3.181 \(\int \frac{\log (c (a+b x)^p)}{(d+e x)^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{\log \left (c (a+b x)^p\right )}{e (d+e x)}+\frac{b p \log (a+b x)}{e (b d-a e)}-\frac{b p \log (d+e x)}{e (b d-a e)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0274055, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2395, 36, 31} \[ -\frac{\log \left (c (a+b x)^p\right )}{e (d+e x)}+\frac{b p \log (a+b x)}{e (b d-a e)}-\frac{b p \log (d+e x)}{e (b d-a e)} \]

Antiderivative was successfully verified.

[In]

Int[Log[c*(a + b*x)^p]/(d + e*x)^2,x]

[Out]

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

Rule 2395

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]

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 31

Int[((a_) + (b_.)*(x_))^(-1), x_Symbol] :> Simp[Log[RemoveContent[a + b*x, x]]/b, x] /; FreeQ[{a, b}, x]

Rubi steps

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

Mathematica [A]  time = 0.0482561, size = 52, normalized size = 0.76 \[ \frac{\frac{b p (\log (a+b x)-\log (d+e x))}{b d-a e}-\frac{\log \left (c (a+b x)^p\right )}{d+e x}}{e} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [C]  time = 0.358, size = 329, normalized size = 4.8 \begin{align*} -{\frac{\ln \left ( \left ( bx+a \right ) ^{p} \right ) }{ \left ( ex+d \right ) e}}-{\frac{-i\pi \,ae{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( bx+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) +i\pi \,ae{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{2}+i\pi \,ae{\it csgn} \left ( i \left ( bx+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{2}-i\pi \,ae \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{3}+i\pi \,bd{\it csgn} \left ( ic \right ){\it csgn} \left ( i \left ( bx+a \right ) ^{p} \right ){\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) -i\pi \,bd{\it csgn} \left ( ic \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{2}-i\pi \,bd{\it csgn} \left ( i \left ( bx+a \right ) ^{p} \right ) \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{2}+i\pi \,bd \left ({\it csgn} \left ( ic \left ( bx+a \right ) ^{p} \right ) \right ) ^{3}-2\,\ln \left ( -ex-d \right ) bepx+2\,\ln \left ( bx+a \right ) bepx-2\,\ln \left ( -ex-d \right ) bdp+2\,\ln \left ( bx+a \right ) bdp+2\,\ln \left ( c \right ) ae-2\,\ln \left ( c \right ) bd}{ \left ( 2\,ex+2\,d \right ) e \left ( ae-bd \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(ln(c*(b*x+a)^p)/(e*x+d)^2,x)

[Out]

-1/e/(e*x+d)*ln((b*x+a)^p)-1/2*(-I*Pi*a*e*csgn(I*c)*csgn(I*(b*x+a)^p)*csgn(I*c*(b*x+a)^p)+I*Pi*a*e*csgn(I*c)*c
sgn(I*c*(b*x+a)^p)^2+I*Pi*a*e*csgn(I*(b*x+a)^p)*csgn(I*c*(b*x+a)^p)^2-I*Pi*a*e*csgn(I*c*(b*x+a)^p)^3+I*Pi*b*d*
csgn(I*c)*csgn(I*(b*x+a)^p)*csgn(I*c*(b*x+a)^p)-I*Pi*b*d*csgn(I*c)*csgn(I*c*(b*x+a)^p)^2-I*Pi*b*d*csgn(I*(b*x+
a)^p)*csgn(I*c*(b*x+a)^p)^2+I*Pi*b*d*csgn(I*c*(b*x+a)^p)^3-2*ln(-e*x-d)*b*e*p*x+2*ln(b*x+a)*b*e*p*x-2*ln(-e*x-
d)*b*d*p+2*ln(b*x+a)*b*d*p+2*ln(c)*a*e-2*ln(c)*b*d)/(e*x+d)/e/(a*e-b*d)

________________________________________________________________________________________

Maxima [A]  time = 1.01787, size = 88, normalized size = 1.29 \begin{align*} \frac{b p{\left (\frac{\log \left (b x + a\right )}{b d - a e} - \frac{\log \left (e x + d\right )}{b d - a e}\right )}}{e} - \frac{\log \left ({\left (b x + a\right )}^{p} c\right )}{{\left (e x + d\right )} e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^p)/(e*x+d)^2,x, algorithm="maxima")

[Out]

b*p*(log(b*x + a)/(b*d - a*e) - log(e*x + d)/(b*d - a*e))/e - log((b*x + a)^p*c)/((e*x + d)*e)

________________________________________________________________________________________

Fricas [A]  time = 1.96902, size = 176, normalized size = 2.59 \begin{align*} \frac{{\left (b e p x + a e p\right )} \log \left (b x + a\right ) -{\left (b e p x + b d p\right )} \log \left (e x + d\right ) -{\left (b d - a e\right )} \log \left (c\right )}{b d^{2} e - a d e^{2} +{\left (b d e^{2} - a e^{3}\right )} x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^p)/(e*x+d)^2,x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

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*(b*x+a)**p)/(e*x+d)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.19767, size = 123, normalized size = 1.81 \begin{align*} \frac{b p x e \log \left (b x + a\right ) - b p x e \log \left (x e + d\right ) + a p e \log \left (b x + a\right ) - b d p \log \left (x e + d\right ) - b d \log \left (c\right ) + a e \log \left (c\right )}{b d x e^{2} + b d^{2} e - a x e^{3} - a d e^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(log(c*(b*x+a)^p)/(e*x+d)^2,x, algorithm="giac")

[Out]

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