\(\int \frac {\log (c (a+b x)^p)}{(d+e x)^4} \, dx\) [183]
Optimal result
Integrand size = 18, antiderivative size = 133 \[
\int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {b p}{6 e (b d-a e) (d+e x)^2}+\frac {b^2 p}{3 e (b d-a e)^2 (d+e x)}+\frac {b^3 p \log (a+b x)}{3 e (b d-a e)^3}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}-\frac {b^3 p \log (d+e x)}{3 e (b d-a e)^3}
\]
[Out]
1/6*b*p/e/(-a*e+b*d)/(e*x+d)^2+1/3*b^2*p/e/(-a*e+b*d)^2/(e*x+d)+1/3*b^3*p*ln(b*x+a)/e/(-a*e+b*d)^3-1/3*ln(c*(b
*x+a)^p)/e/(e*x+d)^3-1/3*b^3*p*ln(e*x+d)/e/(-a*e+b*d)^3
Rubi [A] (verified)
Time = 0.05 (sec) , antiderivative size = 133, 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}
\[
\int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {b^3 p \log (a+b x)}{3 e (b d-a e)^3}-\frac {b^3 p \log (d+e x)}{3 e (b d-a e)^3}+\frac {b^2 p}{3 e (d+e x) (b d-a e)^2}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}+\frac {b p}{6 e (d+e x)^2 (b d-a e)}
\]
[In]
Int[Log[c*(a + b*x)^p]/(d + e*x)^4,x]
[Out]
(b*p)/(6*e*(b*d - a*e)*(d + e*x)^2) + (b^2*p)/(3*e*(b*d - a*e)^2*(d + e*x)) + (b^3*p*Log[a + b*x])/(3*e*(b*d -
a*e)^3) - Log[c*(a + b*x)^p]/(3*e*(d + e*x)^3) - (b^3*p*Log[d + e*x])/(3*e*(b*d - a*e)^3)
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 )}{3 e (d+e x)^3}+\frac {(b p) \int \frac {1}{(a+b x) (d+e x)^3} \, dx}{3 e} \\ & = -\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}+\frac {(b p) \int \left (\frac {b^3}{(b d-a e)^3 (a+b x)}-\frac {e}{(b d-a e) (d+e x)^3}-\frac {b e}{(b d-a e)^2 (d+e x)^2}-\frac {b^2 e}{(b d-a e)^3 (d+e x)}\right ) \, dx}{3 e} \\ & = \frac {b p}{6 e (b d-a e) (d+e x)^2}+\frac {b^2 p}{3 e (b d-a e)^2 (d+e x)}+\frac {b^3 p \log (a+b x)}{3 e (b d-a e)^3}-\frac {\log \left (c (a+b x)^p\right )}{3 e (d+e x)^3}-\frac {b^3 p \log (d+e x)}{3 e (b d-a e)^3} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.08 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.79
\[
\int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {-2 \log \left (c (a+b x)^p\right )+\frac {b p (d+e x) \left ((b d-a e) (3 b d-a e+2 b e x)+2 b^2 (d+e x)^2 \log (a+b x)-2 b^2 (d+e x)^2 \log (d+e x)\right )}{(b d-a e)^3}}{6 e (d+e x)^3}
\]
[In]
Integrate[Log[c*(a + b*x)^p]/(d + e*x)^4,x]
[Out]
(-2*Log[c*(a + b*x)^p] + (b*p*(d + e*x)*((b*d - a*e)*(3*b*d - a*e + 2*b*e*x) + 2*b^2*(d + e*x)^2*Log[a + b*x]
- 2*b^2*(d + e*x)^2*Log[d + e*x]))/(b*d - a*e)^3)/(6*e*(d + e*x)^3)
Maple [A] (verified)
Time = 1.78 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.83
| | |
| method | result | size |
| | |
| parts |
\(-\frac {\ln \left (c \left (b x +a \right )^{p}\right )}{3 e \left (e x +d \right )^{3}}+\frac {p b \left (-\frac {1}{2 \left (a e -b d \right ) \left (e x +d \right )^{2}}+\frac {b^{2} \ln \left (e x +d \right )}{\left (a e -b d \right )^{3}}+\frac {b}{\left (a e -b d \right )^{2} \left (e x +d \right )}-\frac {b^{2} \ln \left (b x +a \right )}{\left (a e -b d \right )^{3}}\right )}{3 e}\) |
\(111\) |
| parallelrisch |
\(-\frac {-6 x a \,b^{3} d \,e^{4} p +6 \ln \left (b x +a \right ) x^{2} b^{4} d \,e^{4} p -6 \ln \left (e x +d \right ) x^{2} b^{4} d \,e^{4} p +6 \ln \left (b x +a \right ) x \,b^{4} d^{2} e^{3} p -6 \ln \left (e x +d \right ) x \,b^{4} d^{2} e^{3} p +2 \ln \left (b x +a \right ) x^{3} b^{4} e^{5} p -2 \ln \left (e x +d \right ) x^{3} b^{4} e^{5} p +2 \ln \left (b x +a \right ) b^{4} d^{3} e^{2} p -2 \ln \left (e x +d \right ) b^{4} d^{3} e^{2} p -2 x^{2} a \,b^{3} e^{5} p +2 x^{2} b^{4} d \,e^{4} p +x \,a^{2} b^{2} e^{5} p +5 x \,b^{4} d^{2} e^{3} p -6 \ln \left (c \left (b x +a \right )^{p}\right ) a^{2} b^{2} d \,e^{4}+6 \ln \left (c \left (b x +a \right )^{p}\right ) a \,b^{3} d^{2} e^{3}-4 a \,b^{3} d^{2} e^{3} p +a^{2} b^{2} d \,e^{4} p +3 b^{4} d^{3} e^{2} p +2 \ln \left (c \left (b x +a \right )^{p}\right ) a^{3} b \,e^{5}-2 \ln \left (c \left (b x +a \right )^{p}\right ) b^{4} d^{3} e^{2}}{6 \left (a^{3} e^{3}-3 a^{2} b d \,e^{2}+3 a \,b^{2} d^{2} e -b^{3} d^{3}\right ) \left (e x +d \right )^{3} e^{3} b}\) |
\(387\) |
| risch |
\(-\frac {\ln \left (\left (b x +a \right )^{p}\right )}{3 e \left (e x +d \right )^{3}}+\frac {-6 \ln \left (b x +a \right ) b^{3} d \,e^{2} p \,x^{2}+6 \ln \left (-e x -d \right ) b^{3} d \,e^{2} p \,x^{2}-6 \ln \left (b x +a \right ) b^{3} d^{2} e p x +6 \ln \left (-e x -d \right ) b^{3} d^{2} e p x -3 b^{3} d^{3} p -2 \ln \left (b x +a \right ) b^{3} d^{3} p +2 a \,b^{2} e^{3} p \,x^{2}-2 \ln \left (b x +a \right ) b^{3} e^{3} p \,x^{3}+2 \ln \left (-e x -d \right ) b^{3} e^{3} p \,x^{3}-a^{2} b d p \,e^{2}+4 a \,b^{2} d^{2} p e +3 i \pi \,a^{2} b d \,e^{2} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )-3 i \pi a \,b^{2} d^{2} 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}+3 i \pi \,a^{2} b d \,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}-3 i \pi a \,b^{2} d^{2} e \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )+i \pi \,a^{3} e^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-i \pi \,b^{3} d^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-2 b^{3} d \,e^{2} p \,x^{2}-a^{2} b \,e^{3} p x -5 b^{3} d^{2} e p x -2 \ln \left (c \right ) a^{3} e^{3}+2 \ln \left (c \right ) b^{3} d^{3}+6 a \,b^{2} d \,e^{2} p x +2 \ln \left (-e x -d \right ) b^{3} d^{3} p -3 i \pi \,a^{2} b d \,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 )+3 i \pi a \,b^{2} d^{2} 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 )+6 \ln \left (c \right ) a^{2} b d \,e^{2}-6 \ln \left (c \right ) a \,b^{2} d^{2} e +i \pi \,a^{3} e^{3} \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 )-3 i \pi \,a^{2} b d \,e^{2} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}+3 i \pi a \,b^{2} d^{2} e \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{3}-i \pi \,b^{3} d^{3} \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 )-i \pi \,a^{3} e^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )-i \pi \,a^{3} e^{3} \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 \,b^{3} d^{3} \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 \,b^{3} d^{3} \operatorname {csgn}\left (i c \left (b x +a \right )^{p}\right )^{2} \operatorname {csgn}\left (i c \right )}{6 \left (e x +d \right )^{3} \left (a^{2} e^{2}-2 a d e b +b^{2} d^{2}\right ) \left (a e -b d \right ) e}\) |
\(873\) |
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[In]
int(ln(c*(b*x+a)^p)/(e*x+d)^4,x,method=_RETURNVERBOSE)
[Out]
-1/3*ln(c*(b*x+a)^p)/e/(e*x+d)^3+1/3*p*b/e*(-1/2/(a*e-b*d)/(e*x+d)^2+b^2/(a*e-b*d)^3*ln(e*x+d)+b/(a*e-b*d)^2/(
e*x+d)-b^2/(a*e-b*d)^3*ln(b*x+a))
Fricas [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 443 vs. \(2 (123) = 246\).
Time = 0.39 (sec) , antiderivative size = 443, normalized size of antiderivative = 3.33
\[
\int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {2 \, {\left (b^{3} d e^{2} - a b^{2} e^{3}\right )} p x^{2} + {\left (5 \, b^{3} d^{2} e - 6 \, a b^{2} d e^{2} + a^{2} b e^{3}\right )} p x + {\left (3 \, b^{3} d^{3} - 4 \, a b^{2} d^{2} e + a^{2} b d e^{2}\right )} p + 2 \, {\left (b^{3} e^{3} p x^{3} + 3 \, b^{3} d e^{2} p x^{2} + 3 \, b^{3} d^{2} e p x + {\left (3 \, a b^{2} d^{2} e - 3 \, a^{2} b d e^{2} + a^{3} e^{3}\right )} p\right )} \log \left (b x + a\right ) - 2 \, {\left (b^{3} e^{3} p x^{3} + 3 \, b^{3} d e^{2} p x^{2} + 3 \, b^{3} d^{2} e p x + b^{3} d^{3} p\right )} \log \left (e x + d\right ) - 2 \, {\left (b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}\right )} \log \left (c\right )}{6 \, {\left (b^{3} d^{6} e - 3 \, a b^{2} d^{5} e^{2} + 3 \, a^{2} b d^{4} e^{3} - a^{3} d^{3} e^{4} + {\left (b^{3} d^{3} e^{4} - 3 \, a b^{2} d^{2} e^{5} + 3 \, a^{2} b d e^{6} - a^{3} e^{7}\right )} x^{3} + 3 \, {\left (b^{3} d^{4} e^{3} - 3 \, a b^{2} d^{3} e^{4} + 3 \, a^{2} b d^{2} e^{5} - a^{3} d e^{6}\right )} x^{2} + 3 \, {\left (b^{3} d^{5} e^{2} - 3 \, a b^{2} d^{4} e^{3} + 3 \, a^{2} b d^{3} e^{4} - a^{3} d^{2} e^{5}\right )} x\right )}}
\]
[In]
integrate(log(c*(b*x+a)^p)/(e*x+d)^4,x, algorithm="fricas")
[Out]
1/6*(2*(b^3*d*e^2 - a*b^2*e^3)*p*x^2 + (5*b^3*d^2*e - 6*a*b^2*d*e^2 + a^2*b*e^3)*p*x + (3*b^3*d^3 - 4*a*b^2*d^
2*e + a^2*b*d*e^2)*p + 2*(b^3*e^3*p*x^3 + 3*b^3*d*e^2*p*x^2 + 3*b^3*d^2*e*p*x + (3*a*b^2*d^2*e - 3*a^2*b*d*e^2
+ a^3*e^3)*p)*log(b*x + a) - 2*(b^3*e^3*p*x^3 + 3*b^3*d*e^2*p*x^2 + 3*b^3*d^2*e*p*x + b^3*d^3*p)*log(e*x + d)
- 2*(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3)*log(c))/(b^3*d^6*e - 3*a*b^2*d^5*e^2 + 3*a^2*b*d^4*e^
3 - a^3*d^3*e^4 + (b^3*d^3*e^4 - 3*a*b^2*d^2*e^5 + 3*a^2*b*d*e^6 - a^3*e^7)*x^3 + 3*(b^3*d^4*e^3 - 3*a*b^2*d^3
*e^4 + 3*a^2*b*d^2*e^5 - a^3*d*e^6)*x^2 + 3*(b^3*d^5*e^2 - 3*a*b^2*d^4*e^3 + 3*a^2*b*d^3*e^4 - a^3*d^2*e^5)*x)
Sympy [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 4571 vs. \(2 (109) = 218\).
Time = 17.52 (sec) , antiderivative size = 4571, normalized size of antiderivative = 34.37
\[
\int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\text {Too large to display}
\]
[In]
integrate(ln(c*(b*x+a)**p)/(e*x+d)**4,x)
[Out]
Piecewise(((a*log(c*(a + b*x)**p)/b - p*x + x*log(c*(a + b*x)**p))/d**4, Eq(e, 0)), (-p/(9*d**3*e + 27*d**2*e*
*2*x + 27*d*e**3*x**2 + 9*e**4*x**3) - 3*log(c*(b*d/e + b*x)**p)/(9*d**3*e + 27*d**2*e**2*x + 27*d*e**3*x**2 +
9*e**4*x**3), Eq(a, b*d/e)), (-2*a**3*e**3*log(c*(a + b*x)**p)/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a
**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 -
18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**
2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) - a**2*b*d
*e**2*p/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3
- 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*
d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b
**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) + 6*a**2*b*d*e**2*log(c*(a + b*x)**p)/(6*a**3*d**3*e**4 + 18*a**3*
d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b
*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x*
*2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**
4*x**3) - a**2*b*e**3*p*x/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 1
8*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5
*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*
d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) + 4*a*b**2*d**2*e*p/(6*a**3*d**3*e**4 + 18*a**3*
d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b
*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x*
*2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**
4*x**3) - 6*a*b**2*d**2*e*log(c*(a + b*x)**p)/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 +
6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x
**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3 - 6*b*
*3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) + 6*a*b**2*d*e**2*p*x/(6*a**
3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d
**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x +
54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*
x**2 - 6*b**3*d**3*e**4*x**3) + 2*a*b**2*e**3*p*x**2/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*
x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a**2*b*d
*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3
- 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) + 2*b**3*d**3*p*log(d
/e + x)/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3
- 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*
d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b
**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) - 3*b**3*d**3*p/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*
d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a
**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e*
*5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) + 6*b**3*d**2*
e*p*x*log(d/e + x)/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*
b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 +
54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e*
*2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) - 5*b**3*d**2*e*p*x/(6*a**3*d**3*e**4 + 18*a**3*d**2*e*
*5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e
**5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18
*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3)
- 6*b**3*d**2*e*x*log(c*(a + b*x)**p)/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*
e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x**3 + 1
8*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6
*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) + 6*b**3*d*e**2*p*x**2*log(d/e + x)
/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a
**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e*
*3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**
4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) - 2*b**3*d*e**2*p*x**2/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*
d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a
**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e*
*5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) - 6*b**3*d*e**
2*x**2*log(c*(a + b*x)**p)/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 -
18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**
5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3
*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3) + 2*b**3*e**3*p*x**3*log(d/e + x)/(6*a**3*d**3*
e**4 + 18*a**3*d**2*e**5*x + 18*a**3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**
4*x - 54*a**2*b*d**2*e**5*x**2 - 18*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b*
*2*d**3*e**4*x**2 + 18*a*b**2*d**2*e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 -
6*b**3*d**3*e**4*x**3) - 2*b**3*e**3*x**3*log(c*(a + b*x)**p)/(6*a**3*d**3*e**4 + 18*a**3*d**2*e**5*x + 18*a**
3*d*e**6*x**2 + 6*a**3*e**7*x**3 - 18*a**2*b*d**4*e**3 - 54*a**2*b*d**3*e**4*x - 54*a**2*b*d**2*e**5*x**2 - 18
*a**2*b*d*e**6*x**3 + 18*a*b**2*d**5*e**2 + 54*a*b**2*d**4*e**3*x + 54*a*b**2*d**3*e**4*x**2 + 18*a*b**2*d**2*
e**5*x**3 - 6*b**3*d**6*e - 18*b**3*d**5*e**2*x - 18*b**3*d**4*e**3*x**2 - 6*b**3*d**3*e**4*x**3), True))
Maxima [A] (verification not implemented)
none
Time = 0.21 (sec) , antiderivative size = 232, normalized size of antiderivative = 1.74
\[
\int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {{\left (\frac {2 \, b^{2} \log \left (b x + a\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} - \frac {2 \, b^{2} \log \left (e x + d\right )}{b^{3} d^{3} - 3 \, a b^{2} d^{2} e + 3 \, a^{2} b d e^{2} - a^{3} e^{3}} + \frac {2 \, b e x + 3 \, b d - a e}{b^{2} d^{4} - 2 \, a b d^{3} e + a^{2} d^{2} e^{2} + {\left (b^{2} d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )} x^{2} + 2 \, {\left (b^{2} d^{3} e - 2 \, a b d^{2} e^{2} + a^{2} d e^{3}\right )} x}\right )} b p}{6 \, e} - \frac {\log \left ({\left (b x + a\right )}^{p} c\right )}{3 \, {\left (e x + d\right )}^{3} e}
\]
[In]
integrate(log(c*(b*x+a)^p)/(e*x+d)^4,x, algorithm="maxima")
[Out]
1/6*(2*b^2*log(b*x + a)/(b^3*d^3 - 3*a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3) - 2*b^2*log(e*x + d)/(b^3*d^3 - 3*
a*b^2*d^2*e + 3*a^2*b*d*e^2 - a^3*e^3) + (2*b*e*x + 3*b*d - a*e)/(b^2*d^4 - 2*a*b*d^3*e + a^2*d^2*e^2 + (b^2*d
^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)*x^2 + 2*(b^2*d^3*e - 2*a*b*d^2*e^2 + a^2*d*e^3)*x))*b*p/e - 1/3*log((b*x + a)^
p*c)/((e*x + d)^3*e)
Giac [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 365 vs. \(2 (123) = 246\).
Time = 0.32 (sec) , antiderivative size = 365, normalized size of antiderivative = 2.74
\[
\int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {b^{3} p \log \left (b x + a\right )}{3 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )}} - \frac {b^{3} p \log \left (e x + d\right )}{3 \, {\left (b^{3} d^{3} e - 3 \, a b^{2} d^{2} e^{2} + 3 \, a^{2} b d e^{3} - a^{3} e^{4}\right )}} - \frac {p \log \left (b x + a\right )}{3 \, {\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}} + \frac {2 \, b^{2} e^{2} p x^{2} + 5 \, b^{2} d e p x - a b e^{2} p x + 3 \, b^{2} d^{2} p - a b d e p - 2 \, b^{2} d^{2} \log \left (c\right ) + 4 \, a b d e \log \left (c\right ) - 2 \, a^{2} e^{2} \log \left (c\right )}{6 \, {\left (b^{2} d^{2} e^{4} x^{3} - 2 \, a b d e^{5} x^{3} + a^{2} e^{6} x^{3} + 3 \, b^{2} d^{3} e^{3} x^{2} - 6 \, a b d^{2} e^{4} x^{2} + 3 \, a^{2} d e^{5} x^{2} + 3 \, b^{2} d^{4} e^{2} x - 6 \, a b d^{3} e^{3} x + 3 \, a^{2} d^{2} e^{4} x + b^{2} d^{5} e - 2 \, a b d^{4} e^{2} + a^{2} d^{3} e^{3}\right )}}
\]
[In]
integrate(log(c*(b*x+a)^p)/(e*x+d)^4,x, algorithm="giac")
[Out]
1/3*b^3*p*log(b*x + a)/(b^3*d^3*e - 3*a*b^2*d^2*e^2 + 3*a^2*b*d*e^3 - a^3*e^4) - 1/3*b^3*p*log(e*x + d)/(b^3*d
^3*e - 3*a*b^2*d^2*e^2 + 3*a^2*b*d*e^3 - a^3*e^4) - 1/3*p*log(b*x + a)/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x +
d^3*e) + 1/6*(2*b^2*e^2*p*x^2 + 5*b^2*d*e*p*x - a*b*e^2*p*x + 3*b^2*d^2*p - a*b*d*e*p - 2*b^2*d^2*log(c) + 4*a
*b*d*e*log(c) - 2*a^2*e^2*log(c))/(b^2*d^2*e^4*x^3 - 2*a*b*d*e^5*x^3 + a^2*e^6*x^3 + 3*b^2*d^3*e^3*x^2 - 6*a*b
*d^2*e^4*x^2 + 3*a^2*d*e^5*x^2 + 3*b^2*d^4*e^2*x - 6*a*b*d^3*e^3*x + 3*a^2*d^2*e^4*x + b^2*d^5*e - 2*a*b*d^4*e
^2 + a^2*d^3*e^3)
Mupad [B] (verification not implemented)
Time = 1.86 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.09
\[
\int \frac {\log \left (c (a+b x)^p\right )}{(d+e x)^4} \, dx=\frac {b^2\,p\,x}{3\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^2}-\frac {\ln \left (c\,{\left (a+b\,x\right )}^p\right )}{3\,e\,{\left (d+e\,x\right )}^3}-\frac {a\,b\,p}{6\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^2}+\frac {b^2\,d\,p}{2\,e\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^2}+\frac {b^3\,p\,\mathrm {atan}\left (\frac {a\,e\,1{}\mathrm {i}+b\,d\,1{}\mathrm {i}+b\,e\,x\,2{}\mathrm {i}}{a\,e-b\,d}\right )\,2{}\mathrm {i}}{3\,e\,{\left (a\,e-b\,d\right )}^3}
\]
[In]
int(log(c*(a + b*x)^p)/(d + e*x)^4,x)
[Out]
(b^2*p*x)/(3*(a*e - b*d)^2*(d + e*x)^2) - log(c*(a + b*x)^p)/(3*e*(d + e*x)^3) + (b^3*p*atan((a*e*1i + b*d*1i
+ b*e*x*2i)/(a*e - b*d))*2i)/(3*e*(a*e - b*d)^3) - (a*b*p)/(6*(a*e - b*d)^2*(d + e*x)^2) + (b^2*d*p)/(2*e*(a*e
- b*d)^2*(d + e*x)^2)