Optimal. Leaf size=218 \[ \frac {32 d^5 (d g+e f)^2}{e^3 (d-e x)}+\frac {16 d^4 (d g+e f) (9 d g+5 e f) \log (d-e x)}{e^3}+\frac {1}{4} e x^4 \left (23 d^2 g^2+14 d e f g+e^2 f^2\right )+\frac {1}{3} d x^3 \left (49 d^2 g^2+46 d e f g+7 e^2 f^2\right )+\frac {d^2 x^2 \left (80 d^2 g^2+98 d e f g+23 e^2 f^2\right )}{2 e}+\frac {d^3 x \left (112 d^2 g^2+160 d e f g+49 e^2 f^2\right )}{e^2}+\frac {1}{5} e^2 g x^5 (7 d g+2 e f)+\frac {1}{6} e^3 g^2 x^6 \]
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Rubi [A] time = 0.28, antiderivative size = 218, 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} \[ \frac {1}{4} e x^4 \left (23 d^2 g^2+14 d e f g+e^2 f^2\right )+\frac {1}{3} d x^3 \left (49 d^2 g^2+46 d e f g+7 e^2 f^2\right )+\frac {d^2 x^2 \left (80 d^2 g^2+98 d e f g+23 e^2 f^2\right )}{2 e}+\frac {d^3 x \left (112 d^2 g^2+160 d e f g+49 e^2 f^2\right )}{e^2}+\frac {32 d^5 (d g+e f)^2}{e^3 (d-e x)}+\frac {16 d^4 (d g+e f) (9 d g+5 e f) \log (d-e x)}{e^3}+\frac {1}{5} e^2 g x^5 (7 d g+2 e f)+\frac {1}{6} e^3 g^2 x^6 \]
Antiderivative was successfully verified.
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Rule 88
Rule 848
Rubi steps
\begin {align*} \int \frac {(d+e x)^7 (f+g x)^2}{\left (d^2-e^2 x^2\right )^2} \, dx &=\int \frac {(d+e x)^5 (f+g x)^2}{(d-e x)^2} \, dx\\ &=\int \left (\frac {d^3 \left (49 e^2 f^2+160 d e f g+112 d^2 g^2\right )}{e^2}+\frac {d^2 \left (23 e^2 f^2+98 d e f g+80 d^2 g^2\right ) x}{e}+d \left (7 e^2 f^2+46 d e f g+49 d^2 g^2\right ) x^2+e \left (e^2 f^2+14 d e f g+23 d^2 g^2\right ) x^3+e^2 g (2 e f+7 d g) x^4+e^3 g^2 x^5+\frac {16 d^4 (-5 e f-9 d g) (e f+d g)}{e^2 (d-e x)}+\frac {32 d^5 (e f+d g)^2}{e^2 (-d+e x)^2}\right ) \, dx\\ &=\frac {d^3 \left (49 e^2 f^2+160 d e f g+112 d^2 g^2\right ) x}{e^2}+\frac {d^2 \left (23 e^2 f^2+98 d e f g+80 d^2 g^2\right ) x^2}{2 e}+\frac {1}{3} d \left (7 e^2 f^2+46 d e f g+49 d^2 g^2\right ) x^3+\frac {1}{4} e \left (e^2 f^2+14 d e f g+23 d^2 g^2\right ) x^4+\frac {1}{5} e^2 g (2 e f+7 d g) x^5+\frac {1}{6} e^3 g^2 x^6+\frac {32 d^5 (e f+d g)^2}{e^3 (d-e x)}+\frac {16 d^4 (e f+d g) (5 e f+9 d g) \log (d-e x)}{e^3}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 226, normalized size = 1.04 \[ -\frac {32 d^5 (d g+e f)^2}{e^3 (e x-d)}+\frac {1}{4} e x^4 \left (23 d^2 g^2+14 d e f g+e^2 f^2\right )+\frac {1}{3} d x^3 \left (49 d^2 g^2+46 d e f g+7 e^2 f^2\right )+\frac {d^2 x^2 \left (80 d^2 g^2+98 d e f g+23 e^2 f^2\right )}{2 e}+\frac {16 d^4 \left (9 d^2 g^2+14 d e f g+5 e^2 f^2\right ) \log (d-e x)}{e^3}+\frac {d^3 x \left (112 d^2 g^2+160 d e f g+49 e^2 f^2\right )}{e^2}+\frac {1}{5} e^2 g x^5 (7 d g+2 e f)+\frac {1}{6} e^3 g^2 x^6 \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 328, normalized size = 1.50 \[ \frac {10 \, e^{7} g^{2} x^{7} - 1920 \, d^{5} e^{2} f^{2} - 3840 \, d^{6} e f g - 1920 \, d^{7} g^{2} + 2 \, {\left (12 \, e^{7} f g + 37 \, d e^{6} g^{2}\right )} x^{6} + 3 \, {\left (5 \, e^{7} f^{2} + 62 \, d e^{6} f g + 87 \, d^{2} e^{5} g^{2}\right )} x^{5} + 5 \, {\left (25 \, d e^{6} f^{2} + 142 \, d^{2} e^{5} f g + 127 \, d^{3} e^{4} g^{2}\right )} x^{4} + 10 \, {\left (55 \, d^{2} e^{5} f^{2} + 202 \, d^{3} e^{4} f g + 142 \, d^{4} e^{3} g^{2}\right )} x^{3} + 90 \, {\left (25 \, d^{3} e^{4} f^{2} + 74 \, d^{4} e^{3} f g + 48 \, d^{5} e^{2} g^{2}\right )} x^{2} - 60 \, {\left (49 \, d^{4} e^{3} f^{2} + 160 \, d^{5} e^{2} f g + 112 \, d^{6} e g^{2}\right )} x - 960 \, {\left (5 \, d^{5} e^{2} f^{2} + 14 \, d^{6} e f g + 9 \, d^{7} g^{2} - {\left (5 \, d^{4} e^{3} f^{2} + 14 \, d^{5} e^{2} f g + 9 \, d^{6} e g^{2}\right )} x\right )} \log \left (e x - d\right )}{60 \, {\left (e^{4} x - d e^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.18, size = 367, normalized size = 1.68 \[ 8 \, {\left (9 \, d^{6} g^{2} e^{7} + 14 \, d^{5} f g e^{8} + 5 \, d^{4} f^{2} e^{9}\right )} e^{\left (-10\right )} \log \left ({\left | x^{2} e^{2} - d^{2} \right |}\right ) + \frac {1}{60} \, {\left (10 \, g^{2} x^{6} e^{27} + 84 \, d g^{2} x^{5} e^{26} + 345 \, d^{2} g^{2} x^{4} e^{25} + 980 \, d^{3} g^{2} x^{3} e^{24} + 2400 \, d^{4} g^{2} x^{2} e^{23} + 6720 \, d^{5} g^{2} x e^{22} + 24 \, f g x^{5} e^{27} + 210 \, d f g x^{4} e^{26} + 920 \, d^{2} f g x^{3} e^{25} + 2940 \, d^{3} f g x^{2} e^{24} + 9600 \, d^{4} f g x e^{23} + 15 \, f^{2} x^{4} e^{27} + 140 \, d f^{2} x^{3} e^{26} + 690 \, d^{2} f^{2} x^{2} e^{25} + 2940 \, d^{3} f^{2} x e^{24}\right )} e^{\left (-24\right )} + \frac {8 \, {\left (9 \, d^{7} g^{2} e^{6} + 14 \, d^{6} f g e^{7} + 5 \, d^{5} f^{2} e^{8}\right )} e^{\left (-9\right )} \log \left (\frac {{\left | 2 \, x e^{2} - 2 \, {\left | d \right |} e \right |}}{{\left | 2 \, x e^{2} + 2 \, {\left | d \right |} e \right |}}\right )}{{\left | d \right |}} - \frac {32 \, {\left (d^{8} g^{2} e^{7} + 2 \, d^{7} f g e^{8} + d^{6} f^{2} e^{9} + {\left (d^{7} g^{2} e^{8} + 2 \, d^{6} f g e^{9} + d^{5} f^{2} e^{10}\right )} x\right )} e^{\left (-10\right )}}{x^{2} e^{2} - d^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 286, normalized size = 1.31 \[ \frac {e^{3} g^{2} x^{6}}{6}+\frac {7 d \,e^{2} g^{2} x^{5}}{5}+\frac {2 e^{3} f g \,x^{5}}{5}+\frac {23 d^{2} e \,g^{2} x^{4}}{4}+\frac {7 d \,e^{2} f g \,x^{4}}{2}+\frac {e^{3} f^{2} x^{4}}{4}+\frac {49 d^{3} g^{2} x^{3}}{3}+\frac {46 d^{2} e f g \,x^{3}}{3}+\frac {7 d \,e^{2} f^{2} x^{3}}{3}+\frac {40 d^{4} g^{2} x^{2}}{e}+49 d^{3} f g \,x^{2}+\frac {23 d^{2} e \,f^{2} x^{2}}{2}-\frac {32 d^{7} g^{2}}{\left (e x -d \right ) e^{3}}-\frac {64 d^{6} f g}{\left (e x -d \right ) e^{2}}+\frac {144 d^{6} g^{2} \ln \left (e x -d \right )}{e^{3}}-\frac {32 d^{5} f^{2}}{\left (e x -d \right ) e}+\frac {224 d^{5} f g \ln \left (e x -d \right )}{e^{2}}+\frac {112 d^{5} g^{2} x}{e^{2}}+\frac {80 d^{4} f^{2} \ln \left (e x -d \right )}{e}+\frac {160 d^{4} f g x}{e}+49 d^{3} f^{2} x \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 258, normalized size = 1.18 \[ -\frac {32 \, {\left (d^{5} e^{2} f^{2} + 2 \, d^{6} e f g + d^{7} g^{2}\right )}}{e^{4} x - d e^{3}} + \frac {10 \, e^{5} g^{2} x^{6} + 12 \, {\left (2 \, e^{5} f g + 7 \, d e^{4} g^{2}\right )} x^{5} + 15 \, {\left (e^{5} f^{2} + 14 \, d e^{4} f g + 23 \, d^{2} e^{3} g^{2}\right )} x^{4} + 20 \, {\left (7 \, d e^{4} f^{2} + 46 \, d^{2} e^{3} f g + 49 \, d^{3} e^{2} g^{2}\right )} x^{3} + 30 \, {\left (23 \, d^{2} e^{3} f^{2} + 98 \, d^{3} e^{2} f g + 80 \, d^{4} e g^{2}\right )} x^{2} + 60 \, {\left (49 \, d^{3} e^{2} f^{2} + 160 \, d^{4} e f g + 112 \, d^{5} g^{2}\right )} x}{60 \, e^{2}} + \frac {16 \, {\left (5 \, d^{4} e^{2} f^{2} + 14 \, d^{5} e f g + 9 \, d^{6} g^{2}\right )} \log \left (e x - d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.64, size = 1029, normalized size = 4.72 \[ x^5\,\left (\frac {e^2\,g\,\left (5\,d\,g+2\,e\,f\right )}{5}+\frac {2\,d\,e^2\,g^2}{5}\right )+x^3\,\left (\frac {5\,d\,\left (2\,d^2\,g^2+4\,d\,e\,f\,g+e^2\,f^2\right )}{3}+\frac {2\,d\,\left (\frac {10\,d^2\,e^3\,g^2+10\,d\,e^4\,f\,g+e^5\,f^2}{e^2}-d^2\,e\,g^2+\frac {2\,d\,\left (e^2\,g\,\left (5\,d\,g+2\,e\,f\right )+2\,d\,e^2\,g^2\right )}{e}\right )}{3\,e}-\frac {d^2\,\left (e^2\,g\,\left (5\,d\,g+2\,e\,f\right )+2\,d\,e^2\,g^2\right )}{3\,e^2}\right )+x^4\,\left (\frac {10\,d^2\,e^3\,g^2+10\,d\,e^4\,f\,g+e^5\,f^2}{4\,e^2}-\frac {d^2\,e\,g^2}{4}+\frac {d\,\left (e^2\,g\,\left (5\,d\,g+2\,e\,f\right )+2\,d\,e^2\,g^2\right )}{2\,e}\right )+x^2\,\left (\frac {5\,d^2\,\left (d^2\,g^2+4\,d\,e\,f\,g+2\,e^2\,f^2\right )}{2\,e}-\frac {d^2\,\left (\frac {10\,d^2\,e^3\,g^2+10\,d\,e^4\,f\,g+e^5\,f^2}{e^2}-d^2\,e\,g^2+\frac {2\,d\,\left (e^2\,g\,\left (5\,d\,g+2\,e\,f\right )+2\,d\,e^2\,g^2\right )}{e}\right )}{2\,e^2}+\frac {d\,\left (5\,d\,\left (2\,d^2\,g^2+4\,d\,e\,f\,g+e^2\,f^2\right )+\frac {2\,d\,\left (\frac {10\,d^2\,e^3\,g^2+10\,d\,e^4\,f\,g+e^5\,f^2}{e^2}-d^2\,e\,g^2+\frac {2\,d\,\left (e^2\,g\,\left (5\,d\,g+2\,e\,f\right )+2\,d\,e^2\,g^2\right )}{e}\right )}{e}-\frac {d^2\,\left (e^2\,g\,\left (5\,d\,g+2\,e\,f\right )+2\,d\,e^2\,g^2\right )}{e^2}\right )}{e}\right )+x\,\left (\frac {d^5\,g^2+10\,d^4\,e\,f\,g+10\,d^3\,e^2\,f^2}{e^2}-\frac {d^2\,\left (5\,d\,\left (2\,d^2\,g^2+4\,d\,e\,f\,g+e^2\,f^2\right )+\frac {2\,d\,\left (\frac {10\,d^2\,e^3\,g^2+10\,d\,e^4\,f\,g+e^5\,f^2}{e^2}-d^2\,e\,g^2+\frac {2\,d\,\left (e^2\,g\,\left (5\,d\,g+2\,e\,f\right )+2\,d\,e^2\,g^2\right )}{e}\right )}{e}-\frac {d^2\,\left (e^2\,g\,\left (5\,d\,g+2\,e\,f\right )+2\,d\,e^2\,g^2\right )}{e^2}\right )}{e^2}+\frac {2\,d\,\left (\frac {5\,d^2\,\left (d^2\,g^2+4\,d\,e\,f\,g+2\,e^2\,f^2\right )}{e}-\frac {d^2\,\left (\frac {10\,d^2\,e^3\,g^2+10\,d\,e^4\,f\,g+e^5\,f^2}{e^2}-d^2\,e\,g^2+\frac {2\,d\,\left (e^2\,g\,\left (5\,d\,g+2\,e\,f\right )+2\,d\,e^2\,g^2\right )}{e}\right )}{e^2}+\frac {2\,d\,\left (5\,d\,\left (2\,d^2\,g^2+4\,d\,e\,f\,g+e^2\,f^2\right )+\frac {2\,d\,\left (\frac {10\,d^2\,e^3\,g^2+10\,d\,e^4\,f\,g+e^5\,f^2}{e^2}-d^2\,e\,g^2+\frac {2\,d\,\left (e^2\,g\,\left (5\,d\,g+2\,e\,f\right )+2\,d\,e^2\,g^2\right )}{e}\right )}{e}-\frac {d^2\,\left (e^2\,g\,\left (5\,d\,g+2\,e\,f\right )+2\,d\,e^2\,g^2\right )}{e^2}\right )}{e}\right )}{e}\right )+\frac {\ln \left (e\,x-d\right )\,\left (144\,d^6\,g^2+224\,d^5\,e\,f\,g+80\,d^4\,e^2\,f^2\right )}{e^3}+\frac {32\,\left (d^7\,g^2+2\,d^6\,e\,f\,g+d^5\,e^2\,f^2\right )}{e\,\left (d\,e^2-e^3\,x\right )}+\frac {e^3\,g^2\,x^6}{6} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.20, size = 250, normalized size = 1.15 \[ \frac {16 d^{4} \left (d g + e f\right ) \left (9 d g + 5 e f\right ) \log {\left (- d + e x \right )}}{e^{3}} + \frac {e^{3} g^{2} x^{6}}{6} + x^{5} \left (\frac {7 d e^{2} g^{2}}{5} + \frac {2 e^{3} f g}{5}\right ) + x^{4} \left (\frac {23 d^{2} e g^{2}}{4} + \frac {7 d e^{2} f g}{2} + \frac {e^{3} f^{2}}{4}\right ) + x^{3} \left (\frac {49 d^{3} g^{2}}{3} + \frac {46 d^{2} e f g}{3} + \frac {7 d e^{2} f^{2}}{3}\right ) + x^{2} \left (\frac {40 d^{4} g^{2}}{e} + 49 d^{3} f g + \frac {23 d^{2} e f^{2}}{2}\right ) + x \left (\frac {112 d^{5} g^{2}}{e^{2}} + \frac {160 d^{4} f g}{e} + 49 d^{3} f^{2}\right ) + \frac {- 32 d^{7} g^{2} - 64 d^{6} e f g - 32 d^{5} e^{2} f^{2}}{- d e^{3} + e^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
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