Optimal. Leaf size=146 \[ \frac {8 d^3 (d g+e f)^2}{e^3 (d-e x)}+\frac {4 d^2 (d g+e f) (7 d g+3 e f) \log (d-e x)}{e^3}+\frac {x^2 \left (12 d^2 g^2+10 d e f g+e^2 f^2\right )}{2 e}+\frac {d x \left (20 d^2 g^2+24 d e f g+5 e^2 f^2\right )}{e^2}+\frac {1}{3} g x^3 (5 d g+2 e f)+\frac {1}{4} e g^2 x^4 \]
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Rubi [A] time = 0.18, antiderivative size = 146, 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} \[ \frac {x^2 \left (12 d^2 g^2+10 d e f g+e^2 f^2\right )}{2 e}+\frac {d x \left (20 d^2 g^2+24 d e f g+5 e^2 f^2\right )}{e^2}+\frac {8 d^3 (d g+e f)^2}{e^3 (d-e x)}+\frac {4 d^2 (d g+e f) (7 d g+3 e f) \log (d-e x)}{e^3}+\frac {1}{3} g x^3 (5 d g+2 e f)+\frac {1}{4} e g^2 x^4 \]
Antiderivative was successfully verified.
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Rule 88
Rule 848
Rubi steps
\begin {align*} \int \frac {(d+e x)^5 (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^3 (f+g x)^2}{(d-e x)^2} \, dx\\ &=\int \left (\frac {d \left (5 e^2 f^2+24 d e f g+20 d^2 g^2\right )}{e^2}+\frac {\left (e^2 f^2+10 d e f g+12 d^2 g^2\right ) x}{e}+g (2 e f+5 d g) x^2+e g^2 x^3+\frac {4 d^2 (-3 e f-7 d g) (e f+d g)}{e^2 (d-e x)}+\frac {8 d^3 (e f+d g)^2}{e^2 (-d+e x)^2}\right ) \, dx\\ &=\frac {d \left (5 e^2 f^2+24 d e f g+20 d^2 g^2\right ) x}{e^2}+\frac {\left (e^2 f^2+10 d e f g+12 d^2 g^2\right ) x^2}{2 e}+\frac {1}{3} g (2 e f+5 d g) x^3+\frac {1}{4} e g^2 x^4+\frac {8 d^3 (e f+d g)^2}{e^3 (d-e x)}+\frac {4 d^2 (e f+d g) (3 e f+7 d g) \log (d-e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 154, normalized size = 1.05 \[ -\frac {8 d^3 (d g+e f)^2}{e^3 (e x-d)}+\frac {x^2 \left (12 d^2 g^2+10 d e f g+e^2 f^2\right )}{2 e}+\frac {d x \left (20 d^2 g^2+24 d e f g+5 e^2 f^2\right )}{e^2}+\frac {4 d^2 \left (7 d^2 g^2+10 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}+\frac {1}{3} g x^3 (5 d g+2 e f)+\frac {1}{4} e g^2 x^4 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 251, normalized size = 1.72 \[ \frac {3 \, e^{5} g^{2} x^{5} - 96 \, d^{3} e^{2} f^{2} - 192 \, d^{4} e f g - 96 \, d^{5} g^{2} + {\left (8 \, e^{5} f g + 17 \, d e^{4} g^{2}\right )} x^{4} + 2 \, {\left (3 \, e^{5} f^{2} + 26 \, d e^{4} f g + 26 \, d^{2} e^{3} g^{2}\right )} x^{3} + 6 \, {\left (9 \, d e^{4} f^{2} + 38 \, d^{2} e^{3} f g + 28 \, d^{3} e^{2} g^{2}\right )} x^{2} - 12 \, {\left (5 \, d^{2} e^{3} f^{2} + 24 \, d^{3} e^{2} f g + 20 \, d^{4} e g^{2}\right )} x - 48 \, {\left (3 \, d^{3} e^{2} f^{2} + 10 \, d^{4} e f g + 7 \, d^{5} g^{2} - {\left (3 \, d^{2} e^{3} f^{2} + 10 \, d^{3} e^{2} f g + 7 \, d^{4} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{12 \, {\left (e^{4} x - d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 291, normalized size = 1.99 \[ 2 \, {\left (7 \, d^{4} g^{2} e^{5} + 10 \, d^{3} f g e^{6} + 3 \, d^{2} f^{2} e^{7}\right )} e^{\left (-8\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac {1}{12} \, {\left (3 \, g^{2} x^{4} e^{17} + 20 \, d g^{2} x^{3} e^{16} + 72 \, d^{2} g^{2} x^{2} e^{15} + 240 \, d^{3} g^{2} x e^{14} + 8 \, f g x^{3} e^{17} + 60 \, d f g x^{2} e^{16} + 288 \, d^{2} f g x e^{15} + 6 \, f^{2} x^{2} e^{17} + 60 \, d f^{2} x e^{16}\right )} e^{\left (-16\right )} + \frac {2 \, {\left (7 \, d^{5} g^{2} e^{4} + 10 \, d^{4} f g e^{5} + 3 \, d^{3} f^{2} e^{6}\right )} e^{\left (-7\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{{\left | d \right |}} - \frac {8 \, {\left (d^{6} g^{2} e^{5} + 2 \, d^{5} f g e^{6} + d^{4} f^{2} e^{7} + {\left (d^{5} g^{2} e^{6} + 2 \, d^{4} f g e^{7} + d^{3} f^{2} e^{8}\right )} x\right )} e^{\left (-8\right )}}{x^{2} e^{2} - d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 204, normalized size = 1.40 \[ \frac {e \,g^{2} x^{4}}{4}+\frac {5 d \,g^{2} x^{3}}{3}+\frac {2 e f g \,x^{3}}{3}+\frac {6 d^{2} g^{2} x^{2}}{e}+5 d f g \,x^{2}+\frac {e \,f^{2} x^{2}}{2}-\frac {8 d^{5} g^{2}}{\left (e x -d \right ) e^{3}}-\frac {16 d^{4} f g}{\left (e x -d \right ) e^{2}}+\frac {28 d^{4} g^{2} \ln \left (e x -d \right )}{e^{3}}-\frac {8 d^{3} f^{2}}{\left (e x -d \right ) e}+\frac {40 d^{3} f g \ln \left (e x -d \right )}{e^{2}}+\frac {20 d^{3} g^{2} x}{e^{2}}+\frac {12 d^{2} f^{2} \ln \left (e x -d \right )}{e}+\frac {24 d^{2} f g x}{e}+5 d \,f^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 182, normalized size = 1.25 \[ -\frac {8 \, {\left (d^{3} e^{2} f^{2} + 2 \, d^{4} e f g + d^{5} g^{2}\right )}}{e^{4} x - d e^{3}} + \frac {3 \, e^{3} g^{2} x^{4} + 4 \, {\left (2 \, e^{3} f g + 5 \, d e^{2} g^{2}\right )} x^{3} + 6 \, {\left (e^{3} f^{2} + 10 \, d e^{2} f g + 12 \, d^{2} e g^{2}\right )} x^{2} + 12 \, {\left (5 \, d e^{2} f^{2} + 24 \, d^{2} e f g + 20 \, d^{3} g^{2}\right )} x}{12 \, e^{2}} + \frac {4 \, {\left (3 \, d^{2} e^{2} f^{2} + 10 \, d^{3} e f g + 7 \, d^{4} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 316, normalized size = 2.16 \[ x\,\left (\frac {d^3\,g^2+6\,d^2\,e\,f\,g+3\,d\,e^2\,f^2}{e^2}-\frac {d^2\,\left (g\,\left (3\,d\,g+2\,e\,f\right )+2\,d\,g^2\right )}{e^2}+\frac {2\,d\,\left (\frac {3\,d^2\,e\,g^2+6\,d\,e^2\,f\,g+e^3\,f^2}{e^2}-\frac {d^2\,g^2}{e}+\frac {2\,d\,\left (g\,\left (3\,d\,g+2\,e\,f\right )+2\,d\,g^2\right )}{e}\right )}{e}\right )+x^2\,\left (\frac {3\,d^2\,e\,g^2+6\,d\,e^2\,f\,g+e^3\,f^2}{2\,e^2}-\frac {d^2\,g^2}{2\,e}+\frac {d\,\left (g\,\left (3\,d\,g+2\,e\,f\right )+2\,d\,g^2\right )}{e}\right )+x^3\,\left (\frac {g\,\left (3\,d\,g+2\,e\,f\right )}{3}+\frac {2\,d\,g^2}{3}\right )+\frac {\ln \left (e\,x-d\right )\,\left (28\,d^4\,g^2+40\,d^3\,e\,f\,g+12\,d^2\,e^2\,f^2\right )}{e^3}+\frac {8\,\left (d^5\,g^2+2\,d^4\,e\,f\,g+d^3\,e^2\,f^2\right )}{e\,\left (d\,e^2-e^3\,x\right )}+\frac {e\,g^2\,x^4}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.85, size = 162, normalized size = 1.11 \[ \frac {4 d^{2} \left (d g + e f\right ) \left (7 d g + 3 e f\right ) \log {\left (- d + e x \right )}}{e^{3}} + \frac {e g^{2} x^{4}}{4} + x^{3} \left (\frac {5 d g^{2}}{3} + \frac {2 e f g}{3}\right ) + x^{2} \left (\frac {6 d^{2} g^{2}}{e} + 5 d f g + \frac {e f^{2}}{2}\right ) + x \left (\frac {20 d^{3} g^{2}}{e^{2}} + \frac {24 d^{2} f g}{e} + 5 d f^{2}\right ) + \frac {- 8 d^{5} g^{2} - 16 d^{4} e f g - 8 d^{3} e^{2} f^{2}}{- d e^{3} + e^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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