3.3.24 \(\int \frac {\log ^p(e (\frac {a+b x}{c+d x})^n)}{(a+b x) (c+d x)} \, dx\) [224]

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

[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 = 0.08, antiderivative size = 41, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {2561, 2339, 30} \begin {gather*} \frac {\log ^{p+1}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{n (p+1) (b c-a d)} \end {gather*}

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 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2339

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)/(x_), x_Symbol] :> Dist[1/(b*n), Subst[Int[x^p, x], x, a + b*L
og[c*x^n]], x] /; FreeQ[{a, b, c, n, p}, x]

Rule 2561

Int[((A_.) + Log[(e_.)*(((a_.) + (b_.)*(x_))/((c_.) + (d_.)*(x_)))^(n_.)]*(B_.))^(p_.)*((f_.) + (g_.)*(x_))^(m
_.)*((h_.) + (i_.)*(x_))^(q_.), x_Symbol] :> Dist[(b*c - a*d)^(m + q + 1)*(g/b)^m*(i/d)^q, Subst[Int[x^m*((A +
B*Log[e*x^n])^p/(b - d*x)^(m + q + 2)), x], x, (a + b*x)/(c + d*x)], x] /; FreeQ[{a, b, c, d, e, f, g, h, i,
A, B, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[b*f - a*g, 0] && EqQ[d*h - c*i, 0] && IntegersQ[m, q]

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 = 0.02, size = 40, normalized size = 0.98 \begin {gather*} \frac {\log ^{1+p}\left (e \left (\frac {a+b x}{c+d x}\right )^n\right )}{(b c n-a d n) (1+p)} \end {gather*}

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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Maple [F]
time = 0.22, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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Fricas [A]
time = 0.40, size = 63, normalized size = 1.54 \begin {gather*} \frac {{\left (n \log \left (\frac {b x + a}{d x + c}\right ) + 1\right )} {\left (n \log \left (\frac {b x + a}{d x + c}\right ) + 1\right )}^{p}}{{\left (b c - a d\right )} n p + {\left (b c - a d\right )} n} \end {gather*}

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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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Giac [A]
time = 3.38, size = 40, normalized size = 0.98 \begin {gather*} \frac {{\left (n \log \left (\frac {b x + a}{d x + c}\right ) + 1\right )}^{p + 1}}{{\left (b c n - a d n\right )} {\left (p + 1\right )}} \end {gather*}

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]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \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 \end {gather*}

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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