Optimal. Leaf size=121 \[ \frac {(3 e f-d g) (d g+e f) \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^4 e^3}+\frac {(d g+e f)^2}{8 d^3 e^3 (d-e x)}-\frac {(e f-d g)^2}{8 d^2 e^3 (d+e x)^2}-\frac {e^2 f^2-d^2 g^2}{4 d^3 e^3 (d+e x)} \]
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Rubi [A] time = 0.14, antiderivative size = 121, 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} \[ -\frac {e^2 f^2-d^2 g^2}{4 d^3 e^3 (d+e x)}-\frac {(e f-d g)^2}{8 d^2 e^3 (d+e x)^2}+\frac {(d g+e f)^2}{8 d^3 e^3 (d-e x)}+\frac {(3 e f-d g) (d g+e f) \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^4 e^3} \]
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) \left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(f+g x)^2}{(d-e x)^2 (d+e x)^3} \, dx\\ &=\int \left (\frac {(e f+d g)^2}{8 d^3 e^2 (d-e x)^2}+\frac {(-e f+d g)^2}{4 d^2 e^2 (d+e x)^3}+\frac {e^2 f^2-d^2 g^2}{4 d^3 e^2 (d+e x)^2}+\frac {(3 e f-d g) (e f+d g)}{8 d^3 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac {(e f+d g)^2}{8 d^3 e^3 (d-e x)}-\frac {(e f-d g)^2}{8 d^2 e^3 (d+e x)^2}-\frac {e^2 f^2-d^2 g^2}{4 d^3 e^3 (d+e x)}+\frac {((3 e f-d g) (e f+d g)) \int \frac {1}{d^2-e^2 x^2} \, dx}{8 d^3 e^2}\\ &=\frac {(e f+d g)^2}{8 d^3 e^3 (d-e x)}-\frac {(e f-d g)^2}{8 d^2 e^3 (d+e x)^2}-\frac {e^2 f^2-d^2 g^2}{4 d^3 e^3 (d+e x)}+\frac {(3 e f-d g) (e f+d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{8 d^4 e^3}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 139, normalized size = 1.15 \[ \frac {\frac {4 d \left (d^2 g^2-e^2 f^2\right )}{d+e x}+\left (d^2 g^2-2 d e f g-3 e^2 f^2\right ) \log (d-e x)+\left (-d^2 g^2+2 d e f g+3 e^2 f^2\right ) \log (d+e x)-\frac {2 d^2 (e f-d g)^2}{(d+e x)^2}+\frac {2 d (d g+e f)^2}{d-e x}}{16 d^4 e^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.80, size = 417, normalized size = 3.45 \[ \frac {4 \, d^{3} e^{2} f^{2} - 8 \, d^{4} e f g - 4 \, d^{5} g^{2} - 2 \, {\left (3 \, d e^{4} f^{2} + 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} - 2 \, {\left (3 \, d^{2} e^{3} f^{2} + 2 \, d^{3} e^{2} f g + 3 \, d^{4} e g^{2}\right )} x - {\left (3 \, d^{3} e^{2} f^{2} + 2 \, d^{4} e f g - d^{5} g^{2} - {\left (3 \, e^{5} f^{2} + 2 \, d e^{4} f g - d^{2} e^{3} g^{2}\right )} x^{3} - {\left (3 \, d e^{4} f^{2} + 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} + {\left (3 \, d^{2} e^{3} f^{2} + 2 \, d^{3} e^{2} f g - d^{4} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + {\left (3 \, d^{3} e^{2} f^{2} + 2 \, d^{4} e f g - d^{5} g^{2} - {\left (3 \, e^{5} f^{2} + 2 \, d e^{4} f g - d^{2} e^{3} g^{2}\right )} x^{3} - {\left (3 \, d e^{4} f^{2} + 2 \, d^{2} e^{3} f g - d^{3} e^{2} g^{2}\right )} x^{2} + {\left (3 \, d^{2} e^{3} f^{2} + 2 \, d^{3} e^{2} f g - d^{4} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{16 \, {\left (d^{4} e^{6} x^{3} + d^{5} e^{5} x^{2} - d^{6} e^{4} x - d^{7} e^{3}\right )}} \]
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 [B] time = 0.01, size = 253, normalized size = 2.09 \[ \frac {f g}{4 \left (e x +d \right )^{2} d \,e^{2}}-\frac {f^{2}}{8 \left (e x +d \right )^{2} d^{2} e}-\frac {g^{2}}{8 \left (e x +d \right )^{2} e^{3}}-\frac {g^{2}}{8 \left (e x -d \right ) d \,e^{3}}+\frac {g^{2}}{4 \left (e x +d \right ) d \,e^{3}}-\frac {f g}{4 \left (e x -d \right ) d^{2} e^{2}}+\frac {g^{2} \ln \left (e x -d \right )}{16 d^{2} e^{3}}-\frac {g^{2} \ln \left (e x +d \right )}{16 d^{2} e^{3}}-\frac {f^{2}}{8 \left (e x -d \right ) d^{3} e}-\frac {f^{2}}{4 \left (e x +d \right ) d^{3} e}-\frac {f g \ln \left (e x -d \right )}{8 d^{3} e^{2}}+\frac {f g \ln \left (e x +d \right )}{8 d^{3} e^{2}}-\frac {3 f^{2} \ln \left (e x -d \right )}{16 d^{4} e}+\frac {3 f^{2} \ln \left (e x +d \right )}{16 d^{4} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.47, size = 212, normalized size = 1.75 \[ \frac {2 \, d^{2} e^{2} f^{2} - 4 \, d^{3} e f g - 2 \, d^{4} g^{2} - {\left (3 \, e^{4} f^{2} + 2 \, d e^{3} f g - d^{2} e^{2} g^{2}\right )} x^{2} - {\left (3 \, d e^{3} f^{2} + 2 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} x}{8 \, {\left (d^{3} e^{6} x^{3} + d^{4} e^{5} x^{2} - d^{5} e^{4} x - d^{6} e^{3}\right )}} + \frac {{\left (3 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{16 \, d^{4} e^{3}} - \frac {{\left (3 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{16 \, d^{4} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 198, normalized size = 1.64 \[ \frac {\frac {d^2\,g^2+2\,d\,e\,f\,g-e^2\,f^2}{4\,d\,e^3}+\frac {x\,\left (3\,d^2\,g^2+2\,d\,e\,f\,g+3\,e^2\,f^2\right )}{8\,d^2\,e^2}+\frac {x^2\,\left (-d^2\,g^2+2\,d\,e\,f\,g+3\,e^2\,f^2\right )}{8\,d^3\,e}}{d^3+d^2\,e\,x-d\,e^2\,x^2-e^3\,x^3}+\frac {\mathrm {atanh}\left (\frac {e\,x\,\left (d\,g+e\,f\right )\,\left (d\,g-3\,e\,f\right )}{d\,\left (-d^2\,g^2+2\,d\,e\,f\,g+3\,e^2\,f^2\right )}\right )\,\left (d\,g+e\,f\right )\,\left (d\,g-3\,e\,f\right )}{8\,d^4\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.26, size = 279, normalized size = 2.31 \[ \frac {- 2 d^{4} g^{2} - 4 d^{3} e f g + 2 d^{2} e^{2} f^{2} + x^{2} \left (d^{2} e^{2} g^{2} - 2 d e^{3} f g - 3 e^{4} f^{2}\right ) + x \left (- 3 d^{3} e g^{2} - 2 d^{2} e^{2} f g - 3 d e^{3} f^{2}\right )}{- 8 d^{6} e^{3} - 8 d^{5} e^{4} x + 8 d^{4} e^{5} x^{2} + 8 d^{3} e^{6} x^{3}} + \frac {\left (d g - 3 e f\right ) \left (d g + e f\right ) \log {\left (- \frac {d \left (d g - 3 e f\right ) \left (d g + e f\right )}{e \left (d^{2} g^{2} - 2 d e f g - 3 e^{2} f^{2}\right )} + x \right )}}{16 d^{4} e^{3}} - \frac {\left (d g - 3 e f\right ) \left (d g + e f\right ) \log {\left (\frac {d \left (d g - 3 e f\right ) \left (d g + e f\right )}{e \left (d^{2} g^{2} - 2 d e f g - 3 e^{2} f^{2}\right )} + x \right )}}{16 d^{4} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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