\(\int \frac {\log (c (a+b x)^p)}{(d+e x)^3} \, dx\) [182]

Optimal result

Integrand size = 18, antiderivative size = 105 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^3} \, dx=\frac {b p}{2 e (b d-a e) (d+e x)}+\frac {b^2 p \log (a+b x)}{2 e (b d-a e)^2}-\frac {\log \left (c (a+b x)^p\right )}{2 e (d+e x)^2}-\frac {b^2 p \log (d+e x)}{2 e (b d-a e)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.04 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2442, 46} \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^3} \, dx=\frac {b^2 p \log (a+b x)}{2 e (b d-a e)^2}-\frac {b^2 p \log (d+e x)}{2 e (b d-a e)^2}-\frac {\log \left (c (a+b x)^p\right )}{2 e (d+e x)^2}+\frac {b p}{2 e (d+e x) (b d-a e)} \]

[In]

Int[Log[c*(a + b*x)^p]/(d + e*x)^3,x]

[Out]

(b*p)/(2*e*(b*d - a*e)*(d + e*x)) + (b^2*p*Log[a + b*x])/(2*e*(b*d - a*e)^2) - Log[c*(a + b*x)^p]/(2*e*(d + e*
x)^2) - (b^2*p*Log[d + e*x])/(2*e*(b*d - a*e)^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 {\log \left (c (a+b x)^p\right )}{2 e (d+e x)^2}+\frac {(b p) \int \frac {1}{(a+b x) (d+e x)^2} \, dx}{2 e} \\ & = -\frac {\log \left (c (a+b x)^p\right )}{2 e (d+e x)^2}+\frac {(b p) \int \left (\frac {b^2}{(b d-a e)^2 (a+b x)}-\frac {e}{(b d-a e) (d+e x)^2}-\frac {b e}{(b d-a e)^2 (d+e x)}\right ) \, dx}{2 e} \\ & = \frac {b p}{2 e (b d-a e) (d+e x)}+\frac {b^2 p \log (a+b x)}{2 e (b d-a e)^2}-\frac {\log \left (c (a+b x)^p\right )}{2 e (d+e x)^2}-\frac {b^2 p \log (d+e x)}{2 e (b d-a e)^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.05 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.76 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^3} \, dx=\frac {-\log \left (c (a+b x)^p\right )+\frac {b p (d+e x) (b d-a e+b (d+e x) \log (a+b x)-b (d+e x) \log (d+e x))}{(b d-a e)^2}}{2 e (d+e x)^2} \]

[In]

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

[Out]

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

Maple [A] (verified)

Time = 1.23 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.84

method	result	size
parts	\(-\frac {\ln \left (c \left (b x +a \right )^{p}\right )}{2 e \left (e x +d \right )^{2}}+\frac {p b \left (-\frac {1}{\left (a e -b d \right ) \left (e x +d \right )}-\frac {b \ln \left (e x +d \right )}{\left (a e -b d \right )^{2}}+\frac {b \ln \left (b x +a \right )}{\left (a e -b d \right )^{2}}\right )}{2 e}\)	\(88\)
parallelrisch	\(-\frac {2 \ln \left (e x +d \right ) x \,b^{3} d \,e^{2} p -2 \ln \left (b x +a \right ) x \,b^{3} d \,e^{2} p +a \,b^{2} d \,e^{2} p +x a \,b^{2} e^{3} p -x \,b^{3} d \,e^{2} p -2 \ln \left (c \left (b x +a \right )^{p}\right ) a \,b^{2} d \,e^{2}-\ln \left (b x +a \right ) x^{2} b^{3} e^{3} p +\ln \left (e x +d \right ) x^{2} b^{3} e^{3} p -\ln \left (b x +a \right ) b^{3} d^{2} e p +\ln \left (e x +d \right ) b^{3} d^{2} e p -b^{3} d^{2} e p +\ln \left (c \left (b x +a \right )^{p}\right ) a^{2} b \,e^{3}+\ln \left (c \left (b x +a \right )^{p}\right ) b^{3} d^{2} e}{2 \left (a^{2} e^{2}-2 a d e b +b^{2} d^{2}\right ) \left (e x +d \right )^{2} e^{2} b}\)	\(237\)
risch	\(-\frac {\ln \left (\left (b x +a \right )^{p}\right )}{2 e \left (e x +d \right )^{2}}-\frac {2 i \pi a b d e \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )-4 \ln \left (-b x -a \right ) b^{2} d e p x +4 \ln \left (e x +d \right ) b^{2} d e p x -i \pi \,b^{2} d^{2} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}+2 a b d p e -2 b^{2} d^{2} p +2 b^{2} d^{2} \ln \left (c \right )+2 \ln \left (c \right ) a^{2} e^{2}-i \pi \,a^{2} e^{2} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}+2 a b \,e^{2} p x -2 b^{2} d e p x -2 i \pi a b d e \,\operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}-2 i \pi a b d e \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )+i \pi \,a^{2} e^{2} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )+i \pi \,b^{2} d^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}-2 \ln \left (-b x -a \right ) b^{2} d^{2} p +2 \ln \left (e x +d \right ) b^{2} d^{2} p -2 \ln \left (-b x -a \right ) b^{2} e^{2} p \,x^{2}+2 \ln \left (e x +d \right ) b^{2} e^{2} p \,x^{2}-4 \ln \left (c \right ) a b d e +i \pi \,b^{2} d^{2} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )+i \pi \,a^{2} e^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2}-i \pi \,a^{2} e^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+2 i \pi a b d e \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-i \pi \,b^{2} d^{2} \operatorname {csgn}\left (i \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{4 \left (e x +d \right )^{2} \left (a e -b d \right )^{2} e}\)	\(582\)

[In]

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

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 236 vs. \(2 (97) = 194\).

Time = 0.32 (sec) , antiderivative size = 236, normalized size of antiderivative = 2.25 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^3} \, dx=\frac {{\left (b^{2} d e - a b e^{2}\right )} p x + {\left (b^{2} d^{2} - a b d e\right )} p + {\left (b^{2} e^{2} p x^{2} + 2 \, b^{2} d e p x + {\left (2 \, a b d e - a^{2} e^{2}\right )} p\right )} \log \left (b x + a\right ) - {\left (b^{2} e^{2} p x^{2} + 2 \, b^{2} d e p x + b^{2} d^{2} p\right )} \log \left (e x + d\right ) - {\left (b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}\right )} \log \left (c\right )}{2 \, {\left (b^{2} d^{4} e - 2 \, a b d^{3} e^{2} + a^{2} d^{2} e^{3} + {\left (b^{2} d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e^{2} - 2 \, a b d^{2} e^{3} + a^{2} d e^{4}\right )} x\right )}} \]

[In]

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

[Out]

1/2*((b^2*d*e - a*b*e^2)*p*x + (b^2*d^2 - a*b*d*e)*p + (b^2*e^2*p*x^2 + 2*b^2*d*e*p*x + (2*a*b*d*e - a^2*e^2)*
p)*log(b*x + a) - (b^2*e^2*p*x^2 + 2*b^2*d*e*p*x + b^2*d^2*p)*log(e*x + d) - (b^2*d^2 - 2*a*b*d*e + a^2*e^2)*l
og(c))/(b^2*d^4*e - 2*a*b*d^3*e^2 + a^2*d^2*e^3 + (b^2*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)*x^2 + 2*(b^2*d^3*e^2 -
2*a*b*d^2*e^3 + a^2*d*e^4)*x)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1518 vs. \(2 (85) = 170\).

Time = 5.25 (sec) , antiderivative size = 1518, normalized size of antiderivative = 14.46 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^3} \, dx=\text {Too large to display} \]

[In]

integrate(ln(c*(b*x+a)**p)/(e*x+d)**3,x)

[Out]

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

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.14 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^3} \, dx=\frac {b p {\left (\frac {b \log \left (b x + a\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} - \frac {b \log \left (e x + d\right )}{b^{2} d^{2} - 2 \, a b d e + a^{2} e^{2}} + \frac {1}{b d^{2} - a d e + {\left (b d e - a e^{2}\right )} x}\right )}}{2 \, e} - \frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{2 \, {\left (e x + d\right )}^{2} e} \]

[In]

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

[Out]

1/2*b*p*(b*log(b*x + a)/(b^2*d^2 - 2*a*b*d*e + a^2*e^2) - b*log(e*x + d)/(b^2*d^2 - 2*a*b*d*e + a^2*e^2) + 1/(
b*d^2 - a*d*e + (b*d*e - a*e^2)*x))/e - 1/2*log((b*x + a)^p*c)/((e*x + d)^2*e)

Giac [A] (verification not implemented)

none

Time = 0.31 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.76 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^3} \, dx=\frac {b^{2} p \log \left (b x + a\right )}{2 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )}} - \frac {b^{2} p \log \left (e x + d\right )}{2 \, {\left (b^{2} d^{2} e - 2 \, a b d e^{2} + a^{2} e^{3}\right )}} - \frac {p \log \left (b x + a\right )}{2 \, {\left (e^{3} x^{2} + 2 \, d e^{2} x + d^{2} e\right )}} + \frac {b e p x + b d p - b d \log \left (c\right ) + a e \log \left (c\right )}{2 \, {\left (b d e^{3} x^{2} - a e^{4} x^{2} + 2 \, b d^{2} e^{2} x - 2 \, a d e^{3} x + b d^{3} e - a d^{2} e^{2}\right )}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 1.64 (sec) , antiderivative size = 96, normalized size of antiderivative = 0.91 \[ \int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^3} \, dx=-\frac {\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{2\,e\,{\left (d+e\,x\right )}^2}-\frac {b\,p}{2\,e\,\left (a\,e-b\,d\right )\,\left (d+e\,x\right )}-\frac {b^2\,p\,\mathrm {atan}\left (\frac {a\,e\,1{}\mathrm {i}+b\,d\,1{}\mathrm {i}+b\,e\,x\,2{}\mathrm {i}}{a\,e-b\,d}\right )\,1{}\mathrm {i}}{e\,{\left (a\,e-b\,d\right )}^2} \]

[In]

int(log(c*(a + b*x)^p)/(d + e*x)^3,x)

[Out]

- log(c*(a + b*x)^p)/(2*e*(d + e*x)^2) - (b*p)/(2*e*(a*e - b*d)*(d + e*x)) - (b^2*p*atan((a*e*1i + b*d*1i + b*
e*x*2i)/(a*e - b*d))*1i)/(e*(a*e - b*d)^2)