Optimal. Leaf size=88 \[ -\frac{\text{Unintegrable}\left (\frac{1}{\left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (c (a+b x)^n\right )},x\right )}{2 \sqrt{-d}}-\frac{\text{Unintegrable}\left (\frac{1}{\left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (c (a+b x)^n\right )},x\right )}{2 \sqrt{-d}} \]
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Rubi [A] time = 0.114693, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin{align*} \int \frac{1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx &=\int \left (\frac{\sqrt{-d}}{2 d \left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (c (a+b x)^n\right )}+\frac{\sqrt{-d}}{2 d \left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (c (a+b x)^n\right )}\right ) \, dx\\ &=-\frac{\int \frac{1}{\left (\sqrt{-d}-\sqrt{e} x\right ) \log \left (c (a+b x)^n\right )} \, dx}{2 \sqrt{-d}}-\frac{\int \frac{1}{\left (\sqrt{-d}+\sqrt{e} x\right ) \log \left (c (a+b x)^n\right )} \, dx}{2 \sqrt{-d}}\\ \end{align*}
Mathematica [A] time = 0.395655, size = 0, normalized size = 0. \[ \int \frac{1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \]
Verification is Not applicable to the result.
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Maple [A] time = 0.838, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( e{x}^{2}+d \right ) \ln \left ( c \left ( bx+a \right ) ^{n} \right ) }}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (e x^{2} + d\right )} \log \left ({\left (b x + a\right )}^{n} c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{{\left (e x^{2} + d\right )} \log \left ({\left (b x + a\right )}^{n} c\right )}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d + e x^{2}\right ) \log{\left (c \left (a + b x\right )^{n} \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (e x^{2} + d\right )} \log \left ({\left (b x + a\right )}^{n} c\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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