Optimal. Leaf size=65 \[ -\frac {2 d (d g+e f)^2 \log (d-e x)}{e^3}-\frac {2 d g x (d g+e f)}{e^2}-\frac {d (f+g x)^2}{e}-\frac {(f+g x)^3}{3 g} \]
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Rubi [A] time = 0.06, antiderivative size = 65, 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} \[ -\frac {2 d g x (d g+e f)}{e^2}-\frac {2 d (d g+e f)^2 \log (d-e x)}{e^3}-\frac {d (f+g x)^2}{e}-\frac {(f+g x)^3}{3 g} \]
Antiderivative was successfully verified.
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Rule 77
Rule 848
Rubi steps
\begin {align*} \int \frac {(d+e x)^2 (f+g x)^2}{d^2-e^2 x^2} \, dx &=\int \frac {(d+e x) (f+g x)^2}{d-e x} \, dx\\ &=\int \left (-\frac {2 d g (e f+d g)}{e^2}-\frac {2 d (e f+d g)^2}{e^2 (-d+e x)}-\frac {2 d g (f+g x)}{e}-(f+g x)^2\right ) \, dx\\ &=-\frac {2 d g (e f+d g) x}{e^2}-\frac {d (f+g x)^2}{e}-\frac {(f+g x)^3}{3 g}-\frac {2 d (e f+d g)^2 \log (d-e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 1.12 \[ -\frac {e x \left (6 d^2 g^2+3 d e g (4 f+g x)+e^2 \left (3 f^2+3 f g x+g^2 x^2\right )\right )+6 d (d g+e f)^2 \log (d-e x)}{3 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.98, size = 98, normalized size = 1.51 \[ -\frac {e^{3} g^{2} x^{3} + 3 \, {\left (e^{3} f g + d e^{2} g^{2}\right )} x^{2} + 3 \, {\left (e^{3} f^{2} + 4 \, d e^{2} f g + 2 \, d^{2} e g^{2}\right )} x + 6 \, {\left (d e^{2} f^{2} + 2 \, d^{2} e f g + d^{3} g^{2}\right )} \log \left (e x - d\right )}{3 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.16, size = 172, normalized size = 2.65 \[ -{\left (d^{3} g^{2} e + 2 \, d^{2} f g e^{2} + d f^{2} e^{3}\right )} e^{\left (-4\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac {1}{3} \, {\left (g^{2} x^{3} e^{6} + 3 \, d g^{2} x^{2} e^{5} + 6 \, d^{2} g^{2} x e^{4} + 3 \, f g x^{2} e^{6} + 12 \, d f g x e^{5} + 3 \, f^{2} x e^{6}\right )} e^{\left (-6\right )} - \frac {{\left (d^{4} g^{2} e^{2} + 2 \, d^{3} f g e^{3} + d^{2} f^{2} e^{4}\right )} e^{\left (-5\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{{\left | d \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 110, normalized size = 1.69 \[ -\frac {g^{2} x^{3}}{3}-\frac {d \,g^{2} x^{2}}{e}-f g \,x^{2}-\frac {2 d^{3} g^{2} \ln \left (e x -d \right )}{e^{3}}-\frac {4 d^{2} f g \ln \left (e x -d \right )}{e^{2}}-\frac {2 d^{2} g^{2} x}{e^{2}}-\frac {2 d \,f^{2} \ln \left (e x -d \right )}{e}-\frac {4 d f g x}{e}-f^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 97, normalized size = 1.49 \[ -\frac {e^{2} g^{2} x^{3} + 3 \, {\left (e^{2} f g + d e g^{2}\right )} x^{2} + 3 \, {\left (e^{2} f^{2} + 4 \, d e f g + 2 \, d^{2} g^{2}\right )} x}{3 \, e^{2}} - \frac {2 \, {\left (d e^{2} f^{2} + 2 \, d^{2} e f g + d^{3} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 127, normalized size = 1.95 \[ -x^2\,\left (\frac {d\,g^2+2\,e\,f\,g}{2\,e}+\frac {d\,g^2}{2\,e}\right )-x\,\left (\frac {e\,f^2+2\,d\,g\,f}{e}+\frac {d\,\left (\frac {d\,g^2+2\,e\,f\,g}{e}+\frac {d\,g^2}{e}\right )}{e}\right )-\frac {g^2\,x^3}{3}-\frac {\ln \left (e\,x-d\right )\,\left (2\,d^3\,g^2+4\,d^2\,e\,f\,g+2\,d\,e^2\,f^2\right )}{e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.38, size = 70, normalized size = 1.08 \[ - \frac {2 d \left (d g + e f\right )^{2} \log {\left (- d + e x \right )}}{e^{3}} - \frac {g^{2} x^{3}}{3} - x^{2} \left (\frac {d g^{2}}{e} + f g\right ) - x \left (\frac {2 d^{2} g^{2}}{e^{2}} + \frac {4 d f g}{e} + f^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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