3.3.4 \(\int \frac {\log (c (a+\frac {b}{x})^p)}{(d+e x)^4} \, dx\) [204]

Optimal. Leaf size=175 \[ \frac {b p}{6 d (a d-b e) (d+e x)^2}+\frac {b (2 a d-b e) p}{3 d^2 (a d-b e)^2 (d+e x)}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e (d+e x)^3}-\frac {p \log (x)}{3 d^3 e}+\frac {a^3 p \log (b+a x)}{3 e (a d-b e)^3}-\frac {b \left (3 a^2 d^2-3 a b d e+b^2 e^2\right ) p \log (d+e x)}{3 d^3 (a d-b e)^3} \]

[Out]

1/6*b*p/d/(a*d-b*e)/(e*x+d)^2+1/3*b*(2*a*d-b*e)*p/d^2/(a*d-b*e)^2/(e*x+d)-1/3*ln(c*(a+b/x)^p)/e/(e*x+d)^3-1/3*
p*ln(x)/d^3/e+1/3*a^3*p*ln(a*x+b)/e/(a*d-b*e)^3-1/3*b*(3*a^2*d^2-3*a*b*d*e+b^2*e^2)*p*ln(e*x+d)/d^3/(a*d-b*e)^
3

________________________________________________________________________________________

Rubi [A]
time = 0.12, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {2513, 528, 84} \begin {gather*} \frac {a^3 p \log (a x+b)}{3 e (a d-b e)^3}-\frac {b p \left (3 a^2 d^2-3 a b d e+b^2 e^2\right ) \log (d+e x)}{3 d^3 (a d-b e)^3}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e (d+e x)^3}+\frac {b p (2 a d-b e)}{3 d^2 (d+e x) (a d-b e)^2}+\frac {b p}{6 d (d+e x)^2 (a d-b e)}-\frac {p \log (x)}{3 d^3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b/x)^p]/(d + e*x)^4,x]

[Out]

(b*p)/(6*d*(a*d - b*e)*(d + e*x)^2) + (b*(2*a*d - b*e)*p)/(3*d^2*(a*d - b*e)^2*(d + e*x)) - Log[c*(a + b/x)^p]
/(3*e*(d + e*x)^3) - (p*Log[x])/(3*d^3*e) + (a^3*p*Log[b + a*x])/(3*e*(a*d - b*e)^3) - (b*(3*a^2*d^2 - 3*a*b*d
*e + b^2*e^2)*p*Log[d + e*x])/(3*d^3*(a*d - b*e)^3)

Rule 84

Int[((e_.) + (f_.)*(x_))^(p_.)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Int[ExpandIntegrand[(
e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && IntegerQ[p]

Rule 528

Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[x^(m - n*q)*
(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ[{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] |
|  !IntegerQ[p])

Rule 2513

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

Rubi steps

\begin {align*} \int \frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^4} \, dx &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e (d+e x)^3}-\frac {(b p) \int \frac {1}{\left (a+\frac {b}{x}\right ) x^2 (d+e x)^3} \, dx}{3 e}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e (d+e x)^3}-\frac {(b p) \int \frac {1}{x (b+a x) (d+e x)^3} \, dx}{3 e}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e (d+e x)^3}-\frac {(b p) \int \left (\frac {1}{b d^3 x}+\frac {a^4}{b (-a d+b e)^3 (b+a x)}+\frac {e^2}{d (a d-b e) (d+e x)^3}+\frac {e^2 (2 a d-b e)}{d^2 (a d-b e)^2 (d+e x)^2}+\frac {e^2 \left (3 a^2 d^2-3 a b d e+b^2 e^2\right )}{d^3 (a d-b e)^3 (d+e x)}\right ) \, dx}{3 e}\\ &=\frac {b p}{6 d (a d-b e) (d+e x)^2}+\frac {b (2 a d-b e) p}{3 d^2 (a d-b e)^2 (d+e x)}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{3 e (d+e x)^3}-\frac {p \log (x)}{3 d^3 e}+\frac {a^3 p \log (b+a x)}{3 e (a d-b e)^3}-\frac {b \left (3 a^2 d^2-3 a b d e+b^2 e^2\right ) p \log (d+e x)}{3 d^3 (a d-b e)^3}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.19, size = 164, normalized size = 0.94 \begin {gather*} \frac {\frac {b e p}{2 d (a d-b e) (d+e x)^2}+\frac {b e (2 a d-b e) p}{d^2 (a d-b e)^2 (d+e x)}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^3}-\frac {p \log (x)}{d^3}+\frac {a^3 p \log (b+a x)}{(a d-b e)^3}-\frac {b e \left (3 a^2 d^2-3 a b d e+b^2 e^2\right ) p \log (d+e x)}{d^3 (a d-b e)^3}}{3 e} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Log[c*(a + b/x)^p]/(d + e*x)^4,x]

[Out]

((b*e*p)/(2*d*(a*d - b*e)*(d + e*x)^2) + (b*e*(2*a*d - b*e)*p)/(d^2*(a*d - b*e)^2*(d + e*x)) - Log[c*(a + b/x)
^p]/(d + e*x)^3 - (p*Log[x])/d^3 + (a^3*p*Log[b + a*x])/(a*d - b*e)^3 - (b*e*(3*a^2*d^2 - 3*a*b*d*e + b^2*e^2)
*p*Log[d + e*x])/(d^3*(a*d - b*e)^3))/(3*e)

________________________________________________________________________________________

Maple [F]
time = 0.22, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{\left (e x +d \right )^{4}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(a+b/x)^p)/(e*x+d)^4,x)

[Out]

int(ln(c*(a+b/x)^p)/(e*x+d)^4,x)

________________________________________________________________________________________

Maxima [A]
time = 0.40, size = 290, normalized size = 1.66 \begin {gather*} \frac {1}{6} \, {\left (\frac {2 \, a^{3} \log \left (a x + b\right )}{a^{3} b d^{3} - 3 \, a^{2} b^{2} d^{2} e + 3 \, a b^{3} d e^{2} - b^{4} e^{3}} - \frac {2 \, {\left (3 \, a^{2} d^{2} e - 3 \, a b d e^{2} + b^{2} e^{3}\right )} \log \left (x e + d\right )}{a^{3} d^{6} - 3 \, a^{2} b d^{5} e + 3 \, a b^{2} d^{4} e^{2} - b^{3} d^{3} e^{3}} + \frac {5 \, a d^{2} e - 3 \, b d e^{2} + 2 \, {\left (2 \, a d e^{2} - b e^{3}\right )} x}{a^{2} d^{6} - 2 \, a b d^{5} e + b^{2} d^{4} e^{2} + {\left (a^{2} d^{4} e^{2} - 2 \, a b d^{3} e^{3} + b^{2} d^{2} e^{4}\right )} x^{2} + 2 \, {\left (a^{2} d^{5} e - 2 \, a b d^{4} e^{2} + b^{2} d^{3} e^{3}\right )} x} - \frac {2 \, \log \left (x\right )}{b d^{3}}\right )} b p e^{\left (-1\right )} - \frac {e^{\left (-1\right )} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{3 \, {\left (x e + d\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b/x)^p)/(e*x+d)^4,x, algorithm="maxima")

[Out]

1/6*(2*a^3*log(a*x + b)/(a^3*b*d^3 - 3*a^2*b^2*d^2*e + 3*a*b^3*d*e^2 - b^4*e^3) - 2*(3*a^2*d^2*e - 3*a*b*d*e^2
+ b^2*e^3)*log(x*e + d)/(a^3*d^6 - 3*a^2*b*d^5*e + 3*a*b^2*d^4*e^2 - b^3*d^3*e^3) + (5*a*d^2*e - 3*b*d*e^2 +
2*(2*a*d*e^2 - b*e^3)*x)/(a^2*d^6 - 2*a*b*d^5*e + b^2*d^4*e^2 + (a^2*d^4*e^2 - 2*a*b*d^3*e^3 + b^2*d^2*e^4)*x^
2 + 2*(a^2*d^5*e - 2*a*b*d^4*e^2 + b^2*d^3*e^3)*x) - 2*log(x)/(b*d^3))*b*p*e^(-1) - 1/3*e^(-1)*log((a + b/x)^p
*c)/(x*e + d)^3

________________________________________________________________________________________

Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 814 vs. \(2 (169) = 338\).
time = 4.14, size = 814, normalized size = 4.65 \begin {gather*} \frac {5 \, a^{2} b d^{5} p e + 2 \, b^{3} d p x^{2} e^{5} - {\left (6 \, a b^{2} d^{2} p x^{2} - 5 \, b^{3} d^{2} p x\right )} e^{4} + {\left (4 \, a^{2} b d^{3} p x^{2} - 14 \, a b^{2} d^{3} p x + 3 \, b^{3} d^{3} p\right )} e^{3} + {\left (9 \, a^{2} b d^{4} p x - 8 \, a b^{2} d^{4} p\right )} e^{2} + 2 \, {\left (a^{3} d^{3} p x^{3} e^{3} + 3 \, a^{3} d^{4} p x^{2} e^{2} + 3 \, a^{3} d^{5} p x e + a^{3} d^{6} p\right )} \log \left (a x + b\right ) - 2 \, {\left (3 \, a^{2} b d^{5} p e + b^{3} p x^{3} e^{6} - 3 \, {\left (a b^{2} d p x^{3} - b^{3} d p x^{2}\right )} e^{5} + 3 \, {\left (a^{2} b d^{2} p x^{3} - 3 \, a b^{2} d^{2} p x^{2} + b^{3} d^{2} p x\right )} e^{4} + {\left (9 \, a^{2} b d^{3} p x^{2} - 9 \, a b^{2} d^{3} p x + b^{3} d^{3} p\right )} e^{3} + 3 \, {\left (3 \, a^{2} b d^{4} p x - a b^{2} d^{4} p\right )} e^{2}\right )} \log \left (x e + d\right ) - 2 \, {\left (a^{3} d^{6} - 3 \, a^{2} b d^{5} e + 3 \, a b^{2} d^{4} e^{2} - b^{3} d^{3} e^{3}\right )} \log \left (c\right ) - 2 \, {\left (a^{3} d^{6} p - b^{3} p x^{3} e^{6} + 3 \, {\left (a b^{2} d p x^{3} - b^{3} d p x^{2}\right )} e^{5} - 3 \, {\left (a^{2} b d^{2} p x^{3} - 3 \, a b^{2} d^{2} p x^{2} + b^{3} d^{2} p x\right )} e^{4} + {\left (a^{3} d^{3} p x^{3} - 9 \, a^{2} b d^{3} p x^{2} + 9 \, a b^{2} d^{3} p x - b^{3} d^{3} p\right )} e^{3} + 3 \, {\left (a^{3} d^{4} p x^{2} - 3 \, a^{2} b d^{4} p x + a b^{2} d^{4} p\right )} e^{2} + 3 \, {\left (a^{3} d^{5} p x - a^{2} b d^{5} p\right )} e\right )} \log \left (x\right ) - 2 \, {\left (a^{3} d^{6} p - 3 \, a^{2} b d^{5} p e + 3 \, a b^{2} d^{4} p e^{2} - b^{3} d^{3} p e^{3}\right )} \log \left (\frac {a x + b}{x}\right )}{6 \, {\left (a^{3} d^{9} e - b^{3} d^{3} x^{3} e^{7} + 3 \, {\left (a b^{2} d^{4} x^{3} - b^{3} d^{4} x^{2}\right )} e^{6} - 3 \, {\left (a^{2} b d^{5} x^{3} - 3 \, a b^{2} d^{5} x^{2} + b^{3} d^{5} x\right )} e^{5} + {\left (a^{3} d^{6} x^{3} - 9 \, a^{2} b d^{6} x^{2} + 9 \, a b^{2} d^{6} x - b^{3} d^{6}\right )} e^{4} + 3 \, {\left (a^{3} d^{7} x^{2} - 3 \, a^{2} b d^{7} x + a b^{2} d^{7}\right )} e^{3} + 3 \, {\left (a^{3} d^{8} x - a^{2} b d^{8}\right )} e^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b/x)^p)/(e*x+d)^4,x, algorithm="fricas")

[Out]

1/6*(5*a^2*b*d^5*p*e + 2*b^3*d*p*x^2*e^5 - (6*a*b^2*d^2*p*x^2 - 5*b^3*d^2*p*x)*e^4 + (4*a^2*b*d^3*p*x^2 - 14*a
*b^2*d^3*p*x + 3*b^3*d^3*p)*e^3 + (9*a^2*b*d^4*p*x - 8*a*b^2*d^4*p)*e^2 + 2*(a^3*d^3*p*x^3*e^3 + 3*a^3*d^4*p*x
^2*e^2 + 3*a^3*d^5*p*x*e + a^3*d^6*p)*log(a*x + b) - 2*(3*a^2*b*d^5*p*e + b^3*p*x^3*e^6 - 3*(a*b^2*d*p*x^3 - b
^3*d*p*x^2)*e^5 + 3*(a^2*b*d^2*p*x^3 - 3*a*b^2*d^2*p*x^2 + b^3*d^2*p*x)*e^4 + (9*a^2*b*d^3*p*x^2 - 9*a*b^2*d^3
*p*x + b^3*d^3*p)*e^3 + 3*(3*a^2*b*d^4*p*x - a*b^2*d^4*p)*e^2)*log(x*e + d) - 2*(a^3*d^6 - 3*a^2*b*d^5*e + 3*a
*b^2*d^4*e^2 - b^3*d^3*e^3)*log(c) - 2*(a^3*d^6*p - b^3*p*x^3*e^6 + 3*(a*b^2*d*p*x^3 - b^3*d*p*x^2)*e^5 - 3*(a
^2*b*d^2*p*x^3 - 3*a*b^2*d^2*p*x^2 + b^3*d^2*p*x)*e^4 + (a^3*d^3*p*x^3 - 9*a^2*b*d^3*p*x^2 + 9*a*b^2*d^3*p*x -
b^3*d^3*p)*e^3 + 3*(a^3*d^4*p*x^2 - 3*a^2*b*d^4*p*x + a*b^2*d^4*p)*e^2 + 3*(a^3*d^5*p*x - a^2*b*d^5*p)*e)*log
(x) - 2*(a^3*d^6*p - 3*a^2*b*d^5*p*e + 3*a*b^2*d^4*p*e^2 - b^3*d^3*p*e^3)*log((a*x + b)/x))/(a^3*d^9*e - b^3*d
^3*x^3*e^7 + 3*(a*b^2*d^4*x^3 - b^3*d^4*x^2)*e^6 - 3*(a^2*b*d^5*x^3 - 3*a*b^2*d^5*x^2 + b^3*d^5*x)*e^5 + (a^3*
d^6*x^3 - 9*a^2*b*d^6*x^2 + 9*a*b^2*d^6*x - b^3*d^6)*e^4 + 3*(a^3*d^7*x^2 - 3*a^2*b*d^7*x + a*b^2*d^7)*e^3 + 3
*(a^3*d^8*x - a^2*b*d^8)*e^2)

________________________________________________________________________________________

Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(ln(c*(a+b/x)**p)/(e*x+d)**4,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1841 vs. \(2 (169) = 338\).
time = 5.62, size = 1841, normalized size = 10.52 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(a+b/x)^p)/(e*x+d)^4,x, algorithm="giac")

[Out]

-1/6*(6*a^5*b^2*d^5*p*log(-a*d + b*e + (a*x + b)*d/x) - 24*a^4*b^3*d^4*p*e*log(-a*d + b*e + (a*x + b)*d/x) - 6
*a^4*b^3*d^4*p*e - 18*(a*x + b)*a^4*b^2*d^5*p*log(-a*d + b*e + (a*x + b)*d/x)/x + 38*a^3*b^4*d^3*p*e^2*log(-a*
d + b*e + (a*x + b)*d/x) + 54*(a*x + b)*a^3*b^3*d^4*p*e*log(-a*d + b*e + (a*x + b)*d/x)/x + 6*a^5*b^2*d^5*log(
c) - 24*a^4*b^3*d^4*e*log(c) + 6*(a*x + b)*a^4*b^2*d^5*p*log((a*x + b)/x)/x - 12*(a*x + b)*a^3*b^3*d^4*p*e*log
((a*x + b)/x)/x + 21*a^3*b^4*d^3*p*e^2 + 12*(a*x + b)*a^3*b^3*d^4*p*e/x + 18*(a*x + b)^2*a^3*b^2*d^5*p*log(-a*
d + b*e + (a*x + b)*d/x)/x^2 - 30*a^2*b^5*d^2*p*e^3*log(-a*d + b*e + (a*x + b)*d/x) - 60*(a*x + b)*a^2*b^4*d^3
*p*e^2*log(-a*d + b*e + (a*x + b)*d/x)/x - 36*(a*x + b)^2*a^2*b^3*d^4*p*e*log(-a*d + b*e + (a*x + b)*d/x)/x^2
- 12*(a*x + b)*a^4*b^2*d^5*log(c)/x + 38*a^3*b^4*d^3*e^2*log(c) + 42*(a*x + b)*a^3*b^3*d^4*e*log(c)/x - 12*(a*
x + b)^2*a^3*b^2*d^5*p*log((a*x + b)/x)/x^2 + 6*(a*x + b)*a^2*b^4*d^3*p*e^2*log((a*x + b)/x)/x + 18*(a*x + b)^
2*a^2*b^3*d^4*p*e*log((a*x + b)/x)/x^2 - 27*a^2*b^5*d^2*p*e^3 - 31*(a*x + b)*a^2*b^4*d^3*p*e^2/x - 6*(a*x + b)
^2*a^2*b^3*d^4*p*e/x^2 - 6*(a*x + b)^3*a^2*b^2*d^5*p*log(-a*d + b*e + (a*x + b)*d/x)/x^3 + 12*a*b^6*d*p*e^4*lo
g(-a*d + b*e + (a*x + b)*d/x) + 30*(a*x + b)*a*b^5*d^2*p*e^3*log(-a*d + b*e + (a*x + b)*d/x)/x + 24*(a*x + b)^
2*a*b^4*d^3*p*e^2*log(-a*d + b*e + (a*x + b)*d/x)/x^2 + 6*(a*x + b)^3*a*b^3*d^4*p*e*log(-a*d + b*e + (a*x + b)
*d/x)/x^3 + 6*(a*x + b)^2*a^3*b^2*d^5*log(c)/x^2 - 30*a^2*b^5*d^2*e^3*log(c) - 54*(a*x + b)*a^2*b^4*d^3*e^2*lo
g(c)/x - 18*(a*x + b)^2*a^2*b^3*d^4*e*log(c)/x^2 + 6*(a*x + b)^3*a^2*b^2*d^5*p*log((a*x + b)/x)/x^3 - 6*(a*x +
b)^2*a*b^4*d^3*p*e^2*log((a*x + b)/x)/x^2 - 6*(a*x + b)^3*a*b^3*d^4*p*e*log((a*x + b)/x)/x^3 + 15*a*b^6*d*p*e
^4 + 26*(a*x + b)*a*b^5*d^2*p*e^3/x + 10*(a*x + b)^2*a*b^4*d^3*p*e^2/x^2 - 2*b^7*p*e^5*log(-a*d + b*e + (a*x +
b)*d/x) - 6*(a*x + b)*b^6*d*p*e^4*log(-a*d + b*e + (a*x + b)*d/x)/x - 6*(a*x + b)^2*b^5*d^2*p*e^3*log(-a*d +
b*e + (a*x + b)*d/x)/x^2 - 2*(a*x + b)^3*b^4*d^3*p*e^2*log(-a*d + b*e + (a*x + b)*d/x)/x^3 + 12*a*b^6*d*e^4*lo
g(c) + 30*(a*x + b)*a*b^5*d^2*e^3*log(c)/x + 18*(a*x + b)^2*a*b^4*d^3*e^2*log(c)/x^2 + 2*(a*x + b)^3*b^4*d^3*p
*e^2*log((a*x + b)/x)/x^3 - 3*b^7*p*e^5 - 7*(a*x + b)*b^6*d*p*e^4/x - 4*(a*x + b)^2*b^5*d^2*p*e^3/x^2 - 2*b^7*
e^5*log(c) - 6*(a*x + b)*b^6*d*e^4*log(c)/x - 6*(a*x + b)^2*b^5*d^2*e^3*log(c)/x^2)/((a^6*d^9 - 6*a^5*b*d^8*e
- 3*(a*x + b)*a^5*d^9/x + 15*a^4*b^2*d^7*e^2 + 15*(a*x + b)*a^4*b*d^8*e/x + 3*(a*x + b)^2*a^4*d^9/x^2 - 20*a^3
*b^3*d^6*e^3 - 30*(a*x + b)*a^3*b^2*d^7*e^2/x - 12*(a*x + b)^2*a^3*b*d^8*e/x^2 - (a*x + b)^3*a^3*d^9/x^3 + 15*
a^2*b^4*d^5*e^4 + 30*(a*x + b)*a^2*b^3*d^6*e^3/x + 18*(a*x + b)^2*a^2*b^2*d^7*e^2/x^2 + 3*(a*x + b)^3*a^2*b*d^
8*e/x^3 - 6*a*b^5*d^4*e^5 - 15*(a*x + b)*a*b^4*d^5*e^4/x - 12*(a*x + b)^2*a*b^3*d^6*e^3/x^2 - 3*(a*x + b)^3*a*
b^2*d^7*e^2/x^3 + b^6*d^3*e^6 + 3*(a*x + b)*b^5*d^4*e^5/x + 3*(a*x + b)^2*b^4*d^5*e^4/x^2 + (a*x + b)^3*b^3*d^
6*e^3/x^3)*b)

________________________________________________________________________________________

Mupad [B]
time = 1.85, size = 662, normalized size = 3.78 \begin {gather*} \frac {p\,\ln \left (d+e\,x\right )}{3\,d^3\,e}-\frac {3\,b^2\,e^2\,p}{2\,\left (3\,a^2\,d^5\,e+6\,a^2\,d^4\,e^2\,x+3\,a^2\,d^3\,e^3\,x^2-6\,a\,b\,d^4\,e^2-12\,a\,b\,d^3\,e^3\,x-6\,a\,b\,d^2\,e^4\,x^2+3\,b^2\,d^3\,e^3+6\,b^2\,d^2\,e^4\,x+3\,b^2\,d\,e^5\,x^2\right )}-\frac {p\,\ln \left (x\right )}{3\,d^3\,e}-\frac {a^3\,p\,\ln \left (b+a\,x\right )}{-3\,a^3\,d^3\,e+9\,a^2\,b\,d^2\,e^2-9\,a\,b^2\,d\,e^3+3\,b^3\,e^4}-\frac {\ln \left (c\,{\left (\frac {b+a\,x}{x}\right )}^p\right )}{3\,\left (d^3\,e+3\,d^2\,e^2\,x+3\,d\,e^3\,x^2+e^4\,x^3\right )}-\frac {b^2\,e^3\,p\,x}{3\,a^2\,d^6\,e+6\,a^2\,d^5\,e^2\,x+3\,a^2\,d^4\,e^3\,x^2-6\,a\,b\,d^5\,e^2-12\,a\,b\,d^4\,e^3\,x-6\,a\,b\,d^3\,e^4\,x^2+3\,b^2\,d^4\,e^3+6\,b^2\,d^3\,e^4\,x+3\,b^2\,d^2\,e^5\,x^2}-\frac {a^3\,d^3\,p\,\ln \left (d+e\,x\right )}{3\,a^3\,d^6\,e-9\,a^2\,b\,d^5\,e^2+9\,a\,b^2\,d^4\,e^3-3\,b^3\,d^3\,e^4}+\frac {5\,a\,b\,d\,e\,p}{2\,\left (3\,a^2\,d^5\,e+6\,a^2\,d^4\,e^2\,x+3\,a^2\,d^3\,e^3\,x^2-6\,a\,b\,d^4\,e^2-12\,a\,b\,d^3\,e^3\,x-6\,a\,b\,d^2\,e^4\,x^2+3\,b^2\,d^3\,e^3+6\,b^2\,d^2\,e^4\,x+3\,b^2\,d\,e^5\,x^2\right )}+\frac {2\,a\,b\,d\,e^2\,p\,x}{3\,a^2\,d^6\,e+6\,a^2\,d^5\,e^2\,x+3\,a^2\,d^4\,e^3\,x^2-6\,a\,b\,d^5\,e^2-12\,a\,b\,d^4\,e^3\,x-6\,a\,b\,d^3\,e^4\,x^2+3\,b^2\,d^4\,e^3+6\,b^2\,d^3\,e^4\,x+3\,b^2\,d^2\,e^5\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(a + b/x)^p)/(d + e*x)^4,x)

[Out]

(p*log(d + e*x))/(3*d^3*e) - (3*b^2*e^2*p)/(2*(3*a^2*d^5*e + 3*b^2*d^3*e^3 + 6*a^2*d^4*e^2*x + 6*b^2*d^2*e^4*x
+ 3*b^2*d*e^5*x^2 + 3*a^2*d^3*e^3*x^2 - 6*a*b*d^4*e^2 - 12*a*b*d^3*e^3*x - 6*a*b*d^2*e^4*x^2)) - (p*log(x))/(
3*d^3*e) - (a^3*p*log(b + a*x))/(3*b^3*e^4 - 3*a^3*d^3*e + 9*a^2*b*d^2*e^2 - 9*a*b^2*d*e^3) - log(c*((b + a*x)
/x)^p)/(3*(d^3*e + e^4*x^3 + 3*d^2*e^2*x + 3*d*e^3*x^2)) - (b^2*e^3*p*x)/(3*a^2*d^6*e + 3*b^2*d^4*e^3 + 6*a^2*
d^5*e^2*x + 6*b^2*d^3*e^4*x + 3*a^2*d^4*e^3*x^2 + 3*b^2*d^2*e^5*x^2 - 6*a*b*d^5*e^2 - 12*a*b*d^4*e^3*x - 6*a*b
*d^3*e^4*x^2) - (a^3*d^3*p*log(d + e*x))/(3*a^3*d^6*e - 3*b^3*d^3*e^4 + 9*a*b^2*d^4*e^3 - 9*a^2*b*d^5*e^2) + (
5*a*b*d*e*p)/(2*(3*a^2*d^5*e + 3*b^2*d^3*e^3 + 6*a^2*d^4*e^2*x + 6*b^2*d^2*e^4*x + 3*b^2*d*e^5*x^2 + 3*a^2*d^3
*e^3*x^2 - 6*a*b*d^4*e^2 - 12*a*b*d^3*e^3*x - 6*a*b*d^2*e^4*x^2)) + (2*a*b*d*e^2*p*x)/(3*a^2*d^6*e + 3*b^2*d^4
*e^3 + 6*a^2*d^5*e^2*x + 6*b^2*d^3*e^4*x + 3*a^2*d^4*e^3*x^2 + 3*b^2*d^2*e^5*x^2 - 6*a*b*d^5*e^2 - 12*a*b*d^4*
e^3*x - 6*a*b*d^3*e^4*x^2)

________________________________________________________________________________________