Optimal. Leaf size=107 \[ \frac {4 d^2 (d g+e f)^2}{e^3 (d-e x)}+\frac {x \left (8 d^2 g^2+8 d e f g+e^2 f^2\right )}{e^2}+\frac {4 d (d g+e f) (3 d g+e f) \log (d-e x)}{e^3}+\frac {g x^2 (2 d g+e f)}{e}+\frac {g^2 x^3}{3} \]
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Rubi [A] time = 0.14, antiderivative size = 107, 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} \[ \frac {x \left (8 d^2 g^2+8 d e f g+e^2 f^2\right )}{e^2}+\frac {4 d^2 (d g+e f)^2}{e^3 (d-e x)}+\frac {4 d (d g+e f) (3 d g+e f) \log (d-e x)}{e^3}+\frac {g x^2 (2 d g+e f)}{e}+\frac {g^2 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 848
Rubi steps
\begin {align*} \int \frac {(d+e x)^4 (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^2 (f+g x)^2}{(d-e x)^2} \, dx\\ &=\int \left (\frac {e^2 f^2+8 d e f g+8 d^2 g^2}{e^2}+\frac {2 g (e f+2 d g) x}{e}+g^2 x^2+\frac {4 d (-e f-3 d g) (e f+d g)}{e^2 (d-e x)}+\frac {4 d^2 (e f+d g)^2}{e^2 (-d+e x)^2}\right ) \, dx\\ &=\frac {\left (e^2 f^2+8 d e f g+8 d^2 g^2\right ) x}{e^2}+\frac {g (e f+2 d g) x^2}{e}+\frac {g^2 x^3}{3}+\frac {4 d^2 (e f+d g)^2}{e^3 (d-e x)}+\frac {4 d (e f+d g) (e f+3 d g) \log (d-e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 115, normalized size = 1.07 \[ -\frac {4 d^2 (d g+e f)^2}{e^3 (e x-d)}+\frac {x \left (8 d^2 g^2+8 d e f g+e^2 f^2\right )}{e^2}+\frac {4 d \left (3 d^2 g^2+4 d e f g+e^2 f^2\right ) \log (d-e x)}{e^3}+\frac {g x^2 (2 d g+e f)}{e}+\frac {g^2 x^3}{3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.75, size = 206, normalized size = 1.93 \[ \frac {e^{4} g^{2} x^{4} - 12 \, d^{2} e^{2} f^{2} - 24 \, d^{3} e f g - 12 \, d^{4} g^{2} + {\left (3 \, e^{4} f g + 5 \, d e^{3} g^{2}\right )} x^{3} + 3 \, {\left (e^{4} f^{2} + 7 \, d e^{3} f g + 6 \, d^{2} e^{2} g^{2}\right )} x^{2} - 3 \, {\left (d e^{3} f^{2} + 8 \, d^{2} e^{2} f g + 8 \, d^{3} e g^{2}\right )} x - 12 \, {\left (d^{2} e^{2} f^{2} + 4 \, d^{3} e f g + 3 \, d^{4} g^{2} - {\left (d e^{3} f^{2} + 4 \, d^{2} e^{2} f g + 3 \, d^{3} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{3 \, {\left (e^{4} x - d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.17, size = 250, normalized size = 2.34 \[ 2 \, {\left (3 \, d^{3} g^{2} e^{3} + 4 \, d^{2} f g e^{4} + d f^{2} e^{5}\right )} e^{\left (-6\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac {1}{3} \, {\left (g^{2} x^{3} e^{12} + 6 \, d g^{2} x^{2} e^{11} + 24 \, d^{2} g^{2} x e^{10} + 3 \, f g x^{2} e^{12} + 24 \, d f g x e^{11} + 3 \, f^{2} x e^{12}\right )} e^{\left (-12\right )} + \frac {2 \, {\left (3 \, d^{4} g^{2} e^{4} + 4 \, d^{3} f g e^{5} + d^{2} f^{2} e^{6}\right )} e^{\left (-7\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{{\left | d \right |}} - \frac {4 \, {\left (d^{5} g^{2} e^{3} + 2 \, d^{4} f g e^{4} + d^{3} f^{2} e^{5} + {\left (d^{4} g^{2} e^{4} + 2 \, d^{3} f g e^{5} + d^{2} f^{2} e^{6}\right )} x\right )} e^{\left (-6\right )}}{x^{2} e^{2} - d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 167, normalized size = 1.56 \[ \frac {g^{2} x^{3}}{3}+\frac {2 d \,g^{2} x^{2}}{e}+f g \,x^{2}-\frac {4 d^{4} g^{2}}{\left (e x -d \right ) e^{3}}-\frac {8 d^{3} f g}{\left (e x -d \right ) e^{2}}+\frac {12 d^{3} g^{2} \ln \left (e x -d \right )}{e^{3}}-\frac {4 d^{2} f^{2}}{\left (e x -d \right ) e}+\frac {16 d^{2} f g \ln \left (e x -d \right )}{e^{2}}+\frac {8 d^{2} g^{2} x}{e^{2}}+\frac {4 d \,f^{2} \ln \left (e x -d \right )}{e}+\frac {8 d f g x}{e}+f^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 141, normalized size = 1.32 \[ -\frac {4 \, {\left (d^{2} e^{2} f^{2} + 2 \, d^{3} e f g + d^{4} g^{2}\right )}}{e^{4} x - d e^{3}} + \frac {e^{2} g^{2} x^{3} + 3 \, {\left (e^{2} f g + 2 \, d e g^{2}\right )} x^{2} + 3 \, {\left (e^{2} f^{2} + 8 \, d e f g + 8 \, d^{2} g^{2}\right )} x}{3 \, e^{2}} + \frac {4 \, {\left (d e^{2} f^{2} + 4 \, d^{2} e f g + 3 \, d^{3} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 185, normalized size = 1.73 \[ x^2\,\left (\frac {g\,\left (d\,g+e\,f\right )}{e}+\frac {d\,g^2}{e}\right )+x\,\left (\frac {d^2\,g^2+4\,d\,e\,f\,g+e^2\,f^2}{e^2}+\frac {2\,d\,\left (\frac {2\,g\,\left (d\,g+e\,f\right )}{e}+\frac {2\,d\,g^2}{e}\right )}{e}-\frac {d^2\,g^2}{e^2}\right )+\frac {g^2\,x^3}{3}+\frac {4\,\left (d^4\,g^2+2\,d^3\,e\,f\,g+d^2\,e^2\,f^2\right )}{e\,\left (d\,e^2-e^3\,x\right )}+\frac {\ln \left (e\,x-d\right )\,\left (12\,d^3\,g^2+16\,d^2\,e\,f\,g+4\,d\,e^2\,f^2\right )}{e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.74, size = 119, normalized size = 1.11 \[ \frac {4 d \left (d g + e f\right ) \left (3 d g + e f\right ) \log {\left (- d + e x \right )}}{e^{3}} + \frac {g^{2} x^{3}}{3} + x^{2} \left (\frac {2 d g^{2}}{e} + f g\right ) + x \left (\frac {8 d^{2} g^{2}}{e^{2}} + \frac {8 d f g}{e} + f^{2}\right ) + \frac {- 4 d^{4} g^{2} - 8 d^{3} e f g - 4 d^{2} e^{2} f^{2}}{- d e^{3} + e^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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