\(\int \frac {a+b \log (c (d+e x)^n)}{(f+g x)^{3/2}} \, dx\) [141]

Optimal result

Integrand size = 24, antiderivative size = 81 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^{3/2}} \, dx=-\frac {4 b \sqrt {e} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f+g x}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2442, 65, 214} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^{3/2}} \, dx=-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f+g x}}-\frac {4 b \sqrt {e} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}} \]

[In]

Int[(a + b*Log[c*(d + e*x)^n])/(f + g*x)^(3/2),x]

[Out]

(-4*b*Sqrt[e]*n*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(g*Sqrt[e*f - d*g]) - (2*(a + b*Log[c*(d + e
*x)^n]))/(g*Sqrt[f + g*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 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f+g x}}+\frac {(2 b e n) \int \frac {1}{(d+e x) \sqrt {f+g x}} \, dx}{g} \\ & = -\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f+g x}}+\frac {(4 b e 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 \sqrt {e} n \tanh ^{-1}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{g \sqrt {e f-d g}}-\frac {2 \left (a+b \log \left (c (d+e x)^n\right )\right )}{g \sqrt {f+g x}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.99 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^{3/2}} \, dx=\frac {2 \left (-\frac {2 b \sqrt {e} n \text {arctanh}\left (\frac {\sqrt {e} \sqrt {f+g x}}{\sqrt {e f-d g}}\right )}{\sqrt {e f-d g}}-\frac {a+b \log \left (c (d+e x)^n\right )}{\sqrt {f+g x}}\right )}{g} \]

[In]

Integrate[(a + b*Log[c*(d + e*x)^n])/(f + g*x)^(3/2),x]

[Out]

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

Maple [F]

\[\int \frac {a +b \ln \left (c \left (e x +d \right )^{n}\right )}{\left (g x +f \right )^{\frac {3}{2}}}d x\]

[In]

int((a+b*ln(c*(e*x+d)^n))/(g*x+f)^(3/2),x)

[Out]

int((a+b*ln(c*(e*x+d)^n))/(g*x+f)^(3/2),x)

Fricas [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 224, normalized size of antiderivative = 2.77 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^{3/2}} \, dx=\left [\frac {2 \, {\left ({\left (b g n x + b f n\right )} \sqrt {\frac {e}{e f - d g}} \log \left (\frac {e g x + 2 \, e f - d g - 2 \, {\left (e f - d g\right )} \sqrt {g x + f} \sqrt {\frac {e}{e f - d g}}}{e x + d}\right ) - {\left (b n \log \left (e x + d\right ) + b \log \left (c\right ) + a\right )} \sqrt {g x + f}\right )}}{g^{2} x + f g}, -\frac {2 \, {\left (2 \, {\left (b g n x + b f n\right )} \sqrt {-\frac {e}{e f - d g}} \arctan \left (-\frac {{\left (e f - d g\right )} \sqrt {g x + f} \sqrt {-\frac {e}{e f - d g}}}{e g x + e f}\right ) + {\left (b n \log \left (e x + d\right ) + b \log \left (c\right ) + a\right )} \sqrt {g x + f}\right )}}{g^{2} x + f g}\right ] \]

[In]

integrate((a+b*log(c*(e*x+d)^n))/(g*x+f)^(3/2),x, algorithm="fricas")

[Out]

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

Sympy [F]

\[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^{3/2}} \, dx=\int \frac {a + b \log {\left (c \left (d + e x\right )^{n} \right )}}{\left (f + g x\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((a+b*ln(c*(e*x+d)**n))/(g*x+f)**(3/2),x)

[Out]

Integral((a + b*log(c*(d + e*x)**n))/(f + g*x)**(3/2), x)

Maxima [F(-2)]

Exception generated. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((a+b*log(c*(e*x+d)^n))/(g*x+f)^(3/2),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 = 0.33 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.25 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^{3/2}} \, dx=\frac {4 \, b e n \arctan \left (\frac {\sqrt {g x + f} e}{\sqrt {-e^{2} f + d e g}}\right )}{\sqrt {-e^{2} f + d e g} g} - \frac {2 \, b n \log \left ({\left (g x + f\right )} e - e f + d g\right )}{\sqrt {g x + f} g} + \frac {2 \, {\left (b n \log \left (g\right ) - b \log \left (c\right ) - a\right )}}{\sqrt {g x + f} g} \]

[In]

integrate((a+b*log(c*(e*x+d)^n))/(g*x+f)^(3/2),x, algorithm="giac")

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^{3/2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{{\left (f+g\,x\right )}^{3/2}} \,d x \]

[In]

int((a + b*log(c*(d + e*x)^n))/(f + g*x)^(3/2),x)

[Out]

int((a + b*log(c*(d + e*x)^n))/(f + g*x)^(3/2), x)