3.239 \(\int \frac{1}{(a c f+(b c+a d) f x+b d f x^2) (A+B \log (e (a+b x)^n (c+d x)^{-n}))} \, dx\)

Optimal. Leaf size=44 \[ \frac{\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B f n (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.0775872, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.02, Rules used = {6684} \[ \frac{\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B f n (b c-a d)} \]

Antiderivative was successfully verified.

[In]

Int[1/((a*c*f + (b*c + a*d)*f*x + b*d*f*x^2)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])),x]

[Out]

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

Rule 6684

Int[(u_)/(y_), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[q*Log[RemoveContent[y, x]], x] /;  !Fa
lseQ[q]]

Rubi steps

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

Mathematica [A]  time = 0.0544395, size = 42, normalized size = 0.95 \[ \frac{\log \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{f (b B c n-a B d n)} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a*c*f + (b*c + a*d)*f*x + b*d*f*x^2)*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])),x]

[Out]

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

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Maple [C]  time = 0.418, size = 371, normalized size = 8.4 \begin{align*} -{\frac{1}{Bfn \left ( ad-bc \right ) }\ln \left ( \ln \left ( \left ( dx+c \right ) ^{n} \right ) -{\frac{1}{2\,B} \left ( -iB\pi \,{\it csgn} \left ( ie \right ){\it csgn} \left ({\frac{i \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ){\it csgn} \left ({\frac{ie \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) +iB\pi \,{\it csgn} \left ( ie \right ) \left ({\it csgn} \left ({\frac{ie \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{2}-iB\pi \,{\it csgn} \left ( i \left ( bx+a \right ) ^{n} \right ){\it csgn} \left ({\frac{i}{ \left ( dx+c \right ) ^{n}}} \right ){\it csgn} \left ({\frac{i \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) +iB\pi \,{\it csgn} \left ( i \left ( bx+a \right ) ^{n} \right ) \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{2}+iB\pi \,{\it csgn} \left ({\frac{i}{ \left ( dx+c \right ) ^{n}}} \right ) \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{2}-iB\pi \, \left ({\it csgn} \left ({\frac{i \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{3}+iB\pi \,{\it csgn} \left ({\frac{i \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \left ({\it csgn} \left ({\frac{ie \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{2}-iB\pi \, \left ({\it csgn} \left ({\frac{ie \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{3}+2\,B\ln \left ( e \right ) +2\,B\ln \left ( \left ( bx+a \right ) ^{n} \right ) +2\,A \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2)/(A+B*ln(e*(b*x+a)^n/((d*x+c)^n))),x)

[Out]

-1/B/f/n/(a*d-b*c)*ln(ln((d*x+c)^n)-1/2*(-I*B*Pi*csgn(I*e)*csgn(I*(b*x+a)^n/((d*x+c)^n))*csgn(I*e/((d*x+c)^n)*
(b*x+a)^n)+I*B*Pi*csgn(I*e)*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-I*B*Pi*csgn(I*(b*x+a)^n)*csgn(I/((d*x+c)^n))*csg
n(I*(b*x+a)^n/((d*x+c)^n))+I*B*Pi*csgn(I*(b*x+a)^n)*csgn(I*(b*x+a)^n/((d*x+c)^n))^2+I*B*Pi*csgn(I/((d*x+c)^n))
*csgn(I*(b*x+a)^n/((d*x+c)^n))^2-I*B*Pi*csgn(I*(b*x+a)^n/((d*x+c)^n))^3+I*B*Pi*csgn(I*(b*x+a)^n/((d*x+c)^n))*c
sgn(I*e/((d*x+c)^n)*(b*x+a)^n)^2-I*B*Pi*csgn(I*e/((d*x+c)^n)*(b*x+a)^n)^3+2*B*ln(e)+2*B*ln((b*x+a)^n)+2*A)/B)

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Maxima [A]  time = 1.78736, size = 69, normalized size = 1.57 \begin{align*} \frac{\log \left (-\frac{B \log \left ({\left (b x + a\right )}^{n}\right ) - B \log \left ({\left (d x + c\right )}^{n}\right ) + B \log \left (e\right ) + A}{B}\right )}{{\left (b c f n - a d f n\right )} B} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2)/(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.487774, size = 108, normalized size = 2.45 \begin{align*} \frac{\log \left (-B n \log \left (b x + a\right ) + B n \log \left (d x + c\right ) - B \log \left (e\right ) - A\right )}{{\left (B b c - B a d\right )} f n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2)/(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="fricas")

[Out]

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

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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(1/(a*c*f+(a*d+b*c)*f*x+b*d*f*x**2)/(A+B*ln(e*(b*x+a)**n/((d*x+c)**n))),x)

[Out]

Timed out

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Giac [A]  time = 1.22325, size = 54, normalized size = 1.23 \begin{align*} \frac{\log \left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + A + B\right )}{B b c f n - B a d f n} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2)/(A+B*log(e*(b*x+a)^n/((d*x+c)^n))),x, algorithm="giac")

[Out]

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