3.12 \(\int \frac {\log (e (f (a+b x)^p (c+d x)^q)^r)}{(a+b x)^2} \, dx\)
Optimal. Leaf size=95 \[ -\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+\frac {d q r \log (a+b x)}{b (b c-a d)}-\frac {d q r \log (c+d x)}{b (b c-a d)}-\frac {p r}{b (a+b x)} \]
[Out]
-p*r/b/(b*x+a)+d*q*r*ln(b*x+a)/b/(-a*d+b*c)-d*q*r*ln(d*x+c)/b/(-a*d+b*c)-ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/b/(b*
x+a)
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 95, normalized size of antiderivative = 1.00,
number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used =
{2495, 32, 36, 31} \[ -\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+\frac {d q r \log (a+b x)}{b (b c-a d)}-\frac {d q r \log (c+d x)}{b (b c-a d)}-\frac {p r}{b (a+b x)} \]
Antiderivative was successfully verified.
[In]
Int[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(a + b*x)^2,x]
[Out]
-((p*r)/(b*(a + b*x))) + (d*q*r*Log[a + b*x])/(b*(b*c - a*d)) - (d*q*r*Log[c + d*x])/(b*(b*c - a*d)) - Log[e*(
f*(a + b*x)^p*(c + d*x)^q)^r]/(b*(a + b*x))
Rule 31
Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]
Rule 32
Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]
Rule 36
Int[1/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :> Dist[b/(b*c - a*d), Int[1/(a + b*x), x], x] -
Dist[d/(b*c - a*d), Int[1/(c + d*x), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0]
Rule 2495
Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*((g_.) + (h_.)*(x_))^(m_.),
x_Symbol] :> Simp[((g + h*x)^(m + 1)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r])/(h*(m + 1)), x] + (-Dist[(b*p*r)/(
h*(m + 1)), Int[(g + h*x)^(m + 1)/(a + b*x), x], x] - Dist[(d*q*r)/(h*(m + 1)), Int[(g + h*x)^(m + 1)/(c + d*x
), x], x]) /; FreeQ[{a, b, c, d, e, f, g, h, m, p, q, r}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1]
Rubi steps
\begin {align*} \int \frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{(a+b x)^2} \, dx &=-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+(p r) \int \frac {1}{(a+b x)^2} \, dx+\frac {(d q r) \int \frac {1}{(a+b x) (c+d x)} \, dx}{b}\\ &=-\frac {p r}{b (a+b x)}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}+\frac {(d q r) \int \frac {1}{a+b x} \, dx}{b c-a d}-\frac {\left (d^2 q r\right ) \int \frac {1}{c+d x} \, dx}{b (b c-a d)}\\ &=-\frac {p r}{b (a+b x)}+\frac {d q r \log (a+b x)}{b (b c-a d)}-\frac {d q r \log (c+d x)}{b (b c-a d)}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.06, size = 89, normalized size = 0.94 \[ \frac {r \left (\frac {d q \log (a+b x)}{b c-a d}-\frac {d q \log (c+d x)}{b c-a d}-\frac {p}{a+b x}\right )}{b}-\frac {\log \left (e \left (f (a+b x)^p (c+d x)^q\right )^r\right )}{b (a+b x)} \]
Antiderivative was successfully verified.
[In]
Integrate[Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]/(a + b*x)^2,x]
[Out]
(r*(-(p/(a + b*x)) + (d*q*Log[a + b*x])/(b*c - a*d) - (d*q*Log[c + d*x])/(b*c - a*d)))/b - Log[e*(f*(a + b*x)^
p*(c + d*x)^q)^r]/(b*(a + b*x))
________________________________________________________________________________________
fricas [A] time = 0.43, size = 120, normalized size = 1.26 \[ -\frac {{\left (b c - a d\right )} p r + {\left (b c - a d\right )} r \log \relax (f) - {\left (b d q r x + {\left (a d q - {\left (b c - a d\right )} p\right )} r\right )} \log \left (b x + a\right ) + {\left (b d q r x + b c q r\right )} \log \left (d x + c\right ) + {\left (b c - a d\right )} \log \relax (e)}{a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(b*x+a)^2,x, algorithm="fricas")
[Out]
-((b*c - a*d)*p*r + (b*c - a*d)*r*log(f) - (b*d*q*r*x + (a*d*q - (b*c - a*d)*p)*r)*log(b*x + a) + (b*d*q*r*x +
b*c*q*r)*log(d*x + c) + (b*c - a*d)*log(e))/(a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x)
________________________________________________________________________________________
giac [A] time = 0.19, size = 112, normalized size = 1.18 \[ \frac {d q r \log \left (b x + a\right )}{b^{2} c - a b d} - \frac {d q r \log \left (d x + c\right )}{b^{2} c - a b d} - \frac {p r \log \left (b x + a\right )}{b^{2} x + a b} - \frac {q r \log \left (d x + c\right )}{b^{2} x + a b} - \frac {p r + r \log \relax (f) + 1}{b^{2} x + a b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(b*x+a)^2,x, algorithm="giac")
[Out]
d*q*r*log(b*x + a)/(b^2*c - a*b*d) - d*q*r*log(d*x + c)/(b^2*c - a*b*d) - p*r*log(b*x + a)/(b^2*x + a*b) - q*r
*log(d*x + c)/(b^2*x + a*b) - (p*r + r*log(f) + 1)/(b^2*x + a*b)
________________________________________________________________________________________
maple [F] time = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (f \left (b x +a \right )^{p} \left (d x +c \right )^{q}\right )^{r}\right )}{\left (b x +a \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(b*x+a)^2,x)
[Out]
int(ln(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(b*x+a)^2,x)
________________________________________________________________________________________
maxima [A] time = 0.71, size = 99, normalized size = 1.04 \[ \frac {{\left (d f q {\left (\frac {\log \left (b x + a\right )}{b c - a d} - \frac {\log \left (d x + c\right )}{b c - a d}\right )} - \frac {b f p}{b^{2} x + a b}\right )} r}{b f} - \frac {\log \left (\left ({\left (b x + a\right )}^{p} {\left (d x + c\right )}^{q} f\right )^{r} e\right )}{{\left (b x + a\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(log(e*(f*(b*x+a)^p*(d*x+c)^q)^r)/(b*x+a)^2,x, algorithm="maxima")
[Out]
(d*f*q*(log(b*x + a)/(b*c - a*d) - log(d*x + c)/(b*c - a*d)) - b*f*p/(b^2*x + a*b))*r/(b*f) - log(((b*x + a)^p
*(d*x + c)^q*f)^r*e)/((b*x + a)*b)
________________________________________________________________________________________
mupad [B] time = 2.16, size = 99, normalized size = 1.04 \[ -\frac {\ln \left (e\,{\left (f\,{\left (a+b\,x\right )}^p\,{\left (c+d\,x\right )}^q\right )}^r\right )\,\left (x+\frac {a}{b}\right )}{{\left (a+b\,x\right )}^2}-\frac {p\,r}{x\,b^2+a\,b}+\frac {d\,q\,r\,\mathrm {atan}\left (\frac {b\,c\,2{}\mathrm {i}+b\,d\,x\,2{}\mathrm {i}}{a\,d-b\,c}+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{b\,\left (a\,d-b\,c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(log(e*(f*(a + b*x)^p*(c + d*x)^q)^r)/(a + b*x)^2,x)
[Out]
(d*q*r*atan((b*c*2i + b*d*x*2i)/(a*d - b*c) + 1i)*2i)/(b*(a*d - b*c)) - (p*r)/(a*b + b^2*x) - (log(e*(f*(a + b
*x)^p*(c + d*x)^q)^r)*(x + a/b))/(a + b*x)^2
________________________________________________________________________________________
sympy [A] time = 139.98, size = 1931, normalized size = 20.33 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(ln(e*(f*(b*x+a)**p*(d*x+c)**q)**r)/(b*x+a)**2,x)
[Out]
Piecewise((zoo*(2*0**p*zoo**(2*p)*c*q*r*log(c + d*x)/(0**p*zoo**p*d - d) - 0**p*zoo**(2*p)*d*q*r*x/(0**p*zoo**
p*d - d) - 3*0**p*zoo**p*c*q*r*log(c + d*x)/(0**p*zoo**p*d - d) + 0**p*zoo*zoo**p*d*p*r*x/(0**p*zoo**p*d - d)
+ 0**p*zoo**p*d*q*r*x*log(c + d*x)/(0**p*zoo**p*d - d) + 0**p*zoo**p*d*q*r*x/(0**p*zoo**p*d - d) + 0**p*zoo**p
*d*r*x*log(f)/(0**p*zoo**p*d - d) + 0**p*zoo**p*d*x*log(e)/(0**p*zoo**p*d - d) + c*q*r*log(c + d*x)/(0**p*zoo*
*p*d - d) + zoo*d*p*r*x/(0**p*zoo**p*d - d) - d*q*r*x*log(c + d*x)/(0**p*zoo**p*d - d) - d*r*x*log(f)/(0**p*zo
o**p*d - d) - d*x*log(e)/(0**p*zoo**p*d - d)), Eq(a, -b*x)), ((c*q*r*log(c + d*x)/d + p*r*x*log(a) + q*r*x*log
(c + d*x) - q*r*x + r*x*log(f) + x*log(e))/a**2, Eq(b, 0)), (-p*r*log(a + b*x)/(a*b + b**2*x) - p*r/(a*b + b**
2*x) - q*r*log(a*d/b + d*x)/(a*b + b**2*x) - q*r/(a*b + b**2*x) - r*log(f)/(a*b + b**2*x) - log(e)/(a*b + b**2
*x), Eq(c, a*d/b)), (-a*d*q*r*log(a/b + x)/(a**2*b*d - a*b**2*c + a*b**2*d*x - b**3*c*x) + b*c*q*r*log(c + d*x
)/(a**2*b*d - a*b**2*c + a*b**2*d*x - b**3*c*x) + b*c*r*log(f)/(a**2*b*d - a*b**2*c + a*b**2*d*x - b**3*c*x) +
b*c*log(e)/(a**2*b*d - a*b**2*c + a*b**2*d*x - b**3*c*x) + b*d*q*r*x*log(c + d*x)/(a**2*b*d - a*b**2*c + a*b*
*2*d*x - b**3*c*x) - b*d*q*r*x*log(a/b + x)/(a**2*b*d - a*b**2*c + a*b**2*d*x - b**3*c*x) + b*d*r*x*log(f)/(a*
*2*b*d - a*b**2*c + a*b**2*d*x - b**3*c*x) + b*d*x*log(e)/(a**2*b*d - a*b**2*c + a*b**2*d*x - b**3*c*x), Eq(p,
0)), (-p*r*log(a + b*x)/(a*b + b**2*x) - p*r/(a*b + b**2*x) - q*r*log(c)/(a*b + b**2*x) - r*log(f)/(a*b + b**
2*x) - log(e)/(a*b + b**2*x), Eq(d, 0)), (-a*d*p**2*r*log(a + b*x)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b
**3*c*p*x) - a*d*p**2*r/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) - a*d*p*q*r*log(a + b*x)/(a**2*b
*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) - a*d*p*q*r*log(c + d*x)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*
x - b**3*c*p*x) + a*d*p*q*r*log(c/d + x)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) - a*d*p*r*log(f
)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) - a*d*p*log(e)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x
- b**3*c*p*x) - a*d*q**2*r*log(c + d*x)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) + a*d*q**2*r*lo
g(c/d + x)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) - a*d*q*r*log(f)/(a**2*b*d*p - a*b**2*c*p + a
*b**2*d*p*x - b**3*c*p*x) - a*d*q*log(e)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) + b*c*p**2*r*lo
g(a + b*x)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) + b*c*p**2*r/(a**2*b*d*p - a*b**2*c*p + a*b**
2*d*p*x - b**3*c*p*x) + b*c*p*q*r*log(c + d*x)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) + b*c*p*r
*log(f)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) + b*c*p*log(e)/(a**2*b*d*p - a*b**2*c*p + a*b**2
*d*p*x - b**3*c*p*x) - b*d*p*q*r*x*log(a + b*x)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) + b*d*p*
q*r*x*log(c/d + x)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) - b*d*q**2*r*x*log(c + d*x)/(a**2*b*d
*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) + b*d*q**2*r*x*log(c/d + x)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p
*x - b**3*c*p*x) - b*d*q*r*x*log(f)/(a**2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x) - b*d*q*x*log(e)/(a*
*2*b*d*p - a*b**2*c*p + a*b**2*d*p*x - b**3*c*p*x), True))
________________________________________________________________________________________