Optimal. Leaf size=81 \[ -\frac {(d g+e f) (5 d g+e f)}{e^3 (d-e x)}+\frac {d (d g+e f)^2}{e^3 (d-e x)^2}-\frac {2 g (2 d g+e f) \log (d-e x)}{e^3}-\frac {g^2 x}{e^2} \]
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Rubi [A] time = 0.10, antiderivative size = 81, 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, 77} \[ -\frac {(d g+e f) (5 d g+e f)}{e^3 (d-e x)}+\frac {d (d g+e f)^2}{e^3 (d-e x)^2}-\frac {2 g (2 d g+e f) \log (d-e x)}{e^3}-\frac {g^2 x}{e^2} \]
Antiderivative was successfully verified.
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Rule 77
Rule 848
Rubi steps
\begin {align*} \int \frac {(d+e x)^4 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac {(d+e x) (f+g x)^2}{(d-e x)^3} \, dx\\ &=\int \left (-\frac {g^2}{e^2}+\frac {(-e f-5 d g) (e f+d g)}{e^2 (d-e x)^2}-\frac {2 d (e f+d g)^2}{e^2 (-d+e x)^3}-\frac {2 g (e f+2 d g)}{e^2 (-d+e x)}\right ) \, dx\\ &=-\frac {g^2 x}{e^2}+\frac {d (e f+d g)^2}{e^3 (d-e x)^2}-\frac {(e f+d g) (e f+5 d g)}{e^3 (d-e x)}-\frac {2 g (e f+2 d g) \log (d-e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 93, normalized size = 1.15 \[ \frac {-4 d^3 g^2+4 d^2 e g (g x-f)+2 d e^2 g x (3 f+g x)-2 g (d-e x)^2 (2 d g+e f) \log (d-e x)+e^3 x \left (f^2-g^2 x^2\right )}{e^3 (d-e x)^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.03, size = 159, normalized size = 1.96 \[ -\frac {e^{3} g^{2} x^{3} - 2 \, d e^{2} g^{2} x^{2} + 4 \, d^{2} e f g + 4 \, d^{3} g^{2} - {\left (e^{3} f^{2} + 6 \, d e^{2} f g + 4 \, d^{2} e g^{2}\right )} x + 2 \, {\left (d^{2} e f g + 2 \, d^{3} g^{2} + {\left (e^{3} f g + 2 \, d e^{2} g^{2}\right )} x^{2} - 2 \, {\left (d e^{2} f g + 2 \, d^{2} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 227, normalized size = 2.80 \[ -g^{2} x e^{\left (-2\right )} - {\left (2 \, d g^{2} e^{3} + f g e^{4}\right )} e^{\left (-6\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac {{\left (2 \, d^{2} g^{2} e^{4} + d f g e^{5}\right )} e^{\left (-7\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{{\left | d \right |}} - \frac {{\left (4 \, d^{5} g^{2} e^{3} + 4 \, d^{4} f g e^{4} - {\left (5 \, d^{2} g^{2} e^{6} + 6 \, d f g e^{7} + f^{2} e^{8}\right )} x^{3} - 2 \, {\left (3 \, d^{3} g^{2} e^{5} + 4 \, d^{2} f g e^{6} + d f^{2} e^{7}\right )} x^{2} + {\left (3 \, d^{4} g^{2} e^{4} + 2 \, d^{3} f g e^{5} - d^{2} f^{2} e^{6}\right )} x\right )} e^{\left (-6\right )}}{{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 151, normalized size = 1.86 \[ \frac {d^{3} g^{2}}{\left (e x -d \right )^{2} e^{3}}+\frac {2 d^{2} f g}{\left (e x -d \right )^{2} e^{2}}+\frac {d \,f^{2}}{\left (e x -d \right )^{2} e}+\frac {5 d^{2} g^{2}}{\left (e x -d \right ) e^{3}}+\frac {6 d f g}{\left (e x -d \right ) e^{2}}-\frac {4 d \,g^{2} \ln \left (e x -d \right )}{e^{3}}+\frac {f^{2}}{\left (e x -d \right ) e}-\frac {2 f g \ln \left (e x -d \right )}{e^{2}}-\frac {g^{2} x}{e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 105, normalized size = 1.30 \[ -\frac {g^{2} x}{e^{2}} - \frac {4 \, d^{2} e f g + 4 \, d^{3} g^{2} - {\left (e^{3} f^{2} + 6 \, d e^{2} f g + 5 \, d^{2} e g^{2}\right )} x}{e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}} - \frac {2 \, {\left (e f g + 2 \, d g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.60, size = 107, normalized size = 1.32 \[ -\frac {\frac {4\,\left (d^3\,g^2+e\,f\,d^2\,g\right )}{e}-x\,\left (5\,d^2\,g^2+6\,d\,e\,f\,g+e^2\,f^2\right )}{d^2\,e^2-2\,d\,e^3\,x+e^4\,x^2}-\frac {g^2\,x}{e^2}-\frac {\ln \left (e\,x-d\right )\,\left (4\,d\,g^2+2\,e\,f\,g\right )}{e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 102, normalized size = 1.26 \[ - \frac {4 d^{3} g^{2} + 4 d^{2} e f g + x \left (- 5 d^{2} e g^{2} - 6 d e^{2} f g - e^{3} f^{2}\right )}{d^{2} e^{3} - 2 d e^{4} x + e^{5} x^{2}} - \frac {g^{2} x}{e^{2}} - \frac {2 g \left (2 d g + e f\right ) \log {\left (- d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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