Optimal. Leaf size=210 \[ -\frac{e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac{(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}+\frac{(d g+e f)^2}{64 d^6 e^3 (d-e x)}-\frac{(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d+e x)}-\frac{f (d g+e f)}{16 d^5 e^2 (d+e x)^2}-\frac{(3 e f-d g) (d g+e f)}{48 d^4 e^3 (d+e x)^3}+\frac{(d g+e f) (d g+3 e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^7 e^3} \]
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Rubi [A] time = 0.241934, antiderivative size = 210, 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} \[ -\frac{e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac{(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}+\frac{(d g+e f)^2}{64 d^6 e^3 (d-e x)}-\frac{(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d+e x)}-\frac{f (d g+e f)}{16 d^5 e^2 (d+e x)^2}-\frac{(3 e f-d g) (d g+e f)}{48 d^4 e^3 (d+e x)^3}+\frac{(d g+e f) (d g+3 e f) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^7 e^3} \]
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)^4 \left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac{(f+g x)^2}{(d-e x)^2 (d+e x)^6} \, dx\\ &=\int \left (\frac{(e f+d g)^2}{64 d^6 e^2 (d-e x)^2}+\frac{(-e f+d g)^2}{4 d^2 e^2 (d+e x)^6}+\frac{e^2 f^2-d^2 g^2}{4 d^3 e^2 (d+e x)^5}+\frac{(3 e f-d g) (e f+d g)}{16 d^4 e^2 (d+e x)^4}+\frac{f (e f+d g)}{8 d^5 e (d+e x)^3}+\frac{(e f+d g) (5 e f+d g)}{64 d^6 e^2 (d+e x)^2}+\frac{(e f+d g) (3 e f+d g)}{32 d^6 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac{(e f+d g)^2}{64 d^6 e^3 (d-e x)}-\frac{(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac{e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac{(3 e f-d g) (e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac{f (e f+d g)}{16 d^5 e^2 (d+e x)^2}-\frac{(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d+e x)}+\frac{((e f+d g) (3 e f+d g)) \int \frac{1}{d^2-e^2 x^2} \, dx}{32 d^6 e^2}\\ &=\frac{(e f+d g)^2}{64 d^6 e^3 (d-e x)}-\frac{(e f-d g)^2}{20 d^2 e^3 (d+e x)^5}-\frac{e^2 f^2-d^2 g^2}{16 d^3 e^3 (d+e x)^4}-\frac{(3 e f-d g) (e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac{f (e f+d g)}{16 d^5 e^2 (d+e x)^2}-\frac{(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d+e x)}+\frac{(e f+d g) (3 e f+d g) \tanh ^{-1}\left (\frac{e x}{d}\right )}{32 d^7 e^3}\\ \end{align*}
Mathematica [A] time = 0.18319, size = 229, normalized size = 1.09 \[ \frac{\frac{60 d^4 \left (d^2 g^2-e^2 f^2\right )}{(d+e x)^4}+\frac{20 d^3 \left (d^2 g^2-2 d e f g-3 e^2 f^2\right )}{(d+e x)^3}-\frac{15 d \left (d^2 g^2+6 d e f g+5 e^2 f^2\right )}{d+e x}-15 \left (d^2 g^2+4 d e f g+3 e^2 f^2\right ) \log (d-e x)+15 \left (d^2 g^2+4 d e f g+3 e^2 f^2\right ) \log (d+e x)-\frac{48 d^5 (e f-d g)^2}{(d+e x)^5}-\frac{60 d^2 e f (d g+e f)}{(d+e x)^2}+\frac{15 d (d g+e f)^2}{d-e x}}{960 d^7 e^3} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.062, size = 394, normalized size = 1.9 \begin{align*} -{\frac{\ln \left ( ex-d \right ){g}^{2}}{64\,{e}^{3}{d}^{5}}}-{\frac{\ln \left ( ex-d \right ) fg}{16\,{e}^{2}{d}^{6}}}-{\frac{3\,\ln \left ( ex-d \right ){f}^{2}}{64\,e{d}^{7}}}-{\frac{{g}^{2}}{64\,{d}^{4}{e}^{3} \left ( ex-d \right ) }}-{\frac{fg}{32\,{e}^{2}{d}^{5} \left ( ex-d \right ) }}-{\frac{{f}^{2}}{64\,e{d}^{6} \left ( ex-d \right ) }}+{\frac{\ln \left ( ex+d \right ){g}^{2}}{64\,{e}^{3}{d}^{5}}}+{\frac{\ln \left ( ex+d \right ) fg}{16\,{e}^{2}{d}^{6}}}+{\frac{3\,\ln \left ( ex+d \right ){f}^{2}}{64\,e{d}^{7}}}-{\frac{{g}^{2}}{64\,{d}^{4}{e}^{3} \left ( ex+d \right ) }}-{\frac{3\,fg}{32\,{e}^{2}{d}^{5} \left ( ex+d \right ) }}-{\frac{5\,{f}^{2}}{64\,e{d}^{6} \left ( ex+d \right ) }}+{\frac{{g}^{2}}{16\,{e}^{3}d \left ( ex+d \right ) ^{4}}}-{\frac{{f}^{2}}{16\,e{d}^{3} \left ( ex+d \right ) ^{4}}}+{\frac{{g}^{2}}{48\,{e}^{3}{d}^{2} \left ( ex+d \right ) ^{3}}}-{\frac{fg}{24\,{e}^{2}{d}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{f}^{2}}{16\,e{d}^{4} \left ( ex+d \right ) ^{3}}}-{\frac{{g}^{2}}{20\,{e}^{3} \left ( ex+d \right ) ^{5}}}+{\frac{fg}{10\,d{e}^{2} \left ( ex+d \right ) ^{5}}}-{\frac{{f}^{2}}{20\,e{d}^{2} \left ( ex+d \right ) ^{5}}}-{\frac{fg}{16\,{e}^{2}{d}^{4} \left ( ex+d \right ) ^{2}}}-{\frac{{f}^{2}}{16\,e{d}^{5} \left ( ex+d \right ) ^{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.08751, size = 462, normalized size = 2.2 \begin{align*} \frac{144 \, d^{5} e^{2} f^{2} + 32 \, d^{6} e f g - 16 \, d^{7} g^{2} - 15 \,{\left (3 \, e^{7} f^{2} + 4 \, d e^{6} f g + d^{2} e^{5} g^{2}\right )} x^{5} - 60 \,{\left (3 \, d e^{6} f^{2} + 4 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 80 \,{\left (3 \, d^{2} e^{5} f^{2} + 4 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} - 20 \,{\left (3 \, d^{3} e^{4} f^{2} + 4 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} +{\left (141 \, d^{4} e^{3} f^{2} + 188 \, d^{5} e^{2} f g - 49 \, d^{6} e g^{2}\right )} x}{480 \,{\left (d^{6} e^{9} x^{6} + 4 \, d^{7} e^{8} x^{5} + 5 \, d^{8} e^{7} x^{4} - 5 \, d^{10} e^{5} x^{2} - 4 \, d^{11} e^{4} x - d^{12} e^{3}\right )}} + \frac{{\left (3 \, e^{2} f^{2} + 4 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{64 \, d^{7} e^{3}} - \frac{{\left (3 \, e^{2} f^{2} + 4 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{64 \, d^{7} e^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 2.13187, size = 1413, normalized size = 6.73 \begin{align*} \frac{288 \, d^{6} e^{2} f^{2} + 64 \, d^{7} e f g - 32 \, d^{8} g^{2} - 30 \,{\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 120 \,{\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} - 160 \,{\left (3 \, d^{3} e^{5} f^{2} + 4 \, d^{4} e^{4} f g + d^{5} e^{3} g^{2}\right )} x^{3} - 40 \,{\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + 2 \,{\left (141 \, d^{5} e^{3} f^{2} + 188 \, d^{6} e^{2} f g - 49 \, d^{7} e g^{2}\right )} x - 15 \,{\left (3 \, d^{6} e^{2} f^{2} + 4 \, d^{7} e f g + d^{8} g^{2} -{\left (3 \, e^{8} f^{2} + 4 \, d e^{7} f g + d^{2} e^{6} g^{2}\right )} x^{6} - 4 \,{\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 5 \,{\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} + 5 \,{\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + 4 \,{\left (3 \, d^{5} e^{3} f^{2} + 4 \, d^{6} e^{2} f g + d^{7} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 15 \,{\left (3 \, d^{6} e^{2} f^{2} + 4 \, d^{7} e f g + d^{8} g^{2} -{\left (3 \, e^{8} f^{2} + 4 \, d e^{7} f g + d^{2} e^{6} g^{2}\right )} x^{6} - 4 \,{\left (3 \, d e^{7} f^{2} + 4 \, d^{2} e^{6} f g + d^{3} e^{5} g^{2}\right )} x^{5} - 5 \,{\left (3 \, d^{2} e^{6} f^{2} + 4 \, d^{3} e^{5} f g + d^{4} e^{4} g^{2}\right )} x^{4} + 5 \,{\left (3 \, d^{4} e^{4} f^{2} + 4 \, d^{5} e^{3} f g + d^{6} e^{2} g^{2}\right )} x^{2} + 4 \,{\left (3 \, d^{5} e^{3} f^{2} + 4 \, d^{6} e^{2} f g + d^{7} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{960 \,{\left (d^{7} e^{9} x^{6} + 4 \, d^{8} e^{8} x^{5} + 5 \, d^{9} e^{7} x^{4} - 5 \, d^{11} e^{5} x^{2} - 4 \, d^{12} e^{4} x - d^{13} e^{3}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 2.58207, size = 420, normalized size = 2. \begin{align*} - \frac{16 d^{7} g^{2} - 32 d^{6} e f g - 144 d^{5} e^{2} f^{2} + x^{5} \left (15 d^{2} e^{5} g^{2} + 60 d e^{6} f g + 45 e^{7} f^{2}\right ) + x^{4} \left (60 d^{3} e^{4} g^{2} + 240 d^{2} e^{5} f g + 180 d e^{6} f^{2}\right ) + x^{3} \left (80 d^{4} e^{3} g^{2} + 320 d^{3} e^{4} f g + 240 d^{2} e^{5} f^{2}\right ) + x^{2} \left (20 d^{5} e^{2} g^{2} + 80 d^{4} e^{3} f g + 60 d^{3} e^{4} f^{2}\right ) + x \left (49 d^{6} e g^{2} - 188 d^{5} e^{2} f g - 141 d^{4} e^{3} f^{2}\right )}{- 480 d^{12} e^{3} - 1920 d^{11} e^{4} x - 2400 d^{10} e^{5} x^{2} + 2400 d^{8} e^{7} x^{4} + 1920 d^{7} e^{8} x^{5} + 480 d^{6} e^{9} x^{6}} - \frac{\left (d g + e f\right ) \left (d g + 3 e f\right ) \log{\left (- \frac{d \left (d g + e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 4 d e f g + 3 e^{2} f^{2}\right )} + x \right )}}{64 d^{7} e^{3}} + \frac{\left (d g + e f\right ) \left (d g + 3 e f\right ) \log{\left (\frac{d \left (d g + e f\right ) \left (d g + 3 e f\right )}{e \left (d^{2} g^{2} + 4 d e f g + 3 e^{2} f^{2}\right )} + x \right )}}{64 d^{7} 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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