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

Optimal. Leaf size=41 \[ \frac {\log ^{p+1}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (p+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.11, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.029, Rules used = {2505} \[ \frac {\log ^{p+1}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (p+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[e*((a + b*x)/(c + d*x))^n]^(1 + p)/((b*c - a*d)*n*(1 + p))

Rule 2505

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.)*(u_), x_Symbol] :> Wi
th[{h = Simplify[u*(a + b*x)*(c + d*x)]}, Simp[(h*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^(s + 1))/(p*r*(s + 1)*(
b*c - a*d)), x] /; FreeQ[h, x]] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] && EqQ[p + q,
0] && NeQ[s, -1]

Rubi steps

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

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Mathematica [A]  time = 0.03, size = 40, normalized size = 0.98 \[ \frac {\log ^{p+1}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(p+1) (b c n-a d n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

Log[e*((a + b*x)/(c + d*x))^n]^(1 + p)/((b*c*n - a*d*n)*(1 + p))

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fricas [A]  time = 0.69, size = 65, normalized size = 1.59 \[ \frac {{\left (n \log \left (\frac {b x + a}{d x + c}\right ) + \log \relax (e)\right )} {\left (n \log \left (\frac {b x + a}{d x + c}\right ) + \log \relax (e)\right )}^{p}}{{\left (b c - a d\right )} n p + {\left (b c - a d\right )} n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{p}}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(e*((b*x + a)/(d*x + c))^n)^p/((b*x + a)*(d*x + c)), x)

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maple [F]  time = 0.50, size = 0, normalized size = 0.00 \[ \int \frac {\ln \left (e \left (\frac {b x +a}{d x +c}\right )^{n}\right )^{p}}{\left (b x +a \right ) \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\log \left (e \left (\frac {b x + a}{d x + c}\right )^{n}\right )^{p}}{{\left (b x + a\right )} {\left (d x + c\right )}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(log(e*((b*x + a)/(d*x + c))^n)^p/((b*x + a)*(d*x + c)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\ln \left (e\,{\left (\frac {a+b\,x}{c+d\,x}\right )}^n\right )}^p}{\left (a+b\,x\right )\,\left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(e*((a + b*x)/(c + d*x))^n)^p/((a + b*x)*(c + d*x)),x)

[Out]

int(log(e*((a + b*x)/(c + d*x))^n)^p/((a + b*x)*(c + d*x)), x)

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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