\(\int \frac {a+b \log (c (d+e x)^n)}{\sqrt {f+g x}} \, dx\) [140]

Optimal result

Integrand size = 24, antiderivative size = 97 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=-\frac {4 b n \sqrt {f+g x}}{g}+\frac {4 b \sqrt {e f-d g} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e} g}+\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2442, 52, 65, 214} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=\frac {2 \sqrt {f+g x} \left (a+b \log \left (c (d+e x)^n\right )\right )}{g}+\frac {4 b n \sqrt {e f-d g} \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e} g}-\frac {4 b n \sqrt {f+g x}}{g} \]

[In]

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

[Out]

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

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 2442

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_))^(q_.), x_Symbol] :> Simp[(f + g*
x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/(g*(q + 1))), x] - Dist[b*e*(n/(g*(q + 1))), Int[(f + g*x)^(q + 1)/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.86 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=\frac {2 \left (\frac {2 b \sqrt {e f-d g} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e}}+\sqrt {f+g x} \left (a-2 b n+b \log \left (c (d+e x)^n\right )\right )\right )}{g} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 0.86 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.16

method	result	size
derivativedivides	\(\frac {2 \sqrt {g x +f}\, a +2 b \left (\ln \left (c \left (\frac {\left (g x +f \right ) e +d g -e f}{g}\right )^{n}\right ) \sqrt {g x +f}-2 e n \left (\frac {\sqrt {g x +f}}{e}+\frac {\left (-d g +e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e \sqrt {\left (d g -e f \right ) e}}\right )\right )}{g}\)	\(113\)
default	\(\frac {2 \sqrt {g x +f}\, a +2 b \left (\ln \left (c \left (\frac {\left (g x +f \right ) e +d g -e f}{g}\right )^{n}\right ) \sqrt {g x +f}-2 e n \left (\frac {\sqrt {g x +f}}{e}+\frac {\left (-d g +e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e \sqrt {\left (d g -e f \right ) e}}\right )\right )}{g}\)	\(113\)
parts	\(\frac {2 a \sqrt {g x +f}}{g}+\frac {2 b \left (\ln \left (c \left (\frac {\left (g x +f \right ) e +d g -e f}{g}\right )^{n}\right ) \sqrt {g x +f}-2 e n \left (\frac {\sqrt {g x +f}}{e}+\frac {\left (-d g +e f \right ) \arctan \left (\frac {e \sqrt {g x +f}}{\sqrt {\left (d g -e f \right ) e}}\right )}{e \sqrt {\left (d g -e f \right ) e}}\right )\right )}{g}\)	\(116\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.91 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=\left [\frac {2 \, {\left (b n \sqrt {\frac {e f - d g}{e}} \log \left (\frac {e g x + 2 \, e f - d g + 2 \, \sqrt {g x + f} e \sqrt {\frac {e f - d g}{e}}}{e x + d}\right ) + {\left (b n \log \left (e x + d\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt {g x + f}\right )}}{g}, \frac {2 \, {\left (2 \, b n \sqrt {-\frac {e f - d g}{e}} \arctan \left (-\frac {\sqrt {g x + f} e \sqrt {-\frac {e f - d g}{e}}}{e f - d g}\right ) + {\left (b n \log \left (e x + d\right ) - 2 \, b n + b \log \left (c\right ) + a\right )} \sqrt {g x + f}\right )}}{g}\right ] \]

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*log(c*(d + e*x)**n))/sqrt(f + g*x), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=\text {Exception raised: ValueError} \]

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e*(d*g-e*f)>0)', see `assume?`
for more de

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.13 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}} \, dx=-\frac {2 \, {\left ({\left (2 \, e {\left (\frac {{\left (e f - d g\right )} \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {-e^{2} f + d e g}}\right )}{\sqrt {-e^{2} f + d e g} e} + \frac {\sqrt {g x + f}}{e}\right )} - \sqrt {g x + f} \log \left (e x + d\right )\right )} b n - \sqrt {g x + f} b \log \left (c\right ) - \sqrt {g x + f} a\right )}}{g} \]

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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