Optimal. Leaf size=53 \[ \frac {\text {Int}\left (\frac {1}{x \log \left (c (a+b x)^n\right )},x\right )}{d}-\frac {e \text {Int}\left (\frac {1}{(d+e x) \log \left (c (a+b x)^n\right )},x\right )}{d} \]
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Rubi [A] time = 0.11, 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 = {} \[ \int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx &=\int \frac {1}{x (d+e x) \log \left (c (a+b x)^n\right )} \, dx\\ &=\int \left (\frac {1}{d x \log \left (c (a+b x)^n\right )}-\frac {e}{d (d+e x) \log \left (c (a+b x)^n\right )}\right ) \, dx\\ &=\frac {\int \frac {1}{x \log \left (c (a+b x)^n\right )} \, dx}{d}-\frac {e \int \frac {1}{(d+e x) \log \left (c (a+b x)^n\right )} \, dx}{d}\\ \end {align*}
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Mathematica [A] time = 0.70, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d x+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{{\left (e x^{2} + d x\right )} \log \left ({\left (b x + a\right )}^{n} c\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (e x^{2} + d x\right )} \log \left ({\left (b x + a\right )}^{n} c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.71, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e \,x^{2}+d x \right ) \ln \left (c \left (b x +a \right )^{n}\right )}\, dx \]
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 \[ \int \frac {1}{{\left (e x^{2} + d x\right )} \log \left ({\left (b x + a\right )}^{n} c\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{\ln \left (c\,{\left (a+b\,x\right )}^n\right )\,\left (e\,x^2+d\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x \left (d + e x\right ) \log {\left (c \left (a + b x\right )^{n} \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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