Optimal. Leaf size=86 \[ \frac {(3 d g+e f) (e f-d g) \log (d+e x)}{4 d^2 e^3}-\frac {(d g+e f)^2 \log (d-e x)}{4 d^2 e^3}-\frac {(e f-d g)^2}{2 d e^3 (d+e x)} \]
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Rubi [A] time = 0.09, antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {848, 88} \[ \frac {(3 d g+e f) (e f-d g) \log (d+e x)}{4 d^2 e^3}-\frac {(d g+e f)^2 \log (d-e x)}{4 d^2 e^3}-\frac {(e f-d g)^2}{2 d e^3 (d+e x)} \]
Antiderivative was successfully verified.
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Rule 88
Rule 848
Rubi steps
\begin {align*} \int \frac {(f+g x)^2}{(d+e x) \left (d^2-e^2 x^2\right )} \, dx &=\int \frac {(f+g x)^2}{(d-e x) (d+e x)^2} \, dx\\ &=\int \left (\frac {(e f+d g)^2}{4 d^2 e^2 (d-e x)}+\frac {(-e f+d g)^2}{2 d e^2 (d+e x)^2}+\frac {(e f-d g) (e f+3 d g)}{4 d^2 e^2 (d+e x)}\right ) \, dx\\ &=-\frac {(e f-d g)^2}{2 d e^3 (d+e x)}-\frac {(e f+d g)^2 \log (d-e x)}{4 d^2 e^3}+\frac {(e f-d g) (e f+3 d g) \log (d+e x)}{4 d^2 e^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 82, normalized size = 0.95 \[ \frac {(e f-d g) ((d+e x) (3 d g+e f) \log (d+e x)+2 d (d g-e f))-(d+e x) (d g+e f)^2 \log (d-e x)}{4 d^2 e^3 (d+e x)} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.89, size = 165, normalized size = 1.92 \[ -\frac {2 \, d e^{2} f^{2} - 4 \, d^{2} e f g + 2 \, d^{3} g^{2} - {\left (d e^{2} f^{2} + 2 \, d^{2} e f g - 3 \, d^{3} g^{2} + {\left (e^{3} f^{2} + 2 \, d e^{2} f g - 3 \, d^{2} e g^{2}\right )} x\right )} \log \left (e x + d\right ) + {\left (d e^{2} f^{2} + 2 \, d^{2} e f g + d^{3} g^{2} + {\left (e^{3} f^{2} + 2 \, d e^{2} f g + d^{2} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{4 \, {\left (d^{2} e^{4} x + d^{3} e^{3}\right )}} \]
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 = 0.01, size = 149, normalized size = 1.73 \[ -\frac {d \,g^{2}}{2 \left (e x +d \right ) e^{3}}-\frac {f^{2}}{2 \left (e x +d \right ) d e}-\frac {f g \ln \left (e x -d \right )}{2 d \,e^{2}}+\frac {f g \ln \left (e x +d \right )}{2 d \,e^{2}}-\frac {f^{2} \ln \left (e x -d \right )}{4 d^{2} e}+\frac {f^{2} \ln \left (e x +d \right )}{4 d^{2} e}+\frac {f g}{\left (e x +d \right ) e^{2}}-\frac {g^{2} \ln \left (e x -d \right )}{4 e^{3}}-\frac {3 g^{2} \ln \left (e x +d \right )}{4 e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 113, normalized size = 1.31 \[ -\frac {e^{2} f^{2} - 2 \, d e f g + d^{2} g^{2}}{2 \, {\left (d e^{4} x + d^{2} e^{3}\right )}} + \frac {{\left (e^{2} f^{2} + 2 \, d e f g - 3 \, d^{2} g^{2}\right )} \log \left (e x + d\right )}{4 \, d^{2} e^{3}} - \frac {{\left (e^{2} f^{2} + 2 \, d e f g + d^{2} g^{2}\right )} \log \left (e x - d\right )}{4 \, d^{2} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.70, size = 109, normalized size = 1.27 \[ \frac {\ln \left (d+e\,x\right )\,\left (-3\,d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{4\,d^2\,e^3}-\frac {\ln \left (d-e\,x\right )\,\left (d^2\,g^2+2\,d\,e\,f\,g+e^2\,f^2\right )}{4\,d^2\,e^3}-\frac {d^2\,g^2-2\,d\,e\,f\,g+e^2\,f^2}{2\,d\,e^3\,\left (d+e\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 1.00, size = 182, normalized size = 2.12 \[ - \frac {d^{2} g^{2} - 2 d e f g + e^{2} f^{2}}{2 d^{2} e^{3} + 2 d e^{4} x} - \frac {\left (d g - e f\right ) \left (3 d g + e f\right ) \log {\left (x + \frac {- 2 d^{3} g^{2} + d \left (d g - e f\right ) \left (3 d g + e f\right )}{d^{2} e g^{2} - 2 d e^{2} f g - e^{3} f^{2}} \right )}}{4 d^{2} e^{3}} - \frac {\left (d g + e f\right )^{2} \log {\left (x + \frac {- 2 d^{3} g^{2} + d \left (d g + e f\right )^{2}}{d^{2} e g^{2} - 2 d e^{2} f g - e^{3} f^{2}} \right )}}{4 d^{2} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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