left (\begin {cases} \frac {2 \sqrt {d + e x}}{e} & \text {for}\: e \neq 0 \\\frac {x}{\sqrt {d}} & \text {otherwise} \end {cases}\right ) - b n \left (\begin {cases} - \frac {4 \sqrt {d} \operatorname {asinh}{\left (\frac {\sqrt {d}}{\sqrt {e} \sqrt {x}} \right )}}{e} + \frac {4 d}{e^{\frac {3}{2}} \sqrt {x} \sqrt {\frac {d}{e x} + 1}} + \frac {4 \sqrt {x}}{\sqrt {e} \sqrt {\frac {d}{e x} + 1}} & \text {for}\: e > -\infty \wedge e < \infty \wedge e \neq 0 \\\frac {x}{\sqrt {d}} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} \frac {2 \sqrt {d + e x}}{e} & \text {for}\: e \neq 0 \\\frac {x}{\sqrt {d}} & \text {otherwise} \end {cases}\right ) \log {\left (c x^{n} \right )} \]

[In]

integrate((a+b*ln(c*x**n))/(e*x+d)**(1/2),x)

[Out]

a*Piecewise((2*sqrt(d + e*x)/e, Ne(e, 0)), (x/sqrt(d), True)) - b*n*Piecewise((-4*sqrt(d)*asinh(sqrt(d)/(sqrt(
e)*sqrt(x)))/e + 4*d/(e**(3/2)*sqrt(x)*sqrt(d/(e*x) + 1)) + 4*sqrt(x)/(sqrt(e)*sqrt(d/(e*x) + 1)), (e > -oo) &
(e < oo) & Ne(e, 0)), (x/sqrt(d), True)) + b*Piecewise((2*sqrt(d + e*x)/e, Ne(e, 0)), (x/sqrt(d), True))*log(
c*x**n)

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 82, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}} \, dx=-\frac {2 \, {\left (\sqrt {d} \log \left (\frac {\sqrt {e x + d} - \sqrt {d}}{\sqrt {e x + d} + \sqrt {d}}\right ) + 2 \, \sqrt {e x + d}\right )} b n}{e} + \frac {2 \, \sqrt {e x + d} b \log \left (c x^{n}\right )}{e} + \frac {2 \, \sqrt {e x + d} a}{e} \]

[In]

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

[Out]

-2*(sqrt(d)*log((sqrt(e*x + d) - sqrt(d))/(sqrt(e*x + d) + sqrt(d))) + 2*sqrt(e*x + d))*b*n/e + 2*sqrt(e*x + d
)*b*log(c*x^n)/e + 2*sqrt(e*x + d)*a/e

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.07 \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}} \, dx=-\frac {2 \, {\left ({\left (\frac {2 \, d \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-d}}\right )}{\sqrt {-d}} - \sqrt {e x + d} \log \left (x\right ) + 2 \, \sqrt {e x + d}\right )} b n - \sqrt {e x + d} b \log \left (c\right ) - \sqrt {e x + d} a\right )}}{e} \]

[In]

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

[Out]

-2*((2*d*arctan(sqrt(e*x + d)/sqrt(-d))/sqrt(-d) - sqrt(e*x + d)*log(x) + 2*sqrt(e*x + d))*b*n - sqrt(e*x + d)
*b*log(c) - sqrt(e*x + d)*a)/e

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c x^n\right )}{\sqrt {d+e x}} \, dx=\int \frac {a+b\,\ln \left (c\,x^n\right )}{\sqrt {d+e\,x}} \,d x \]

[In]

int((a + b*log(c*x^n))/(d + e*x)^(1/2),x)

[Out]

int((a + b*log(c*x^n))/(d + e*x)^(1/2), x)