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

Optimal. Leaf size=45 \[ \frac{\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{3 B n (b c-a d)} \]

[Out]

(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(3*B*(b*c - a*d)*n)

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Rubi [A]  time = 0.112263, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.025, Rules used = {6686} \[ \frac{\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{3 B n (b c-a d)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(3*B*(b*c - a*d)*n)

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 )^2}{(a+b x) (c+d x)} \, dx &=\frac{\left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3}{3 B (b c-a d) n}\\ \end{align*}

Mathematica [A]  time = 0.0154685, size = 43, normalized size = 0.96 \[ \frac{\left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^3}{3 (b B c n-a B d n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^3/(3*(b*B*c*n - a*B*d*n))

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Maple [C]  time = 2.427, size = 11062, normalized size = 245.8 \begin{align*} \text{output too large to display} \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)))^2/(b*x+a)/(d*x+c),x)

[Out]

result too large to display

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Maxima [B]  time = 1.36915, size = 522, normalized size = 11.6 \begin{align*} B^{2}{\left (\frac{\log \left (b x + a\right )}{b c - a d} - \frac{\log \left (d x + c\right )}{b c - a d}\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )^{2} + 2 \, A B{\left (\frac{\log \left (b x + a\right )}{b c - a d} - \frac{\log \left (d x + c\right )}{b c - a d}\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A^{2}{\left (\frac{\log \left (b x + a\right )}{b c - a d} - \frac{\log \left (d x + c\right )}{b c - a d}\right )} - \frac{1}{3} \, B^{2}{\left (\frac{3 \,{\left (e n \log \left (b x + a\right )^{2} - 2 \, e n \log \left (b x + a\right ) \log \left (d x + c\right ) + e n \log \left (d x + c\right )^{2}\right )} \log \left (\frac{{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right )}{{\left (b c - a d\right )} e} - \frac{e^{2} n^{2} \log \left (b x + a\right )^{3} - 3 \, e^{2} n^{2} \log \left (b x + a\right )^{2} \log \left (d x + c\right ) + 3 \, e^{2} n^{2} \log \left (b x + a\right ) \log \left (d x + c\right )^{2} - e^{2} n^{2} \log \left (d x + c\right )^{3}}{{\left (b c - a d\right )} e^{2}}\right )} - \frac{{\left (e n \log \left (b x + a\right )^{2} - 2 \, e n \log \left (b x + a\right ) \log \left (d x + c\right ) + e n \log \left (d x + c\right )^{2}\right )} A B}{{\left (b c - a d\right )} 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)))^2/(b*x+a)/(d*x+c),x, algorithm="maxima")

[Out]

B^2*(log(b*x + a)/(b*c - a*d) - log(d*x + c)/(b*c - a*d))*log((b*x + a)^n*e/(d*x + c)^n)^2 + 2*A*B*(log(b*x +
a)/(b*c - a*d) - log(d*x + c)/(b*c - a*d))*log((b*x + a)^n*e/(d*x + c)^n) + A^2*(log(b*x + a)/(b*c - a*d) - lo
g(d*x + c)/(b*c - a*d)) - 1/3*B^2*(3*(e*n*log(b*x + a)^2 - 2*e*n*log(b*x + a)*log(d*x + c) + e*n*log(d*x + c)^
2)*log((b*x + a)^n*e/(d*x + c)^n)/((b*c - a*d)*e) - (e^2*n^2*log(b*x + a)^3 - 3*e^2*n^2*log(b*x + a)^2*log(d*x
+ c) + 3*e^2*n^2*log(b*x + a)*log(d*x + c)^2 - e^2*n^2*log(d*x + c)^3)/((b*c - a*d)*e^2)) - (e*n*log(b*x + a)
^2 - 2*e*n*log(b*x + a)*log(d*x + c) + e*n*log(d*x + c)^2)*A*B/((b*c - a*d)*e)

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Fricas [B]  time = 0.484184, size = 466, normalized size = 10.36 \begin{align*} \frac{B^{2} n^{2} \log \left (b x + a\right )^{3} - B^{2} n^{2} \log \left (d x + c\right )^{3} + 3 \,{\left (B^{2} n \log \left (e\right ) + A B n\right )} \log \left (b x + a\right )^{2} + 3 \,{\left (B^{2} n^{2} \log \left (b x + a\right ) + B^{2} n \log \left (e\right ) + A B n\right )} \log \left (d x + c\right )^{2} + 3 \,{\left (B^{2} \log \left (e\right )^{2} + 2 \, A B \log \left (e\right ) + A^{2}\right )} \log \left (b x + a\right ) - 3 \,{\left (B^{2} n^{2} \log \left (b x + a\right )^{2} + B^{2} \log \left (e\right )^{2} + 2 \, A B \log \left (e\right ) + A^{2} + 2 \,{\left (B^{2} n \log \left (e\right ) + A B n\right )} \log \left (b x + a\right )\right )} \log \left (d x + c\right )}{3 \,{\left (b c - a d\right )}} \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)))^2/(b*x+a)/(d*x+c),x, algorithm="fricas")

[Out]

1/3*(B^2*n^2*log(b*x + a)^3 - B^2*n^2*log(d*x + c)^3 + 3*(B^2*n*log(e) + A*B*n)*log(b*x + a)^2 + 3*(B^2*n^2*lo
g(b*x + a) + B^2*n*log(e) + A*B*n)*log(d*x + c)^2 + 3*(B^2*log(e)^2 + 2*A*B*log(e) + A^2)*log(b*x + a) - 3*(B^
2*n^2*log(b*x + a)^2 + B^2*log(e)^2 + 2*A*B*log(e) + A^2 + 2*(B^2*n*log(e) + A*B*n)*log(b*x + a))*log(d*x + c)
)/(b*c - a*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)))**2/(b*x+a)/(d*x+c),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 )}^{2}}{{\left (b x + a\right )}{\left (d x + c\right )}}\,{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)))^2/(b*x+a)/(d*x+c),x, algorithm="giac")

[Out]

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