Optimal. Leaf size=146 \[ \frac {f (d g+e f) \tanh ^{-1}\left (\frac {e x}{d}\right )}{4 d^5 e^2}+\frac {(d g+e f)^2}{16 d^4 e^3 (d-e x)}-\frac {(3 e f-d g) (d g+e f)}{16 d^4 e^3 (d+e x)}-\frac {(e f-d g)^2}{12 d^2 e^3 (d+e x)^3}-\frac {e^2 f^2-d^2 g^2}{8 d^3 e^3 (d+e x)^2} \]
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Rubi [A] time = 0.16, antiderivative size = 146, normalized size of antiderivative = 1.00, 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} \[ -\frac {e^2 f^2-d^2 g^2}{8 d^3 e^3 (d+e x)^2}-\frac {(e f-d g)^2}{12 d^2 e^3 (d+e x)^3}+\frac {(d g+e f)^2}{16 d^4 e^3 (d-e x)}-\frac {(3 e f-d g) (d g+e f)}{16 d^4 e^3 (d+e x)}+\frac {f (d g+e f) \tanh ^{-1}\left (\frac {e x}{d}\right )}{4 d^5 e^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 208
Rule 848
Rubi steps
\begin {align*} \int \frac {(f+g x)^2}{(d+e x)^2 \left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(f+g x)^2}{(d-e x)^2 (d+e x)^4} \, dx\\ &=\int \left (\frac {(e f+d g)^2}{16 d^4 e^2 (d-e x)^2}+\frac {(-e f+d g)^2}{4 d^2 e^2 (d+e x)^4}+\frac {e^2 f^2-d^2 g^2}{4 d^3 e^2 (d+e x)^3}+\frac {(3 e f-d g) (e f+d g)}{16 d^4 e^2 (d+e x)^2}+\frac {f (e f+d g)}{4 d^4 e \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac {(e f+d g)^2}{16 d^4 e^3 (d-e x)}-\frac {(e f-d g)^2}{12 d^2 e^3 (d+e x)^3}-\frac {e^2 f^2-d^2 g^2}{8 d^3 e^3 (d+e x)^2}-\frac {(3 e f-d g) (e f+d g)}{16 d^4 e^3 (d+e x)}+\frac {(f (e f+d g)) \int \frac {1}{d^2-e^2 x^2} \, dx}{4 d^4 e}\\ &=\frac {(e f+d g)^2}{16 d^4 e^3 (d-e x)}-\frac {(e f-d g)^2}{12 d^2 e^3 (d+e x)^3}-\frac {e^2 f^2-d^2 g^2}{8 d^3 e^3 (d+e x)^2}-\frac {(3 e f-d g) (e f+d g)}{16 d^4 e^3 (d+e x)}+\frac {f (e f+d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{4 d^5 e^2}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 171, normalized size = 1.17 \[ \frac {2 d \left (2 d^5 g^2+2 d^4 e g (f+2 g x)+d^3 e^2 f (g x-4 f)+d^2 e^3 f x (f+6 g x)+3 d e^4 f x^2 (2 f+g x)+3 e^5 f^2 x^3\right )+3 e f (e x-d) (d+e x)^3 (d g+e f) \log (d-e x)+3 e f (d-e x) (d+e x)^3 (d g+e f) \log (d+e x)}{24 d^5 e^3 (d-e x) (d+e x)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.84, size = 337, normalized size = 2.31 \[ \frac {8 \, d^{4} e^{2} f^{2} - 4 \, d^{5} e f g - 4 \, d^{6} g^{2} - 6 \, {\left (d e^{5} f^{2} + d^{2} e^{4} f g\right )} x^{3} - 12 \, {\left (d^{2} e^{4} f^{2} + d^{3} e^{3} f g\right )} x^{2} - 2 \, {\left (d^{3} e^{3} f^{2} + d^{4} e^{2} f g + 4 \, d^{5} e g^{2}\right )} x - 3 \, {\left (d^{4} e^{2} f^{2} + d^{5} e f g - {\left (e^{6} f^{2} + d e^{5} f g\right )} x^{4} - 2 \, {\left (d e^{5} f^{2} + d^{2} e^{4} f g\right )} x^{3} + 2 \, {\left (d^{3} e^{3} f^{2} + d^{4} e^{2} f g\right )} x\right )} \log \left (e x + d\right ) + 3 \, {\left (d^{4} e^{2} f^{2} + d^{5} e f g - {\left (e^{6} f^{2} + d e^{5} f g\right )} x^{4} - 2 \, {\left (d e^{5} f^{2} + d^{2} e^{4} f g\right )} x^{3} + 2 \, {\left (d^{3} e^{3} f^{2} + d^{4} e^{2} f g\right )} x\right )} \log \left (e x - d\right )}{24 \, {\left (d^{5} e^{7} x^{4} + 2 \, d^{6} e^{6} x^{3} - 2 \, d^{8} e^{4} x - d^{9} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.02, size = 270, normalized size = 1.85 \[ \frac {f g}{6 \left (e x +d \right )^{3} d \,e^{2}}-\frac {f^{2}}{12 \left (e x +d \right )^{3} d^{2} e}-\frac {g^{2}}{12 \left (e x +d \right )^{3} e^{3}}+\frac {g^{2}}{8 \left (e x +d \right )^{2} d \,e^{3}}-\frac {f^{2}}{8 \left (e x +d \right )^{2} d^{3} e}-\frac {g^{2}}{16 \left (e x -d \right ) d^{2} e^{3}}+\frac {g^{2}}{16 \left (e x +d \right ) d^{2} e^{3}}-\frac {f g}{8 \left (e x -d \right ) d^{3} e^{2}}-\frac {f g}{8 \left (e x +d \right ) d^{3} e^{2}}-\frac {f^{2}}{16 \left (e x -d \right ) d^{4} e}-\frac {3 f^{2}}{16 \left (e x +d \right ) d^{4} e}-\frac {f g \ln \left (e x -d \right )}{8 d^{4} e^{2}}+\frac {f g \ln \left (e x +d \right )}{8 d^{4} e^{2}}-\frac {f^{2} \ln \left (e x -d \right )}{8 d^{5} e}+\frac {f^{2} \ln \left (e x +d \right )}{8 d^{5} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 197, normalized size = 1.35 \[ \frac {4 \, d^{3} e^{2} f^{2} - 2 \, d^{4} e f g - 2 \, d^{5} g^{2} - 3 \, {\left (e^{5} f^{2} + d e^{4} f g\right )} x^{3} - 6 \, {\left (d e^{4} f^{2} + d^{2} e^{3} f g\right )} x^{2} - {\left (d^{2} e^{3} f^{2} + d^{3} e^{2} f g + 4 \, d^{4} e g^{2}\right )} x}{12 \, {\left (d^{4} e^{7} x^{4} + 2 \, d^{5} e^{6} x^{3} - 2 \, d^{7} e^{4} x - d^{8} e^{3}\right )}} + \frac {{\left (e f^{2} + d f g\right )} \log \left (e x + d\right )}{8 \, d^{5} e^{2}} - \frac {{\left (e f^{2} + d f g\right )} \log \left (e x - d\right )}{8 \, d^{5} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.63, size = 148, normalized size = 1.01 \[ \frac {\frac {d^2\,g^2+d\,e\,f\,g-2\,e^2\,f^2}{6\,d\,e^3}+\frac {f\,x^2\,\left (d\,g+e\,f\right )}{2\,d^3}+\frac {x\,\left (4\,d^2\,g^2+d\,e\,f\,g+e^2\,f^2\right )}{12\,d^2\,e^2}+\frac {e\,f\,x^3\,\left (d\,g+e\,f\right )}{4\,d^4}}{d^4+2\,d^3\,e\,x-2\,d\,e^3\,x^3-e^4\,x^4}+\frac {f\,\mathrm {atanh}\left (\frac {e\,x}{d}\right )\,\left (d\,g+e\,f\right )}{4\,d^5\,e^2} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.36, size = 241, normalized size = 1.65 \[ \frac {- 2 d^{5} g^{2} - 2 d^{4} e f g + 4 d^{3} e^{2} f^{2} + x^{3} \left (- 3 d e^{4} f g - 3 e^{5} f^{2}\right ) + x^{2} \left (- 6 d^{2} e^{3} f g - 6 d e^{4} f^{2}\right ) + x \left (- 4 d^{4} e g^{2} - d^{3} e^{2} f g - d^{2} e^{3} f^{2}\right )}{- 12 d^{8} e^{3} - 24 d^{7} e^{4} x + 24 d^{5} e^{6} x^{3} + 12 d^{4} e^{7} x^{4}} - \frac {f \left (d g + e f\right ) \log {\left (- \frac {d f \left (d g + e f\right )}{e \left (d f g + e f^{2}\right )} + x \right )}}{8 d^{5} e^{2}} + \frac {f \left (d g + e f\right ) \log {\left (\frac {d f \left (d g + e f\right )}{e \left (d f g + e f^{2}\right )} + x \right )}}{8 d^{5} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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