Optimal. Leaf size=87 \[ \frac {(d g+e f)^2 \tanh ^{-1}\left (\frac {e x}{d}\right )}{4 d^3 e^3}-\frac {(3 d g+e f) (e f-d g)}{4 d^2 e^3 (d+e x)}-\frac {(e f-d g)^2}{4 d e^3 (d+e x)^2} \]
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Rubi [A] time = 0.10, antiderivative size = 87, 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} \[ -\frac {(3 d g+e f) (e f-d g)}{4 d^2 e^3 (d+e x)}+\frac {(d g+e f)^2 \tanh ^{-1}\left (\frac {e x}{d}\right )}{4 d^3 e^3}-\frac {(e f-d g)^2}{4 d e^3 (d+e x)^2} \]
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 )} \, dx &=\int \frac {(f+g x)^2}{(d-e x) (d+e x)^3} \, dx\\ &=\int \left (\frac {(-e f+d g)^2}{2 d e^2 (d+e x)^3}+\frac {(e f-d g) (e f+3 d g)}{4 d^2 e^2 (d+e x)^2}+\frac {(e f+d g)^2}{4 d^2 e^2 \left (d^2-e^2 x^2\right )}\right ) \, dx\\ &=-\frac {(e f-d g)^2}{4 d e^3 (d+e x)^2}-\frac {(e f-d g) (e f+3 d g)}{4 d^2 e^3 (d+e x)}+\frac {(e f+d g)^2 \int \frac {1}{d^2-e^2 x^2} \, dx}{4 d^2 e^2}\\ &=-\frac {(e f-d g)^2}{4 d e^3 (d+e x)^2}-\frac {(e f-d g) (e f+3 d g)}{4 d^2 e^3 (d+e x)}+\frac {(e f+d g)^2 \tanh ^{-1}\left (\frac {e x}{d}\right )}{4 d^3 e^3}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 87, normalized size = 1.00 \[ \frac {\frac {2 d (d g-e f) \left (2 d^2 g+d e (2 f+3 g x)+e^2 f x\right )}{(d+e x)^2}+(d g+e f)^2 (-\log (d-e x))+(d g+e f)^2 \log (d+e x)}{8 d^3 e^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.77, size = 271, normalized size = 3.11 \[ -\frac {4 \, d^{2} e^{2} f^{2} - 4 \, d^{4} g^{2} + 2 \, {\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g - 3 \, d^{3} e g^{2}\right )} x - {\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2} + {\left (e^{4} f^{2} + 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + {\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2} + {\left (e^{4} f^{2} + 2 \, d e^{3} f g + d^{2} e^{2} g^{2}\right )} x^{2} + 2 \, {\left (d e^{3} f^{2} + 2 \, d^{2} e^{2} f g + d^{3} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{8 \, {\left (d^{3} e^{5} x^{2} + 2 \, d^{4} e^{4} x + d^{5} e^{3}\right )}} \]
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 = 206, normalized size = 2.37 \[ -\frac {d \,g^{2}}{4 \left (e x +d \right )^{2} e^{3}}-\frac {f^{2}}{4 \left (e x +d \right )^{2} d e}+\frac {f g}{2 \left (e x +d \right )^{2} e^{2}}-\frac {f g}{2 \left (e x +d \right ) d \,e^{2}}-\frac {g^{2} \ln \left (e x -d \right )}{8 d \,e^{3}}+\frac {g^{2} \ln \left (e x +d \right )}{8 d \,e^{3}}-\frac {f^{2}}{4 \left (e x +d \right ) d^{2} e}-\frac {f g \ln \left (e x -d \right )}{4 d^{2} e^{2}}+\frac {f g \ln \left (e x +d \right )}{4 d^{2} e^{2}}-\frac {f^{2} \ln \left (e x -d \right )}{8 d^{3} e}+\frac {f^{2} \ln \left (e x +d \right )}{8 d^{3} e}+\frac {3 g^{2}}{4 \left (e x +d \right ) e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 149, normalized size = 1.71 \[ -\frac {2 \, d e^{2} f^{2} - 2 \, d^{3} g^{2} + {\left (e^{3} f^{2} + 2 \, d e^{2} f g - 3 \, d^{2} e g^{2}\right )} x}{4 \, {\left (d^{2} e^{5} x^{2} + 2 \, d^{3} e^{4} x + d^{4} e^{3}\right )}} + \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x + d\right )}{8 \, d^{3} e^{3}} - \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{8 \, d^{3} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.13, size = 100, normalized size = 1.15 \[ \frac {\frac {d^2\,g^2-e^2\,f^2}{2\,d\,e^3}-\frac {x\,\left (-3\,d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{4\,d^2\,e^2}}{d^2+2\,d\,e\,x+e^2\,x^2}+\frac {\mathrm {atanh}\left (\frac {e\,x}{d}\right )\,{\left (d\,g+e\,f\right )}^2}{4\,d^3\,e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.03, size = 185, normalized size = 2.13 \[ - \frac {- 2 d^{3} g^{2} + 2 d e^{2} f^{2} + x \left (- 3 d^{2} e g^{2} + 2 d e^{2} f g + e^{3} f^{2}\right )}{4 d^{4} e^{3} + 8 d^{3} e^{4} x + 4 d^{2} e^{5} x^{2}} - \frac {\left (d g + e f\right )^{2} \log {\left (- \frac {d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{8 d^{3} e^{3}} + \frac {\left (d g + e f\right )^{2} \log {\left (\frac {d \left (d g + e f\right )^{2}}{e \left (d^{2} g^{2} + 2 d e f g + e^{2} f^{2}\right )} + x \right )}}{8 d^{3} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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