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

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Rubi [A]  time = 0.03, antiderivative size = 57, normalized size of antiderivative = 1.00, 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 31

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

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]

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*}

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Mathematica [A]  time = 0.01, size = 57, normalized size = 1.00 \[ \frac {B (a+b x) \log \left (e (a+b x)^n (c+d x)^{-n}\right )}{b}-\frac {B n (b c-a d) \log (c+d x)}{b d}+A x \]

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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fricas [A]  time = 0.90, size = 59, normalized size = 1.04 \[ \frac {B b d x \log \relax (e) + A b d x + {\left (B b d n x + B a d n\right )} \log \left (b x + a\right ) - {\left (B b d n x + B b c n\right )} \log \left (d x + c\right )}{b d} \]

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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giac [A]  time = 0.20, size = 55, normalized size = 0.96 \[ {\left (n x \log \left (b x + a\right ) - n x \log \left (d x + c\right ) + \frac {a n \log \left (b x + a\right )}{b} - \frac {c n \log \left (-d x - c\right )}{d} + x\right )} B + A x \]

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

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maple [B]  time = 0.05, size = 123, normalized size = 2.16 \[ \frac {B \,a^{2} d n \ln \left (b x +a \right )}{\left (a d -b c \right ) b}-\frac {B a c n \ln \left (b x +a \right )}{a d -b c}-\frac {B a c n \ln \left (d x +c \right )}{a d -b c}+\frac {B b \,c^{2} n \ln \left (d x +c \right )}{\left (a d -b c \right ) d}+B x \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+A x \]

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*x*ln(e*(b*x+a)^n/((d*x+c)^n))-1/(a*d-b*c)*B*a*c*n*ln(d*x+c)+1/(a*d-b*c)*B*b*c^2/d*n*ln(d*x+c)+1/(a*d-b*c
)*B*a^2/b*d*n*ln(b*x+a)-1/(a*d-b*c)*B*a*c*n*ln(b*x+a)

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maxima [A]  time = 0.82, size = 59, normalized size = 1.04 \[ B x \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A x + \frac {{\left (\frac {a e n \log \left (b x + a\right )}{b} - \frac {c e n \log \left (d x + c\right )}{d}\right )} B}{e} \]

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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mupad [B]  time = 4.11, size = 53, normalized size = 0.93 \[ A\,x+B\,x\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )+\frac {B\,a\,n\,\ln \left (a+b\,x\right )}{b}-\frac {B\,c\,n\,\ln \left (c+d\,x\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: HeuristicGCDFailed} \]

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]

Exception raised: HeuristicGCDFailed

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