Optimal. Leaf size=83 \[ \frac {\text {Subst}\left (\text {Int}\left (\frac {1}{(a+b x) (f+g x) \left (A+B \log \left (e \left (\frac {a+b x}{c+d x}\right )^n\right )\right )},x\right ),e \left (\frac {a+b x}{c+d x}\right )^n,e (a+b x)^n (c+d x)^{-n}\right )}{h} \]
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Rubi [A]
time = 0.19, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps
used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {}
\begin {gather*} \int \frac {1}{\left (a f h+b g h x^2+h (b f x+a g x)\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (a f h+b g h x^2+h (b f x+a g x)\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx &=\int \frac {1}{h (a+b x) (f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx\\ &=\frac {\int \frac {1}{(a+b x) (f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx}{h}\\ &=\frac {\int \left (\frac {b}{(b f-a g) (a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}-\frac {g}{(b f-a g) (f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}\right ) \, dx}{h}\\ &=\frac {b \int \frac {1}{(a+b x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx}{(b f-a g) h}-\frac {g \int \frac {1}{(f+g x) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx}{(b f-a g) h}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a f h+b g h x^2+h (b f x+a g x)\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.47, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a f h +b g h \,x^{2}+h \left (a g x +b f x \right )\right ) \left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )\,\left (h\,\left (a\,g\,x+b\,f\,x\right )+a\,f\,h+b\,g\,h\,x^2\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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