Optimal. Leaf size=109 \[ -\frac {4 d^2 (d g+e f)^2 \log (d-e x)}{e^3}-\frac {x^2 \left (4 d^2 g^2+6 d e f g+e^2 f^2\right )}{2 e}-\frac {d x (2 d g+e f) (2 d g+3 e f)}{e^2}-\frac {1}{3} g x^3 (3 d g+2 e f)-\frac {1}{4} e g^2 x^4 \]
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Rubi [A] time = 0.14, antiderivative size = 109, 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} \[ -\frac {x^2 \left (4 d^2 g^2+6 d e f g+e^2 f^2\right )}{2 e}-\frac {4 d^2 (d g+e f)^2 \log (d-e x)}{e^3}-\frac {d x (2 d g+e f) (2 d g+3 e f)}{e^2}-\frac {1}{3} g x^3 (3 d g+2 e f)-\frac {1}{4} e g^2 x^4 \]
Antiderivative was successfully verified.
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Rule 88
Rule 848
Rubi steps
\begin {align*} \int \frac {(d+e x)^3 (f+g x)^2}{d^2-e^2 x^2} \, dx &=\int \frac {(d+e x)^2 (f+g x)^2}{d-e x} \, dx\\ &=\int \left (\frac {d (-3 e f-2 d g) (e f+2 d g)}{e^2}-\frac {\left (e^2 f^2+6 d e f g+4 d^2 g^2\right ) x}{e}-g (2 e f+3 d g) x^2-e g^2 x^3-\frac {4 d^2 (e f+d g)^2}{e^2 (-d+e x)}\right ) \, dx\\ &=-\frac {d (e f+2 d g) (3 e f+2 d g) x}{e^2}-\frac {\left (e^2 f^2+6 d e f g+4 d^2 g^2\right ) x^2}{2 e}-\frac {1}{3} g (2 e f+3 d g) x^3-\frac {1}{4} e g^2 x^4-\frac {4 d^2 (e f+d g)^2 \log (d-e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 103, normalized size = 0.94 \[ -\frac {48 d^2 (d g+e f)^2 \log (d-e x)+e x \left (48 d^3 g^2+24 d^2 e g (4 f+g x)+12 d e^2 \left (3 f^2+3 f g x+g^2 x^2\right )+e^3 x \left (6 f^2+8 f g x+3 g^2 x^2\right )\right )}{12 e^3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.68, size = 139, normalized size = 1.28 \[ -\frac {3 \, e^{4} g^{2} x^{4} + 4 \, {\left (2 \, e^{4} f g + 3 \, d e^{3} g^{2}\right )} x^{3} + 6 \, {\left (e^{4} f^{2} + 6 \, d e^{3} f g + 4 \, d^{2} e^{2} g^{2}\right )} x^{2} + 12 \, {\left (3 \, d e^{3} f^{2} + 8 \, d^{2} e^{2} f g + 4 \, d^{3} e g^{2}\right )} x + 48 \, {\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2}\right )} \log \left (e x - d\right )}{12 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.21, size = 211, normalized size = 1.94 \[ -2 \, {\left (d^{4} g^{2} e^{3} + 2 \, d^{3} f g e^{4} + d^{2} f^{2} e^{5}\right )} e^{\left (-6\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac {1}{12} \, {\left (3 \, g^{2} x^{4} e^{9} + 12 \, d g^{2} x^{3} e^{8} + 24 \, d^{2} g^{2} x^{2} e^{7} + 48 \, d^{3} g^{2} x e^{6} + 8 \, f g x^{3} e^{9} + 36 \, d f g x^{2} e^{8} + 96 \, d^{2} f g x e^{7} + 6 \, f^{2} x^{2} e^{9} + 36 \, d f^{2} x e^{8}\right )} e^{\left (-8\right )} - \frac {2 \, {\left (d^{5} g^{2} e^{2} + 2 \, d^{4} f g e^{3} + d^{3} f^{2} e^{4}\right )} e^{\left (-5\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{{\left | d \right |}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 145, normalized size = 1.33 \[ -\frac {e \,g^{2} x^{4}}{4}-d \,g^{2} x^{3}-\frac {2 e f g \,x^{3}}{3}-\frac {2 d^{2} g^{2} x^{2}}{e}-3 d f g \,x^{2}-\frac {e \,f^{2} x^{2}}{2}-\frac {4 d^{4} g^{2} \ln \left (e x -d \right )}{e^{3}}-\frac {8 d^{3} f g \ln \left (e x -d \right )}{e^{2}}-\frac {4 d^{3} g^{2} x}{e^{2}}-\frac {4 d^{2} f^{2} \ln \left (e x -d \right )}{e}-\frac {8 d^{2} f g x}{e}-3 d \,f^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 138, normalized size = 1.27 \[ -\frac {3 \, e^{3} g^{2} x^{4} + 4 \, {\left (2 \, e^{3} f g + 3 \, d e^{2} g^{2}\right )} x^{3} + 6 \, {\left (e^{3} f^{2} + 6 \, d e^{2} f g + 4 \, d^{2} e g^{2}\right )} x^{2} + 12 \, {\left (3 \, d e^{2} f^{2} + 8 \, d^{2} e f g + 4 \, d^{3} g^{2}\right )} x}{12 \, e^{2}} - \frac {4 \, {\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.59, size = 197, normalized size = 1.81 \[ -x^3\,\left (\frac {2\,g\,\left (d\,g+e\,f\right )}{3}+\frac {d\,g^2}{3}\right )-x^2\,\left (\frac {d^2\,g^2+4\,d\,e\,f\,g+e^2\,f^2}{2\,e}+\frac {d\,\left (2\,g\,\left (d\,g+e\,f\right )+d\,g^2\right )}{2\,e}\right )-x\,\left (\frac {d\,\left (\frac {d^2\,g^2+4\,d\,e\,f\,g+e^2\,f^2}{e}+\frac {d\,\left (2\,g\,\left (d\,g+e\,f\right )+d\,g^2\right )}{e}\right )}{e}+\frac {2\,d\,f\,\left (d\,g+e\,f\right )}{e}\right )-\frac {\ln \left (e\,x-d\right )\,\left (4\,d^4\,g^2+8\,d^3\,e\,f\,g+4\,d^2\,e^2\,f^2\right )}{e^3}-\frac {e\,g^2\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.48, size = 109, normalized size = 1.00 \[ - \frac {4 d^{2} \left (d g + e f\right )^{2} \log {\left (- d + e x \right )}}{e^{3}} - \frac {e g^{2} x^{4}}{4} - x^{3} \left (d g^{2} + \frac {2 e f g}{3}\right ) - x^{2} \left (\frac {2 d^{2} g^{2}}{e} + 3 d f g + \frac {e f^{2}}{2}\right ) - x \left (\frac {4 d^{3} g^{2}}{e^{2}} + \frac {8 d^{2} f g}{e} + 3 d f^{2}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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