3.297 \(\int (A+B \log (e (a+b x)^n (c+d x)^{-n})) \, dx\)

Optimal. Leaf size=57 \[ \frac{B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{B n (b c-a d) \log (c+d x)}{b d}+A x \]

[Out]

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

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Rubi [A]  time = 0.0299171, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {2486, 31} \[ \frac{B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{B n (b c-a d) \log (c+d x)}{b d}+A x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 2486

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]^(s_.), x_Symbol] :> Simp[((
a + b*x)*Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r]^s)/b, x] + Dist[(q*r*s*(b*c - a*d))/b, Int[Log[e*(f*(a + b*x)^p*
(c + d*x)^q)^r]^(s - 1)/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, p, q, r, s}, x] && NeQ[b*c - a*d, 0] &&
EqQ[p + q, 0] && IGtQ[s, 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 \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right ) \, dx &=A x+B \int \log \left (e (a+b x)^n (c+d x)^{-n}\right ) \, dx\\ &=A x+\frac{B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{(B (b c-a d) n) \int \frac{1}{c+d x} \, dx}{b}\\ &=A x-\frac{B (b c-a d) n \log (c+d x)}{b d}+\frac{B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}\\ \end{align*}

Mathematica [A]  time = 0.0107576, size = 57, normalized size = 1. \[ \frac{B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac{B n (b c-a d) \log (c+d x)}{b d}+A x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.058, size = 123, normalized size = 2.2 \begin{align*} Ax+B\ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) x+{\frac{Bn{a}^{2}\ln \left ( bx+a \right ) d}{b \left ( ad-bc \right ) }}-{\frac{Bna\ln \left ( bx+a \right ) c}{ad-bc}}-{\frac{Bnc\ln \left ( dx+c \right ) a}{ad-bc}}+{\frac{Bn{c}^{2}\ln \left ( dx+c \right ) b}{d \left ( ad-bc \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

A*x+B*ln(e*(b*x+a)^n/((d*x+c)^n))*x+B*n*a^2/(a*d-b*c)/b*ln(b*x+a)*d-B*n*a/(a*d-b*c)*ln(b*x+a)*c-B*n*c/(a*d-b*c
)*ln(d*x+c)*a+B*n*c^2/(a*d-b*c)/d*ln(d*x+c)*b

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Maxima [A]  time = 1.12728, size = 80, normalized size = 1.4 \begin{align*} B x \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A x + \frac{{\left (\frac{a e n \log \left (b x + a\right )}{b} - \frac{c e n \log \left (d x + c\right )}{d}\right )} B}{e} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.03051, size = 146, normalized size = 2.56 \begin{align*} \frac{B b d x \log \left (e\right ) + A b d x +{\left (B b d n x + B a d n\right )} \log \left (b x + a\right ) -{\left (B b d n x + B b c n\right )} \log \left (d x + c\right )}{b d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(B*b*d*x*log(e) + A*b*d*x + (B*b*d*n*x + B*a*d*n)*log(b*x + a) - (B*b*d*n*x + B*b*c*n)*log(d*x + c))/(b*d)

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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(A+B*ln(e*(b*x+a)**n/((d*x+c)**n)),x)

[Out]

Timed out

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Giac [A]  time = 1.32551, size = 74, normalized size = 1.3 \begin{align*}{\left (n x \log \left (b x + a\right ) - n x \log \left (d x + c\right ) + \frac{a n \log \left (b x + a\right )}{b} - \frac{c n \log \left (-d x - c\right )}{d} + x\right )} B + A x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(n*x*log(b*x + a) - n*x*log(d*x + c) + a*n*log(b*x + a)/b - c*n*log(-d*x - c)/d + x)*B + A*x