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

Optimal. Leaf size=52 \[ \frac{\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^{p+1}}{B f n (p+1) (b c-a d)} \]

[Out]

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

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Rubi [A]  time = 0.106263, antiderivative size = 52, 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 = {6686} \[ \frac{\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^{p+1}}{B f n (p+1) (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 6686

Int[(u_)*(y_)^(m_.), x_Symbol] :> With[{q = DerivativeDivides[y, u, x]}, Simp[(q*y^(m + 1))/(m + 1), x] /;  !F
alseQ[q]] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

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

Mathematica [A]  time = 0.0116366, size = 50, normalized size = 0.96 \[ \frac{\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^{p+1}}{f (p+1) (b B c n-a B d n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [F]  time = 2.884, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{acf+ \left ( ad+bc \right ) fx+bdf{x}^{2}} \left ( A+B\ln \left ({\frac{e \left ( bx+a \right ) ^{n}}{ \left ( dx+c \right ) ^{n}}} \right ) \right ) ^{p}}\, dx \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)))^p/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p}}{b d f x^{2} + a c f +{\left (b c + a d\right )} f x}\,{d 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)))^p/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2),x, algorithm="maxima")

[Out]

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^p/(b*d*f*x^2 + a*c*f + (b*c + a*d)*f*x), x)

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Fricas [A]  time = 0.546326, size = 209, normalized size = 4.02 \begin{align*} \frac{{\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 n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + B \log \left (e\right ) + A\right )}^{p}}{{\left (B b c - B a d\right )} f n p +{\left (B b c - B a d\right )} f n} \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)))^p/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (B \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p}}{b d f x^{2} + a c f +{\left (b c + a d\right )} f x}\,{d 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)))^p/(a*c*f+(a*d+b*c)*f*x+b*d*f*x^2),x, algorithm="giac")

[Out]

integrate((B*log((b*x + a)^n*e/(d*x + c)^n) + A)^p/(b*d*f*x^2 + a*c*f + (b*c + a*d)*f*x), x)