Optimal. Leaf size=149 \[ \frac {4 d^3 (d g+e f)^2}{e^3 (d-e x)^2}-\frac {4 d^2 (d g+e f) (7 d g+3 e f)}{e^3 (d-e x)}-\frac {x \left (18 d^2 g^2+12 d e f g+e^2 f^2\right )}{e^2}-\frac {2 d \left (19 d^2 g^2+18 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}-\frac {g x^2 (3 d g+e f)}{e}-\frac {g^2 x^3}{3} \]
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Rubi [A] time = 0.20, antiderivative size = 149, 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} \[ -\frac {x \left (18 d^2 g^2+12 d e f g+e^2 f^2\right )}{e^2}-\frac {2 d \left (19 d^2 g^2+18 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}+\frac {4 d^3 (d g+e f)^2}{e^3 (d-e x)^2}-\frac {4 d^2 (d g+e f) (7 d g+3 e f)}{e^3 (d-e x)}-\frac {g x^2 (3 d g+e f)}{e}-\frac {g^2 x^3}{3} \]
Antiderivative was successfully verified.
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Rule 88
Rule 848
Rubi steps
\begin {align*} \int \frac {(d+e x)^6 (f+g x)^2}{\left (d^2-e^2 x^2\right )^3} \, dx &=\int \frac {(d+e x)^3 (f+g x)^2}{(d-e x)^3} \, dx\\ &=\int \left (\frac {-e^2 f^2-12 d e f g-18 d^2 g^2}{e^2}-\frac {2 g (e f+3 d g) x}{e}-g^2 x^2+\frac {4 d^2 (-3 e f-7 d g) (e f+d g)}{e^2 (d-e x)^2}-\frac {8 d^3 (e f+d g)^2}{e^2 (-d+e x)^3}-\frac {2 d \left (3 e^2 f^2+18 d e f g+19 d^2 g^2\right )}{e^2 (-d+e x)}\right ) \, dx\\ &=-\frac {\left (e^2 f^2+12 d e f g+18 d^2 g^2\right ) x}{e^2}-\frac {g (e f+3 d g) x^2}{e}-\frac {g^2 x^3}{3}+\frac {4 d^3 (e f+d g)^2}{e^3 (d-e x)^2}-\frac {4 d^2 (e f+d g) (3 e f+7 d g)}{e^3 (d-e x)}-\frac {2 d \left (3 e^2 f^2+18 d e f g+19 d^2 g^2\right ) \log (d-e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 157, normalized size = 1.05 \[ \frac {4 d^3 (d g+e f)^2}{e^3 (d-e x)^2}-\frac {x \left (18 d^2 g^2+12 d e f g+e^2 f^2\right )}{e^2}+\frac {4 d^2 \left (7 d^2 g^2+10 d e f g+3 e^2 f^2\right )}{e^3 (e x-d)}-\frac {2 d \left (19 d^2 g^2+18 d e f g+3 e^2 f^2\right ) \log (d-e x)}{e^3}-\frac {g x^2 (3 d g+e f)}{e}-\frac {g^2 x^3}{3} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 294, normalized size = 1.97 \[ -\frac {e^{5} g^{2} x^{5} + 24 \, d^{3} e^{2} f^{2} + 96 \, d^{4} e f g + 72 \, d^{5} g^{2} + {\left (3 \, e^{5} f g + 7 \, d e^{4} g^{2}\right )} x^{4} + {\left (3 \, e^{5} f^{2} + 30 \, d e^{4} f g + 37 \, d^{2} e^{3} g^{2}\right )} x^{3} - 3 \, {\left (2 \, d e^{4} f^{2} + 23 \, d^{2} e^{3} f g + 33 \, d^{3} e^{2} g^{2}\right )} x^{2} - 3 \, {\left (11 \, d^{2} e^{3} f^{2} + 28 \, d^{3} e^{2} f g + 10 \, d^{4} e g^{2}\right )} x + 6 \, {\left (3 \, d^{3} e^{2} f^{2} + 18 \, d^{4} e f g + 19 \, d^{5} g^{2} + {\left (3 \, d e^{4} f^{2} + 18 \, d^{2} e^{3} f g + 19 \, d^{3} e^{2} g^{2}\right )} x^{2} - 2 \, {\left (3 \, d^{2} e^{3} f^{2} + 18 \, d^{3} e^{2} f g + 19 \, d^{4} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{3 \, {\left (e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 324, normalized size = 2.17 \[ -{\left (19 \, d^{3} g^{2} e^{5} + 18 \, d^{2} f g e^{6} + 3 \, d f^{2} e^{7}\right )} e^{\left (-8\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) - \frac {1}{3} \, {\left (g^{2} x^{3} e^{18} + 9 \, d g^{2} x^{2} e^{17} + 54 \, d^{2} g^{2} x e^{16} + 3 \, f g x^{2} e^{18} + 36 \, d f g x e^{17} + 3 \, f^{2} x e^{18}\right )} e^{\left (-18\right )} - \frac {{\left (19 \, d^{4} g^{2} e^{6} + 18 \, d^{3} f g e^{7} + 3 \, d^{2} f^{2} e^{8}\right )} e^{\left (-9\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{{\left | d \right |}} - \frac {4 \, {\left (6 \, d^{7} g^{2} e^{5} + 8 \, d^{6} f g e^{6} + 2 \, d^{5} f^{2} e^{7} - {\left (7 \, d^{4} g^{2} e^{8} + 10 \, d^{3} f g e^{9} + 3 \, d^{2} f^{2} e^{10}\right )} x^{3} - 4 \, {\left (2 \, d^{5} g^{2} e^{7} + 3 \, d^{4} f g e^{8} + d^{3} f^{2} e^{9}\right )} x^{2} + {\left (5 \, d^{6} g^{2} e^{6} + 6 \, d^{5} f g e^{7} + d^{4} f^{2} e^{8}\right )} x\right )} e^{\left (-8\right )}}{{\left (x^{2} e^{2} - d^{2}\right )}^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 228, normalized size = 1.53 \[ -\frac {g^{2} x^{3}}{3}+\frac {4 d^{5} g^{2}}{\left (e x -d \right )^{2} e^{3}}+\frac {8 d^{4} f g}{\left (e x -d \right )^{2} e^{2}}+\frac {4 d^{3} f^{2}}{\left (e x -d \right )^{2} e}-\frac {3 d \,g^{2} x^{2}}{e}-f g \,x^{2}+\frac {28 d^{4} g^{2}}{\left (e x -d \right ) e^{3}}+\frac {40 d^{3} f g}{\left (e x -d \right ) e^{2}}-\frac {38 d^{3} g^{2} \ln \left (e x -d \right )}{e^{3}}+\frac {12 d^{2} f^{2}}{\left (e x -d \right ) e}-\frac {36 d^{2} f g \ln \left (e x -d \right )}{e^{2}}-\frac {18 d^{2} g^{2} x}{e^{2}}-\frac {6 d \,f^{2} \ln \left (e x -d \right )}{e}-\frac {12 d f g x}{e}-f^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 188, normalized size = 1.26 \[ -\frac {4 \, {\left (2 \, d^{3} e^{2} f^{2} + 8 \, d^{4} e f g + 6 \, d^{5} g^{2} - {\left (3 \, d^{2} e^{3} f^{2} + 10 \, d^{3} e^{2} f g + 7 \, d^{4} e g^{2}\right )} x\right )}}{e^{5} x^{2} - 2 \, d e^{4} x + d^{2} e^{3}} - \frac {e^{2} g^{2} x^{3} + 3 \, {\left (e^{2} f g + 3 \, d e g^{2}\right )} x^{2} + 3 \, {\left (e^{2} f^{2} + 12 \, d e f g + 18 \, d^{2} g^{2}\right )} x}{3 \, e^{2}} - \frac {2 \, {\left (3 \, d e^{2} f^{2} + 18 \, d^{2} e f g + 19 \, d^{3} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.10, size = 240, normalized size = 1.61 \[ \frac {x\,\left (28\,d^4\,g^2+40\,d^3\,e\,f\,g+12\,d^2\,e^2\,f^2\right )-\frac {8\,\left (3\,d^5\,g^2+4\,d^4\,e\,f\,g+d^3\,e^2\,f^2\right )}{e}}{d^2\,e^2-2\,d\,e^3\,x+e^4\,x^2}-x\,\left (\frac {3\,d^2\,e\,g^2+6\,d\,e^2\,f\,g+e^3\,f^2}{e^3}+\frac {3\,d\,\left (\frac {g\,\left (3\,d\,g+2\,e\,f\right )}{e}+\frac {3\,d\,g^2}{e}\right )}{e}-\frac {3\,d^2\,g^2}{e^2}\right )-x^2\,\left (\frac {g\,\left (3\,d\,g+2\,e\,f\right )}{2\,e}+\frac {3\,d\,g^2}{2\,e}\right )-\frac {g^2\,x^3}{3}-\frac {\ln \left (e\,x-d\right )\,\left (38\,d^3\,g^2+36\,d^2\,e\,f\,g+6\,d\,e^2\,f^2\right )}{e^3} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.37, size = 178, normalized size = 1.19 \[ - \frac {2 d \left (19 d^{2} g^{2} + 18 d e f g + 3 e^{2} f^{2}\right ) \log {\left (- d + e x \right )}}{e^{3}} - \frac {g^{2} x^{3}}{3} - x^{2} \left (\frac {3 d g^{2}}{e} + f g\right ) - x \left (\frac {18 d^{2} g^{2}}{e^{2}} + \frac {12 d f g}{e} + f^{2}\right ) - \frac {24 d^{5} g^{2} + 32 d^{4} e f g + 8 d^{3} e^{2} f^{2} + x \left (- 28 d^{4} e g^{2} - 40 d^{3} e^{2} f g - 12 d^{2} e^{3} f^{2}\right )}{d^{2} e^{3} - 2 d e^{4} x + e^{5} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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