Optimal. Leaf size=344 \[ -\frac{2 e \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]
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Rubi [A] time = 0.652194, antiderivative size = 344, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {740, 800, 634, 618, 206, 628} \[ -\frac{2 e \left (-c e (3 a e+b d)+b^2 e^2+c^2 d^2\right )}{\left (b^2-4 a c\right ) (d+e x) \left (a e^2-b d e+c d^2\right )^2}+\frac{2 \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (a e^2-b d e+c d^2\right )^3}-\frac{2 a c e+b^2 (-e)+c x (2 c d-b e)+b c d}{\left (b^2-4 a c\right ) (d+e x) \left (a+b x+c x^2\right ) \left (a e^2-b d e+c d^2\right )}-\frac{e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{\left (a e^2-b d e+c d^2\right )^3}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{\left (a e^2-b d e+c d^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 740
Rule 800
Rule 634
Rule 618
Rule 206
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(d+e x)^2 \left (a+b x+c x^2\right )^2} \, dx &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}-\frac{\int \frac{2 \left (c^2 d^2-b^2 e^2+3 a c e^2\right )+2 c e (2 c d-b e) x}{(d+e x)^2 \left (a+b x+c x^2\right )} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}-\frac{\int \left (\frac{2 e^2 \left (-c^2 d^2-b^2 e^2+c e (b d+3 a e)\right )}{\left (c d^2-b d e+a e^2\right ) (d+e x)^2}+\frac{2 \left (b^2-4 a c\right ) e^4 (-2 c d+b e)}{\left (c d^2-b d e+a e^2\right )^2 (d+e x)}+\frac{2 \left (c^4 d^4-b^4 e^4-2 c^3 d^2 e (b d-3 a e)-a c^2 e^3 (10 b d+3 a e)+b^2 c e^3 (2 b d+5 a e)+c \left (b^2-4 a c\right ) e^3 (2 c d-b e) x\right )}{\left (c d^2-b d e+a e^2\right )^2 \left (a+b x+c x^2\right )}\right ) \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )}\\ &=-\frac{2 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac{2 \int \frac{c^4 d^4-b^4 e^4-2 c^3 d^2 e (b d-3 a e)-a c^2 e^3 (10 b d+3 a e)+b^2 c e^3 (2 b d+5 a e)+c \left (b^2-4 a c\right ) e^3 (2 c d-b e) x}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac{\left (e^3 (2 c d-b e)\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{\left (c d^2-b d e+a e^2\right )^3}-\frac{\left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \int \frac{1}{a+b x+c x^2} \, dx}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac{e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{\left (c d^2-b d e+a e^2\right )^3}+\frac{\left (2 \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^3}\\ &=-\frac{2 e \left (c^2 d^2+b^2 e^2-c e (b d+3 a e)\right )}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right )^2 (d+e x)}-\frac{b c d-b^2 e+2 a c e+c (2 c d-b e) x}{\left (b^2-4 a c\right ) \left (c d^2-b d e+a e^2\right ) (d+e x) \left (a+b x+c x^2\right )}+\frac{2 \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{\left (b^2-4 a c\right )^{3/2} \left (c d^2-b d e+a e^2\right )^3}+\frac{2 e^3 (2 c d-b e) \log (d+e x)}{\left (c d^2-b d e+a e^2\right )^3}-\frac{e^3 (2 c d-b e) \log \left (a+b x+c x^2\right )}{\left (c d^2-b d e+a e^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.74381, size = 339, normalized size = 0.99 \[ \frac{b c \left (3 a e^2-c d (d-2 e x)\right )-2 c^2 \left (a e (2 d-e x)+c d^2 x\right )+b^2 c e (2 d-e x)+b^3 \left (-e^2\right )}{\left (b^2-4 a c\right ) (a+x (b+c x)) \left (e (a e-b d)+c d^2\right )^2}-\frac{2 \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2} \left (e (b d-a e)-c d^2\right )^3}-\frac{e^3}{(d+e x) \left (e (a e-b d)+c d^2\right )^2}-\frac{2 e^3 (b e-2 c d) \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^3}+\frac{e^3 (b e-2 c d) \log (a+x (b+c x))}{\left (e (a e-b d)+c d^2\right )^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.174, size = 1617, normalized size = 4.7 \begin{align*} \text{result 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 [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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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 = 1.18058, size = 1220, normalized size = 3.55 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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