Optimal. Leaf size=153 \[ \frac{x \left (a C e^2+c \left (3 C d^2-e (2 B d-A e)\right )\right )}{e^4}-\frac{\left (a e^2+c d^2\right ) \left (A e^2-B d e+C d^2\right )}{e^5 (d+e x)}-\frac{\log (d+e x) \left (a e^2 (2 C d-B e)+c d \left (4 C d^2-e (3 B d-2 A e)\right )\right )}{e^5}-\frac{c x^2 (2 C d-B e)}{2 e^3}+\frac{c C x^3}{3 e^2} \]
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Rubi [A] time = 0.204897, antiderivative size = 151, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {1628} \[ \frac{x \left (a C e^2-c e (2 B d-A e)+3 c C d^2\right )}{e^4}-\frac{\left (a e^2+c d^2\right ) \left (A e^2-B d e+C d^2\right )}{e^5 (d+e x)}-\frac{\log (d+e x) \left (a e^2 (2 C d-B e)-c d e (3 B d-2 A e)+4 c C d^3\right )}{e^5}-\frac{c x^2 (2 C d-B e)}{2 e^3}+\frac{c C x^3}{3 e^2} \]
Antiderivative was successfully verified.
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Rule 1628
Rubi steps
\begin{align*} \int \frac{\left (a+c x^2\right ) \left (A+B x+C x^2\right )}{(d+e x)^2} \, dx &=\int \left (\frac{3 c C d^2+a C e^2-c e (2 B d-A e)}{e^4}+\frac{c (-2 C d+B e) x}{e^3}+\frac{c C x^2}{e^2}+\frac{\left (c d^2+a e^2\right ) \left (C d^2-B d e+A e^2\right )}{e^4 (d+e x)^2}+\frac{-4 c C d^3+c d e (3 B d-2 A e)-a e^2 (2 C d-B e)}{e^4 (d+e x)}\right ) \, dx\\ &=\frac{\left (3 c C d^2+a C e^2-c e (2 B d-A e)\right ) x}{e^4}-\frac{c (2 C d-B e) x^2}{2 e^3}+\frac{c C x^3}{3 e^2}-\frac{\left (c d^2+a e^2\right ) \left (C d^2-B d e+A e^2\right )}{e^5 (d+e x)}-\frac{\left (4 c C d^3-c d e (3 B d-2 A e)+a e^2 (2 C d-B e)\right ) \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.158209, size = 142, normalized size = 0.93 \[ \frac{6 e x \left (a C e^2+c e (A e-2 B d)+3 c C d^2\right )-\frac{6 \left (a e^2+c d^2\right ) \left (e (A e-B d)+C d^2\right )}{d+e x}+6 \log (d+e x) \left (a e^2 (B e-2 C d)+c d e (3 B d-2 A e)-4 c C d^3\right )+3 c e^2 x^2 (B e-2 C d)+2 c C e^3 x^3}{6 e^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.056, size = 234, normalized size = 1.5 \begin{align*}{\frac{Cc{x}^{3}}{3\,{e}^{2}}}+{\frac{Bc{x}^{2}}{2\,{e}^{2}}}-{\frac{C{x}^{2}cd}{{e}^{3}}}+{\frac{Acx}{{e}^{2}}}-2\,{\frac{Bcdx}{{e}^{3}}}+{\frac{aCx}{{e}^{2}}}+3\,{\frac{Cc{d}^{2}x}{{e}^{4}}}-2\,{\frac{\ln \left ( ex+d \right ) Acd}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) Ba}{{e}^{2}}}+3\,{\frac{\ln \left ( ex+d \right ) Bc{d}^{2}}{{e}^{4}}}-2\,{\frac{\ln \left ( ex+d \right ) Cad}{{e}^{3}}}-4\,{\frac{\ln \left ( ex+d \right ) Cc{d}^{3}}{{e}^{5}}}-{\frac{aA}{e \left ( ex+d \right ) }}-{\frac{Ac{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}+{\frac{Bda}{{e}^{2} \left ( ex+d \right ) }}+{\frac{Bc{d}^{3}}{{e}^{4} \left ( ex+d \right ) }}-{\frac{aC{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-{\frac{Cc{d}^{4}}{{e}^{5} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.998793, size = 228, normalized size = 1.49 \begin{align*} -\frac{C c d^{4} - B c d^{3} e - B a d e^{3} + A a e^{4} +{\left (C a + A c\right )} d^{2} e^{2}}{e^{6} x + d e^{5}} + \frac{2 \, C c e^{2} x^{3} - 3 \,{\left (2 \, C c d e - B c e^{2}\right )} x^{2} + 6 \,{\left (3 \, C c d^{2} - 2 \, B c d e +{\left (C a + A c\right )} e^{2}\right )} x}{6 \, e^{4}} - \frac{{\left (4 \, C c d^{3} - 3 \, B c d^{2} e - B a e^{3} + 2 \,{\left (C a + A c\right )} d e^{2}\right )} \log \left (e x + d\right )}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.58326, size = 545, normalized size = 3.56 \begin{align*} \frac{2 \, C c e^{4} x^{4} - 6 \, C c d^{4} + 6 \, B c d^{3} e + 6 \, B a d e^{3} - 6 \, A a e^{4} - 6 \,{\left (C a + A c\right )} d^{2} e^{2} -{\left (4 \, C c d e^{3} - 3 \, B c e^{4}\right )} x^{3} + 3 \,{\left (4 \, C c d^{2} e^{2} - 3 \, B c d e^{3} + 2 \,{\left (C a + A c\right )} e^{4}\right )} x^{2} + 6 \,{\left (3 \, C c d^{3} e - 2 \, B c d^{2} e^{2} +{\left (C a + A c\right )} d e^{3}\right )} x - 6 \,{\left (4 \, C c d^{4} - 3 \, B c d^{3} e - B a d e^{3} + 2 \,{\left (C a + A c\right )} d^{2} e^{2} +{\left (4 \, C c d^{3} e - 3 \, B c d^{2} e^{2} - B a e^{4} + 2 \,{\left (C a + A c\right )} d e^{3}\right )} x\right )} \log \left (e x + d\right )}{6 \,{\left (e^{6} x + d e^{5}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 1.56747, size = 184, normalized size = 1.2 \begin{align*} \frac{C c x^{3}}{3 e^{2}} - \frac{A a e^{4} + A c d^{2} e^{2} - B a d e^{3} - B c d^{3} e + C a d^{2} e^{2} + C c d^{4}}{d e^{5} + e^{6} x} - \frac{x^{2} \left (- B c e + 2 C c d\right )}{2 e^{3}} + \frac{x \left (A c e^{2} - 2 B c d e + C a e^{2} + 3 C c d^{2}\right )}{e^{4}} - \frac{\left (2 A c d e^{2} - B a e^{3} - 3 B c d^{2} e + 2 C a d e^{2} + 4 C c d^{3}\right ) \log{\left (d + e x \right )}}{e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14848, size = 324, normalized size = 2.12 \begin{align*} \frac{1}{6} \,{\left (2 \, C c - \frac{3 \,{\left (4 \, C c d e - B c e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac{6 \,{\left (6 \, C c d^{2} e^{2} - 3 \, B c d e^{3} + C a e^{4} + A c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )}{\left (x e + d\right )}^{3} e^{\left (-5\right )} +{\left (4 \, C c d^{3} - 3 \, B c d^{2} e + 2 \, C a d e^{2} + 2 \, A c d e^{2} - B a e^{3}\right )} e^{\left (-5\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{C c d^{4} e^{3}}{x e + d} - \frac{B c d^{3} e^{4}}{x e + d} + \frac{C a d^{2} e^{5}}{x e + d} + \frac{A c d^{2} e^{5}}{x e + d} - \frac{B a d e^{6}}{x e + d} + \frac{A a e^{7}}{x e + d}\right )} e^{\left (-8\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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