Optimal. Leaf size=188 \[ -\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}+\frac{\left (-d^2 g^2+2 d e f g+5 e^2 f^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{16 d^6 e^3}-\frac{(e f-d g)^2}{24 d^3 e^3 (d+e x)^3}-\frac{(d g+3 e f) (e f-d g)}{32 d^4 e^3 (d+e x)^2}+\frac{f (d g+e f)}{8 d^5 e^2 (d-e x)}+\frac{(d g+e f)^2}{32 d^4 e^3 (d-e x)^2} \]
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Rubi [A] time = 0.209689, antiderivative size = 188, 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} \[ -\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}+\frac{\left (-d^2 g^2+2 d e f g+5 e^2 f^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{16 d^6 e^3}-\frac{(e f-d g)^2}{24 d^3 e^3 (d+e x)^3}-\frac{(d g+3 e f) (e f-d g)}{32 d^4 e^3 (d+e x)^2}+\frac{f (d g+e f)}{8 d^5 e^2 (d-e x)}+\frac{(d g+e f)^2}{32 d^4 e^3 (d-e x)^2} \]
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) \left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac{(f+g x)^2}{(d-e x)^3 (d+e x)^4} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{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)^2}{8 d^3 e^2 (d+e x)^4}+\frac{(e f-d g) (3 e f+d g)}{16 d^4 e^2 (d+e x)^3}+\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^2 (d+e x)^2}+\frac{-5 e^2 f^2-2 d e f g+d^2 g^2}{16 d^5 e^2 \left (-d^2+e^2 x^2\right )}\right ) \, dx\\ &=\frac{(e f+d g)^2}{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)^2}{24 d^3 e^3 (d+e x)^3}-\frac{(e f-d g) (3 e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}-\frac{\left (5 e^2 f^2+2 d e f g-d^2 g^2\right ) \int \frac{1}{-d^2+e^2 x^2} \, dx}{16 d^5 e^2}\\ &=\frac{(e f+d g)^2}{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)^2}{24 d^3 e^3 (d+e x)^3}-\frac{(e f-d g) (3 e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac{3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}+\frac{\left (5 e^2 f^2+2 d e f g-d^2 g^2\right ) \tanh ^{-1}\left (\frac{e x}{d}\right )}{16 d^6 e^3}\\ \end{align*}
Mathematica [A] time = 0.157574, size = 197, normalized size = 1.05 \[ \frac{\frac{3 d^2 \left (d^2 g^2+2 d e f g-3 e^2 f^2\right )}{(d+e x)^2}+\frac{6 d \left (d^2 g^2-3 e^2 f^2\right )}{d+e x}+3 \left (d^2 g^2-2 d e f g-5 e^2 f^2\right ) \log (d-e x)+3 \left (-d^2 g^2+2 d e f g+5 e^2 f^2\right ) \log (d+e x)-\frac{4 d^3 (e f-d g)^2}{(d+e x)^3}+\frac{3 d^2 (d g+e f)^2}{(d-e x)^2}+\frac{12 d e f (d g+e f)}{d-e x}}{96 d^6 e^3} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.058, size = 348, normalized size = 1.9 \begin{align*}{\frac{{g}^{2}}{32\,{d}^{2}{e}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{fg}{16\,{e}^{2}{d}^{3} \left ( ex-d \right ) ^{2}}}+{\frac{{f}^{2}}{32\,e{d}^{4} \left ( ex-d \right ) ^{2}}}+{\frac{\ln \left ( ex-d \right ){g}^{2}}{32\,{e}^{3}{d}^{4}}}-{\frac{\ln \left ( ex-d \right ) fg}{16\,{e}^{2}{d}^{5}}}-{\frac{5\,\ln \left ( ex-d \right ){f}^{2}}{32\,e{d}^{6}}}-{\frac{fg}{8\,{e}^{2}{d}^{4} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{8\,e{d}^{5} \left ( ex-d \right ) }}+{\frac{{g}^{2}}{16\,{e}^{3}{d}^{3} \left ( ex+d \right ) }}-{\frac{3\,{f}^{2}}{16\,e{d}^{5} \left ( ex+d \right ) }}+{\frac{{g}^{2}}{32\,{d}^{2}{e}^{3} \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{\ln \left ( ex+d \right ){g}^{2}}{32\,{e}^{3}{d}^{4}}}+{\frac{\ln \left ( ex+d \right ) fg}{16\,{e}^{2}{d}^{5}}}+{\frac{5\,\ln \left ( ex+d \right ){f}^{2}}{32\,e{d}^{6}}}-{\frac{{g}^{2}}{24\,d{e}^{3} \left ( ex+d \right ) ^{3}}}+{\frac{fg}{12\,{d}^{2}{e}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{24\,e{d}^{3} \left ( ex+d \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.07283, size = 416, normalized size = 2.21 \begin{align*} -\frac{8 \, d^{4} e^{2} f^{2} - 16 \, d^{5} e f g - 4 \, d^{6} g^{2} + 3 \,{\left (5 \, e^{6} f^{2} + 2 \, d e^{5} f g - d^{2} e^{4} g^{2}\right )} x^{4} + 3 \,{\left (5 \, d e^{5} f^{2} + 2 \, d^{2} e^{4} f g - d^{3} e^{3} g^{2}\right )} x^{3} - 5 \,{\left (5 \, d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g - d^{4} e^{2} g^{2}\right )} x^{2} -{\left (25 \, d^{3} e^{3} f^{2} + 10 \, d^{4} e^{2} f g + 7 \, d^{5} e g^{2}\right )} x}{48 \,{\left (d^{5} e^{8} x^{5} + d^{6} e^{7} x^{4} - 2 \, d^{7} e^{6} x^{3} - 2 \, d^{8} e^{5} x^{2} + d^{9} e^{4} x + d^{10} e^{3}\right )}} + \frac{{\left (5 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{32 \, d^{6} e^{3}} - \frac{{\left (5 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{32 \, d^{6} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.79197, size = 1307, normalized size = 6.95 \begin{align*} -\frac{16 \, d^{5} e^{2} f^{2} - 32 \, d^{6} e f g - 8 \, d^{7} g^{2} + 6 \,{\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} + 6 \,{\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 10 \,{\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} - 2 \,{\left (25 \, d^{4} e^{3} f^{2} + 10 \, d^{5} e^{2} f g + 7 \, d^{6} e g^{2}\right )} x - 3 \,{\left (5 \, d^{5} e^{2} f^{2} + 2 \, d^{6} e f g - d^{7} g^{2} +{\left (5 \, e^{7} f^{2} + 2 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} +{\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 2 \,{\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} +{\left (5 \, d^{4} e^{3} f^{2} + 2 \, 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} + 2 \, d^{6} e f g - d^{7} g^{2} +{\left (5 \, e^{7} f^{2} + 2 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} +{\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \,{\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 2 \,{\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} +{\left (5 \, d^{4} e^{3} f^{2} + 2 \, d^{5} e^{2} f g - d^{6} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{96 \,{\left (d^{6} e^{8} x^{5} + d^{7} e^{7} x^{4} - 2 \, d^{8} e^{6} x^{3} - 2 \, d^{9} e^{5} x^{2} + 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 [A] time = 2.1155, size = 320, normalized size = 1.7 \begin{align*} \frac{4 d^{6} g^{2} + 16 d^{5} e f g - 8 d^{4} e^{2} f^{2} + x^{4} \left (3 d^{2} e^{4} g^{2} - 6 d e^{5} f g - 15 e^{6} f^{2}\right ) + x^{3} \left (3 d^{3} e^{3} g^{2} - 6 d^{2} e^{4} f g - 15 d e^{5} f^{2}\right ) + x^{2} \left (- 5 d^{4} e^{2} g^{2} + 10 d^{3} e^{3} f g + 25 d^{2} e^{4} f^{2}\right ) + x \left (7 d^{5} e g^{2} + 10 d^{4} e^{2} f g + 25 d^{3} e^{3} f^{2}\right )}{48 d^{10} e^{3} + 48 d^{9} e^{4} x - 96 d^{8} e^{5} x^{2} - 96 d^{7} e^{6} x^{3} + 48 d^{6} e^{7} x^{4} + 48 d^{5} e^{8} x^{5}} + \frac{\left (d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}\right ) \log{\left (- \frac{d}{e} + x \right )}}{32 d^{6} e^{3}} - \frac{\left (d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}\right ) \log{\left (\frac{d}{e} + x \right )}}{32 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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