Optimal. Leaf size=240 \[ -\frac{e x^2 \left (a C e^2-c \left (e (A e+3 B d)+3 C d^2\right )\right )}{2 c^2}+\frac{\log \left (a+c x^2\right ) \left (e (A c-a C) \left (3 c d^2-a e^2\right )+B c d \left (c d^2-3 a e^2\right )\right )}{2 c^3}-\frac{x \left (a e^2 (B e+3 C d)-c d \left (3 e (A e+B d)+C d^2\right )\right )}{c^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )+a \left (a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right )\right )}{\sqrt{a} c^{5/2}}+\frac{e^2 x^3 (B e+3 C d)}{3 c}+\frac{C e^3 x^4}{4 c} \]
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Rubi [A] time = 0.471217, antiderivative size = 237, 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} \[ \frac{e x^2 \left (-a C e^2+c e (A e+3 B d)+3 c C d^2\right )}{2 c^2}+\frac{\log \left (a+c x^2\right ) \left (e (A c-a C) \left (3 c d^2-a e^2\right )+B c d \left (c d^2-3 a e^2\right )\right )}{2 c^3}+\frac{x \left (-a e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3\right )}{c^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )+a \left (a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right )\right )}{\sqrt{a} c^{5/2}}+\frac{e^2 x^3 (B e+3 C d)}{3 c}+\frac{C e^3 x^4}{4 c} \]
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)^3 \left (A+B x+C x^2\right )}{a+c x^2} \, dx &=\int \left (\frac{c C d^3+3 c d e (B d+A e)-a e^2 (3 C d+B e)}{c^2}+\frac{e \left (3 c C d^2-a C e^2+c e (3 B d+A e)\right ) x}{c^2}+\frac{e^2 (3 C d+B e) x^2}{c}+\frac{C e^3 x^3}{c}+\frac{A c d \left (c d^2-3 a e^2\right )+a \left (a e^2 (3 C d+B e)-c d^2 (C d+3 B e)\right )+\left (B c d \left (c d^2-3 a e^2\right )+(A c-a C) e \left (3 c d^2-a e^2\right )\right ) x}{c^2 \left (a+c x^2\right )}\right ) \, dx\\ &=\frac{\left (c C d^3+3 c d e (B d+A e)-a e^2 (3 C d+B e)\right ) x}{c^2}+\frac{e \left (3 c C d^2-a C e^2+c e (3 B d+A e)\right ) x^2}{2 c^2}+\frac{e^2 (3 C d+B e) x^3}{3 c}+\frac{C e^3 x^4}{4 c}+\frac{\int \frac{A c d \left (c d^2-3 a e^2\right )+a \left (a e^2 (3 C d+B e)-c d^2 (C d+3 B e)\right )+\left (B c d \left (c d^2-3 a e^2\right )+(A c-a C) e \left (3 c d^2-a e^2\right )\right ) x}{a+c x^2} \, dx}{c^2}\\ &=\frac{\left (c C d^3+3 c d e (B d+A e)-a e^2 (3 C d+B e)\right ) x}{c^2}+\frac{e \left (3 c C d^2-a C e^2+c e (3 B d+A e)\right ) x^2}{2 c^2}+\frac{e^2 (3 C d+B e) x^3}{3 c}+\frac{C e^3 x^4}{4 c}+\frac{\left (B c d \left (c d^2-3 a e^2\right )+(A c-a C) e \left (3 c d^2-a e^2\right )\right ) \int \frac{x}{a+c x^2} \, dx}{c^2}+\frac{\left (A c d \left (c d^2-3 a e^2\right )+a \left (a e^2 (3 C d+B e)-c d^2 (C d+3 B e)\right )\right ) \int \frac{1}{a+c x^2} \, dx}{c^2}\\ &=\frac{\left (c C d^3+3 c d e (B d+A e)-a e^2 (3 C d+B e)\right ) x}{c^2}+\frac{e \left (3 c C d^2-a C e^2+c e (3 B d+A e)\right ) x^2}{2 c^2}+\frac{e^2 (3 C d+B e) x^3}{3 c}+\frac{C e^3 x^4}{4 c}+\frac{\left (A c d \left (c d^2-3 a e^2\right )+a \left (a e^2 (3 C d+B e)-c d^2 (C d+3 B e)\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{\sqrt{a} c^{5/2}}+\frac{\left (B c d \left (c d^2-3 a e^2\right )+(A c-a C) e \left (3 c d^2-a e^2\right )\right ) \log \left (a+c x^2\right )}{2 c^3}\\ \end{align*}
Mathematica [A] time = 0.224381, size = 223, normalized size = 0.93 \[ \frac{c x \left (-6 a e^2 (2 B e+6 C d+C e x)+2 c e \left (3 A e (6 d+e x)+B \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 c C \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )\right )+6 \log \left (a+c x^2\right ) \left (e (A c-a C) \left (3 c d^2-a e^2\right )+B c d \left (c d^2-3 a e^2\right )\right )}{12 c^3}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )+a \left (a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right )\right )}{\sqrt{a} c^{5/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.052, size = 399, normalized size = 1.7 \begin{align*}{\frac{C{e}^{3}{x}^{4}}{4\,c}}+{\frac{B{x}^{3}{e}^{3}}{3\,c}}+{\frac{C{x}^{3}d{e}^{2}}{c}}+{\frac{A{x}^{2}{e}^{3}}{2\,c}}+{\frac{3\,B{x}^{2}d{e}^{2}}{2\,c}}-{\frac{C{x}^{2}a{e}^{3}}{2\,{c}^{2}}}+{\frac{3\,C{x}^{2}{d}^{2}e}{2\,c}}+3\,{\frac{Ad{e}^{2}x}{c}}-{\frac{Ba{e}^{3}x}{{c}^{2}}}+3\,{\frac{B{d}^{2}ex}{c}}-3\,{\frac{Cad{e}^{2}x}{{c}^{2}}}+{\frac{C{d}^{3}x}{c}}-{\frac{\ln \left ( c{x}^{2}+a \right ) aA{e}^{3}}{2\,{c}^{2}}}+{\frac{3\,\ln \left ( c{x}^{2}+a \right ) A{d}^{2}e}{2\,c}}-{\frac{3\,\ln \left ( c{x}^{2}+a \right ) aBd{e}^{2}}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+a \right ) B{d}^{3}}{2\,c}}+{\frac{\ln \left ( c{x}^{2}+a \right ) C{a}^{2}{e}^{3}}{2\,{c}^{3}}}-{\frac{3\,\ln \left ( c{x}^{2}+a \right ) Ca{d}^{2}e}{2\,{c}^{2}}}-3\,{\frac{Aad{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{A{d}^{3}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{a}^{2}B{e}^{3}}{{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-3\,{\frac{Ba{d}^{2}e}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+3\,{\frac{C{a}^{2}d{e}^{2}}{{c}^{2}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }-{\frac{{d}^{3}aC}{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 = 1.88152, size = 1239, normalized size = 5.16 \begin{align*} \left [\frac{3 \, C a c^{2} e^{3} x^{4} + 4 \,{\left (3 \, C a c^{2} d e^{2} + B a c^{2} e^{3}\right )} x^{3} + 6 \,{\left (3 \, C a c^{2} d^{2} e + 3 \, B a c^{2} d e^{2} -{\left (C a^{2} c - A a c^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (3 \, B a c d^{2} e - B a^{2} e^{3} +{\left (C a c - A c^{2}\right )} d^{3} - 3 \,{\left (C a^{2} - A a c\right )} d e^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) + 12 \,{\left (C a c^{2} d^{3} + 3 \, B a c^{2} d^{2} e - B a^{2} c e^{3} - 3 \,{\left (C a^{2} c - A a c^{2}\right )} d e^{2}\right )} x + 6 \,{\left (B a c^{2} d^{3} - 3 \, B a^{2} c d e^{2} - 3 \,{\left (C a^{2} c - A a c^{2}\right )} d^{2} e +{\left (C a^{3} - A a^{2} c\right )} e^{3}\right )} \log \left (c x^{2} + a\right )}{12 \, a c^{3}}, \frac{3 \, C a c^{2} e^{3} x^{4} + 4 \,{\left (3 \, C a c^{2} d e^{2} + B a c^{2} e^{3}\right )} x^{3} + 6 \,{\left (3 \, C a c^{2} d^{2} e + 3 \, B a c^{2} d e^{2} -{\left (C a^{2} c - A a c^{2}\right )} e^{3}\right )} x^{2} - 12 \,{\left (3 \, B a c d^{2} e - B a^{2} e^{3} +{\left (C a c - A c^{2}\right )} d^{3} - 3 \,{\left (C a^{2} - A a c\right )} d e^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) + 12 \,{\left (C a c^{2} d^{3} + 3 \, B a c^{2} d^{2} e - B a^{2} c e^{3} - 3 \,{\left (C a^{2} c - A a c^{2}\right )} d e^{2}\right )} x + 6 \,{\left (B a c^{2} d^{3} - 3 \, B a^{2} c d e^{2} - 3 \,{\left (C a^{2} c - A a c^{2}\right )} d^{2} e +{\left (C a^{3} - A a^{2} c\right )} e^{3}\right )} \log \left (c x^{2} + a\right )}{12 \, a c^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 6.76913, size = 1000, normalized size = 4.17 \begin{align*} \frac{C e^{3} x^{4}}{4 c} + \left (\frac{- A a c e^{3} + 3 A c^{2} d^{2} e - 3 B a c d e^{2} + B c^{2} d^{3} + C a^{2} e^{3} - 3 C a c d^{2} e}{2 c^{3}} - \frac{\sqrt{- a c^{7}} \left (- 3 A a c d e^{2} + A c^{2} d^{3} + B a^{2} e^{3} - 3 B a c d^{2} e + 3 C a^{2} d e^{2} - C a c d^{3}\right )}{2 a c^{6}}\right ) \log{\left (x + \frac{A a^{2} c e^{3} - 3 A a c^{2} d^{2} e + 3 B a^{2} c d e^{2} - B a c^{2} d^{3} - C a^{3} e^{3} + 3 C a^{2} c d^{2} e + 2 a c^{3} \left (\frac{- A a c e^{3} + 3 A c^{2} d^{2} e - 3 B a c d e^{2} + B c^{2} d^{3} + C a^{2} e^{3} - 3 C a c d^{2} e}{2 c^{3}} - \frac{\sqrt{- a c^{7}} \left (- 3 A a c d e^{2} + A c^{2} d^{3} + B a^{2} e^{3} - 3 B a c d^{2} e + 3 C a^{2} d e^{2} - C a c d^{3}\right )}{2 a c^{6}}\right )}{- 3 A a c^{2} d e^{2} + A c^{3} d^{3} + B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 3 C a^{2} c d e^{2} - C a c^{2} d^{3}} \right )} + \left (\frac{- A a c e^{3} + 3 A c^{2} d^{2} e - 3 B a c d e^{2} + B c^{2} d^{3} + C a^{2} e^{3} - 3 C a c d^{2} e}{2 c^{3}} + \frac{\sqrt{- a c^{7}} \left (- 3 A a c d e^{2} + A c^{2} d^{3} + B a^{2} e^{3} - 3 B a c d^{2} e + 3 C a^{2} d e^{2} - C a c d^{3}\right )}{2 a c^{6}}\right ) \log{\left (x + \frac{A a^{2} c e^{3} - 3 A a c^{2} d^{2} e + 3 B a^{2} c d e^{2} - B a c^{2} d^{3} - C a^{3} e^{3} + 3 C a^{2} c d^{2} e + 2 a c^{3} \left (\frac{- A a c e^{3} + 3 A c^{2} d^{2} e - 3 B a c d e^{2} + B c^{2} d^{3} + C a^{2} e^{3} - 3 C a c d^{2} e}{2 c^{3}} + \frac{\sqrt{- a c^{7}} \left (- 3 A a c d e^{2} + A c^{2} d^{3} + B a^{2} e^{3} - 3 B a c d^{2} e + 3 C a^{2} d e^{2} - C a c d^{3}\right )}{2 a c^{6}}\right )}{- 3 A a c^{2} d e^{2} + A c^{3} d^{3} + B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 3 C a^{2} c d e^{2} - C a c^{2} d^{3}} \right )} + \frac{x^{3} \left (B e^{3} + 3 C d e^{2}\right )}{3 c} - \frac{x^{2} \left (- A c e^{3} - 3 B c d e^{2} + C a e^{3} - 3 C c d^{2} e\right )}{2 c^{2}} - \frac{x \left (- 3 A c d e^{2} + B a e^{3} - 3 B c d^{2} e + 3 C a d e^{2} - C c d^{3}\right )}{c^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.14414, size = 377, normalized size = 1.57 \begin{align*} -\frac{{\left (C a c d^{3} - A c^{2} d^{3} + 3 \, B a c d^{2} e - 3 \, C a^{2} d e^{2} + 3 \, A a c d e^{2} - B a^{2} e^{3}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c^{2}} + \frac{{\left (B c^{2} d^{3} - 3 \, C a c d^{2} e + 3 \, A c^{2} d^{2} e - 3 \, B a c d e^{2} + C a^{2} e^{3} - A a c e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{3 \, C c^{3} x^{4} e^{3} + 12 \, C c^{3} d x^{3} e^{2} + 18 \, C c^{3} d^{2} x^{2} e + 12 \, C c^{3} d^{3} x + 4 \, B c^{3} x^{3} e^{3} + 18 \, B c^{3} d x^{2} e^{2} + 36 \, B c^{3} d^{2} x e - 6 \, C a c^{2} x^{2} e^{3} + 6 \, A c^{3} x^{2} e^{3} - 36 \, C a c^{2} d x e^{2} + 36 \, A c^{3} d x e^{2} - 12 \, B a c^{2} x e^{3}}{12 \, c^{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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