Optimal. Leaf size=338 \[ -\frac{n \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{8 c^4 e}-\frac{e n x^2 \left (-2 c e (a e+2 b d)+b^2 e^2+12 c^2 d^2\right )}{8 c^2}-\frac{n x \left (-2 c^2 d e (4 a e+3 b d)+b c e^2 (3 a e+4 b d)-b^3 e^3+8 c^3 d^3\right )}{4 c^3}+\frac{n \sqrt{b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{4 c^4}+\frac{(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}-\frac{e^2 n x^3 (8 c d-b e)}{12 c}-\frac{1}{8} e^3 n x^4 \]
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Rubi [A] time = 0.515817, antiderivative size = 338, 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} \[ -\frac{n \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{8 c^4 e}-\frac{e n x^2 \left (-2 c e (a e+2 b d)+b^2 e^2+12 c^2 d^2\right )}{8 c^2}-\frac{n x \left (-2 c^2 d e (4 a e+3 b d)+b c e^2 (3 a e+4 b d)-b^3 e^3+8 c^3 d^3\right )}{4 c^3}+\frac{n \sqrt{b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{4 c^4}+\frac{(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}-\frac{e^2 n x^3 (8 c d-b e)}{12 c}-\frac{1}{8} e^3 n x^4 \]
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)^3 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac{(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}-\frac{n \int \frac{(b+2 c x) (d+e x)^4}{a+b x+c x^2} \, dx}{4 e}\\ &=\frac{(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}-\frac{n \int \left (\frac{e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right )}{c^3}+\frac{e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x}{c^2}+\frac{e^3 (8 c d-b e) x^2}{c}+2 e^4 x^3+\frac{-4 a b^2 c d e^3+a b^3 e^4-8 a c^2 d e \left (c d^2-a e^2\right )+b c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx}{4 e}\\ &=-\frac{\left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) n x}{4 c^3}-\frac{e \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) n x^2}{8 c^2}-\frac{e^2 (8 c d-b e) n x^3}{12 c}-\frac{1}{8} e^3 n x^4+\frac{(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}-\frac{n \int \frac{-4 a b^2 c d e^3+a b^3 e^4-8 a c^2 d e \left (c d^2-a e^2\right )+b c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{a+b x+c x^2} \, dx}{4 c^3 e}\\ &=-\frac{\left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) n x}{4 c^3}-\frac{e \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) n x^2}{8 c^2}-\frac{e^2 (8 c d-b e) n x^3}{12 c}-\frac{1}{8} e^3 n x^4+\frac{(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}-\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n\right ) \int \frac{1}{a+b x+c x^2} \, dx}{8 c^4}-\frac{\left (\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{8 c^4 e}\\ &=-\frac{\left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) n x}{4 c^3}-\frac{e \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) n x^2}{8 c^2}-\frac{e^2 (8 c d-b e) n x^3}{12 c}-\frac{1}{8} e^3 n x^4-\frac{\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{8 c^4 e}+\frac{(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}+\frac{\left (\left (b^2-4 a c\right ) (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (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 )}{4 c^4}\\ &=-\frac{\left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) n x}{4 c^3}-\frac{e \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) n x^2}{8 c^2}-\frac{e^2 (8 c d-b e) n x^3}{12 c}-\frac{1}{8} e^3 n x^4+\frac{\sqrt{b^2-4 a c} (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{4 c^4}-\frac{\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{8 c^4 e}+\frac{(d+e x)^4 \log \left (d \left (a+b x+c x^2\right )^n\right )}{4 e}\\ \end{align*}
Mathematica [A] time = 0.504436, size = 324, normalized size = 0.96 \[ \frac{(d+e x)^4 \log \left (d (a+x (b+c x))^n\right )-\frac{n \left (3 \left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log (a+x (b+c x))+3 c^2 e^2 x^2 \left (-2 c e (a e+2 b d)+b^2 e^2+12 c^2 d^2\right )+6 c e x \left (-2 c^2 d e (4 a e+3 b d)+b c e^2 (3 a e+4 b d)-b^3 e^3+8 c^3 d^3\right )-6 e \sqrt{b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+2 c^3 e^3 x^3 (8 c d-b e)+3 c^4 e^4 x^4\right )}{6 c^4}}{4 e} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.196, size = 16059, normalized size = 47.5 \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 = 2.4354, size = 1858, normalized size = 5.5 \begin{align*} \text{result too large to display} \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 = 1.39151, size = 923, normalized size = 2.73 \begin{align*} \frac{b d^{3} n \log \left (c x^{2} + b x + a\right )}{2 \, c} - \frac{{\left (b^{2} d^{3} n - 4 \, a c d^{3} n\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c} - \frac{3 \,{\left (b^{2} d^{2} n e - 2 \, a c d^{2} n e\right )} \log \left (c x^{2} + b x + a\right )}{4 \, c^{2}} + \frac{3 \,{\left (b^{3} d^{2} n e - 4 \, a b c d^{2} n e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{2}} + \frac{{\left (b^{3} d n e^{2} - 3 \, a b c d n e^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} - \frac{{\left (b^{4} d n e^{2} - 5 \, a b^{2} c d n e^{2} + 4 \, a^{2} c^{2} d n e^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{3}} + \frac{6 \, c^{3} n x^{4} e^{3} \log \left (c x^{2} + b x + a\right ) + 24 \, c^{3} d n x^{3} e^{2} \log \left (c x^{2} + b x + a\right ) + 36 \, c^{3} d^{2} n x^{2} e \log \left (c x^{2} + b x + a\right ) - 3 \, c^{3} n x^{4} e^{3} - 16 \, c^{3} d n x^{3} e^{2} - 36 \, c^{3} d^{2} n x^{2} e + 24 \, c^{3} d^{3} n x \log \left (c x^{2} + b x + a\right ) + 6 \, c^{3} x^{4} e^{3} \log \left (d\right ) + 24 \, c^{3} d x^{3} e^{2} \log \left (d\right ) + 36 \, c^{3} d^{2} x^{2} e \log \left (d\right ) - 48 \, c^{3} d^{3} n x + 2 \, b c^{2} n x^{3} e^{3} + 12 \, b c^{2} d n x^{2} e^{2} + 36 \, b c^{2} d^{2} n x e + 24 \, c^{3} d^{3} x \log \left (d\right ) - 3 \, b^{2} c n x^{2} e^{3} + 6 \, a c^{2} n x^{2} e^{3} - 24 \, b^{2} c d n x e^{2} + 48 \, a c^{2} d n x e^{2} + 6 \, b^{3} n x e^{3} - 18 \, a b c n x e^{3}}{24 \, c^{3}} - \frac{{\left (b^{4} n e^{3} - 4 \, a b^{2} c n e^{3} + 2 \, a^{2} c^{2} n e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{8 \, c^{4}} + \frac{{\left (b^{5} n e^{3} - 6 \, a b^{3} c n e^{3} + 8 \, a^{2} b c^{2} n e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{4 \, \sqrt{-b^{2} + 4 \, a c} c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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