Optimal. Leaf size=168 \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2-2 a C d e+2 A c d e+B c d^2\right )}{2 c^2}-\frac{x \left (a C e^2-c \left (e (A e+2 B d)+C d^2\right )\right )}{c^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (2 B e+C d)\right )\right )}{\sqrt{a} c^{5/2}}+\frac{e x^2 (B e+2 C d)}{2 c}+\frac{C e^2 x^3}{3 c} \]
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Rubi [A] time = 0.261574, antiderivative size = 166, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148, Rules used = {1629, 635, 205, 260} \[ \frac{\log \left (a+c x^2\right ) \left (-a B e^2-2 a C d e+2 A c d e+B c d^2\right )}{2 c^2}+\frac{x \left (-a C e^2+c e (A e+2 B d)+c C d^2\right )}{c^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (2 B e+C d)\right )\right )}{\sqrt{a} c^{5/2}}+\frac{e x^2 (B e+2 C d)}{2 c}+\frac{C e^2 x^3}{3 c} \]
Antiderivative was successfully verified.
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Rule 1629
Rule 635
Rule 205
Rule 260
Rubi steps
\begin{align*} \int \frac{(d+e x)^2 \left (A+B x+C x^2\right )}{a+c x^2} \, dx &=\int \left (\frac{c C d^2-a C e^2+c e (2 B d+A e)}{c^2}+\frac{e (2 C d+B e) x}{c}+\frac{C e^2 x^2}{c}+\frac{A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d+2 B e)\right )+c \left (B c d^2+2 A c d e-2 a C d e-a B e^2\right ) x}{c^2 \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{\left (c C d^2-a C e^2+c e (2 B d+A e)\right ) x}{c^2}+\frac{e (2 C d+B e) x^2}{2 c}+\frac{C e^2 x^3}{3 c}+\frac{\int \frac{A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d+2 B e)\right )+c \left (B c d^2+2 A c d e-2 a C d e-a B e^2\right ) x}{a+c x^2} \, dx}{c^2}\\ &=\frac{\left (c C d^2-a C e^2+c e (2 B d+A e)\right ) x}{c^2}+\frac{e (2 C d+B e) x^2}{2 c}+\frac{C e^2 x^3}{3 c}+\frac{\left (B c d^2+2 A c d e-2 a C d e-a B e^2\right ) \int \frac{x}{a+c x^2} \, dx}{c}+\frac{\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d+2 B e)\right )\right ) \int \frac{1}{a+c x^2} \, dx}{c^2}\\ &=\frac{\left (c C d^2-a C e^2+c e (2 B d+A e)\right ) x}{c^2}+\frac{e (2 C d+B e) x^2}{2 c}+\frac{C e^2 x^3}{3 c}+\frac{\left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (C d+2 B e)\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{5/2}}+\frac{\left (B c d^2+2 A c d e-2 a C d e-a B e^2\right ) \log \left (a+c x^2\right )}{2 c^2}\\ \end{align*}
Mathematica [A] time = 0.193036, size = 155, normalized size = 0.92 \[ \frac{x \left (-6 a C e^2+3 c e (2 A e+4 B d+B e x)+2 c C \left (3 d^2+3 d e x+e^2 x^2\right )\right )+3 \log \left (a+c x^2\right ) \left (-a B e^2-2 a C d e+2 A c d e+B c d^2\right )}{6 c^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c \left (c d^2-a e^2\right )+a \left (a C e^2-c d (2 B e+C d)\right )\right )}{\sqrt{a} c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.069, size = 256, normalized size = 1.5 \begin{align*}{\frac{C{e}^{2}{x}^{3}}{3\,c}}+{\frac{B{x}^{2}{e}^{2}}{2\,c}}+{\frac{C{x}^{2}de}{c}}+{\frac{A{e}^{2}x}{c}}+2\,{\frac{Bdex}{c}}-{\frac{aC{e}^{2}x}{{c}^{2}}}+{\frac{C{d}^{2}x}{c}}+{\frac{\ln \left ( c{x}^{2}+a \right ) Ade}{c}}-{\frac{\ln \left ( c{x}^{2}+a \right ) Ba{e}^{2}}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+a \right ) B{d}^{2}}{2\,c}}-{\frac{\ln \left ( c{x}^{2}+a \right ) Cade}{{c}^{2}}}-{\frac{Aa{e}^{2}}{c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{A{d}^{2}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-2\,{\frac{aBde}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{\frac{{a}^{2}C{e}^{2}}{{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{aC{d}^{2}}{c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \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 = 1.8533, size = 855, normalized size = 5.09 \begin{align*} \left [\frac{2 \, C a c^{2} e^{2} x^{3} + 3 \,{\left (2 \, C a c^{2} d e + B a c^{2} e^{2}\right )} x^{2} - 3 \,{\left (2 \, B a c d e +{\left (C a c - A c^{2}\right )} d^{2} -{\left (C a^{2} - A a c\right )} e^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} + 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) + 6 \,{\left (C a c^{2} d^{2} + 2 \, B a c^{2} d e -{\left (C a^{2} c - A a c^{2}\right )} e^{2}\right )} x + 3 \,{\left (B a c^{2} d^{2} - B a^{2} c e^{2} - 2 \,{\left (C a^{2} c - A a c^{2}\right )} d e\right )} \log \left (c x^{2} + a\right )}{6 \, a c^{3}}, \frac{2 \, C a c^{2} e^{2} x^{3} + 3 \,{\left (2 \, C a c^{2} d e + B a c^{2} e^{2}\right )} x^{2} - 6 \,{\left (2 \, B a c d e +{\left (C a c - A c^{2}\right )} d^{2} -{\left (C a^{2} - A a c\right )} e^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) + 6 \,{\left (C a c^{2} d^{2} + 2 \, B a c^{2} d e -{\left (C a^{2} c - A a c^{2}\right )} e^{2}\right )} x + 3 \,{\left (B a c^{2} d^{2} - B a^{2} c e^{2} - 2 \,{\left (C a^{2} c - A a c^{2}\right )} d e\right )} \log \left (c x^{2} + a\right )}{6 \, a c^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 4.46252, size = 638, normalized size = 3.8 \begin{align*} \frac{C e^{2} x^{3}}{3 c} + \left (- \frac{- 2 A c d e + B a e^{2} - B c d^{2} + 2 C a d e}{2 c^{2}} - \frac{\sqrt{- a c^{5}} \left (- A a c e^{2} + A c^{2} d^{2} - 2 B a c d e + C a^{2} e^{2} - C a c d^{2}\right )}{2 a c^{5}}\right ) \log{\left (x + \frac{- 2 A a c d e + B a^{2} e^{2} - B a c d^{2} + 2 C a^{2} d e + 2 a c^{2} \left (- \frac{- 2 A c d e + B a e^{2} - B c d^{2} + 2 C a d e}{2 c^{2}} - \frac{\sqrt{- a c^{5}} \left (- A a c e^{2} + A c^{2} d^{2} - 2 B a c d e + C a^{2} e^{2} - C a c d^{2}\right )}{2 a c^{5}}\right )}{- A a c e^{2} + A c^{2} d^{2} - 2 B a c d e + C a^{2} e^{2} - C a c d^{2}} \right )} + \left (- \frac{- 2 A c d e + B a e^{2} - B c d^{2} + 2 C a d e}{2 c^{2}} + \frac{\sqrt{- a c^{5}} \left (- A a c e^{2} + A c^{2} d^{2} - 2 B a c d e + C a^{2} e^{2} - C a c d^{2}\right )}{2 a c^{5}}\right ) \log{\left (x + \frac{- 2 A a c d e + B a^{2} e^{2} - B a c d^{2} + 2 C a^{2} d e + 2 a c^{2} \left (- \frac{- 2 A c d e + B a e^{2} - B c d^{2} + 2 C a d e}{2 c^{2}} + \frac{\sqrt{- a c^{5}} \left (- A a c e^{2} + A c^{2} d^{2} - 2 B a c d e + C a^{2} e^{2} - C a c d^{2}\right )}{2 a c^{5}}\right )}{- A a c e^{2} + A c^{2} d^{2} - 2 B a c d e + C a^{2} e^{2} - C a c d^{2}} \right )} + \frac{x^{2} \left (B e^{2} + 2 C d e\right )}{2 c} - \frac{x \left (- A c e^{2} - 2 B c d e + C a e^{2} - C c d^{2}\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.1366, size = 238, normalized size = 1.42 \begin{align*} \frac{{\left (B c d^{2} - 2 \, C a d e + 2 \, A c d e - B a e^{2}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{2}} - \frac{{\left (C a c d^{2} - A c^{2} d^{2} + 2 \, B a c d e - C a^{2} e^{2} + A a c e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c^{2}} + \frac{2 \, C c^{2} x^{3} e^{2} + 6 \, C c^{2} d x^{2} e + 6 \, C c^{2} d^{2} x + 3 \, B c^{2} x^{2} e^{2} + 12 \, B c^{2} d x e - 6 \, C a c x e^{2} + 6 \, A c^{2} x e^{2}}{6 \, c^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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