Optimal. Leaf size=139 \[ \frac {(d g+e f)^2 \tanh ^{-1}\left (\frac {e x}{d}\right )}{16 d^5 e^3}-\frac {(d g+e f)^2}{16 d^4 e^3 (d+e x)}-\frac {(d g+e f)^2}{16 d^3 e^3 (d+e x)^2}-\frac {(3 d g+e f) (e f-d g)}{12 d^2 e^3 (d+e x)^3}-\frac {(e f-d g)^2}{8 d e^3 (d+e x)^4} \]
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Rubi [A] time = 0.13, antiderivative size = 139, 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} \[ -\frac {(3 d g+e f) (e f-d g)}{12 d^2 e^3 (d+e x)^3}-\frac {(d g+e f)^2}{16 d^4 e^3 (d+e x)}-\frac {(d g+e f)^2}{16 d^3 e^3 (d+e x)^2}+\frac {(d g+e f)^2 \tanh ^{-1}\left (\frac {e x}{d}\right )}{16 d^5 e^3}-\frac {(e f-d g)^2}{8 d e^3 (d+e x)^4} \]
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)^4 \left (d^2-e^2 x^2\right )} \, dx &=\int \frac {(f+g x)^2}{(d-e x) (d+e x)^5} \, dx\\ &=\int \left (\frac {(-e f+d g)^2}{2 d e^2 (d+e x)^5}+\frac {(e f-d g) (e f+3 d g)}{4 d^2 e^2 (d+e x)^4}+\frac {(e f+d g)^2}{8 d^3 e^2 (d+e x)^3}+\frac {(e f+d g)^2}{16 d^4 e^2 (d+e x)^2}+\frac {(e f+d g)^2}{16 d^4 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=-\frac {(e f-d g)^2}{8 d e^3 (d+e x)^4}-\frac {(e f-d g) (e f+3 d g)}{12 d^2 e^3 (d+e x)^3}-\frac {(e f+d g)^2}{16 d^3 e^3 (d+e x)^2}-\frac {(e f+d g)^2}{16 d^4 e^3 (d+e x)}+\frac {(e f+d g)^2 \int \frac {1}{d^2-e^2 x^2} \, dx}{16 d^4 e^2}\\ &=-\frac {(e f-d g)^2}{8 d e^3 (d+e x)^4}-\frac {(e f-d g) (e f+3 d g)}{12 d^2 e^3 (d+e x)^3}-\frac {(e f+d g)^2}{16 d^3 e^3 (d+e x)^2}-\frac {(e f+d g)^2}{16 d^4 e^3 (d+e x)}+\frac {(e f+d g)^2 \tanh ^{-1}\left (\frac {e x}{d}\right )}{16 d^5 e^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 142, normalized size = 1.02 \[ -\frac {\frac {12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac {6 d^2 (d g+e f)^2}{(d+e x)^2}+\frac {8 d^3 \left (-3 d^2 g^2+2 d e f g+e^2 f^2\right )}{(d+e x)^3}+\frac {6 d (d g+e f)^2}{d+e x}+3 (d g+e f)^2 \log (d-e x)-3 (d g+e f)^2 \log (d+e x)}{96 d^5 e^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 511, normalized size = 3.68 \[ -\frac {32 \, d^{4} e^{2} f^{2} + 16 \, d^{5} e f g + 6 \, {\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 24 \, {\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (19 \, d^{3} e^{3} f^{2} + 38 \, d^{4} e^{2} f g + 3 \, d^{5} e g^{2}\right )} x - 3 \, {\left (d^{4} e^{2} f^{2} + 2 \, d^{5} e f g + d^{6} g^{2} + {\left (e^{6} f^{2} + 2 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{4} + 4 \, {\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 6 \, {\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 4 \, {\left (d^{3} e^{3} f^{2} + 2 \, d^{4} e^{2} f g + d^{5} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \, {\left (d^{4} e^{2} f^{2} + 2 \, d^{5} e f g + d^{6} g^{2} + {\left (e^{6} f^{2} + 2 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{4} + 4 \, {\left (d e^{5} f^{2} + 2 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} + 6 \, {\left (d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 4 \, {\left (d^{3} e^{3} f^{2} + 2 \, d^{4} e^{2} f g + d^{5} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{96 \, {\left (d^{5} e^{7} x^{4} + 4 \, d^{6} e^{6} x^{3} + 6 \, d^{7} e^{5} x^{2} + 4 \, d^{8} e^{4} x + d^{9} e^{3}\right )}} \]
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 [B] time = 0.01, size = 312, normalized size = 2.24 \[ -\frac {d \,g^{2}}{8 \left (e x +d \right )^{4} e^{3}}-\frac {f^{2}}{8 \left (e x +d \right )^{4} d e}+\frac {f g}{4 \left (e x +d \right )^{4} e^{2}}-\frac {f g}{6 \left (e x +d \right )^{3} d \,e^{2}}-\frac {f^{2}}{12 \left (e x +d \right )^{3} d^{2} e}+\frac {g^{2}}{4 \left (e x +d \right )^{3} e^{3}}-\frac {g^{2}}{16 \left (e x +d \right )^{2} d \,e^{3}}-\frac {f g}{8 \left (e x +d \right )^{2} d^{2} e^{2}}-\frac {f^{2}}{16 \left (e x +d \right )^{2} d^{3} e}-\frac {g^{2}}{16 \left (e x +d \right ) d^{2} e^{3}}-\frac {f g}{8 \left (e x +d \right ) d^{3} e^{2}}-\frac {g^{2} \ln \left (e x -d \right )}{32 d^{3} e^{3}}+\frac {g^{2} \ln \left (e x +d \right )}{32 d^{3} e^{3}}-\frac {f^{2}}{16 \left (e x +d \right ) d^{4} e}-\frac {f g \ln \left (e x -d \right )}{16 d^{4} e^{2}}+\frac {f g \ln \left (e x +d \right )}{16 d^{4} e^{2}}-\frac {f^{2} \ln \left (e x -d \right )}{32 d^{5} e}+\frac {f^{2} \ln \left (e x +d \right )}{32 d^{5} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.49, size = 236, normalized size = 1.70 \[ -\frac {16 \, d^{3} e f^{2} + 8 \, d^{4} f g + 3 \, {\left (e^{4} f^{2} + 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{3} + 12 \, {\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x^{2} + {\left (19 \, d^{2} e^{2} f^{2} + 38 \, d^{3} e f g + 3 \, d^{4} g^{2}\right )} x}{48 \, {\left (d^{4} e^{6} x^{4} + 4 \, d^{5} e^{5} x^{3} + 6 \, d^{6} e^{4} x^{2} + 4 \, d^{7} e^{3} x + d^{8} e^{2}\right )}} + \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{32 \, d^{5} e^{3}} - \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{32 \, d^{5} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 180, normalized size = 1.29 \[ \frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )\,{\left (d\,g+e\,f\right )}^2}{16\,d^5\,e^3}-\frac {\frac {x^3\,\left (d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{16\,d^4}+\frac {2\,e\,f^2+d\,g\,f}{6\,d\,e^2}+\frac {x\,\left (3\,d^2\,g^2+38\,d\,e\,f\,g+19\,e^2\,f^2\right )}{48\,d^2\,e^2}+\frac {x^2\,\left (d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{4\,d^3\,e}}{d^4+4\,d^3\,e\,x+6\,d^2\,e^2\,x^2+4\,d\,e^3\,x^3+e^4\,x^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.93, size = 282, normalized size = 2.03 \[ - \frac {8 d^{4} f g + 16 d^{3} e f^{2} + x^{3} \left (3 d^{2} e^{2} g^{2} + 6 d e^{3} f g + 3 e^{4} f^{2}\right ) + x^{2} \left (12 d^{3} e g^{2} + 24 d^{2} e^{2} f g + 12 d e^{3} f^{2}\right ) + x \left (3 d^{4} g^{2} + 38 d^{3} e f g + 19 d^{2} e^{2} f^{2}\right )}{48 d^{8} e^{2} + 192 d^{7} e^{3} x + 288 d^{6} e^{4} x^{2} + 192 d^{5} e^{5} x^{3} + 48 d^{4} e^{6} x^{4}} - \frac {\left (d g + e f\right )^{2} \log {\left (- \frac {d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{32 d^{5} e^{3}} + \frac {\left (d g + e f\right )^{2} \log {\left (\frac {d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{32 d^{5} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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