Optimal. Leaf size=45 \[ -\frac{1}{2 B n (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2} \]
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Rubi [A] time = 0.122359, 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} \[ -\frac{1}{2 B n (b c-a d) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2} \]
Antiderivative was successfully verified.
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Rule 6686
Rubi steps
\begin{align*} \int \frac{1}{(a+b x) (c+d x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^3} \, dx &=-\frac{1}{2 B (b c-a d) n \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^2}\\ \end{align*}
Mathematica [A] time = 0.0196387, size = 43, normalized size = 0.96 \[ -\frac{1}{2 (b B c n-a B d n) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^2} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.404, size = 366, normalized size = 8.1 \begin{align*} 2\,{\frac{1}{Bn \left ( ad-bc \right ) } \left ( 2\,A+2\,B\ln \left ( e \right ) +2\,B\ln \left ( \left ( bx+a \right ) ^{n} \right ) -2\,B\ln \left ( \left ( dx+c \right ) ^{n} \right ) -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 ( 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 ({\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} \right ) ^{-2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 2.09508, size = 297, normalized size = 6.6 \begin{align*} -\frac{1}{2 \,{\left ({\left (b c n - a d n\right )} B^{3} \log \left ({\left (b x + a\right )}^{n}\right )^{2} +{\left (b c n - a d n\right )} B^{3} \log \left ({\left (d x + c\right )}^{n}\right )^{2} +{\left (b c n - a d n\right )} A^{2} B + 2 \,{\left (b c n \log \left (e\right ) - a d n \log \left (e\right )\right )} A B^{2} +{\left (b c n \log \left (e\right )^{2} - a d n \log \left (e\right )^{2}\right )} B^{3} + 2 \,{\left ({\left (b c n - a d n\right )} A B^{2} +{\left (b c n \log \left (e\right ) - a d n \log \left (e\right )\right )} B^{3}\right )} \log \left ({\left (b x + a\right )}^{n}\right ) - 2 \,{\left ({\left (b c n - a d n\right )} B^{3} \log \left ({\left (b x + a\right )}^{n}\right ) +{\left (b c n - a d n\right )} A B^{2} +{\left (b c n \log \left (e\right ) - a d n \log \left (e\right )\right )} B^{3}\right )} \log \left ({\left (d x + c\right )}^{n}\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 0.495569, size = 520, normalized size = 11.56 \begin{align*} -\frac{1}{2 \,{\left ({\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (b x + a\right )^{2} +{\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (d x + c\right )^{2} +{\left (B^{3} b c - B^{3} a d\right )} n \log \left (e\right )^{2} + 2 \,{\left (A B^{2} b c - A B^{2} a d\right )} n \log \left (e\right ) +{\left (A^{2} B b c - A^{2} B a d\right )} n + 2 \,{\left ({\left (B^{3} b c - B^{3} a d\right )} n^{2} \log \left (e\right ) +{\left (A B^{2} b c - A B^{2} a d\right )} n^{2}\right )} \log \left (b x + a\right ) - 2 \,{\left ({\left (B^{3} b c - B^{3} a d\right )} n^{3} \log \left (b x + a\right ) +{\left (B^{3} b c - B^{3} a d\right )} n^{2} \log \left (e\right ) +{\left (A B^{2} b c - A B^{2} a d\right )} n^{2}\right )} \log \left (d x + c\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [B] time = 1.23947, size = 406, normalized size = 9.02 \begin{align*} -\frac{1}{2 \,{\left (B^{3} b c n^{3} \log \left (b x + a\right )^{2} - B^{3} a d n^{3} \log \left (b x + a\right )^{2} - 2 \, B^{3} b c n^{3} \log \left (b x + a\right ) \log \left (d x + c\right ) + 2 \, B^{3} a d n^{3} \log \left (b x + a\right ) \log \left (d x + c\right ) + B^{3} b c n^{3} \log \left (d x + c\right )^{2} - B^{3} a d n^{3} \log \left (d x + c\right )^{2} + 2 \, A B^{2} b c n^{2} \log \left (b x + a\right ) + 2 \, B^{3} b c n^{2} \log \left (b x + a\right ) - 2 \, A B^{2} a d n^{2} \log \left (b x + a\right ) - 2 \, B^{3} a d n^{2} \log \left (b x + a\right ) - 2 \, A B^{2} b c n^{2} \log \left (d x + c\right ) - 2 \, B^{3} b c n^{2} \log \left (d x + c\right ) + 2 \, A B^{2} a d n^{2} \log \left (d x + c\right ) + 2 \, B^{3} a d n^{2} \log \left (d x + c\right ) + A^{2} B b c n + 2 \, A B^{2} b c n + B^{3} b c n - A^{2} B a d n - 2 \, A B^{2} a d n - B^{3} a d n\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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