Optimal. Leaf size=178 \[ -\frac{e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac{(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}+\frac{(d g+e f)^2}{32 d^5 e^3 (d-e x)}-\frac{f (d g+e f)}{8 d^5 e^2 (d+e x)}-\frac{(3 e f-d g) (d g+e f)}{32 d^4 e^3 (d+e x)^2}+\frac{(d g+e f) (d g+5 e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^6 e^3} \]
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Rubi [A] time = 0.199561, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103, Rules used = {848, 88, 208} \[ -\frac{e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac{(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}+\frac{(d g+e f)^2}{32 d^5 e^3 (d-e x)}-\frac{f (d g+e f)}{8 d^5 e^2 (d+e x)}-\frac{(3 e f-d g) (d g+e f)}{32 d^4 e^3 (d+e x)^2}+\frac{(d g+e f) (d g+5 e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^6 e^3} \]
Antiderivative was successfully verified.
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Rule 848
Rule 88
Rule 208
Rubi steps
\begin{align*} \int \frac{(f+g x)^2}{(d+e x)^3 \left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac{(f+g x)^2}{(d-e x)^2 (d+e x)^5} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{32 d^5 e^2 (d-e x)^2}+\frac{(-e f+d g)^2}{4 d^2 e^2 (d+e x)^5}+\frac{e^2 f^2-d^2 g^2}{4 d^3 e^2 (d+e x)^4}+\frac{(3 e f-d g) (e f+d g)}{16 d^4 e^2 (d+e x)^3}+\frac{f (e f+d g)}{8 d^5 e (d+e x)^2}+\frac{(e f+d g) (5 e f+d g)}{32 d^5 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac{(e f+d g)^2}{32 d^5 e^3 (d-e x)}-\frac{(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac{e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac{(3 e f-d g) (e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac{f (e f+d g)}{8 d^5 e^2 (d+e x)}+\frac{((e f+d g) (5 e f+d g)) \int \frac{1}{d^2-e^2 x^2} \, dx}{32 d^5 e^2}\\ &=\frac{(e f+d g)^2}{32 d^5 e^3 (d-e x)}-\frac{(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac{e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac{(3 e f-d g) (e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac{f (e f+d g)}{8 d^5 e^2 (d+e x)}+\frac{(e f+d g) (5 e f+d g) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^6 e^3}\\ \end{align*}
Mathematica [A] time = 0.149004, size = 195, normalized size = 1.1 \[ \frac{\frac{16 d^3 \left (d^2 g^2-e^2 f^2\right )}{(d+e x)^3}+\frac{6 d^2 \left (d^2 g^2-2 d e f g-3 e^2 f^2\right )}{(d+e x)^2}-3 \left (d^2 g^2+6 d e f g+5 e^2 f^2\right ) \log (d-e x)+3 \left (d^2 g^2+6 d e f g+5 e^2 f^2\right ) \log (d+e x)-\frac{12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac{6 d (d g+e f)^2}{d-e x}-\frac{24 d e f (d g+e f)}{d+e x}}{192 d^6 e^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.059, size = 341, normalized size = 1.9 \begin{align*} -{\frac{\ln \left ( ex-d \right ){g}^{2}}{64\,{e}^{3}{d}^{4}}}-{\frac{3\,\ln \left ( ex-d \right ) fg}{32\,{e}^{2}{d}^{5}}}-{\frac{5\,\ln \left ( ex-d \right ){f}^{2}}{64\,e{d}^{6}}}-{\frac{{g}^{2}}{32\,{d}^{3}{e}^{3} \left ( ex-d \right ) }}-{\frac{fg}{16\,{e}^{2}{d}^{4} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{32\,e{d}^{5} \left ( ex-d \right ) }}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{64\,{e}^{3}{d}^{4}}}+{\frac{3\,\ln \left ( ex+d \right ) fg}{32\,{e}^{2}{d}^{5}}}+{\frac{5\,\ln \left ( ex+d \right ){f}^{2}}{64\,e{d}^{6}}}+{\frac{{g}^{2}}{12\,{e}^{3}d \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{12\,e{d}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{{g}^{2}}{32\,{e}^{3}{d}^{2} \left ( ex+d \right ) ^{2}}}-{\frac{fg}{16\,{e}^{2}{d}^{3} \left ( ex+d \right ) ^{2}}}-{\frac{3\,{f}^{2}}{32\,e{d}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{g}^{2}}{16\,{e}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{fg}{8\,d{e}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{{f}^{2}}{16\,e{d}^{2} \left ( ex+d \right ) ^{4}}}-{\frac{fg}{8\,{e}^{2}{d}^{4} \left ( ex+d \right ) }}-{\frac{{f}^{2}}{8\,e{d}^{5} \left ( ex+d \right ) }} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.03643, size = 402, normalized size = 2.26 \begin{align*} \frac{32 \, d^{4} e^{2} f^{2} - 8 \, d^{6} g^{2} - 3 \,{\left (5 \, e^{6} f^{2} + 6 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{4} - 9 \,{\left (5 \, d e^{5} f^{2} + 6 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} - 7 \,{\left (5 \, d^{2} e^{4} f^{2} + 6 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 3 \,{\left (5 \, d^{3} e^{3} f^{2} + 6 \, d^{4} e^{2} f g - 7 \, d^{5} e g^{2}\right )} x}{96 \,{\left (d^{5} e^{8} x^{5} + 3 \, d^{6} e^{7} x^{4} + 2 \, d^{7} e^{6} x^{3} - 2 \, d^{8} e^{5} x^{2} - 3 \, d^{9} e^{4} x - d^{10} e^{3}\right )}} + \frac{{\left (5 \, e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{64 \, d^{6} e^{3}} - \frac{{\left (5 \, e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{64 \, d^{6} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.02957, size = 1303, normalized size = 7.32 \begin{align*} \frac{64 \, d^{5} e^{2} f^{2} - 16 \, d^{7} g^{2} - 6 \,{\left (5 \, d e^{6} f^{2} + 6 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 18 \,{\left (5 \, d^{2} e^{5} f^{2} + 6 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} - 14 \,{\left (5 \, d^{3} e^{4} f^{2} + 6 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + 6 \,{\left (5 \, d^{4} e^{3} f^{2} + 6 \, d^{5} e^{2} f g - 7 \, d^{6} e g^{2}\right )} x - 3 \,{\left (5 \, d^{5} e^{2} f^{2} + 6 \, d^{6} e f g + d^{7} g^{2} -{\left (5 \, e^{7} f^{2} + 6 \, d e^{6} f g + d^{2} e^{5} g^{2}\right )} x^{5} - 3 \,{\left (5 \, d e^{6} f^{2} + 6 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (5 \, d^{2} e^{5} f^{2} + 6 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} + 2 \,{\left (5 \, d^{3} e^{4} f^{2} + 6 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + 3 \,{\left (5 \, d^{4} e^{3} f^{2} + 6 \, d^{5} e^{2} f g + d^{6} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \,{\left (5 \, d^{5} e^{2} f^{2} + 6 \, d^{6} e f g + d^{7} g^{2} -{\left (5 \, e^{7} f^{2} + 6 \, d e^{6} f g + d^{2} e^{5} g^{2}\right )} x^{5} - 3 \,{\left (5 \, d e^{6} f^{2} + 6 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (5 \, d^{2} e^{5} f^{2} + 6 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} + 2 \,{\left (5 \, d^{3} e^{4} f^{2} + 6 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + 3 \,{\left (5 \, d^{4} e^{3} f^{2} + 6 \, d^{5} e^{2} f g + d^{6} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{192 \,{\left (d^{6} e^{8} x^{5} + 3 \, d^{7} e^{7} x^{4} + 2 \, d^{8} e^{6} x^{3} - 2 \, d^{9} e^{5} x^{2} - 3 \, d^{10} e^{4} x - d^{11} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.15653, size = 371, normalized size = 2.08 \begin{align*} - \frac{8 d^{6} g^{2} - 32 d^{4} e^{2} f^{2} + x^{4} \left (3 d^{2} e^{4} g^{2} + 18 d e^{5} f g + 15 e^{6} f^{2}\right ) + x^{3} \left (9 d^{3} e^{3} g^{2} + 54 d^{2} e^{4} f g + 45 d e^{5} f^{2}\right ) + x^{2} \left (7 d^{4} e^{2} g^{2} + 42 d^{3} e^{3} f g + 35 d^{2} e^{4} f^{2}\right ) + x \left (21 d^{5} e g^{2} - 18 d^{4} e^{2} f g - 15 d^{3} e^{3} f^{2}\right )}{- 96 d^{10} e^{3} - 288 d^{9} e^{4} x - 192 d^{8} e^{5} x^{2} + 192 d^{7} e^{6} x^{3} + 288 d^{6} e^{7} x^{4} + 96 d^{5} e^{8} x^{5}} - \frac{\left (d g + e f\right ) \left (d g + 5 e f\right ) \log{\left (- \frac{d \left (d g + e f\right ) \left (d g + 5 e f\right )}{e \left (d^{2} g^{2} + 6 d e f g + 5 e^{2} f^{2}\right )} + x \right )}}{64 d^{6} e^{3}} + \frac{\left (d g + e f\right ) \left (d g + 5 e f\right ) \log{\left (\frac{d \left (d g + e f\right ) \left (d g + 5 e f\right )}{e \left (d^{2} g^{2} + 6 d e f g + 5 e^{2} f^{2}\right )} + x \right )}}{64 d^{6} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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