3.3.3 \(\int \frac {\log (c (a+\frac {b}{x})^p)}{(d+e x)^3} \, dx\) [203]
Optimal. Leaf size=127 \[ \frac {b p}{2 d (a d-b e) (d+e x)}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e (d+e x)^2}-\frac {p \log (x)}{2 d^2 e}+\frac {a^2 p \log (b+a x)}{2 e (a d-b e)^2}-\frac {b (2 a d-b e) p \log (d+e x)}{2 d^2 (a d-b e)^2} \]
[Out]
1/2*b*p/d/(a*d-b*e)/(e*x+d)-1/2*ln(c*(a+b/x)^p)/e/(e*x+d)^2-1/2*p*ln(x)/d^2/e+1/2*a^2*p*ln(a*x+b)/e/(a*d-b*e)^
2-1/2*b*(2*a*d-b*e)*p*ln(e*x+d)/d^2/(a*d-b*e)^2
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Rubi [A]
time = 0.08, antiderivative size = 127, 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^2 p \log (a x+b)}{2 e (a d-b e)^2}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e (d+e x)^2}-\frac {b p (2 a d-b e) \log (d+e x)}{2 d^2 (a d-b e)^2}+\frac {b p}{2 d (d+e x) (a d-b e)}-\frac {p \log (x)}{2 d^2 e} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[Log[c*(a + b/x)^p]/(d + e*x)^3,x]
[Out]
(b*p)/(2*d*(a*d - b*e)*(d + e*x)) - Log[c*(a + b/x)^p]/(2*e*(d + e*x)^2) - (p*Log[x])/(2*d^2*e) + (a^2*p*Log[b
+ a*x])/(2*e*(a*d - b*e)^2) - (b*(2*a*d - b*e)*p*Log[d + e*x])/(2*d^2*(a*d - b*e)^2)
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)^3} \, dx &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e (d+e x)^2}-\frac {(b p) \int \frac {1}{\left (a+\frac {b}{x}\right ) x^2 (d+e x)^2} \, dx}{2 e}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e (d+e x)^2}-\frac {(b p) \int \frac {1}{x (b+a x) (d+e x)^2} \, dx}{2 e}\\ &=-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e (d+e x)^2}-\frac {(b p) \int \left (\frac {1}{b d^2 x}-\frac {a^3}{b (-a d+b e)^2 (b+a x)}+\frac {e^2}{d (a d-b e) (d+e x)^2}+\frac {e^2 (2 a d-b e)}{d^2 (a d-b e)^2 (d+e x)}\right ) \, dx}{2 e}\\ &=\frac {b p}{2 d (a d-b e) (d+e x)}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{2 e (d+e x)^2}-\frac {p \log (x)}{2 d^2 e}+\frac {a^2 p \log (b+a x)}{2 e (a d-b e)^2}-\frac {b (2 a d-b e) p \log (d+e x)}{2 d^2 (a d-b e)^2}\\ \end {align*}
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Mathematica [A]
time = 0.13, size = 113, normalized size = 0.89 \begin {gather*} \frac {\frac {b e p}{d (a d-b e) (d+e x)}-\frac {\log \left (c \left (a+\frac {b}{x}\right )^p\right )}{(d+e x)^2}-\frac {p \log (x)}{d^2}+\frac {a^2 p \log (b+a x)}{(a d-b e)^2}+\frac {b e (-2 a d+b e) p \log (d+e x)}{d^2 (a d-b e)^2}}{2 e} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[Log[c*(a + b/x)^p]/(d + e*x)^3,x]
[Out]
((b*e*p)/(d*(a*d - b*e)*(d + e*x)) - Log[c*(a + b/x)^p]/(d + e*x)^2 - (p*Log[x])/d^2 + (a^2*p*Log[b + a*x])/(a
*d - b*e)^2 + (b*e*(-2*a*d + b*e)*p*Log[d + e*x])/(d^2*(a*d - b*e)^2))/(2*e)
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Maple [F]
time = 0.24, size = 0, normalized size = 0.00 \[\int \frac {\ln \left (c \left (a +\frac {b}{x}\right )^{p}\right )}{\left (e x +d \right )^{3}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(ln(c*(a+b/x)^p)/(e*x+d)^3,x)
[Out]
int(ln(c*(a+b/x)^p)/(e*x+d)^3,x)
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Maxima [A]
time = 0.43, size = 162, normalized size = 1.28 \begin {gather*} \frac {1}{2} \, {\left (\frac {a^{2} \log \left (a x + b\right )}{a^{2} b d^{2} - 2 \, a b^{2} d e + b^{3} e^{2}} - \frac {{\left (2 \, a d e - b e^{2}\right )} \log \left (x e + d\right )}{a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2}} + \frac {e}{a d^{3} - b d^{2} e + {\left (a d^{2} e - b d e^{2}\right )} x} - \frac {\log \left (x\right )}{b d^{2}}\right )} b p e^{\left (-1\right )} - \frac {e^{\left (-1\right )} \log \left ({\left (a + \frac {b}{x}\right )}^{p} c\right )}{2 \, {\left (x e + d\right )}^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(c*(a+b/x)^p)/(e*x+d)^3,x, algorithm="maxima")
[Out]
1/2*(a^2*log(a*x + b)/(a^2*b*d^2 - 2*a*b^2*d*e + b^3*e^2) - (2*a*d*e - b*e^2)*log(x*e + d)/(a^2*d^4 - 2*a*b*d^
3*e + b^2*d^2*e^2) + e/(a*d^3 - b*d^2*e + (a*d^2*e - b*d*e^2)*x) - log(x)/(b*d^2))*b*p*e^(-1) - 1/2*e^(-1)*log
((a + b/x)^p*c)/(x*e + d)^2
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 423 vs.
\(2 (121) = 242\).
time = 0.86, size = 423, normalized size = 3.33 \begin {gather*} \frac {a b d^{3} p e - b^{2} d p x e^{3} + {\left (a b d^{2} p x - b^{2} d^{2} p\right )} e^{2} + {\left (a^{2} d^{2} p x^{2} e^{2} + 2 \, a^{2} d^{3} p x e + a^{2} d^{4} p\right )} \log \left (a x + b\right ) - {\left (2 \, a b d^{3} p e - b^{2} p x^{2} e^{4} + 2 \, {\left (a b d p x^{2} - b^{2} d p x\right )} e^{3} + {\left (4 \, a b d^{2} p x - b^{2} d^{2} p\right )} e^{2}\right )} \log \left (x e + d\right ) - {\left (a^{2} d^{4} - 2 \, a b d^{3} e + b^{2} d^{2} e^{2}\right )} \log \left (c\right ) - {\left (a^{2} d^{4} p + b^{2} p x^{2} e^{4} - 2 \, {\left (a b d p x^{2} - b^{2} d p x\right )} e^{3} + {\left (a^{2} d^{2} p x^{2} - 4 \, a b d^{2} p x + b^{2} d^{2} p\right )} e^{2} + 2 \, {\left (a^{2} d^{3} p x - a b d^{3} p\right )} e\right )} \log \left (x\right ) - {\left (a^{2} d^{4} p - 2 \, a b d^{3} p e + b^{2} d^{2} p e^{2}\right )} \log \left (\frac {a x + b}{x}\right )}{2 \, {\left (a^{2} d^{6} e + b^{2} d^{2} x^{2} e^{5} - 2 \, {\left (a b d^{3} x^{2} - b^{2} d^{3} x\right )} e^{4} + {\left (a^{2} d^{4} x^{2} - 4 \, a b d^{4} x + b^{2} d^{4}\right )} e^{3} + 2 \, {\left (a^{2} d^{5} x - a b d^{5}\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)^3,x, algorithm="fricas")
[Out]
1/2*(a*b*d^3*p*e - b^2*d*p*x*e^3 + (a*b*d^2*p*x - b^2*d^2*p)*e^2 + (a^2*d^2*p*x^2*e^2 + 2*a^2*d^3*p*x*e + a^2*
d^4*p)*log(a*x + b) - (2*a*b*d^3*p*e - b^2*p*x^2*e^4 + 2*(a*b*d*p*x^2 - b^2*d*p*x)*e^3 + (4*a*b*d^2*p*x - b^2*
d^2*p)*e^2)*log(x*e + d) - (a^2*d^4 - 2*a*b*d^3*e + b^2*d^2*e^2)*log(c) - (a^2*d^4*p + b^2*p*x^2*e^4 - 2*(a*b*
d*p*x^2 - b^2*d*p*x)*e^3 + (a^2*d^2*p*x^2 - 4*a*b*d^2*p*x + b^2*d^2*p)*e^2 + 2*(a^2*d^3*p*x - a*b*d^3*p)*e)*lo
g(x) - (a^2*d^4*p - 2*a*b*d^3*p*e + b^2*d^2*p*e^2)*log((a*x + b)/x))/(a^2*d^6*e + b^2*d^2*x^2*e^5 - 2*(a*b*d^3
*x^2 - b^2*d^3*x)*e^4 + (a^2*d^4*x^2 - 4*a*b*d^4*x + b^2*d^4)*e^3 + 2*(a^2*d^5*x - a*b*d^5)*e^2)
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 3512 vs.
\(2 (105) = 210\).
time = 48.70, size = 3512, normalized size = 27.65 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(ln(c*(a+b/x)**p)/(e*x+d)**3,x)
[Out]
Piecewise((d**2*p*log(d/e + x)/(2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3*x**2) - d**2*p/(2*d**4*e + 4*d**3*e**2*
x + 2*d**2*e**3*x**2) + 2*d*e*p*x*log(d/e + x)/(2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3*x**2) - d*e*p*x/(2*d**4
*e + 4*d**3*e**2*x + 2*d**2*e**3*x**2) + 2*d*e*x*log(c*(b/x)**p)/(2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3*x**2)
+ e**2*p*x**2*log(d/e + x)/(2*d**4*e + 4*d**3*e**2*x + 2*d**2*e**3*x**2) + e**2*x**2*log(c*(b/x)**p)/(2*d**4*
e + 4*d**3*e**2*x + 2*d**2*e**3*x**2), Eq(a, 0)), (-3*d**2*p/(4*d**4*e + 8*d**3*e**2*x + 4*d**2*e**3*x**2) - 2
*d*e*p*x/(4*d**4*e + 8*d**3*e**2*x + 4*d**2*e**3*x**2) + 4*d*e*x*log(c*(b/x + b*e/d)**p)/(4*d**4*e + 8*d**3*e*
*2*x + 4*d**2*e**3*x**2) + 2*e**2*x**2*log(c*(b/x + b*e/d)**p)/(4*d**4*e + 8*d**3*e**2*x + 4*d**2*e**3*x**2),
Eq(a, b*e/d)), ((a**2*log(c*(a + b/x)**p)/(2*b**2) - a*p/(2*b*x) + p/(4*x**2) - log(c*(a + b/x)**p)/(2*x**2))/
e**3, Eq(d, 0)), (zoo*(x*log(c*(a + b/x)**p) + b*p*log(a*x + b)/a), Eq(d, -e*x)), ((x*log(c*(a + b/x)**p) + b*
p*log(a*x + b)/a)/d**3, Eq(e, 0)), (2*a**2*d**3*x*log(c*(a + b/x)**p)/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*
d**4*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3
*x + 2*b**2*d**2*e**4*x**2) + a**2*d**2*e*x**2*log(c*(a + b/x)**p)/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**
4*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3*x
+ 2*b**2*d**2*e**4*x**2) + 2*a*b*d**3*p*log(x + b/a)/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**4*e**2*x**2 -
4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3*x + 2*b**2*d**2*
e**4*x**2) - 2*a*b*d**3*p*log(d/e + x)/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**4*e**2*x**2 - 4*a*b*d**5*e -
8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3*x + 2*b**2*d**2*e**4*x**2) + a
*b*d**3*p/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**4*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d*
*3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3*x + 2*b**2*d**2*e**4*x**2) + 4*a*b*d**2*e*p*x*log(x + b/a)/
(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**4*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x*
*2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3*x + 2*b**2*d**2*e**4*x**2) - 4*a*b*d**2*e*p*x*log(d/e + x)/(2*a**2*d*
*6 + 4*a**2*d**5*e*x + 2*a**2*d**4*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**
2*d**4*e**2 + 4*b**2*d**3*e**3*x + 2*b**2*d**2*e**4*x**2) + a*b*d**2*e*p*x/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*
a**2*d**4*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3
*e**3*x + 2*b**2*d**2*e**4*x**2) - 4*a*b*d**2*e*x*log(c*(a + b/x)**p)/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*
d**4*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3
*x + 2*b**2*d**2*e**4*x**2) + 2*a*b*d*e**2*p*x**2*log(x + b/a)/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**4*e*
*2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3*x + 2*
b**2*d**2*e**4*x**2) - 2*a*b*d*e**2*p*x**2*log(d/e + x)/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**4*e**2*x**2
- 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3*x + 2*b**2*d*
*2*e**4*x**2) - 2*a*b*d*e**2*x**2*log(c*(a + b/x)**p)/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**4*e**2*x**2 -
4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3*x + 2*b**2*d**2
*e**4*x**2) - b**2*d**2*e*p*log(x + b/a)/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**4*e**2*x**2 - 4*a*b*d**5*e
- 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3*x + 2*b**2*d**2*e**4*x**2) +
b**2*d**2*e*p*log(d/e + x)/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**4*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4
*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3*x + 2*b**2*d**2*e**4*x**2) - b**2*d**2*e*
p/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**4*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*
x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3*x + 2*b**2*d**2*e**4*x**2) - 2*b**2*d*e**2*p*x*log(x + b/a)/(2*a**2
*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**4*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*
b**2*d**4*e**2 + 4*b**2*d**3*e**3*x + 2*b**2*d**2*e**4*x**2) + 2*b**2*d*e**2*p*x*log(d/e + x)/(2*a**2*d**6 + 4
*a**2*d**5*e*x + 2*a**2*d**4*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4
*e**2 + 4*b**2*d**3*e**3*x + 2*b**2*d**2*e**4*x**2) - b**2*d*e**2*p*x/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*
d**4*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3
*x + 2*b**2*d**2*e**4*x**2) + 2*b**2*d*e**2*x*log(c*(a + b/x)**p)/(2*a**2*d**6 + 4*a**2*d**5*e*x + 2*a**2*d**4
*e**2*x**2 - 4*a*b*d**5*e - 8*a*b*d**4*e**2*x - 4*a*b*d**3*e**3*x**2 + 2*b**2*d**4*e**2 + 4*b**2*d**3*e**3*x +
2*b**2*d**2*e**4*x**2) - b**2*e**3*p*x**2*log(...
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 805 vs.
\(2 (121) = 242\).
time = 4.52, size = 805, normalized size = 6.34 \begin {gather*} -\frac {2 \, a^{3} b^{2} d^{3} p \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right ) - 5 \, a^{2} b^{3} d^{2} p e \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right ) - a^{2} b^{3} d^{2} p e - \frac {4 \, {\left (a x + b\right )} a^{2} b^{2} d^{3} p \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right )}{x} + 4 \, a b^{4} d p e^{2} \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right ) + \frac {6 \, {\left (a x + b\right )} a b^{3} d^{2} p e \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right )}{x} + 2 \, a^{3} b^{2} d^{3} \log \left (c\right ) - 5 \, a^{2} b^{3} d^{2} e \log \left (c\right ) + \frac {2 \, {\left (a x + b\right )} a^{2} b^{2} d^{3} p \log \left (\frac {a x + b}{x}\right )}{x} - \frac {2 \, {\left (a x + b\right )} a b^{3} d^{2} p e \log \left (\frac {a x + b}{x}\right )}{x} + 2 \, a b^{4} d p e^{2} + \frac {{\left (a x + b\right )} a b^{3} d^{2} p e}{x} + \frac {2 \, {\left (a x + b\right )}^{2} a b^{2} d^{3} p \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right )}{x^{2}} - b^{5} p e^{3} \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right ) - \frac {2 \, {\left (a x + b\right )} b^{4} d p e^{2} \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right )}{x} - \frac {{\left (a x + b\right )}^{2} b^{3} d^{2} p e \log \left (-a d + b e + \frac {{\left (a x + b\right )} d}{x}\right )}{x^{2}} - \frac {2 \, {\left (a x + b\right )} a^{2} b^{2} d^{3} \log \left (c\right )}{x} + 4 \, a b^{4} d e^{2} \log \left (c\right ) + \frac {4 \, {\left (a x + b\right )} a b^{3} d^{2} e \log \left (c\right )}{x} - \frac {2 \, {\left (a x + b\right )}^{2} a b^{2} d^{3} p \log \left (\frac {a x + b}{x}\right )}{x^{2}} + \frac {{\left (a x + b\right )}^{2} b^{3} d^{2} p e \log \left (\frac {a x + b}{x}\right )}{x^{2}} - b^{5} p e^{3} - \frac {{\left (a x + b\right )} b^{4} d p e^{2}}{x} - b^{5} e^{3} \log \left (c\right ) - \frac {2 \, {\left (a x + b\right )} b^{4} d e^{2} \log \left (c\right )}{x}}{2 \, {\left (a^{4} d^{6} - 4 \, a^{3} b d^{5} e - \frac {2 \, {\left (a x + b\right )} a^{3} d^{6}}{x} + 6 \, a^{2} b^{2} d^{4} e^{2} + \frac {6 \, {\left (a x + b\right )} a^{2} b d^{5} e}{x} + \frac {{\left (a x + b\right )}^{2} a^{2} d^{6}}{x^{2}} - 4 \, a b^{3} d^{3} e^{3} - \frac {6 \, {\left (a x + b\right )} a b^{2} d^{4} e^{2}}{x} - \frac {2 \, {\left (a x + b\right )}^{2} a b d^{5} e}{x^{2}} + b^{4} d^{2} e^{4} + \frac {2 \, {\left (a x + b\right )} b^{3} d^{3} e^{3}}{x} + \frac {{\left (a x + b\right )}^{2} b^{2} d^{4} e^{2}}{x^{2}}\right )} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(c*(a+b/x)^p)/(e*x+d)^3,x, algorithm="giac")
[Out]
-1/2*(2*a^3*b^2*d^3*p*log(-a*d + b*e + (a*x + b)*d/x) - 5*a^2*b^3*d^2*p*e*log(-a*d + b*e + (a*x + b)*d/x) - a^
2*b^3*d^2*p*e - 4*(a*x + b)*a^2*b^2*d^3*p*log(-a*d + b*e + (a*x + b)*d/x)/x + 4*a*b^4*d*p*e^2*log(-a*d + b*e +
(a*x + b)*d/x) + 6*(a*x + b)*a*b^3*d^2*p*e*log(-a*d + b*e + (a*x + b)*d/x)/x + 2*a^3*b^2*d^3*log(c) - 5*a^2*b
^3*d^2*e*log(c) + 2*(a*x + b)*a^2*b^2*d^3*p*log((a*x + b)/x)/x - 2*(a*x + b)*a*b^3*d^2*p*e*log((a*x + b)/x)/x
+ 2*a*b^4*d*p*e^2 + (a*x + b)*a*b^3*d^2*p*e/x + 2*(a*x + b)^2*a*b^2*d^3*p*log(-a*d + b*e + (a*x + b)*d/x)/x^2
- b^5*p*e^3*log(-a*d + b*e + (a*x + b)*d/x) - 2*(a*x + b)*b^4*d*p*e^2*log(-a*d + b*e + (a*x + b)*d/x)/x - (a*x
+ b)^2*b^3*d^2*p*e*log(-a*d + b*e + (a*x + b)*d/x)/x^2 - 2*(a*x + b)*a^2*b^2*d^3*log(c)/x + 4*a*b^4*d*e^2*log
(c) + 4*(a*x + b)*a*b^3*d^2*e*log(c)/x - 2*(a*x + b)^2*a*b^2*d^3*p*log((a*x + b)/x)/x^2 + (a*x + b)^2*b^3*d^2*
p*e*log((a*x + b)/x)/x^2 - b^5*p*e^3 - (a*x + b)*b^4*d*p*e^2/x - b^5*e^3*log(c) - 2*(a*x + b)*b^4*d*e^2*log(c)
/x)/((a^4*d^6 - 4*a^3*b*d^5*e - 2*(a*x + b)*a^3*d^6/x + 6*a^2*b^2*d^4*e^2 + 6*(a*x + b)*a^2*b*d^5*e/x + (a*x +
b)^2*a^2*d^6/x^2 - 4*a*b^3*d^3*e^3 - 6*(a*x + b)*a*b^2*d^4*e^2/x - 2*(a*x + b)^2*a*b*d^5*e/x^2 + b^4*d^2*e^4
+ 2*(a*x + b)*b^3*d^3*e^3/x + (a*x + b)^2*b^2*d^4*e^2/x^2)*b)
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Mupad [B]
time = 1.08, size = 217, normalized size = 1.71 \begin {gather*} \frac {a^2\,p\,\ln \left (b+a\,x\right )}{2\,a^2\,d^2\,e-4\,a\,b\,d\,e^2+2\,b^2\,e^3}-\frac {\ln \left (c\,{\left (\frac {b+a\,x}{x}\right )}^p\right )}{2\,\left (d^2\,e+2\,d\,e^2\,x+e^3\,x^2\right )}-\frac {p\,\ln \left (x\right )}{2\,d^2\,e}-\frac {b\,e\,p}{2\,b\,d^2\,e^2-2\,a\,d^3\,e+2\,b\,d\,e^3\,x-2\,a\,d^2\,e^2\,x}+\frac {b^2\,e\,p\,\ln \left (d+e\,x\right )}{2\,a^2\,d^4-4\,a\,b\,d^3\,e+2\,b^2\,d^2\,e^2}-\frac {2\,a\,b\,d\,p\,\ln \left (d+e\,x\right )}{2\,a^2\,d^4-4\,a\,b\,d^3\,e+2\,b^2\,d^2\,e^2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(log(c*(a + b/x)^p)/(d + e*x)^3,x)
[Out]
(a^2*p*log(b + a*x))/(2*b^2*e^3 + 2*a^2*d^2*e - 4*a*b*d*e^2) - log(c*((b + a*x)/x)^p)/(2*(d^2*e + e^3*x^2 + 2*
d*e^2*x)) - (p*log(x))/(2*d^2*e) - (b*e*p)/(2*b*d^2*e^2 - 2*a*d^3*e + 2*b*d*e^3*x - 2*a*d^2*e^2*x) + (b^2*e*p*
log(d + e*x))/(2*a^2*d^4 + 2*b^2*d^2*e^2 - 4*a*b*d^3*e) - (2*a*b*d*p*log(d + e*x))/(2*a^2*d^4 + 2*b^2*d^2*e^2
- 4*a*b*d^3*e)
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