Optimal. Leaf size=485 \[ -\frac{n (2 c d-b e) \left (c^2 e^2 \left (5 a^2 e^2+10 a b d e+4 b^2 d^2\right )-b^2 c e^3 (5 a e+3 b d)-2 c^3 d^2 e (5 a e+b d)+b^4 e^4+c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{10 c^5 e}-\frac{n x \left (c^2 e^2 \left (2 a^2 e^2+15 a b d e+10 b^2 d^2\right )-b^2 c e^3 (4 a e+5 b d)-10 c^3 d^2 e (2 a e+b d)+b^4 e^4+10 c^4 d^4\right )}{5 c^4}+\frac{n \sqrt{b^2-4 a c} \left (c^2 e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )-b^2 c e^3 (3 a e+5 b d)-10 c^3 d^2 e (a e+b d)+b^4 e^4+5 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{5 c^5}-\frac{e^2 n x^3 \left (-c e (2 a e+5 b d)+b^2 e^2+20 c^2 d^2\right )}{15 c^2}-\frac{e n x^2 \left (-10 c^2 d e (a e+b d)+b c e^2 (3 a e+5 b d)-b^3 e^3+20 c^3 d^3\right )}{10 c^3}+\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac{e^3 n x^4 (10 c d-b e)}{20 c}-\frac{2}{25} e^4 n x^5 \]
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Rubi [A] time = 2.05768, antiderivative size = 485, 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} \[ -\frac{n (2 c d-b e) \left (c^2 e^2 \left (5 a^2 e^2+10 a b d e+4 b^2 d^2\right )-b^2 c e^3 (5 a e+3 b d)-2 c^3 d^2 e (5 a e+b d)+b^4 e^4+c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{10 c^5 e}-\frac{n x \left (c^2 e^2 \left (2 a^2 e^2+15 a b d e+10 b^2 d^2\right )-b^2 c e^3 (4 a e+5 b d)-10 c^3 d^2 e (2 a e+b d)+b^4 e^4+10 c^4 d^4\right )}{5 c^4}+\frac{n \sqrt{b^2-4 a c} \left (c^2 e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )-b^2 c e^3 (3 a e+5 b d)-10 c^3 d^2 e (a e+b d)+b^4 e^4+5 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{5 c^5}-\frac{e^2 n x^3 \left (-c e (2 a e+5 b d)+b^2 e^2+20 c^2 d^2\right )}{15 c^2}-\frac{e n x^2 \left (-10 c^2 d e (a e+b d)+b c e^2 (3 a e+5 b d)-b^3 e^3+20 c^3 d^3\right )}{10 c^3}+\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac{e^3 n x^4 (10 c d-b e)}{20 c}-\frac{2}{25} e^4 n x^5 \]
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)^4 \log \left (d \left (a+b x+c x^2\right )^n\right ) \, dx &=\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac{n \int \frac{(b+2 c x) (d+e x)^5}{a+b x+c x^2} \, dx}{5 e}\\ &=\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac{n \int \left (\frac{e \left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right )}{c^4}+\frac{e^2 \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) x}{c^3}+\frac{e^3 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) x^2}{c^2}+\frac{e^4 (10 c d-b e) x^3}{c}+2 e^5 x^4+\frac{5 a b^3 c d e^4-a b^4 e^5-2 a b^2 c e^3 \left (5 c d^2-2 a e^2\right )+b c^2 d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c^2 e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )+(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) x}{c^4 \left (a+b x+c x^2\right )}\right ) \, dx}{5 e}\\ &=-\frac{\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac{e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac{e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac{e^3 (10 c d-b e) n x^4}{20 c}-\frac{2}{25} e^4 n x^5+\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac{n \int \frac{5 a b^3 c d e^4-a b^4 e^5-2 a b^2 c e^3 \left (5 c d^2-2 a e^2\right )+b c^2 d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c^2 e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )+(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) x}{a+b x+c x^2} \, dx}{5 c^4 e}\\ &=-\frac{\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac{e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac{e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac{e^3 (10 c d-b e) n x^4}{20 c}-\frac{2}{25} e^4 n x^5+\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}-\frac{\left ((2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) n\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{10 c^5 e}-\frac{\left (\left (-b (2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right )+2 c \left (5 a b^3 c d e^4-a b^4 e^5-2 a b^2 c e^3 \left (5 c d^2-2 a e^2\right )+b c^2 d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c^2 e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )\right )\right ) n\right ) \int \frac{1}{a+b x+c x^2} \, dx}{10 c^5 e}\\ &=-\frac{\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac{e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac{e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac{e^3 (10 c d-b e) n x^4}{20 c}-\frac{2}{25} e^4 n x^5-\frac{(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{10 c^5 e}+\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}+\frac{\left (\left (-b (2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right )+2 c \left (5 a b^3 c d e^4-a b^4 e^5-2 a b^2 c e^3 \left (5 c d^2-2 a e^2\right )+b c^2 d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )-2 a c^2 e \left (5 c^2 d^4-10 a c d^2 e^2+a^2 e^4\right )\right )\right ) n\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{5 c^5 e}\\ &=-\frac{\left (10 c^4 d^4+b^4 e^4-10 c^3 d^2 e (b d+2 a e)-b^2 c e^3 (5 b d+4 a e)+c^2 e^2 \left (10 b^2 d^2+15 a b d e+2 a^2 e^2\right )\right ) n x}{5 c^4}-\frac{e \left (20 c^3 d^3-b^3 e^3-10 c^2 d e (b d+a e)+b c e^2 (5 b d+3 a e)\right ) n x^2}{10 c^3}-\frac{e^2 \left (20 c^2 d^2+b^2 e^2-c e (5 b d+2 a e)\right ) n x^3}{15 c^2}-\frac{e^3 (10 c d-b e) n x^4}{20 c}-\frac{2}{25} e^4 n x^5+\frac{\sqrt{b^2-4 a c} \left (5 c^4 d^4-10 b c^3 d^3 e+10 b^2 c^2 d^2 e^2-10 a c^3 d^2 e^2-5 b^3 c d e^3+10 a b c^2 d e^3+b^4 e^4-3 a b^2 c e^4+a^2 c^2 e^4\right ) n \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{5 c^5}-\frac{(2 c d-b e) \left (c^4 d^4+b^4 e^4-2 c^3 d^2 e (b d+5 a e)-b^2 c e^3 (3 b d+5 a e)+c^2 e^2 \left (4 b^2 d^2+10 a b d e+5 a^2 e^2\right )\right ) n \log \left (a+b x+c x^2\right )}{10 c^5 e}+\frac{(d+e x)^5 \log \left (d \left (a+b x+c x^2\right )^n\right )}{5 e}\\ \end{align*}
Mathematica [A] time = 1.90509, size = 468, normalized size = 0.96 \[ \frac{(d+e x)^5 \log \left (d (a+x (b+c x))^n\right )-\frac{n \left (60 c e x \left (c^2 e^2 \left (2 a^2 e^2+15 a b d e+10 b^2 d^2\right )-b^2 c e^3 (4 a e+5 b d)-10 c^3 d^2 e (2 a e+b d)+b^4 e^4+10 c^4 d^4\right )+30 (2 c d-b e) \left (c^2 e^2 \left (5 a^2 e^2+10 a b d e+4 b^2 d^2\right )-b^2 c e^3 (5 a e+3 b d)-2 c^3 d^2 e (5 a e+b d)+b^4 e^4+c^4 d^4\right ) \log (a+x (b+c x))-60 e \sqrt{b^2-4 a c} \left (c^2 e^2 \left (a^2 e^2+10 a b d e+10 b^2 d^2\right )-b^2 c e^3 (3 a e+5 b d)-10 c^3 d^2 e (a e+b d)+b^4 e^4+5 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )+20 c^3 e^3 x^3 \left (-c e (2 a e+5 b d)+b^2 e^2+20 c^2 d^2\right )+30 c^2 e^2 x^2 \left (-10 c^2 d e (a e+b d)+b c e^2 (3 a e+5 b d)-b^3 e^3+20 c^3 d^3\right )+15 c^4 e^4 x^4 (10 c d-b e)+24 c^5 e^5 x^5\right )}{60 c^5}}{5 e} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.23, size = 31895, normalized size = 65.8 \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 = 2.99206, size = 2700, normalized size = 5.57 \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 = 1.38521, size = 1338, normalized size = 2.76 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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