Optimal. Leaf size=61 \[ -\frac {2 g (d g+e f)}{e^3 (d-e x)}+\frac {(d g+e f)^2}{2 e^3 (d-e x)^2}-\frac {g^2 \log (d-e x)}{e^3} \]
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Rubi [A] time = 0.06, antiderivative size = 61, 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, 43} \[ -\frac {2 g (d g+e f)}{e^3 (d-e x)}+\frac {(d g+e f)^2}{2 e^3 (d-e x)^2}-\frac {g^2 \log (d-e x)}{e^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 848
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac {(f+g x)^2}{(d-e x)^3} \, dx\\ &=\int \left (\frac {(e f+d g)^2}{e^2 (d-e x)^3}-\frac {2 g (e f+d g)}{e^2 (d-e x)^2}+\frac {g^2}{e^2 (d-e x)}\right ) \, dx\\ &=\frac {(e f+d g)^2}{2 e^3 (d-e x)^2}-\frac {2 g (e f+d g)}{e^3 (d-e x)}-\frac {g^2 \log (d-e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 49, normalized size = 0.80 \[ \frac {\frac {(d g+e f) (e (f+4 g x)-3 d g)}{(d-e x)^2}-2 g^2 \log (d-e x)}{2 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.19, size = 100, normalized size = 1.64 \[ \frac {e^{2} f^{2} - 2 \, d e f g - 3 \, d^{2} g^{2} + 4 \, {\left (e^{2} f g + d e g^{2}\right )} x - 2 \, {\left (e^{2} g^{2} x^{2} - 2 \, d e g^{2} x + d^{2} g^{2}\right )} \log \left (e x - d\right )}{2 \, {\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 195, normalized size = 3.20 \[ -\frac {d g^{2} e^{\left (-3\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{2 \, {\left | d \right |}} - \frac {1}{2} \, g^{2} e^{\left (-3\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac {{\left (4 \, {\left (d^{2} g^{2} e^{4} + d f g e^{5}\right )} x^{3} + {\left (5 \, d^{3} g^{2} e^{3} + 6 \, d^{2} f g e^{4} + d f^{2} e^{5}\right )} x^{2} - 2 \, {\left (d^{4} g^{2} e^{2} - d^{2} f^{2} e^{4}\right )} x - {\left (3 \, d^{5} g^{2} e^{3} + 2 \, d^{4} f g e^{4} - d^{3} f^{2} e^{5}\right )} e^{\left (-2\right )}\right )} e^{\left (-4\right )}}{2 \, {\left (x^{2} e^{2} - d^{2}\right )}^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 105, normalized size = 1.72 \[ \frac {d^{2} g^{2}}{2 \left (e x -d \right )^{2} e^{3}}+\frac {d f g}{\left (e x -d \right )^{2} e^{2}}+\frac {f^{2}}{2 \left (e x -d \right )^{2} e}+\frac {2 d \,g^{2}}{\left (e x -d \right ) e^{3}}+\frac {2 f g}{\left (e x -d \right ) e^{2}}-\frac {g^{2} \ln \left (e x -d \right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 81, normalized size = 1.33 \[ \frac {e^{2} f^{2} - 2 \, d e f g - 3 \, d^{2} g^{2} + 4 \, {\left (e^{2} f g + d e g^{2}\right )} x}{2 \, {\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} - \frac {g^{2} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 80, normalized size = 1.31 \[ -\frac {\frac {3\,d^2\,g^2+2\,d\,e\,f\,g-e^2\,f^2}{2\,e^3}-\frac {2\,g\,x\,\left (d\,g+e\,f\right )}{e^2}}{d^2-2\,d\,e\,x+e^2\,x^2}-\frac {g^2\,\ln \left (e\,x-d\right )}{e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.54, size = 83, normalized size = 1.36 \[ - \frac {3 d^{2} g^{2} + 2 d e f g - e^{2} f^{2} + x \left (- 4 d e g^{2} - 4 e^{2} f g\right )}{2 d^{2} e^{3} - 4 d e^{4} x + 2 e^{5} x^{2}} - \frac {g^{2} \log {\left (- d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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