3.204 \(\int \frac{\log (c (a+\frac{b}{x})^p)}{(d+e x)^4} \, dx\)
Optimal. Leaf size=175 \[ -\frac{b p \left (3 a^2 d^2-3 a b d e+b^2 e^2\right ) \log (d+e x)}{3 d^3 (a d-b e)^3}+\frac{a^3 p \log (a x+b)}{3 e (a d-b e)^3}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{3 e (d+e x)^3}+\frac{b p (2 a d-b e)}{3 d^2 (d+e x) (a d-b e)^2}+\frac{b p}{6 d (d+e x)^2 (a d-b e)}-\frac{p \log (x)}{3 d^3 e} \]
[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)
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Rubi [A] time = 0.171521, antiderivative size = 175, normalized size of antiderivative = 1.,
number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used =
{2463, 514, 72} \[ -\frac{b p \left (3 a^2 d^2-3 a b d e+b^2 e^2\right ) \log (d+e x)}{3 d^3 (a d-b e)^3}+\frac{a^3 p \log (a x+b)}{3 e (a d-b e)^3}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{3 e (d+e x)^3}+\frac{b p (2 a d-b e)}{3 d^2 (d+e x) (a d-b e)^2}+\frac{b p}{6 d (d+e x)^2 (a d-b e)}-\frac{p \log (x)}{3 d^3 e} \]
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 2463
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]
Rule 514
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 72
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]
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 = 0.282941, size = 164, normalized size = 0.94 \[ \frac{-\frac{b e p \left (3 a^2 d^2-3 a b d e+b^2 e^2\right ) \log (d+e x)}{d^3 (a d-b e)^3}+\frac{a^3 p \log (a x+b)}{(a d-b e)^3}-\frac{\log \left (c \left (a+\frac{b}{x}\right )^p\right )}{(d+e x)^3}+\frac{b e p (2 a d-b e)}{d^2 (d+e x) (a d-b e)^2}+\frac{b e p}{2 d (d+e x)^2 (a d-b e)}-\frac{p \log (x)}{d^3}}{3 e} \]
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)
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Maple [F] time = 0.539, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ex+d \right ) ^{4}}\ln \left ( c \left ( a+{\frac{b}{x}} \right ) ^{p} \right ) }\, dx \end{align*}
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)
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Maxima [A] time = 1.0883, size = 404, normalized size = 2.31 \begin{align*} \frac{{\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 (e x + 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}{6 \, e} - \frac{\log \left ({\left (a + \frac{b}{x}\right )}^{p} c\right )}{3 \,{\left (e x + d\right )}^{3} e} \end{align*}
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(e*x + 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 - 1/3*log((a + b/x)^p*c)/((e*x +
d)^3*e)
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Fricas [B] time = 43.0068, size = 1629, normalized size = 9.31 \begin{align*} \text{result too large to display} \end{align*}
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*(2*(2*a^2*b*d^3*e^3 - 3*a*b^2*d^2*e^4 + b^3*d*e^5)*p*x^2 + (9*a^2*b*d^4*e^2 - 14*a*b^2*d^3*e^3 + 5*b^3*d^2
*e^4)*p*x - 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)*p*log((a*x + b)/x) + (5*a^2*b*d^5*e -
8*a*b^2*d^4*e^2 + 3*b^3*d^3*e^3)*p + 2*(a^3*d^3*e^3*p*x^3 + 3*a^3*d^4*e^2*p*x^2 + 3*a^3*d^5*e*p*x + a^3*d^6*p)
*log(a*x + b) - 2*((3*a^2*b*d^2*e^4 - 3*a*b^2*d*e^5 + b^3*e^6)*p*x^3 + 3*(3*a^2*b*d^3*e^3 - 3*a*b^2*d^2*e^4 +
b^3*d*e^5)*p*x^2 + 3*(3*a^2*b*d^4*e^2 - 3*a*b^2*d^3*e^3 + b^3*d^2*e^4)*p*x + (3*a^2*b*d^5*e - 3*a*b^2*d^4*e^2
+ b^3*d^3*e^3)*p)*log(e*x + 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^3*e^3 - 3*a^2*b*d^2*e^4 + 3*a*b^2*d*e^5 - b^3*e^6)*p*x^3 + 3*(a^3*d^4*e^2 - 3*a^2*b*d^3*e^3 + 3*a*b^2*d^2*e^
4 - b^3*d*e^5)*p*x^2 + 3*(a^3*d^5*e - 3*a^2*b*d^4*e^2 + 3*a*b^2*d^3*e^3 - b^3*d^2*e^4)*p*x + (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)*p)*log(x))/(a^3*d^9*e - 3*a^2*b*d^8*e^2 + 3*a*b^2*d^7*e^3 - b^3*d^6*e
^4 + (a^3*d^6*e^4 - 3*a^2*b*d^5*e^5 + 3*a*b^2*d^4*e^6 - b^3*d^3*e^7)*x^3 + 3*(a^3*d^7*e^3 - 3*a^2*b*d^6*e^4 +
3*a*b^2*d^5*e^5 - b^3*d^4*e^6)*x^2 + 3*(a^3*d^8*e^2 - 3*a^2*b*d^7*e^3 + 3*a*b^2*d^6*e^4 - b^3*d^5*e^5)*x)
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
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
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Giac [B] time = 1.2848, size = 1208, normalized size = 6.9 \begin{align*} \text{result too large to display} \end{align*}
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*(2*a^3*d^3*p*x^3*e^3*log(a*x + b) + 6*a^3*d^4*p*x^2*e^2*log(a*x + b) + 6*a^3*d^5*p*x*e*log(a*x + b) - 2*a^
3*d^3*p*x^3*e^3*log(x) - 6*a^3*d^4*p*x^2*e^2*log(x) - 6*a^3*d^5*p*x*e*log(x) + 6*a^2*b*d^5*p*e*log(a*x + b) -
6*a^2*b*d^2*p*x^3*e^4*log(x*e + d) - 18*a^2*b*d^3*p*x^2*e^3*log(x*e + d) - 18*a^2*b*d^4*p*x*e^2*log(x*e + d) -
6*a^2*b*d^5*p*e*log(x*e + d) + 6*a^2*b*d^2*p*x^3*e^4*log(x) + 18*a^2*b*d^3*p*x^2*e^3*log(x) + 18*a^2*b*d^4*p*
x*e^2*log(x) + 4*a^2*b*d^3*p*x^2*e^3 + 9*a^2*b*d^4*p*x*e^2 + 5*a^2*b*d^5*p*e - 6*a*b^2*d^4*p*e^2*log(a*x + b)
+ 6*a*b^2*d*p*x^3*e^5*log(x*e + d) + 18*a*b^2*d^2*p*x^2*e^4*log(x*e + d) + 18*a*b^2*d^3*p*x*e^3*log(x*e + d) +
6*a*b^2*d^4*p*e^2*log(x*e + d) - 2*a^3*d^6*log(c) + 6*a^2*b*d^5*e*log(c) - 6*a*b^2*d*p*x^3*e^5*log(x) - 18*a*
b^2*d^2*p*x^2*e^4*log(x) - 18*a*b^2*d^3*p*x*e^3*log(x) - 6*a*b^2*d^2*p*x^2*e^4 - 14*a*b^2*d^3*p*x*e^3 - 8*a*b^
2*d^4*p*e^2 + 2*b^3*d^3*p*e^3*log(a*x + b) - 2*b^3*p*x^3*e^6*log(x*e + d) - 6*b^3*d*p*x^2*e^5*log(x*e + d) - 6
*b^3*d^2*p*x*e^4*log(x*e + d) - 2*b^3*d^3*p*e^3*log(x*e + d) - 6*a*b^2*d^4*e^2*log(c) + 2*b^3*p*x^3*e^6*log(x)
+ 6*b^3*d*p*x^2*e^5*log(x) + 6*b^3*d^2*p*x*e^4*log(x) + 2*b^3*d*p*x^2*e^5 + 5*b^3*d^2*p*x*e^4 + 3*b^3*d^3*p*e
^3 + 2*b^3*d^3*e^3*log(c))/(a^3*d^6*x^3*e^4 + 3*a^3*d^7*x^2*e^3 + 3*a^3*d^8*x*e^2 + a^3*d^9*e - 3*a^2*b*d^5*x^
3*e^5 - 9*a^2*b*d^6*x^2*e^4 - 9*a^2*b*d^7*x*e^3 - 3*a^2*b*d^8*e^2 + 3*a*b^2*d^4*x^3*e^6 + 9*a*b^2*d^5*x^2*e^5
+ 9*a*b^2*d^6*x*e^4 + 3*a*b^2*d^7*e^3 - b^3*d^3*x^3*e^7 - 3*b^3*d^4*x^2*e^6 - 3*b^3*d^5*x*e^5 - b^3*d^6*e^4)