Optimal. Leaf size=78 \[ \frac {2 d (d g+e f)^2}{e^3 (d-e x)}+\frac {(5 d g+e f) (d g+e f) \log (d-e x)}{e^3}+\frac {g x (3 d g+2 e f)}{e^2}+\frac {g^2 x^2}{2 e} \]
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Rubi [A] time = 0.10, antiderivative size = 78, 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} \[ \frac {2 d (d g+e f)^2}{e^3 (d-e x)}+\frac {g x (3 d g+2 e f)}{e^2}+\frac {(5 d g+e f) (d g+e f) \log (d-e x)}{e^3}+\frac {g^2 x^2}{2 e} \]
Antiderivative was successfully verified.
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Rule 77
Rule 848
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(d+e x) (f+g x)^2}{(d-e x)^2} \, dx\\ &=\int \left (\frac {g (2 e f+3 d g)}{e^2}+\frac {g^2 x}{e}+\frac {(-e f-5 d g) (e f+d g)}{e^2 (d-e x)}+\frac {2 d (e f+d g)^2}{e^2 (-d+e x)^2}\right ) \, dx\\ &=\frac {g (2 e f+3 d g) x}{e^2}+\frac {g^2 x^2}{2 e}+\frac {2 d (e f+d g)^2}{e^3 (d-e x)}+\frac {(e f+d g) (e f+5 d g) \log (d-e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 83, normalized size = 1.06 \[ \frac {2 \left (5 d^2 g^2+6 d e f g+e^2 f^2\right ) \log (d-e x)+\frac {4 d (d g+e f)^2}{d-e x}+2 e g x (3 d g+2 e f)+e^2 g^2 x^2}{2 e^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.72, size = 157, normalized size = 2.01 \[ \frac {e^{3} g^{2} x^{3} - 4 \, d e^{2} f^{2} - 8 \, d^{2} e f g - 4 \, d^{3} g^{2} + {\left (4 \, e^{3} f g + 5 \, d e^{2} g^{2}\right )} x^{2} - 2 \, {\left (2 \, d e^{2} f g + 3 \, d^{2} e g^{2}\right )} x - 2 \, {\left (d e^{2} f^{2} + 6 \, d^{2} e f g + 5 \, d^{3} g^{2} - {\left (e^{3} f^{2} + 6 \, d e^{2} f g + 5 \, d^{2} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{2 \, {\left (e^{4} x - d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.18, size = 212, normalized size = 2.72 \[ \frac {1}{2} \, {\left (5 \, d^{2} g^{2} e^{3} + 6 \, d f g e^{4} + f^{2} e^{5}\right )} e^{\left (-6\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac {1}{2} \, {\left (g^{2} x^{2} e^{7} + 6 \, d g^{2} x e^{6} + 4 \, f g x e^{7}\right )} e^{\left (-8\right )} + \frac {{\left (5 \, d^{3} g^{2} e^{2} + 6 \, d^{2} f g e^{3} + d f^{2} e^{4}\right )} e^{\left (-5\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{2 \, {\left | d \right |}} - \frac {2 \, {\left (d^{4} g^{2} e^{3} + 2 \, d^{3} f g e^{4} + d^{2} f^{2} e^{5} + {\left (d^{3} g^{2} e^{4} + 2 \, d^{2} f g e^{5} + d f^{2} e^{6}\right )} x\right )} e^{\left (-6\right )}}{x^{2} e^{2} - d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 138, normalized size = 1.77 \[ \frac {g^{2} x^{2}}{2 e}-\frac {2 d^{3} g^{2}}{\left (e x -d \right ) e^{3}}-\frac {4 d^{2} f g}{\left (e x -d \right ) e^{2}}+\frac {5 d^{2} g^{2} \ln \left (e x -d \right )}{e^{3}}-\frac {2 d \,f^{2}}{\left (e x -d \right ) e}+\frac {6 d f g \ln \left (e x -d \right )}{e^{2}}+\frac {3 d \,g^{2} x}{e^{2}}+\frac {f^{2} \ln \left (e x -d \right )}{e}+\frac {2 f g x}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.44, size = 104, normalized size = 1.33 \[ -\frac {2 \, {\left (d e^{2} f^{2} + 2 \, d^{2} e f g + d^{3} g^{2}\right )}}{e^{4} x - d e^{3}} + \frac {e g^{2} x^{2} + 2 \, {\left (2 \, e f g + 3 \, d g^{2}\right )} x}{2 \, e^{2}} + \frac {{\left (e^{2} f^{2} + 6 \, d e f g + 5 \, d^{2} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.53, size = 116, normalized size = 1.49 \[ x\,\left (\frac {d\,g^2+2\,e\,f\,g}{e^2}+\frac {2\,d\,g^2}{e^2}\right )+\frac {\ln \left (e\,x-d\right )\,\left (5\,d^2\,g^2+6\,d\,e\,f\,g+e^2\,f^2\right )}{e^3}+\frac {g^2\,x^2}{2\,e}+\frac {2\,\left (d^3\,g^2+2\,d^2\,e\,f\,g+d\,e^2\,f^2\right )}{e\,\left (d\,e^2-e^3\,x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.59, size = 94, normalized size = 1.21 \[ x \left (\frac {3 d g^{2}}{e^{2}} + \frac {2 f g}{e}\right ) + \frac {- 2 d^{3} g^{2} - 4 d^{2} e f g - 2 d e^{2} f^{2}}{- d e^{3} + e^{4} x} + \frac {g^{2} x^{2}}{2 e} + \frac {\left (d g + e f\right ) \left (5 d g + e f\right ) \log {\left (- d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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