Optimal. Leaf size=157 \[ -\frac{2 c n x^{m+2} \, _2F_1\left (1,m+2;m+3;-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{(m+1) (m+2) \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c n x^{m+2} \, _2F_1\left (1,m+2;m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{(m+1) (m+2) \left (\sqrt{b^2-4 a c}+b\right )}+\frac{x^{m+1} \log \left (d \left (a+b x+c x^2\right )^n\right )}{m+1} \]
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Rubi [A] time = 0.220772, antiderivative size = 157, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {2525, 830, 64} \[ -\frac{2 c n x^{m+2} \, _2F_1\left (1,m+2;m+3;-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{(m+1) (m+2) \left (b-\sqrt{b^2-4 a c}\right )}-\frac{2 c n x^{m+2} \, _2F_1\left (1,m+2;m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{(m+1) (m+2) \left (\sqrt{b^2-4 a c}+b\right )}+\frac{x^{m+1} \log \left (d \left (a+b x+c x^2\right )^n\right )}{m+1} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 830
Rule 64
Rubi steps
\begin{align*} \int x^m \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac{x^{1+m} \log \left (d \left (a+b x+c x^2\right )^n\right )}{1+m}-\frac{n \int \frac{x^{1+m} (b+2 c x)}{a+b x+c x^2} \, dx}{1+m}\\ &=\frac{x^{1+m} \log \left (d \left (a+b x+c x^2\right )^n\right )}{1+m}-\frac{n \int \left (\frac{2 c x^{1+m}}{b-\sqrt{b^2-4 a c}+2 c x}+\frac{2 c x^{1+m}}{b+\sqrt{b^2-4 a c}+2 c x}\right ) \, dx}{1+m}\\ &=\frac{x^{1+m} \log \left (d \left (a+b x+c x^2\right )^n\right )}{1+m}-\frac{(2 c n) \int \frac{x^{1+m}}{b-\sqrt{b^2-4 a c}+2 c x} \, dx}{1+m}-\frac{(2 c n) \int \frac{x^{1+m}}{b+\sqrt{b^2-4 a c}+2 c x} \, dx}{1+m}\\ &=-\frac{2 c n x^{2+m} \, _2F_1\left (1,2+m;3+m;-\frac{2 c x}{b-\sqrt{b^2-4 a c}}\right )}{\left (b-\sqrt{b^2-4 a c}\right ) (1+m) (2+m)}-\frac{2 c n x^{2+m} \, _2F_1\left (1,2+m;3+m;-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )}{\left (b+\sqrt{b^2-4 a c}\right ) (1+m) (2+m)}+\frac{x^{1+m} \log \left (d \left (a+b x+c x^2\right )^n\right )}{1+m}\\ \end{align*}
Mathematica [A] time = 0.186494, size = 137, normalized size = 0.87 \[ -\frac{x^{m+1} \left (n x \left (\sqrt{b^2-4 a c}+b\right ) \, _2F_1\left (1,m+2;m+3;\frac{2 c x}{\sqrt{b^2-4 a c}-b}\right )+n x \left (b-\sqrt{b^2-4 a c}\right ) \, _2F_1\left (1,m+2;m+3;-\frac{2 c x}{b+\sqrt{b^2-4 a c}}\right )-2 a (m+2) \log \left (d (a+x (b+c x))^n\right )\right )}{2 a \left (m^2+3 m+2\right )} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.306, size = 0, normalized size = 0. \begin{align*} \int{x}^{m}\ln \left ( d \left ( c{x}^{2}+bx+a \right ) ^{n} \right ) \, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (x^{m} \log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right ), x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{m} \log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right )\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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