Optimal. Leaf size=90 \[ -\frac {\text {Int}\left (\frac {1}{\left (\sqrt {-d}-\sqrt {e} x\right ) \log \left (c (a+b x)^n\right )},x\right )}{2 \sqrt {-d}}-\frac {\text {Int}\left (\frac {1}{\left (\sqrt {-d}+\sqrt {e} x\right ) \log \left (c (a+b x)^n\right )},x\right )}{2 \sqrt {-d}} \]
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Rubi [A]
time = 0.08, 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 (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \end {gather*}
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*}
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Mathematica [A]
time = 0.29, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d+e x^2\right ) \log \left (c (a+b x)^n\right )} \, dx \end {gather*}
Verification is not applicable to the result.
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Maple [A]
time = 0.18, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e \,x^{2}+d \right ) \ln \left (c \left (b x +a \right )^{n}\right )}\, dx\]
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 \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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (d + e x^{2}\right ) \log {\left (c \left (a + b x\right )^{n} \right )}}\, dx \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}{\ln \left (c\,{\left (a+b\,x\right )}^n\right )\,\left (e\,x^2+d\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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