Optimal. Leaf size=178 \[ \frac {(d g+e f) (d g+5 e f) \tanh ^{-1}\left (\frac {e x}{d}\right )}{32 d^6 e^3}+\frac {(d g+e f)^2}{32 d^5 e^3 (d-e x)}-\frac {f (d g+e f)}{8 d^5 e^2 (d+e x)}-\frac {(3 e f-d g) (d g+e f)}{32 d^4 e^3 (d+e x)^2}-\frac {(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac {e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3} \]
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Rubi [A] time = 0.20, antiderivative size = 178, 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} \[ -\frac {e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac {(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}+\frac {(d g+e f)^2}{32 d^5 e^3 (d-e x)}-\frac {f (d g+e f)}{8 d^5 e^2 (d+e x)}-\frac {(3 e f-d g) (d g+e f)}{32 d^4 e^3 (d+e x)^2}+\frac {(d g+e f) (d g+5 e f) \tanh ^{-1}\left (\frac {e x}{d}\right )}{32 d^6 e^3} \]
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)^3 \left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(f+g x)^2}{(d-e x)^2 (d+e x)^5} \, dx\\ &=\int \left (\frac {(e f+d g)^2}{32 d^5 e^2 (d-e x)^2}+\frac {(-e f+d g)^2}{4 d^2 e^2 (d+e x)^5}+\frac {e^2 f^2-d^2 g^2}{4 d^3 e^2 (d+e x)^4}+\frac {(3 e f-d g) (e f+d g)}{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) (5 e f+d g)}{32 d^5 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=\frac {(e f+d g)^2}{32 d^5 e^3 (d-e x)}-\frac {(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac {e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac {(3 e f-d g) (e f+d g)}{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) (5 e f+d g)) \int \frac {1}{d^2-e^2 x^2} \, dx}{32 d^5 e^2}\\ &=\frac {(e f+d g)^2}{32 d^5 e^3 (d-e x)}-\frac {(e f-d g)^2}{16 d^2 e^3 (d+e x)^4}-\frac {e^2 f^2-d^2 g^2}{12 d^3 e^3 (d+e x)^3}-\frac {(3 e f-d g) (e f+d g)}{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) (5 e f+d g) \tanh ^{-1}\left (\frac {e x}{d}\right )}{32 d^6 e^3}\\ \end {align*}
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Mathematica [A] time = 0.14, size = 195, normalized size = 1.10 \[ \frac {-\frac {12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac {6 d^2 \left (d^2 g^2-2 d e f g-3 e^2 f^2\right )}{(d+e x)^2}-3 \left (d^2 g^2+6 d e f g+5 e^2 f^2\right ) \log (d-e x)+3 \left (d^2 g^2+6 d e f g+5 e^2 f^2\right ) \log (d+e x)+\frac {16 d^3 \left (d^2 g^2-e^2 f^2\right )}{(d+e x)^3}+\frac {6 d (d g+e f)^2}{d-e x}-\frac {24 d e f (d g+e f)}{d+e x}}{192 d^6 e^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.85, size = 648, normalized size = 3.64 \[ \frac {64 \, d^{5} e^{2} f^{2} - 16 \, d^{7} g^{2} - 6 \, {\left (5 \, d e^{6} f^{2} + 6 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 18 \, {\left (5 \, d^{2} e^{5} f^{2} + 6 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} - 14 \, {\left (5 \, d^{3} e^{4} f^{2} + 6 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + 6 \, {\left (5 \, d^{4} e^{3} f^{2} + 6 \, d^{5} e^{2} f g - 7 \, d^{6} e g^{2}\right )} x - 3 \, {\left (5 \, d^{5} e^{2} f^{2} + 6 \, d^{6} e f g + d^{7} g^{2} - {\left (5 \, e^{7} f^{2} + 6 \, d e^{6} f g + d^{2} e^{5} g^{2}\right )} x^{5} - 3 \, {\left (5 \, d e^{6} f^{2} + 6 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 2 \, {\left (5 \, d^{2} e^{5} f^{2} + 6 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} + 2 \, {\left (5 \, d^{3} e^{4} f^{2} + 6 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + 3 \, {\left (5 \, d^{4} e^{3} f^{2} + 6 \, d^{5} e^{2} f g + d^{6} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \, {\left (5 \, d^{5} e^{2} f^{2} + 6 \, d^{6} e f g + d^{7} g^{2} - {\left (5 \, e^{7} f^{2} + 6 \, d e^{6} f g + d^{2} e^{5} g^{2}\right )} x^{5} - 3 \, {\left (5 \, d e^{6} f^{2} + 6 \, d^{2} e^{5} f g + d^{3} e^{4} g^{2}\right )} x^{4} - 2 \, {\left (5 \, d^{2} e^{5} f^{2} + 6 \, d^{3} e^{4} f g + d^{4} e^{3} g^{2}\right )} x^{3} + 2 \, {\left (5 \, d^{3} e^{4} f^{2} + 6 \, d^{4} e^{3} f g + d^{5} e^{2} g^{2}\right )} x^{2} + 3 \, {\left (5 \, d^{4} e^{3} f^{2} + 6 \, d^{5} e^{2} f g + d^{6} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{192 \, {\left (d^{6} e^{8} x^{5} + 3 \, d^{7} e^{7} x^{4} + 2 \, d^{8} e^{6} x^{3} - 2 \, d^{9} e^{5} x^{2} - 3 \, d^{10} e^{4} x - d^{11} e^{3}\right )}} \]
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 = 0.02, size = 341, normalized size = 1.92 \[ \frac {f g}{8 \left (e x +d \right )^{4} d \,e^{2}}-\frac {f^{2}}{16 \left (e x +d \right )^{4} d^{2} e}-\frac {g^{2}}{16 \left (e x +d \right )^{4} e^{3}}+\frac {g^{2}}{12 \left (e x +d \right )^{3} d \,e^{3}}-\frac {f^{2}}{12 \left (e x +d \right )^{3} d^{3} e}+\frac {g^{2}}{32 \left (e x +d \right )^{2} d^{2} e^{3}}-\frac {f g}{16 \left (e x +d \right )^{2} d^{3} e^{2}}-\frac {3 f^{2}}{32 \left (e x +d \right )^{2} d^{4} e}-\frac {g^{2}}{32 \left (e x -d \right ) d^{3} e^{3}}-\frac {f g}{16 \left (e x -d \right ) d^{4} e^{2}}-\frac {f g}{8 \left (e x +d \right ) d^{4} e^{2}}-\frac {g^{2} \ln \left (e x -d \right )}{64 d^{4} e^{3}}+\frac {g^{2} \ln \left (e x +d \right )}{64 d^{4} e^{3}}-\frac {f^{2}}{32 \left (e x -d \right ) d^{5} e}-\frac {f^{2}}{8 \left (e x +d \right ) d^{5} e}-\frac {3 f g \ln \left (e x -d \right )}{32 d^{5} e^{2}}+\frac {3 f g \ln \left (e x +d \right )}{32 d^{5} e^{2}}-\frac {5 f^{2} \ln \left (e x -d \right )}{64 d^{6} e}+\frac {5 f^{2} \ln \left (e x +d \right )}{64 d^{6} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 298, normalized size = 1.67 \[ \frac {32 \, d^{4} e^{2} f^{2} - 8 \, d^{6} g^{2} - 3 \, {\left (5 \, e^{6} f^{2} + 6 \, d e^{5} f g + d^{2} e^{4} g^{2}\right )} x^{4} - 9 \, {\left (5 \, d e^{5} f^{2} + 6 \, d^{2} e^{4} f g + d^{3} e^{3} g^{2}\right )} x^{3} - 7 \, {\left (5 \, d^{2} e^{4} f^{2} + 6 \, d^{3} e^{3} f g + d^{4} e^{2} g^{2}\right )} x^{2} + 3 \, {\left (5 \, d^{3} e^{3} f^{2} + 6 \, d^{4} e^{2} f g - 7 \, d^{5} e g^{2}\right )} x}{96 \, {\left (d^{5} e^{8} x^{5} + 3 \, d^{6} e^{7} x^{4} + 2 \, d^{7} e^{6} x^{3} - 2 \, d^{8} e^{5} x^{2} - 3 \, d^{9} e^{4} x - d^{10} e^{3}\right )}} + \frac {{\left (5 \, e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{64 \, d^{6} e^{3}} - \frac {{\left (5 \, e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{64 \, d^{6} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.70, size = 274, normalized size = 1.54 \[ \frac {\frac {d^2\,g^2-4\,e^2\,f^2}{12\,d\,e^3}+\frac {3\,x^3\,\left (d^2\,g^2+6\,d\,e\,f\,g+5\,e^2\,f^2\right )}{32\,d^4}+\frac {e\,x^4\,\left (d^2\,g^2+6\,d\,e\,f\,g+5\,e^2\,f^2\right )}{32\,d^5}-\frac {x\,\left (-7\,d^2\,g^2+6\,d\,e\,f\,g+5\,e^2\,f^2\right )}{32\,d^2\,e^2}+\frac {7\,x^2\,\left (d^2\,g^2+6\,d\,e\,f\,g+5\,e^2\,f^2\right )}{96\,d^3\,e}}{d^5+3\,d^4\,e\,x+2\,d^3\,e^2\,x^2-2\,d^2\,e^3\,x^3-3\,d\,e^4\,x^4-e^5\,x^5}+\frac {\mathrm {atanh}\left (\frac {e\,x\,\left (d\,g+e\,f\right )\,\left (d\,g+5\,e\,f\right )}{d\,\left (d^2\,g^2+6\,d\,e\,f\,g+5\,e^2\,f^2\right )}\right )\,\left (d\,g+e\,f\right )\,\left (d\,g+5\,e\,f\right )}{32\,d^6\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.93, size = 376, normalized size = 2.11 \[ \frac {- 8 d^{6} g^{2} + 32 d^{4} e^{2} f^{2} + x^{4} \left (- 3 d^{2} e^{4} g^{2} - 18 d e^{5} f g - 15 e^{6} f^{2}\right ) + x^{3} \left (- 9 d^{3} e^{3} g^{2} - 54 d^{2} e^{4} f g - 45 d e^{5} f^{2}\right ) + x^{2} \left (- 7 d^{4} e^{2} g^{2} - 42 d^{3} e^{3} f g - 35 d^{2} e^{4} f^{2}\right ) + x \left (- 21 d^{5} e g^{2} + 18 d^{4} e^{2} f g + 15 d^{3} e^{3} f^{2}\right )}{- 96 d^{10} e^{3} - 288 d^{9} e^{4} x - 192 d^{8} e^{5} x^{2} + 192 d^{7} e^{6} x^{3} + 288 d^{6} e^{7} x^{4} + 96 d^{5} e^{8} x^{5}} - \frac {\left (d g + e f\right ) \left (d g + 5 e f\right ) \log {\left (- \frac {d \left (d g + e f\right ) \left (d g + 5 e f\right )}{e \left (d^{2} g^{2} + 6 d e f g + 5 e^{2} f^{2}\right )} + x \right )}}{64 d^{6} e^{3}} + \frac {\left (d g + e f\right ) \left (d g + 5 e f\right ) \log {\left (\frac {d \left (d g + e f\right ) \left (d g + 5 e f\right )}{e \left (d^{2} g^{2} + 6 d e f g + 5 e^{2} f^{2}\right )} + x \right )}}{64 d^{6} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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