Optimal. Leaf size=66 \[ \frac {x^{m+1} \log \left (d \left (b x+c x^2\right )^n\right )}{m+1}+\frac {n x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {c x}{b}\right )}{(m+1)^2}-\frac {2 n x^{m+1}}{(m+1)^2} \]
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Rubi [A] time = 0.06, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2525, 80, 64} \[ \frac {x^{m+1} \log \left (d \left (b x+c x^2\right )^n\right )}{m+1}+\frac {n x^{m+1} \, _2F_1\left (1,m+1;m+2;-\frac {c x}{b}\right )}{(m+1)^2}-\frac {2 n x^{m+1}}{(m+1)^2} \]
Antiderivative was successfully verified.
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Rule 64
Rule 80
Rule 2525
Rubi steps
\begin {align*} \int x^m \log \left (d \left (b x+c x^2\right )^n\right ) \, dx &=\frac {x^{1+m} \log \left (d \left (b x+c x^2\right )^n\right )}{1+m}-\frac {n \int \frac {x^m (b+2 c x)}{b+c x} \, dx}{1+m}\\ &=-\frac {2 n x^{1+m}}{(1+m)^2}+\frac {x^{1+m} \log \left (d \left (b x+c x^2\right )^n\right )}{1+m}+\frac {(b n) \int \frac {x^m}{b+c x} \, dx}{1+m}\\ &=-\frac {2 n x^{1+m}}{(1+m)^2}+\frac {n x^{1+m} \, _2F_1\left (1,1+m;2+m;-\frac {c x}{b}\right )}{(1+m)^2}+\frac {x^{1+m} \log \left (d \left (b x+c x^2\right )^n\right )}{1+m}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 48, normalized size = 0.73 \[ \frac {x^{m+1} \left ((m+1) \log \left (d (x (b+c x))^n\right )+n \, _2F_1\left (1,m+1;m+2;-\frac {c x}{b}\right )-2 n\right )}{(m+1)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right ), x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \log \left ({\left (c x^{2} + b x\right )}^{n} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F(-1)] time = 180.00, size = 0, normalized size = 0.00 \[ \int x^{m} \ln \left (d \left (c \,x^{2}+b x \right )^{n}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {x x^{m} \log \left ({\left (c x + b\right )}^{n}\right ) + x x^{m} \log \left (x^{n}\right )}{m + 1} + \int \frac {{\left ({\left ({\left (m + 1\right )} \log \relax (d) - 2 \, n\right )} c x + {\left ({\left (m + 1\right )} \log \relax (d) - n\right )} b\right )} x^{m}}{c {\left (m + 1\right )} x + b {\left (m + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int x^m\,\ln \left (d\,{\left (c\,x^2+b\,x\right )}^n\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{m} \log {\left (d \left (b x + c x^{2}\right )^{n} \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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