\(\int \frac {a+b \log (c (d+e x)^n)}{(f+g x)^3} \, dx\) [42]

Optimal result

Integrand size = 22, antiderivative size = 112 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^3} \, dx=\frac {b e n}{2 g (e f-d g) (f+g x)}+\frac {b e^2 n \log (d+e x)}{2 g (e f-d g)^2}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}-\frac {b e^2 n \log (f+g x)}{2 g (e f-d g)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {2442, 46} \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^3} \, dx=-\frac {a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}+\frac {b e^2 n \log (d+e x)}{2 g (e f-d g)^2}-\frac {b e^2 n \log (f+g x)}{2 g (e f-d g)^2}+\frac {b e n}{2 g (f+g x) (e f-d g)} \]

[In]

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

[Out]

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

Rule 46

Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*x
)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && Lt
Q[m + n + 2, 0])

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 {a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}+\frac {(b e n) \int \frac {1}{(d+e x) (f+g x)^2} \, dx}{2 g} \\ & = -\frac {a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}+\frac {(b e n) \int \left (\frac {e^2}{(e f-d g)^2 (d+e x)}-\frac {g}{(e f-d g) (f+g x)^2}-\frac {e g}{(e f-d g)^2 (f+g x)}\right ) \, dx}{2 g} \\ & = \frac {b e n}{2 g (e f-d g) (f+g x)}+\frac {b e^2 n \log (d+e x)}{2 g (e f-d g)^2}-\frac {a+b \log \left (c (d+e x)^n\right )}{2 g (f+g x)^2}-\frac {b e^2 n \log (f+g x)}{2 g (e f-d g)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.74 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^3} \, dx=-\frac {a+b \log \left (c (d+e x)^n\right )-\frac {b e n (f+g x) (e f-d g+e (f+g x) \log (d+e x)-e (f+g x) \log (f+g x))}{(e f-d g)^2}}{2 g (f+g x)^2} \]

[In]

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

[Out]

-1/2*(a + b*Log[c*(d + e*x)^n] - (b*e*n*(f + g*x)*(e*f - d*g + e*(f + g*x)*Log[d + e*x] - e*(f + g*x)*Log[f +
g*x]))/(e*f - d*g)^2)/(g*(f + g*x)^2)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(282\) vs. \(2(107)=214\).

Time = 1.08 (sec) , antiderivative size = 283, normalized size of antiderivative = 2.53

method	result	size
parallelrisch	\(\frac {-x b d \,e^{2} g^{3} n +x b \,e^{3} f \,g^{2} n +b \,e^{3} f^{2} g n +2 a d \,e^{2} f \,g^{2}-\ln \left (c \left (e x +d \right )^{n}\right ) b \,d^{2} e \,g^{3}-\ln \left (c \left (e x +d \right )^{n}\right ) b \,e^{3} f^{2} g -b d \,e^{2} f \,g^{2} n +\ln \left (e x +d \right ) b \,e^{3} f^{2} g n -\ln \left (g x +f \right ) b \,e^{3} f^{2} g n +2 \ln \left (c \left (e x +d \right )^{n}\right ) b d \,e^{2} f \,g^{2}+\ln \left (e x +d \right ) x^{2} b \,e^{3} g^{3} n -\ln \left (g x +f \right ) x^{2} b \,e^{3} g^{3} n +2 \ln \left (e x +d \right ) x b \,e^{3} f \,g^{2} n -2 \ln \left (g x +f \right ) x b \,e^{3} f \,g^{2} n -a \,d^{2} e \,g^{3}-a \,e^{3} f^{2} g}{2 \left (g^{2} d^{2}-2 d e f g +e^{2} f^{2}\right ) \left (g x +f \right )^{2} e \,g^{2}}\)	\(283\)
risch	\(-\frac {b \ln \left (\left (e x +d \right )^{n}\right )}{2 g \left (g x +f \right )^{2}}-\frac {2 \ln \left (g x +f \right ) b \,e^{2} f^{2} n -2 \ln \left (-e x -d \right ) b \,e^{2} f^{2} n +2 a \,e^{2} f^{2}+2 \ln \left (g x +f \right ) b \,e^{2} g^{2} n \,x^{2}-2 \ln \left (-e x -d \right ) b \,e^{2} g^{2} n \,x^{2}-4 \ln \left (c \right ) b d e f g +2 i \pi b d e f g \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-i \pi b \,e^{2} f^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )-i \pi b \,d^{2} g^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )+2 g b d e f n +2 i \pi b d e f g \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )-2 i \pi b d e f g \,\operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-2 i \pi b d e f g \,\operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}-2 b \,e^{2} f^{2} n -4 \ln \left (-e x -d \right ) b \,e^{2} f g n x +4 \ln \left (g x +f \right ) b \,e^{2} f g n x -4 a d e f g +2 a \,d^{2} g^{2}+2 \ln \left (c \right ) b \,d^{2} g^{2}+2 \ln \left (c \right ) b \,e^{2} f^{2}+2 b d e \,g^{2} n x -2 b \,e^{2} f g n x -i \pi b \,d^{2} g^{2} \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}-i \pi b \,e^{2} f^{2} \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{3}+i \pi b \,e^{2} f^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,e^{2} f^{2} \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,d^{2} g^{2} \operatorname {csgn}\left (i c \right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}+i \pi b \,d^{2} g^{2} \operatorname {csgn}\left (i \left (e x +d \right )^{n}\right ) \operatorname {csgn}\left (i c \left (e x +d \right )^{n}\right )^{2}}{4 \left (g x +f \right )^{2} \left (d g -e f \right )^{2} g}\)	\(633\)

[In]

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

[Out]

1/2*(-x*b*d*e^2*g^3*n+x*b*e^3*f*g^2*n+b*e^3*f^2*g*n+2*a*d*e^2*f*g^2-ln(c*(e*x+d)^n)*b*d^2*e*g^3-ln(c*(e*x+d)^n
)*b*e^3*f^2*g-b*d*e^2*f*g^2*n+ln(e*x+d)*b*e^3*f^2*g*n-ln(g*x+f)*b*e^3*f^2*g*n+2*ln(c*(e*x+d)^n)*b*d*e^2*f*g^2+
ln(e*x+d)*x^2*b*e^3*g^3*n-ln(g*x+f)*x^2*b*e^3*g^3*n+2*ln(e*x+d)*x*b*e^3*f*g^2*n-2*ln(g*x+f)*x*b*e^3*f*g^2*n-a*
d^2*e*g^3-a*e^3*f^2*g)/(d^2*g^2-2*d*e*f*g+e^2*f^2)/(g*x+f)^2/e/g^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (104) = 208\).

Time = 0.30 (sec) , antiderivative size = 274, normalized size of antiderivative = 2.45 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^3} \, dx=-\frac {a e^{2} f^{2} - 2 \, a d e f g + a d^{2} g^{2} - {\left (b e^{2} f g - b d e g^{2}\right )} n x - {\left (b e^{2} f^{2} - b d e f g\right )} n - {\left (b e^{2} g^{2} n x^{2} + 2 \, b e^{2} f g n x + {\left (2 \, b d e f g - b d^{2} g^{2}\right )} n\right )} \log \left (e x + d\right ) + {\left (b e^{2} g^{2} n x^{2} + 2 \, b e^{2} f g n x + b e^{2} f^{2} n\right )} \log \left (g x + f\right ) + {\left (b e^{2} f^{2} - 2 \, b d e f g + b d^{2} g^{2}\right )} \log \left (c\right )}{2 \, {\left (e^{2} f^{4} g - 2 \, d e f^{3} g^{2} + d^{2} f^{2} g^{3} + {\left (e^{2} f^{2} g^{3} - 2 \, d e f g^{4} + d^{2} g^{5}\right )} x^{2} + 2 \, {\left (e^{2} f^{3} g^{2} - 2 \, d e f^{2} g^{3} + d^{2} f g^{4}\right )} x\right )}} \]

[In]

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

[Out]

-1/2*(a*e^2*f^2 - 2*a*d*e*f*g + a*d^2*g^2 - (b*e^2*f*g - b*d*e*g^2)*n*x - (b*e^2*f^2 - b*d*e*f*g)*n - (b*e^2*g
^2*n*x^2 + 2*b*e^2*f*g*n*x + (2*b*d*e*f*g - b*d^2*g^2)*n)*log(e*x + d) + (b*e^2*g^2*n*x^2 + 2*b*e^2*f*g*n*x +
b*e^2*f^2*n)*log(g*x + f) + (b*e^2*f^2 - 2*b*d*e*f*g + b*d^2*g^2)*log(c))/(e^2*f^4*g - 2*d*e*f^3*g^2 + d^2*f^2
*g^3 + (e^2*f^2*g^3 - 2*d*e*f*g^4 + d^2*g^5)*x^2 + 2*(e^2*f^3*g^2 - 2*d*e*f^2*g^3 + d^2*f*g^4)*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1945 vs. \(2 (97) = 194\).

Time = 13.16 (sec) , antiderivative size = 1945, normalized size of antiderivative = 17.37 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^3} \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise(((a*x + b*d*log(c*(d + e*x)**n)/e - b*n*x + b*x*log(c*(d + e*x)**n))/f**3, Eq(g, 0)), (-2*a/(4*f**2*
g + 8*f*g**2*x + 4*g**3*x**2) - b*n/(4*f**2*g + 8*f*g**2*x + 4*g**3*x**2) - 2*b*log(c*(e*f/g + e*x)**n)/(4*f**
2*g + 8*f*g**2*x + 4*g**3*x**2), Eq(d, e*f/g)), (-a*d**2*g**2/(2*d**2*f**2*g**3 + 4*d**2*f*g**4*x + 2*d**2*g**
5*x**2 - 4*d*e*f**3*g**2 - 8*d*e*f**2*g**3*x - 4*d*e*f*g**4*x**2 + 2*e**2*f**4*g + 4*e**2*f**3*g**2*x + 2*e**2
*f**2*g**3*x**2) + 2*a*d*e*f*g/(2*d**2*f**2*g**3 + 4*d**2*f*g**4*x + 2*d**2*g**5*x**2 - 4*d*e*f**3*g**2 - 8*d*
e*f**2*g**3*x - 4*d*e*f*g**4*x**2 + 2*e**2*f**4*g + 4*e**2*f**3*g**2*x + 2*e**2*f**2*g**3*x**2) - a*e**2*f**2/
(2*d**2*f**2*g**3 + 4*d**2*f*g**4*x + 2*d**2*g**5*x**2 - 4*d*e*f**3*g**2 - 8*d*e*f**2*g**3*x - 4*d*e*f*g**4*x*
*2 + 2*e**2*f**4*g + 4*e**2*f**3*g**2*x + 2*e**2*f**2*g**3*x**2) - b*d**2*g**2*log(c*(d + e*x)**n)/(2*d**2*f**
2*g**3 + 4*d**2*f*g**4*x + 2*d**2*g**5*x**2 - 4*d*e*f**3*g**2 - 8*d*e*f**2*g**3*x - 4*d*e*f*g**4*x**2 + 2*e**2
*f**4*g + 4*e**2*f**3*g**2*x + 2*e**2*f**2*g**3*x**2) - b*d*e*f*g*n/(2*d**2*f**2*g**3 + 4*d**2*f*g**4*x + 2*d*
*2*g**5*x**2 - 4*d*e*f**3*g**2 - 8*d*e*f**2*g**3*x - 4*d*e*f*g**4*x**2 + 2*e**2*f**4*g + 4*e**2*f**3*g**2*x +
2*e**2*f**2*g**3*x**2) + 2*b*d*e*f*g*log(c*(d + e*x)**n)/(2*d**2*f**2*g**3 + 4*d**2*f*g**4*x + 2*d**2*g**5*x**
2 - 4*d*e*f**3*g**2 - 8*d*e*f**2*g**3*x - 4*d*e*f*g**4*x**2 + 2*e**2*f**4*g + 4*e**2*f**3*g**2*x + 2*e**2*f**2
*g**3*x**2) - b*d*e*g**2*n*x/(2*d**2*f**2*g**3 + 4*d**2*f*g**4*x + 2*d**2*g**5*x**2 - 4*d*e*f**3*g**2 - 8*d*e*
f**2*g**3*x - 4*d*e*f*g**4*x**2 + 2*e**2*f**4*g + 4*e**2*f**3*g**2*x + 2*e**2*f**2*g**3*x**2) - b*e**2*f**2*n*
log(f/g + x)/(2*d**2*f**2*g**3 + 4*d**2*f*g**4*x + 2*d**2*g**5*x**2 - 4*d*e*f**3*g**2 - 8*d*e*f**2*g**3*x - 4*
d*e*f*g**4*x**2 + 2*e**2*f**4*g + 4*e**2*f**3*g**2*x + 2*e**2*f**2*g**3*x**2) + b*e**2*f**2*n/(2*d**2*f**2*g**
3 + 4*d**2*f*g**4*x + 2*d**2*g**5*x**2 - 4*d*e*f**3*g**2 - 8*d*e*f**2*g**3*x - 4*d*e*f*g**4*x**2 + 2*e**2*f**4
*g + 4*e**2*f**3*g**2*x + 2*e**2*f**2*g**3*x**2) - 2*b*e**2*f*g*n*x*log(f/g + x)/(2*d**2*f**2*g**3 + 4*d**2*f*
g**4*x + 2*d**2*g**5*x**2 - 4*d*e*f**3*g**2 - 8*d*e*f**2*g**3*x - 4*d*e*f*g**4*x**2 + 2*e**2*f**4*g + 4*e**2*f
**3*g**2*x + 2*e**2*f**2*g**3*x**2) + b*e**2*f*g*n*x/(2*d**2*f**2*g**3 + 4*d**2*f*g**4*x + 2*d**2*g**5*x**2 -
4*d*e*f**3*g**2 - 8*d*e*f**2*g**3*x - 4*d*e*f*g**4*x**2 + 2*e**2*f**4*g + 4*e**2*f**3*g**2*x + 2*e**2*f**2*g**
3*x**2) + 2*b*e**2*f*g*x*log(c*(d + e*x)**n)/(2*d**2*f**2*g**3 + 4*d**2*f*g**4*x + 2*d**2*g**5*x**2 - 4*d*e*f*
*3*g**2 - 8*d*e*f**2*g**3*x - 4*d*e*f*g**4*x**2 + 2*e**2*f**4*g + 4*e**2*f**3*g**2*x + 2*e**2*f**2*g**3*x**2)
- b*e**2*g**2*n*x**2*log(f/g + x)/(2*d**2*f**2*g**3 + 4*d**2*f*g**4*x + 2*d**2*g**5*x**2 - 4*d*e*f**3*g**2 - 8
*d*e*f**2*g**3*x - 4*d*e*f*g**4*x**2 + 2*e**2*f**4*g + 4*e**2*f**3*g**2*x + 2*e**2*f**2*g**3*x**2) + b*e**2*g*
*2*x**2*log(c*(d + e*x)**n)/(2*d**2*f**2*g**3 + 4*d**2*f*g**4*x + 2*d**2*g**5*x**2 - 4*d*e*f**3*g**2 - 8*d*e*f
**2*g**3*x - 4*d*e*f*g**4*x**2 + 2*e**2*f**4*g + 4*e**2*f**3*g**2*x + 2*e**2*f**2*g**3*x**2), True))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.49 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^3} \, dx=\frac {1}{2} \, b e n {\left (\frac {e \log \left (e x + d\right )}{e^{2} f^{2} g - 2 \, d e f g^{2} + d^{2} g^{3}} - \frac {e \log \left (g x + f\right )}{e^{2} f^{2} g - 2 \, d e f g^{2} + d^{2} g^{3}} + \frac {1}{e f^{2} g - d f g^{2} + {\left (e f g^{2} - d g^{3}\right )} x}\right )} - \frac {b \log \left ({\left (e x + d\right )}^{n} c\right )}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} - \frac {a}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} \]

[In]

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

[Out]

1/2*b*e*n*(e*log(e*x + d)/(e^2*f^2*g - 2*d*e*f*g^2 + d^2*g^3) - e*log(g*x + f)/(e^2*f^2*g - 2*d*e*f*g^2 + d^2*
g^3) + 1/(e*f^2*g - d*f*g^2 + (e*f*g^2 - d*g^3)*x)) - 1/2*b*log((e*x + d)^n*c)/(g^3*x^2 + 2*f*g^2*x + f^2*g) -
1/2*a/(g^3*x^2 + 2*f*g^2*x + f^2*g)

Giac [A] (verification not implemented)

none

Time = 0.33 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.79 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^3} \, dx=\frac {b e^{2} n \log \left (e x + d\right )}{2 \, {\left (e^{2} f^{2} g - 2 \, d e f g^{2} + d^{2} g^{3}\right )}} - \frac {b e^{2} n \log \left (g x + f\right )}{2 \, {\left (e^{2} f^{2} g - 2 \, d e f g^{2} + d^{2} g^{3}\right )}} - \frac {b n \log \left (e x + d\right )}{2 \, {\left (g^{3} x^{2} + 2 \, f g^{2} x + f^{2} g\right )}} + \frac {b e g n x + b e f n - b e f \log \left (c\right ) + b d g \log \left (c\right ) - a e f + a d g}{2 \, {\left (e f g^{3} x^{2} - d g^{4} x^{2} + 2 \, e f^{2} g^{2} x - 2 \, d f g^{3} x + e f^{3} g - d f^{2} g^{2}\right )}} \]

[In]

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

[Out]

1/2*b*e^2*n*log(e*x + d)/(e^2*f^2*g - 2*d*e*f*g^2 + d^2*g^3) - 1/2*b*e^2*n*log(g*x + f)/(e^2*f^2*g - 2*d*e*f*g
^2 + d^2*g^3) - 1/2*b*n*log(e*x + d)/(g^3*x^2 + 2*f*g^2*x + f^2*g) + 1/2*(b*e*g*n*x + b*e*f*n - b*e*f*log(c) +
b*d*g*log(c) - a*e*f + a*d*g)/(e*f*g^3*x^2 - d*g^4*x^2 + 2*e*f^2*g^2*x - 2*d*f*g^3*x + e*f^3*g - d*f^2*g^2)

Mupad [B] (verification not implemented)

Time = 1.02 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.54 \[ \int \frac {a+b \log \left (c (d+e x)^n\right )}{(f+g x)^3} \, dx=\frac {b\,e^2\,n\,\mathrm {atanh}\left (\frac {2\,d^2\,g^3-2\,e^2\,f^2\,g}{2\,g\,{\left (d\,g-e\,f\right )}^2}+\frac {2\,e\,g\,x}{d\,g-e\,f}\right )}{g\,{\left (d\,g-e\,f\right )}^2}-\frac {b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )}{2\,g\,\left (f^2+2\,f\,g\,x+g^2\,x^2\right )}-\frac {\frac {a\,d\,g-a\,e\,f+b\,e\,f\,n}{d\,g-e\,f}+\frac {b\,e\,g\,n\,x}{d\,g-e\,f}}{2\,f^2\,g+4\,f\,g^2\,x+2\,g^3\,x^2} \]

[In]

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

[Out]

(b*e^2*n*atanh((2*d^2*g^3 - 2*e^2*f^2*g)/(2*g*(d*g - e*f)^2) + (2*e*g*x)/(d*g - e*f)))/(g*(d*g - e*f)^2) - (b*
log(c*(d + e*x)^n))/(2*g*(f^2 + g^2*x^2 + 2*f*g*x)) - ((a*d*g - a*e*f + b*e*f*n)/(d*g - e*f) + (b*e*g*n*x)/(d*
g - e*f))/(2*f^2*g + 2*g^3*x^2 + 4*f*g^2*x)