Optimal. Leaf size=165 \[ \frac{n \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a e^2-b d e+c d^2}+\frac{n (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 e \left (a e^2-b d e+c d^2\right )}-\frac{n (2 c d-b e) \log (d+e x)}{e \left (a e^2-b d e+c d^2\right )}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{e (d+e x)} \]
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Rubi [A] time = 0.245178, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.261, Rules used = {2525, 800, 634, 618, 206, 628} \[ \frac{n \sqrt{b^2-4 a c} \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{a e^2-b d e+c d^2}+\frac{n (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 e \left (a e^2-b d e+c d^2\right )}-\frac{n (2 c d-b e) \log (d+e x)}{e \left (a e^2-b d e+c d^2\right )}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{e (d+e x)} \]
Antiderivative was successfully verified.
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Rule 2525
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{(d+e x)^2} \, dx &=-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{e (d+e x)}+\frac{n \int \frac{b+2 c x}{(d+e x) \left (a+b x+c x^2\right )} \, dx}{e}\\ &=-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{e (d+e x)}+\frac{n \int \left (\frac{e (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right ) (d+e x)}+\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (c d^2-b d e+a e^2\right ) \left (a+b x+c x^2\right )}\right ) \, dx}{e}\\ &=-\frac{(2 c d-b e) n \log (d+e x)}{e \left (c d^2-b d e+a e^2\right )}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{e (d+e x)}+\frac{n \int \frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{a+b x+c x^2} \, dx}{e \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{(2 c d-b e) n \log (d+e x)}{e \left (c d^2-b d e+a e^2\right )}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{e (d+e x)}-\frac{\left (\left (b^2-4 a c\right ) n\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 \left (c d^2-b d e+a e^2\right )}+\frac{((2 c d-b e) n) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 e \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{(2 c d-b e) n \log (d+e x)}{e \left (c d^2-b d e+a e^2\right )}+\frac{(2 c d-b e) n \log \left (a+b x+c x^2\right )}{2 e \left (c d^2-b d e+a e^2\right )}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{e (d+e x)}+\frac{\left (\left (b^2-4 a c\right ) n\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c d^2-b d e+a e^2}\\ &=\frac{\sqrt{b^2-4 a c} n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c d^2-b d e+a e^2}-\frac{(2 c d-b e) n \log (d+e x)}{e \left (c d^2-b d e+a e^2\right )}+\frac{(2 c d-b e) n \log \left (a+b x+c x^2\right )}{2 e \left (c d^2-b d e+a e^2\right )}-\frac{\log \left (d \left (a+b x+c x^2\right )^n\right )}{e (d+e x)}\\ \end{align*}
Mathematica [A] time = 0.333308, size = 166, normalized size = 1.01 \[ -\frac{n \sqrt{4 a c-b^2} \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{e (b d-a e)-c d^2}+\frac{n (b e-2 c d) \log (d+e x)}{e \left (e (a e-b d)+c d^2\right )}-\frac{n (b e-2 c d) \log (a+x (b+c x))}{2 e \left (e (a e-b d)+c d^2\right )}-\frac{\log \left (d (a+x (b+c x))^n\right )}{e (d+e x)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.148, size = 6540, normalized size = 39.6 \begin{align*} \text{output too large to display} \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 [A] time = 3.22275, size = 950, normalized size = 5.76 \begin{align*} \left [\frac{{\left (e^{2} n x + d e n\right )} \sqrt{b^{2} - 4 \, a c} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) +{\left ({\left (2 \, c d e - b e^{2}\right )} n x +{\left (b d e - 2 \, a e^{2}\right )} n\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left ({\left (2 \, c d e - b e^{2}\right )} n x +{\left (2 \, c d^{2} - b d e\right )} n\right )} \log \left (e x + d\right ) - 2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \log \left (d\right )}{2 \,{\left (c d^{3} e - b d^{2} e^{2} + a d e^{3} +{\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} x\right )}}, \frac{2 \,{\left (e^{2} n x + d e n\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) +{\left ({\left (2 \, c d e - b e^{2}\right )} n x +{\left (b d e - 2 \, a e^{2}\right )} n\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left ({\left (2 \, c d e - b e^{2}\right )} n x +{\left (2 \, c d^{2} - b d e\right )} n\right )} \log \left (e x + d\right ) - 2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \log \left (d\right )}{2 \,{\left (c d^{3} e - b d^{2} e^{2} + a d e^{3} +{\left (c d^{2} e^{2} - b d e^{3} + a e^{4}\right )} x\right )}}\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 [A] time = 1.39488, size = 383, normalized size = 2.32 \begin{align*} \frac{{\left (2 \, c d n - b n e\right )} \log \left (c x^{2} + b x + a\right )}{2 \,{\left (c d^{2} e - b d e^{2} + a e^{3}\right )}} - \frac{{\left (b^{2} n - 4 \, a c n\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} - \frac{2 \, c d n x e \log \left (x e + d\right ) + c d^{2} n \log \left (c x^{2} + b x + a\right ) - b d n e \log \left (c x^{2} + b x + a\right ) + 2 \, c d^{2} n \log \left (x e + d\right ) - b n x e^{2} \log \left (x e + d\right ) - b d n e \log \left (x e + d\right ) + a n e^{2} \log \left (c x^{2} + b x + a\right ) + c d^{2} \log \left (d\right ) - b d e \log \left (d\right ) + a e^{2} \log \left (d\right )}{c d^{2} x e^{2} + c d^{3} e - b d x e^{3} - b d^{2} e^{2} + a x e^{4} + a d e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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