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

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Rubi [A]  time = 0.11, antiderivative size = 52, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.020, 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*}

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Mathematica [A]  time = 0.01, 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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fricas [A]  time = 1.03, size = 83, normalized size = 1.60 \[ \frac {{\left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + B \log \relax (e) + A\right )} {\left (B n \log \left (b x + a\right ) - B n \log \left (d x + c\right ) + B \log \relax (e) + A\right )}^{p}}{{\left (B b c - B a d\right )} f n p + {\left (B b c - B a d\right )} f n} \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p}}{b d f x^{2} + a c f + {\left (b c + a d\right )} f x}\,{d x} \]

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)

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maple [F]  time = 14.17, size = 0, normalized size = 0.00 \[ \int \frac {\left (B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+A \right )^{p}}{b d f \,x^{2}+a c f +\left (a d +b c \right ) f x}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p}}{b d f x^{2} + a c f + {\left (b c + a d\right )} f x}\,{d x} \]

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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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^p}{b\,d\,f\,x^2+f\,\left (a\,d+b\,c\right )\,x+a\,c\,f} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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