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 = 0.22, antiderivative size = 157, normalized size of antiderivative = 1.00, 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 64
Rule 830
Rule 2525
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*}
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Mathematica [A] time = 0.19, 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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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (x^{m} \log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right ), x\right ) \]
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 \[ \int x^{m} \log \left ({\left (c x^{2} + b x + a\right )}^{n} d\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.40, size = 0, normalized size = 0.00 \[ \int x^{m} \ln \left (d \left (c \,x^{2}+b x +a \right )^{n}\right )\, dx \]
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 \[ \frac {x x^{m} \log \left ({\left (c x^{2} + b x + a\right )}^{n}\right )}{m + 1} + \int \frac {{\left ({\left ({\left (m + 1\right )} \log \relax (d) - 2 \, n\right )} c x^{2} + {\left ({\left (m + 1\right )} \log \relax (d) - n\right )} b x + a {\left (m + 1\right )} \log \relax (d)\right )} x^{m}}{c {\left (m + 1\right )} x^{2} + b {\left (m + 1\right )} x + a {\left (m + 1\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int x^m\,\ln \left (d\,{\left (c\,x^2+b\,x+a\right )}^n\right ) \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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