Optimal. Leaf size=226 \[ -\frac{n (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{6 c^3 e}-\frac{n x \left (-c e (2 a e+3 b d)+b^2 e^2+6 c^2 d^2\right )}{3 c^2}+\frac{n \sqrt{b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{3 c^3}+\frac{(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}-\frac{e n x^2 (6 c d-b e)}{6 c}-\frac{2}{9} e^2 n x^3 \]
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Rubi [A] time = 0.319069, antiderivative size = 226, 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} \[ -\frac{n (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{6 c^3 e}-\frac{n x \left (-c e (2 a e+3 b d)+b^2 e^2+6 c^2 d^2\right )}{3 c^2}+\frac{n \sqrt{b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{3 c^3}+\frac{(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}-\frac{e n x^2 (6 c d-b e)}{6 c}-\frac{2}{9} e^2 n x^3 \]
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 (d+e x)^2 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac{(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}-\frac{n \int \frac{(b+2 c x) (d+e x)^3}{a+b x+c x^2} \, dx}{3 e}\\ &=\frac{(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}-\frac{n \int \left (\frac{e \left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right )}{c^2}+\frac{e^2 (6 c d-b e) x}{c}+2 e^3 x^2+\frac{-a b^2 e^3-2 a c e \left (3 c d^2-a e^2\right )+b c d \left (c d^2+3 a e^2\right )+(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx}{3 e}\\ &=-\frac{\left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right ) n x}{3 c^2}-\frac{e (6 c d-b e) n x^2}{6 c}-\frac{2}{9} e^2 n x^3+\frac{(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}-\frac{n \int \frac{-a b^2 e^3-2 a c e \left (3 c d^2-a e^2\right )+b c d \left (c d^2+3 a e^2\right )+(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) x}{a+b x+c x^2} \, dx}{3 c^2 e}\\ &=-\frac{\left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right ) n x}{3 c^2}-\frac{e (6 c d-b e) n x^2}{6 c}-\frac{2}{9} e^2 n x^3+\frac{(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}-\frac{\left (\left (b^2-4 a c\right ) \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) n\right ) \int \frac{1}{a+b x+c x^2} \, dx}{6 c^3}-\frac{\left ((2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{6 c^3 e}\\ &=-\frac{\left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right ) n x}{3 c^2}-\frac{e (6 c d-b e) n x^2}{6 c}-\frac{2}{9} e^2 n x^3-\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n \log \left (a+b x+c x^2\right )}{6 c^3 e}+\frac{(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}+\frac{\left (\left (b^2-4 a c\right ) \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) n\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{3 c^3}\\ &=-\frac{\left (6 c^2 d^2+b^2 e^2-c e (3 b d+2 a e)\right ) n x}{3 c^2}-\frac{e (6 c d-b e) n x^2}{6 c}-\frac{2}{9} e^2 n x^3+\frac{\sqrt{b^2-4 a c} \left (3 c^2 d^2+b^2 e^2-c e (3 b d+a e)\right ) n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{3 c^3}-\frac{(2 c d-b e) \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right ) n \log \left (a+b x+c x^2\right )}{6 c^3 e}+\frac{(d+e x)^3 \log \left (d \left (a+b x+c x^2\right )^n\right )}{3 e}\\ \end{align*}
Mathematica [A] time = 0.377775, size = 204, normalized size = 0.9 \[ \frac{(d+e x)^3 \log \left (d (a+x (b+c x))^n\right )-\frac{n \left (c e x \left (-3 c e (4 a e+6 b d+b e x)+6 b^2 e^2+2 c^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 (2 c d-b e) \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right ) \log (a+x (b+c x))-6 e \sqrt{b^2-4 a c} \left (-c e (a e+3 b d)+b^2 e^2+3 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )\right )}{6 c^3}}{3 e} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.148, size = 7155, normalized size = 31.7 \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 = 2.25016, size = 1242, normalized size = 5.5 \begin{align*} \left [-\frac{4 \, c^{3} e^{2} n x^{3} + 3 \,{\left (6 \, c^{3} d e - b c^{2} e^{2}\right )} n x^{2} + 3 \,{\left (3 \, c^{2} d^{2} - 3 \, b c d e +{\left (b^{2} - a c\right )} e^{2}\right )} \sqrt{b^{2} - 4 \, a c} n \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 ) + 6 \,{\left (6 \, c^{3} d^{2} - 3 \, b c^{2} d e +{\left (b^{2} c - 2 \, a c^{2}\right )} e^{2}\right )} n x - 3 \,{\left (2 \, c^{3} e^{2} n x^{3} + 6 \, c^{3} d e n x^{2} + 6 \, c^{3} d^{2} n x +{\left (3 \, b c^{2} d^{2} - 3 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d e +{\left (b^{3} - 3 \, a b c\right )} e^{2}\right )} n\right )} \log \left (c x^{2} + b x + a\right ) - 6 \,{\left (c^{3} e^{2} x^{3} + 3 \, c^{3} d e x^{2} + 3 \, c^{3} d^{2} x\right )} \log \left (d\right )}{18 \, c^{3}}, -\frac{4 \, c^{3} e^{2} n x^{3} + 3 \,{\left (6 \, c^{3} d e - b c^{2} e^{2}\right )} n x^{2} - 6 \,{\left (3 \, c^{2} d^{2} - 3 \, b c d e +{\left (b^{2} - a c\right )} e^{2}\right )} \sqrt{-b^{2} + 4 \, a c} n \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 6 \,{\left (6 \, c^{3} d^{2} - 3 \, b c^{2} d e +{\left (b^{2} c - 2 \, a c^{2}\right )} e^{2}\right )} n x - 3 \,{\left (2 \, c^{3} e^{2} n x^{3} + 6 \, c^{3} d e n x^{2} + 6 \, c^{3} d^{2} n x +{\left (3 \, b c^{2} d^{2} - 3 \,{\left (b^{2} c - 2 \, a c^{2}\right )} d e +{\left (b^{3} - 3 \, a b c\right )} e^{2}\right )} n\right )} \log \left (c x^{2} + b x + a\right ) - 6 \,{\left (c^{3} e^{2} x^{3} + 3 \, c^{3} d e x^{2} + 3 \, c^{3} d^{2} x\right )} \log \left (d\right )}{18 \, c^{3}}\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 [B] time = 1.34656, size = 591, normalized size = 2.62 \begin{align*} \frac{b d^{2} n \log \left (c x^{2} + b x + a\right )}{2 \, c} - \frac{{\left (b^{2} d^{2} n - 4 \, a c d^{2} n\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c} - \frac{{\left (b^{2} d n e - 2 \, a c d n e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac{{\left (b^{3} d n e - 4 \, a b c d n e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{2}} + \frac{6 \, c^{2} n x^{3} e^{2} \log \left (c x^{2} + b x + a\right ) + 18 \, c^{2} d n x^{2} e \log \left (c x^{2} + b x + a\right ) - 4 \, c^{2} n x^{3} e^{2} - 18 \, c^{2} d n x^{2} e + 18 \, c^{2} d^{2} n x \log \left (c x^{2} + b x + a\right ) + 6 \, c^{2} x^{3} e^{2} \log \left (d\right ) + 18 \, c^{2} d x^{2} e \log \left (d\right ) - 36 \, c^{2} d^{2} n x + 3 \, b c n x^{2} e^{2} + 18 \, b c d n x e + 18 \, c^{2} d^{2} x \log \left (d\right ) - 6 \, b^{2} n x e^{2} + 12 \, a c n x e^{2}}{18 \, c^{2}} + \frac{{\left (b^{3} n e^{2} - 3 \, a b c n e^{2}\right )} \log \left (c x^{2} + b x + a\right )}{6 \, c^{3}} - \frac{{\left (b^{4} n e^{2} - 5 \, a b^{2} c n e^{2} + 4 \, a^{2} c^{2} n e^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{3 \, \sqrt{-b^{2} + 4 \, a c} c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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