Optimal. Leaf size=145 \[ \frac{x^2 \left (a C e^2+c \left (C d^2-e (B d-A e)\right )\right )}{2 e^3}-\frac{x \left (a e^2 (C d-B e)+c d \left (C d^2-e (B d-A e)\right )\right )}{e^4}+\frac{\left (a e^2+c d^2\right ) \log (d+e x) \left (A e^2-B d e+C d^2\right )}{e^5}-\frac{c x^3 (C d-B e)}{3 e^2}+\frac{c C x^4}{4 e} \]
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Rubi [A] time = 0.245168, antiderivative size = 143, 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} \[ \frac{x^2 \left (a C e^2-c e (B d-A e)+c C d^2\right )}{2 e^3}-\frac{x \left (a e^2 (C d-B e)-c d e (B d-A e)+c C d^3\right )}{e^4}+\frac{\left (a e^2+c d^2\right ) \log (d+e x) \left (A e^2-B d e+C d^2\right )}{e^5}-\frac{c x^3 (C d-B e)}{3 e^2}+\frac{c C x^4}{4 e} \]
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} \, dx &=\int \left (\frac{-a e^2 (C d-B e)-c \left (C d^3-d e (B d-A e)\right )}{e^4}+\frac{\left (c C d^2+a C e^2-c e (B d-A e)\right ) x}{e^3}+\frac{c (-C d+B e) x^2}{e^2}+\frac{c C x^3}{e}+\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)}\right ) \, dx\\ &=-\frac{\left (c C d^3-c d e (B d-A e)+a e^2 (C d-B e)\right ) x}{e^4}+\frac{\left (c C d^2+a C e^2-c e (B d-A e)\right ) x^2}{2 e^3}-\frac{c (C d-B e) x^3}{3 e^2}+\frac{c C x^4}{4 e}+\frac{\left (c d^2+a e^2\right ) \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{e^5}\\ \end{align*}
Mathematica [A] time = 0.0782833, size = 136, normalized size = 0.94 \[ \frac{e x \left (6 a e^2 (2 B e-2 C d+C e x)+2 c e \left (3 A e (e x-2 d)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )+c C \left (6 d^2 e x-12 d^3-4 d e^2 x^2+3 e^3 x^3\right )\right )+12 \left (a e^2+c d^2\right ) \log (d+e x) \left (e (A e-B d)+C d^2\right )}{12 e^5} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.049, size = 210, normalized size = 1.5 \begin{align*}{\frac{Cc{x}^{4}}{4\,e}}+{\frac{Bc{x}^{3}}{3\,e}}-{\frac{C{x}^{3}cd}{3\,{e}^{2}}}+{\frac{A{x}^{2}c}{2\,e}}-{\frac{Bc{x}^{2}d}{2\,{e}^{2}}}+{\frac{C{x}^{2}a}{2\,e}}+{\frac{C{x}^{2}c{d}^{2}}{2\,{e}^{3}}}-{\frac{Acdx}{{e}^{2}}}+{\frac{aBx}{e}}+{\frac{Bc{d}^{2}x}{{e}^{3}}}-{\frac{aCdx}{{e}^{2}}}-{\frac{Cc{d}^{3}x}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ) Aa}{e}}+{\frac{\ln \left ( ex+d \right ) Ac{d}^{2}}{{e}^{3}}}-{\frac{\ln \left ( ex+d \right ) Bad}{{e}^{2}}}-{\frac{\ln \left ( ex+d \right ) Bc{d}^{3}}{{e}^{4}}}+{\frac{\ln \left ( ex+d \right ) Ca{d}^{2}}{{e}^{3}}}+{\frac{\ln \left ( ex+d \right ) Cc{d}^{4}}{{e}^{5}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.00087, size = 215, normalized size = 1.48 \begin{align*} \frac{3 \, C c e^{3} x^{4} - 4 \,{\left (C c d e^{2} - B c e^{3}\right )} x^{3} + 6 \,{\left (C c d^{2} e - B c d e^{2} +{\left (C a + A c\right )} e^{3}\right )} x^{2} - 12 \,{\left (C c d^{3} - B c d^{2} e - B a e^{3} +{\left (C a + A c\right )} d e^{2}\right )} x}{12 \, e^{4}} + \frac{{\left (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}\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 = 1.68902, size = 344, normalized size = 2.37 \begin{align*} \frac{3 \, C c e^{4} x^{4} - 4 \,{\left (C c d e^{3} - B c e^{4}\right )} x^{3} + 6 \,{\left (C c d^{2} e^{2} - B c d e^{3} +{\left (C a + A c\right )} e^{4}\right )} x^{2} - 12 \,{\left (C c d^{3} e - B c d^{2} e^{2} - B a e^{4} +{\left (C a + A c\right )} d e^{3}\right )} x + 12 \,{\left (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}\right )} \log \left (e x + d\right )}{12 \, e^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.766002, size = 143, normalized size = 0.99 \begin{align*} \frac{C c x^{4}}{4 e} - \frac{x^{3} \left (- B c e + C c d\right )}{3 e^{2}} + \frac{x^{2} \left (A c e^{2} - B c d e + C a e^{2} + C c d^{2}\right )}{2 e^{3}} - \frac{x \left (A c d e^{2} - B a e^{3} - B c d^{2} e + C a d e^{2} + C c d^{3}\right )}{e^{4}} + \frac{\left (a e^{2} + c d^{2}\right ) \left (A e^{2} - B d e + C d^{2}\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 = 1.12377, size = 230, normalized size = 1.59 \begin{align*}{\left (C c d^{4} - B c d^{3} e + C a d^{2} e^{2} + A c d^{2} e^{2} - B a d e^{3} + A a e^{4}\right )} e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + \frac{1}{12} \,{\left (3 \, C c x^{4} e^{3} - 4 \, C c d x^{3} e^{2} + 6 \, C c d^{2} x^{2} e - 12 \, C c d^{3} x + 4 \, B c x^{3} e^{3} - 6 \, B c d x^{2} e^{2} + 12 \, B c d^{2} x e + 6 \, C a x^{2} e^{3} + 6 \, A c x^{2} e^{3} - 12 \, C a d x e^{2} - 12 \, A c d x e^{2} + 12 \, B a x e^{3}\right )} e^{\left (-4\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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