Optimal. Leaf size=235 \[ \frac {(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d-e x)}+\frac {(d g+e f)^2}{64 d^5 e^3 (d-e x)^2}-\frac {(d g+3 e f) (e f-d g)}{48 d^4 e^3 (d+e x)^3}-\frac {(e f-d g)^2}{32 d^3 e^3 (d+e x)^4}+\frac {\left (-d^2 g^2+10 d e f g+15 e^2 f^2\right ) \tanh ^{-1}\left (\frac {e x}{d}\right )}{64 d^7 e^3}-\frac {-d^2 g^2+2 d e f g+5 e^2 f^2}{32 d^6 e^3 (d+e x)}-\frac {3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2} \]
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Rubi [A] time = 0.27, antiderivative size = 235, 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} \[ -\frac {-d^2 g^2+2 d e f g+5 e^2 f^2}{32 d^6 e^3 (d+e x)}-\frac {3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2}+\frac {\left (-d^2 g^2+10 d e f g+15 e^2 f^2\right ) \tanh ^{-1}\left (\frac {e x}{d}\right )}{64 d^7 e^3}-\frac {(e f-d g)^2}{32 d^3 e^3 (d+e x)^4}-\frac {(d g+3 e f) (e f-d g)}{48 d^4 e^3 (d+e x)^3}+\frac {(d g+e f) (d g+5 e f)}{64 d^6 e^3 (d-e x)}+\frac {(d g+e f)^2}{64 d^5 e^3 (d-e x)^2} \]
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)^2 \left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac {(f+g x)^2}{(d-e x)^3 (d+e x)^5} \, dx\\ &=\int \left (\frac {(e f+d g)^2}{32 d^5 e^2 (d-e x)^3}+\frac {(e f+d g) (5 e f+d g)}{64 d^6 e^2 (d-e x)^2}+\frac {(-e f+d g)^2}{8 d^3 e^2 (d+e x)^5}+\frac {(e f-d g) (3 e f+d g)}{16 d^4 e^2 (d+e x)^4}+\frac {3 e^2 f^2-d^2 g^2}{16 d^5 e^2 (d+e x)^3}+\frac {5 e^2 f^2+2 d e f g-d^2 g^2}{32 d^6 e^2 (d+e x)^2}+\frac {-15 e^2 f^2-10 d e f g+d^2 g^2}{64 d^6 e^2 \left (-d^2+e^2 x^2\right )}\right ) \, dx\\ &=\frac {(e f+d g)^2}{64 d^5 e^3 (d-e x)^2}+\frac {(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d-e x)}-\frac {(e f-d g)^2}{32 d^3 e^3 (d+e x)^4}-\frac {(e f-d g) (3 e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac {3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2}-\frac {5 e^2 f^2+2 d e f g-d^2 g^2}{32 d^6 e^3 (d+e x)}-\frac {\left (15 e^2 f^2+10 d e f g-d^2 g^2\right ) \int \frac {1}{-d^2+e^2 x^2} \, dx}{64 d^6 e^2}\\ &=\frac {(e f+d g)^2}{64 d^5 e^3 (d-e x)^2}+\frac {(e f+d g) (5 e f+d g)}{64 d^6 e^3 (d-e x)}-\frac {(e f-d g)^2}{32 d^3 e^3 (d+e x)^4}-\frac {(e f-d g) (3 e f+d g)}{48 d^4 e^3 (d+e x)^3}-\frac {3 e^2 f^2-d^2 g^2}{32 d^5 e^3 (d+e x)^2}-\frac {5 e^2 f^2+2 d e f g-d^2 g^2}{32 d^6 e^3 (d+e x)}+\frac {\left (15 e^2 f^2+10 d e f g-d^2 g^2\right ) \tanh ^{-1}\left (\frac {e x}{d}\right )}{64 d^7 e^3}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 244, normalized size = 1.04 \[ \frac {-\frac {12 d^4 (e f-d g)^2}{(d+e x)^4}+\frac {12 d^2 \left (d^2 g^2-3 e^2 f^2\right )}{(d+e x)^2}+\frac {6 d \left (d^2 g^2+6 d e f g+5 e^2 f^2\right )}{d-e x}+\frac {12 d \left (d^2 g^2-2 d e f g-5 e^2 f^2\right )}{d+e x}+3 \left (d^2 g^2-10 d e f g-15 e^2 f^2\right ) \log (d-e x)+3 \left (-d^2 g^2+10 d e f g+15 e^2 f^2\right ) \log (d+e x)+\frac {6 d^2 (d g+e f)^2}{(d-e x)^2}+\frac {8 d^3 \left (d^2 g^2+2 d e f g-3 e^2 f^2\right )}{(d+e x)^3}}{384 d^7 e^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.91, size = 793, normalized size = 3.37 \[ -\frac {96 \, d^{6} e^{2} f^{2} - 64 \, d^{7} e f g - 32 \, d^{8} g^{2} + 6 \, {\left (15 \, d e^{7} f^{2} + 10 \, d^{2} e^{6} f g - d^{3} e^{5} g^{2}\right )} x^{5} + 12 \, {\left (15 \, d^{2} e^{6} f^{2} + 10 \, d^{3} e^{5} f g - d^{4} e^{4} g^{2}\right )} x^{4} - 4 \, {\left (15 \, d^{3} e^{5} f^{2} + 10 \, d^{4} e^{4} f g - d^{5} e^{3} g^{2}\right )} x^{3} - 20 \, {\left (15 \, d^{4} e^{4} f^{2} + 10 \, d^{5} e^{3} f g - d^{6} e^{2} g^{2}\right )} x^{2} - 2 \, {\left (51 \, d^{5} e^{3} f^{2} + 34 \, d^{6} e^{2} f g + 35 \, d^{7} e g^{2}\right )} x - 3 \, {\left (15 \, d^{6} e^{2} f^{2} + 10 \, d^{7} e f g - d^{8} g^{2} + {\left (15 \, e^{8} f^{2} + 10 \, d e^{7} f g - d^{2} e^{6} g^{2}\right )} x^{6} + 2 \, {\left (15 \, d e^{7} f^{2} + 10 \, d^{2} e^{6} f g - d^{3} e^{5} g^{2}\right )} x^{5} - {\left (15 \, d^{2} e^{6} f^{2} + 10 \, d^{3} e^{5} f g - d^{4} e^{4} g^{2}\right )} x^{4} - 4 \, {\left (15 \, d^{3} e^{5} f^{2} + 10 \, d^{4} e^{4} f g - d^{5} e^{3} g^{2}\right )} x^{3} - {\left (15 \, d^{4} e^{4} f^{2} + 10 \, d^{5} e^{3} f g - d^{6} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (15 \, d^{5} e^{3} f^{2} + 10 \, d^{6} e^{2} f g - d^{7} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + 3 \, {\left (15 \, d^{6} e^{2} f^{2} + 10 \, d^{7} e f g - d^{8} g^{2} + {\left (15 \, e^{8} f^{2} + 10 \, d e^{7} f g - d^{2} e^{6} g^{2}\right )} x^{6} + 2 \, {\left (15 \, d e^{7} f^{2} + 10 \, d^{2} e^{6} f g - d^{3} e^{5} g^{2}\right )} x^{5} - {\left (15 \, d^{2} e^{6} f^{2} + 10 \, d^{3} e^{5} f g - d^{4} e^{4} g^{2}\right )} x^{4} - 4 \, {\left (15 \, d^{3} e^{5} f^{2} + 10 \, d^{4} e^{4} f g - d^{5} e^{3} g^{2}\right )} x^{3} - {\left (15 \, d^{4} e^{4} f^{2} + 10 \, d^{5} e^{3} f g - d^{6} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (15 \, d^{5} e^{3} f^{2} + 10 \, d^{6} e^{2} f g - d^{7} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{384 \, {\left (d^{7} e^{9} x^{6} + 2 \, d^{8} e^{8} x^{5} - d^{9} e^{7} x^{4} - 4 \, d^{10} e^{6} x^{3} - d^{11} e^{5} x^{2} + 2 \, d^{12} e^{4} x + d^{13} 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 = 421, normalized size = 1.79 \[ -\frac {g^{2}}{32 \left (e x +d \right )^{4} d \,e^{3}}+\frac {f g}{16 \left (e x +d \right )^{4} d^{2} e^{2}}-\frac {f^{2}}{32 \left (e x +d \right )^{4} d^{3} e}+\frac {g^{2}}{48 \left (e x +d \right )^{3} d^{2} e^{3}}+\frac {f g}{24 \left (e x +d \right )^{3} d^{3} e^{2}}-\frac {f^{2}}{16 \left (e x +d \right )^{3} d^{4} e}+\frac {g^{2}}{64 \left (e x -d \right )^{2} d^{3} e^{3}}+\frac {g^{2}}{32 \left (e x +d \right )^{2} d^{3} e^{3}}+\frac {f g}{32 \left (e x -d \right )^{2} d^{4} e^{2}}+\frac {f^{2}}{64 \left (e x -d \right )^{2} d^{5} e}-\frac {3 f^{2}}{32 \left (e x +d \right )^{2} d^{5} e}-\frac {g^{2}}{64 \left (e x -d \right ) d^{4} e^{3}}+\frac {g^{2}}{32 \left (e x +d \right ) d^{4} e^{3}}-\frac {3 f g}{32 \left (e x -d \right ) d^{5} e^{2}}-\frac {f g}{16 \left (e x +d \right ) d^{5} e^{2}}+\frac {g^{2} \ln \left (e x -d \right )}{128 d^{5} e^{3}}-\frac {g^{2} \ln \left (e x +d \right )}{128 d^{5} e^{3}}-\frac {5 f^{2}}{64 \left (e x -d \right ) d^{6} e}-\frac {5 f^{2}}{32 \left (e x +d \right ) d^{6} e}-\frac {5 f g \ln \left (e x -d \right )}{64 d^{6} e^{2}}+\frac {5 f g \ln \left (e x +d \right )}{64 d^{6} e^{2}}-\frac {15 f^{2} \ln \left (e x -d \right )}{128 d^{7} e}+\frac {15 f^{2} \ln \left (e x +d \right )}{128 d^{7} e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.51, size = 359, normalized size = 1.53 \[ -\frac {48 \, d^{5} e^{2} f^{2} - 32 \, d^{6} e f g - 16 \, d^{7} g^{2} + 3 \, {\left (15 \, e^{7} f^{2} + 10 \, d e^{6} f g - d^{2} e^{5} g^{2}\right )} x^{5} + 6 \, {\left (15 \, d e^{6} f^{2} + 10 \, d^{2} e^{5} f g - d^{3} e^{4} g^{2}\right )} x^{4} - 2 \, {\left (15 \, d^{2} e^{5} f^{2} + 10 \, d^{3} e^{4} f g - d^{4} e^{3} g^{2}\right )} x^{3} - 10 \, {\left (15 \, d^{3} e^{4} f^{2} + 10 \, d^{4} e^{3} f g - d^{5} e^{2} g^{2}\right )} x^{2} - {\left (51 \, d^{4} e^{3} f^{2} + 34 \, d^{5} e^{2} f g + 35 \, d^{6} e g^{2}\right )} x}{192 \, {\left (d^{6} e^{9} x^{6} + 2 \, d^{7} e^{8} x^{5} - d^{8} e^{7} x^{4} - 4 \, d^{9} e^{6} x^{3} - d^{10} e^{5} x^{2} + 2 \, d^{11} e^{4} x + d^{12} e^{3}\right )}} + \frac {{\left (15 \, e^{2} f^{2} + 10 \, d e f g - d^{2} g^{2}\right )} \log \left (e x + d\right )}{128 \, d^{7} e^{3}} - \frac {{\left (15 \, e^{2} f^{2} + 10 \, d e f g - d^{2} g^{2}\right )} \log \left (e x - d\right )}{128 \, d^{7} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.64, size = 296, normalized size = 1.26 \[ \frac {\frac {d^2\,g^2+2\,d\,e\,f\,g-3\,e^2\,f^2}{12\,d\,e^3}+\frac {x^3\,\left (-d^2\,g^2+10\,d\,e\,f\,g+15\,e^2\,f^2\right )}{96\,d^4}-\frac {e\,x^4\,\left (-d^2\,g^2+10\,d\,e\,f\,g+15\,e^2\,f^2\right )}{32\,d^5}+\frac {x\,\left (35\,d^2\,g^2+34\,d\,e\,f\,g+51\,e^2\,f^2\right )}{192\,d^2\,e^2}+\frac {5\,x^2\,\left (-d^2\,g^2+10\,d\,e\,f\,g+15\,e^2\,f^2\right )}{96\,d^3\,e}-\frac {e^2\,x^5\,\left (-d^2\,g^2+10\,d\,e\,f\,g+15\,e^2\,f^2\right )}{64\,d^6}}{d^6+2\,d^5\,e\,x-d^4\,e^2\,x^2-4\,d^3\,e^3\,x^3-d^2\,e^4\,x^4+2\,d\,e^5\,x^5+e^6\,x^6}+\frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )\,\left (-d^2\,g^2+10\,d\,e\,f\,g+15\,e^2\,f^2\right )}{64\,d^7\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.15, size = 372, normalized size = 1.58 \[ - \frac {- 16 d^{7} g^{2} - 32 d^{6} e f g + 48 d^{5} e^{2} f^{2} + x^{5} \left (- 3 d^{2} e^{5} g^{2} + 30 d e^{6} f g + 45 e^{7} f^{2}\right ) + x^{4} \left (- 6 d^{3} e^{4} g^{2} + 60 d^{2} e^{5} f g + 90 d e^{6} f^{2}\right ) + x^{3} \left (2 d^{4} e^{3} g^{2} - 20 d^{3} e^{4} f g - 30 d^{2} e^{5} f^{2}\right ) + x^{2} \left (10 d^{5} e^{2} g^{2} - 100 d^{4} e^{3} f g - 150 d^{3} e^{4} f^{2}\right ) + x \left (- 35 d^{6} e g^{2} - 34 d^{5} e^{2} f g - 51 d^{4} e^{3} f^{2}\right )}{192 d^{12} e^{3} + 384 d^{11} e^{4} x - 192 d^{10} e^{5} x^{2} - 768 d^{9} e^{6} x^{3} - 192 d^{8} e^{7} x^{4} + 384 d^{7} e^{8} x^{5} + 192 d^{6} e^{9} x^{6}} + \frac {\left (d^{2} g^{2} - 10 d e f g - 15 e^{2} f^{2}\right ) \log {\left (- \frac {d}{e} + x \right )}}{128 d^{7} e^{3}} - \frac {\left (d^{2} g^{2} - 10 d e f g - 15 e^{2} f^{2}\right ) \log {\left (\frac {d}{e} + x \right )}}{128 d^{7} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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