Optimal. Leaf size=109 \[ -\frac{x^2 \left (4 d^2 g^2+6 d e f g+e^2 f^2\right )}{2 e}-\frac{4 d^2 (d g+e f)^2 \log (d-e x)}{e^3}-\frac{d x (2 d g+e f) (2 d g+3 e f)}{e^2}-\frac{1}{3} g x^3 (3 d g+2 e f)-\frac{1}{4} e g^2 x^4 \]
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Rubi [A] time = 0.135804, antiderivative size = 109, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.069, Rules used = {848, 88} \[ -\frac{x^2 \left (4 d^2 g^2+6 d e f g+e^2 f^2\right )}{2 e}-\frac{4 d^2 (d g+e f)^2 \log (d-e x)}{e^3}-\frac{d x (2 d g+e f) (2 d g+3 e f)}{e^2}-\frac{1}{3} g x^3 (3 d g+2 e f)-\frac{1}{4} e g^2 x^4 \]
Antiderivative was successfully verified.
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Rule 848
Rule 88
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 (f+g x)^2}{d^2-e^2 x^2} \, dx &=\int \frac{(d+e x)^2 (f+g x)^2}{d-e x} \, dx\\ &=\int \left (\frac{d (-3 e f-2 d g) (e f+2 d g)}{e^2}-\frac{\left (e^2 f^2+6 d e f g+4 d^2 g^2\right ) x}{e}-g (2 e f+3 d g) x^2-e g^2 x^3-\frac{4 d^2 (e f+d g)^2}{e^2 (-d+e x)}\right ) \, dx\\ &=-\frac{d (e f+2 d g) (3 e f+2 d g) x}{e^2}-\frac{\left (e^2 f^2+6 d e f g+4 d^2 g^2\right ) x^2}{2 e}-\frac{1}{3} g (2 e f+3 d g) x^3-\frac{1}{4} e g^2 x^4-\frac{4 d^2 (e f+d g)^2 \log (d-e x)}{e^3}\\ \end{align*}
Mathematica [A] time = 0.0500337, size = 103, normalized size = 0.94 \[ -\frac{e x \left (24 d^2 e g (4 f+g x)+48 d^3 g^2+12 d e^2 \left (3 f^2+3 f g x+g^2 x^2\right )+e^3 x \left (6 f^2+8 f g x+3 g^2 x^2\right )\right )+48 d^2 (d g+e f)^2 \log (d-e x)}{12 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.046, size = 145, normalized size = 1.3 \begin{align*} -{\frac{e{g}^{2}{x}^{4}}{4}}-{x}^{3}d{g}^{2}-{\frac{2\,e{x}^{3}fg}{3}}-2\,{\frac{{x}^{2}{d}^{2}{g}^{2}}{e}}-3\,{x}^{2}dfg-{\frac{e{x}^{2}{f}^{2}}{2}}-4\,{\frac{{d}^{3}{g}^{2}x}{{e}^{2}}}-8\,{\frac{{d}^{2}fgx}{e}}-3\,d{f}^{2}x-4\,{\frac{{d}^{4}\ln \left ( ex-d \right ){g}^{2}}{{e}^{3}}}-8\,{\frac{{d}^{3}\ln \left ( ex-d \right ) fg}{{e}^{2}}}-4\,{\frac{{d}^{2}\ln \left ( ex-d \right ){f}^{2}}{e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 0.96125, size = 186, normalized size = 1.71 \begin{align*} -\frac{3 \, e^{3} g^{2} x^{4} + 4 \,{\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x^{3} + 6 \,{\left (e^{3} f^{2} + 6 \, d e^{2} f g + 4 \, d^{2} e g^{2}\right )} x^{2} + 12 \,{\left (3 \, d e^{2} f^{2} + 8 \, d^{2} e f g + 4 \, d^{3} g^{2}\right )} x}{12 \, e^{2}} - \frac{4 \,{\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.70013, size = 292, normalized size = 2.68 \begin{align*} -\frac{3 \, e^{4} g^{2} x^{4} + 4 \,{\left (2 \, e^{4} f g + 3 \, d e^{3} g^{2}\right )} x^{3} + 6 \,{\left (e^{4} f^{2} + 6 \, d e^{3} f g + 4 \, d^{2} e^{2} g^{2}\right )} x^{2} + 12 \,{\left (3 \, d e^{3} f^{2} + 8 \, d^{2} e^{2} f g + 4 \, d^{3} e g^{2}\right )} x + 48 \,{\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2}\right )} \log \left (e x - d\right )}{12 \, e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 0.706913, size = 116, normalized size = 1.06 \begin{align*} - \frac{4 d^{2} \left (d g + e f\right )^{2} \log{\left (- d + e x \right )}}{e^{3}} - \frac{e g^{2} x^{4}}{4} - x^{3} \left (d g^{2} + \frac{2 e f g}{3}\right ) - \frac{x^{2} \left (4 d^{2} g^{2} + 6 d e f g + e^{2} f^{2}\right )}{2 e} - \frac{x \left (4 d^{3} g^{2} + 8 d^{2} e f g + 3 d e^{2} f^{2}\right )}{e^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.16436, size = 285, normalized size = 2.61 \begin{align*} -2 \,{\left (d^{4} g^{2} e^{3} + 2 \, d^{3} f g e^{4} + d^{2} f^{2} e^{5}\right )} e^{\left (-6\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac{1}{12} \,{\left (3 \, g^{2} x^{4} e^{9} + 12 \, d g^{2} x^{3} e^{8} + 24 \, d^{2} g^{2} x^{2} e^{7} + 48 \, d^{3} g^{2} x e^{6} + 8 \, f g x^{3} e^{9} + 36 \, d f g x^{2} e^{8} + 96 \, d^{2} f g x e^{7} + 6 \, f^{2} x^{2} e^{9} + 36 \, d f^{2} x e^{8}\right )} e^{\left (-8\right )} - \frac{2 \,{\left (d^{5} g^{2} e^{2} + 2 \, d^{4} f g e^{3} + d^{3} f^{2} e^{4}\right )} e^{\left (-5\right )} \log \left (\frac{{\left | 2 \, x e^{2} - 2 \,{\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \,{\left | d \right |} e \right |}}\right )}{{\left | d \right |}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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