Optimal. Leaf size=188 \[ \frac {f (d g+e f)}{8 d^5 e^2 (d-e x)}-\frac {(d g+3 e f) (e f-d g)}{32 d^4 e^3 (d+e x)^2}+\frac {(d g+e f)^2}{32 d^4 e^3 (d-e x)^2}-\frac {(e f-d g)^2}{24 d^3 e^3 (d+e x)^3}+\frac {\left (-d^2 g^2+2 d e f g+5 e^2 f^2\right ) \tanh ^{-1}\left (\frac {e x}{d}\right )}{16 d^6 e^3}-\frac {3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)} \]
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Rubi [A] time = 0.21, antiderivative size = 188, 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} \[ -\frac {3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}+\frac {\left (-d^2 g^2+2 d e f g+5 e^2 f^2\right ) \tanh ^{-1}\left (\frac {e x}{d}\right )}{16 d^6 e^3}-\frac {(e f-d g)^2}{24 d^3 e^3 (d+e x)^3}-\frac {(d g+3 e f) (e f-d g)}{32 d^4 e^3 (d+e x)^2}+\frac {f (d g+e f)}{8 d^5 e^2 (d-e x)}+\frac {(d g+e f)^2}{32 d^4 e^3 (d-e x)^2} \]
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) \left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac {(f+g x)^2}{(d-e x)^3 (d+e x)^4} \, dx\\ &=\int \left (\frac {(e f+d g)^2}{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)^2}{8 d^3 e^2 (d+e x)^4}+\frac {(e f-d g) (3 e f+d g)}{16 d^4 e^2 (d+e x)^3}+\frac {3 e^2 f^2-d^2 g^2}{16 d^5 e^2 (d+e x)^2}+\frac {-5 e^2 f^2-2 d e f g+d^2 g^2}{16 d^5 e^2 \left (-d^2+e^2 x^2\right )}\right ) \, dx\\ &=\frac {(e f+d g)^2}{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)^2}{24 d^3 e^3 (d+e x)^3}-\frac {(e f-d g) (3 e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac {3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}-\frac {\left (5 e^2 f^2+2 d e f g-d^2 g^2\right ) \int \frac {1}{-d^2+e^2 x^2} \, dx}{16 d^5 e^2}\\ &=\frac {(e f+d g)^2}{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)^2}{24 d^3 e^3 (d+e x)^3}-\frac {(e f-d g) (3 e f+d g)}{32 d^4 e^3 (d+e x)^2}-\frac {3 e^2 f^2-d^2 g^2}{16 d^5 e^3 (d+e x)}+\frac {\left (5 e^2 f^2+2 d e f g-d^2 g^2\right ) \tanh ^{-1}\left (\frac {e x}{d}\right )}{16 d^6 e^3}\\ \end {align*}
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Mathematica [A] time = 0.15, size = 197, normalized size = 1.05 \[ \frac {-\frac {4 d^3 (e f-d g)^2}{(d+e x)^3}+\frac {3 d^2 \left (d^2 g^2+2 d e f g-3 e^2 f^2\right )}{(d+e x)^2}+\frac {6 d \left (d^2 g^2-3 e^2 f^2\right )}{d+e x}+3 \left (d^2 g^2-2 d e f g-5 e^2 f^2\right ) \log (d-e x)+3 \left (-d^2 g^2+2 d e f g+5 e^2 f^2\right ) \log (d+e x)+\frac {3 d^2 (d g+e f)^2}{(d-e x)^2}+\frac {12 d e f (d g+e f)}{d-e x}}{96 d^6 e^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.68, size = 662, normalized size = 3.52 \[ -\frac {16 \, d^{5} e^{2} f^{2} - 32 \, d^{6} e f g - 8 \, d^{7} g^{2} + 6 \, {\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} + 6 \, {\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 10 \, {\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} - 2 \, {\left (25 \, d^{4} e^{3} f^{2} + 10 \, d^{5} e^{2} f g + 7 \, d^{6} e g^{2}\right )} x - 3 \, {\left (5 \, d^{5} e^{2} f^{2} + 2 \, d^{6} e f g - d^{7} g^{2} + {\left (5 \, e^{7} f^{2} + 2 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} + {\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \, {\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 2 \, {\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} + {\left (5 \, d^{4} e^{3} f^{2} + 2 \, d^{5} e^{2} f g - d^{6} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \, {\left (5 \, d^{5} e^{2} f^{2} + 2 \, d^{6} e f g - d^{7} g^{2} + {\left (5 \, e^{7} f^{2} + 2 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} + {\left (5 \, d e^{6} f^{2} + 2 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \, {\left (5 \, d^{2} e^{5} f^{2} + 2 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 2 \, {\left (5 \, d^{3} e^{4} f^{2} + 2 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} + {\left (5 \, d^{4} e^{3} f^{2} + 2 \, d^{5} e^{2} f g - d^{6} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{96 \, {\left (d^{6} e^{8} x^{5} + d^{7} e^{7} x^{4} - 2 \, d^{8} e^{6} x^{3} - 2 \, d^{9} e^{5} x^{2} + d^{10} e^{4} x + d^{11} e^{3}\right )}} \]
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 [A] time = 0.02, size = 348, normalized size = 1.85 \[ -\frac {g^{2}}{24 \left (e x +d \right )^{3} d \,e^{3}}+\frac {f g}{12 \left (e x +d \right )^{3} d^{2} e^{2}}-\frac {f^{2}}{24 \left (e x +d \right )^{3} d^{3} e}+\frac {g^{2}}{32 \left (e x -d \right )^{2} d^{2} e^{3}}+\frac {g^{2}}{32 \left (e x +d \right )^{2} d^{2} e^{3}}+\frac {f g}{16 \left (e x -d \right )^{2} d^{3} e^{2}}+\frac {f g}{16 \left (e x +d \right )^{2} d^{3} e^{2}}+\frac {f^{2}}{32 \left (e x -d \right )^{2} d^{4} e}-\frac {3 f^{2}}{32 \left (e x +d \right )^{2} d^{4} e}+\frac {g^{2}}{16 \left (e x +d \right ) d^{3} e^{3}}-\frac {f g}{8 \left (e x -d \right ) d^{4} e^{2}}+\frac {g^{2} \ln \left (e x -d \right )}{32 d^{4} e^{3}}-\frac {g^{2} \ln \left (e x +d \right )}{32 d^{4} e^{3}}-\frac {f^{2}}{8 \left (e x -d \right ) d^{5} e}-\frac {3 f^{2}}{16 \left (e x +d \right ) d^{5} e}-\frac {f g \ln \left (e x -d \right )}{16 d^{5} e^{2}}+\frac {f g \ln \left (e x +d \right )}{16 d^{5} e^{2}}-\frac {5 f^{2} \ln \left (e x -d \right )}{32 d^{6} e}+\frac {5 f^{2} \ln \left (e x +d \right )}{32 d^{6} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 308, normalized size = 1.64 \[ -\frac {8 \, d^{4} e^{2} f^{2} - 16 \, d^{5} e f g - 4 \, d^{6} g^{2} + 3 \, {\left (5 \, e^{6} f^{2} + 2 \, d e^{5} f g - d^{2} e^{4} g^{2}\right )} x^{4} + 3 \, {\left (5 \, d e^{5} f^{2} + 2 \, d^{2} e^{4} f g - d^{3} e^{3} g^{2}\right )} x^{3} - 5 \, {\left (5 \, d^{2} e^{4} f^{2} + 2 \, d^{3} e^{3} f g - d^{4} e^{2} g^{2}\right )} x^{2} - {\left (25 \, d^{3} e^{3} f^{2} + 10 \, d^{4} e^{2} f g + 7 \, d^{5} e g^{2}\right )} x}{48 \, {\left (d^{5} e^{8} x^{5} + d^{6} e^{7} x^{4} - 2 \, d^{7} e^{6} x^{3} - 2 \, d^{8} e^{5} x^{2} + d^{9} e^{4} x + d^{10} e^{3}\right )}} + \frac {{\left (5 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{32 \, d^{6} e^{3}} - \frac {{\left (5 \, e^{2} f^{2} + 2 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{32 \, d^{6} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.68, size = 249, normalized size = 1.32 \[ \frac {\frac {d^2\,g^2+4\,d\,e\,f\,g-2\,e^2\,f^2}{12\,d\,e^3}-\frac {x^3\,\left (-d^2\,g^2+2\,d\,e\,f\,g+5\,e^2\,f^2\right )}{16\,d^4}-\frac {e\,x^4\,\left (-d^2\,g^2+2\,d\,e\,f\,g+5\,e^2\,f^2\right )}{16\,d^5}+\frac {x\,\left (7\,d^2\,g^2+10\,d\,e\,f\,g+25\,e^2\,f^2\right )}{48\,d^2\,e^2}+\frac {5\,x^2\,\left (-d^2\,g^2+2\,d\,e\,f\,g+5\,e^2\,f^2\right )}{48\,d^3\,e}}{d^5+d^4\,e\,x-2\,d^3\,e^2\,x^2-2\,d^2\,e^3\,x^3+d\,e^4\,x^4+e^5\,x^5}+\frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )\,\left (-d^2\,g^2+2\,d\,e\,f\,g+5\,e^2\,f^2\right )}{16\,d^6\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.83, size = 321, normalized size = 1.71 \[ - \frac {- 4 d^{6} g^{2} - 16 d^{5} e f g + 8 d^{4} e^{2} f^{2} + x^{4} \left (- 3 d^{2} e^{4} g^{2} + 6 d e^{5} f g + 15 e^{6} f^{2}\right ) + x^{3} \left (- 3 d^{3} e^{3} g^{2} + 6 d^{2} e^{4} f g + 15 d e^{5} f^{2}\right ) + x^{2} \left (5 d^{4} e^{2} g^{2} - 10 d^{3} e^{3} f g - 25 d^{2} e^{4} f^{2}\right ) + x \left (- 7 d^{5} e g^{2} - 10 d^{4} e^{2} f g - 25 d^{3} e^{3} f^{2}\right )}{48 d^{10} e^{3} + 48 d^{9} e^{4} x - 96 d^{8} e^{5} x^{2} - 96 d^{7} e^{6} x^{3} + 48 d^{6} e^{7} x^{4} + 48 d^{5} e^{8} x^{5}} + \frac {\left (d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}\right ) \log {\left (- \frac {d}{e} + x \right )}}{32 d^{6} e^{3}} - \frac {\left (d^{2} g^{2} - 2 d e f g - 5 e^{2} f^{2}\right ) \log {\left (\frac {d}{e} + x \right )}}{32 d^{6} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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