Optimal. Leaf size=221 \[ \frac{e^6 (a e+c d x)^4}{4 c^7 d^7}-\frac{6 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^6}{2 c^7 d^7 (a e+c d x)^2}+\frac{15 e^2 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^7 d^7}+\frac{2 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^3}{c^7 d^7}+\frac{15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{2 c^7 d^7}+\frac{20 e^3 x \left (c d^2-a e^2\right )^3}{c^6 d^6} \]
[Out]
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Rubi [A] time = 0.603846, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.057 \[ \frac{e^6 (a e+c d x)^4}{4 c^7 d^7}-\frac{6 e \left (c d^2-a e^2\right )^5}{c^7 d^7 (a e+c d x)}-\frac{\left (c d^2-a e^2\right )^6}{2 c^7 d^7 (a e+c d x)^2}+\frac{15 e^2 \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^7 d^7}+\frac{2 e^5 \left (c d^2-a e^2\right ) (a e+c d x)^3}{c^7 d^7}+\frac{15 e^4 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{2 c^7 d^7}+\frac{20 e^3 x \left (c d^2-a e^2\right )^3}{c^6 d^6} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
[Out]
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Rubi in Sympy [A] time = 96.6979, size = 209, normalized size = 0.95 \[ - \frac{20 e^{3} x \left (a e^{2} - c d^{2}\right )^{3}}{c^{6} d^{6}} + \frac{e^{6} \left (a e + c d x\right )^{4}}{4 c^{7} d^{7}} - \frac{2 e^{5} \left (a e + c d x\right )^{3} \left (a e^{2} - c d^{2}\right )}{c^{7} d^{7}} + \frac{15 e^{4} \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2}}{2 c^{7} d^{7}} + \frac{15 e^{2} \left (a e^{2} - c d^{2}\right )^{4} \log{\left (a e + c d x \right )}}{c^{7} d^{7}} + \frac{6 e \left (a e^{2} - c d^{2}\right )^{5}}{c^{7} d^{7} \left (a e + c d x\right )} - \frac{\left (a e^{2} - c d^{2}\right )^{6}}{2 c^{7} d^{7} \left (a e + c d x\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**9/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
[Out]
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Mathematica [A] time = 0.255575, size = 337, normalized size = 1.52 \[ \frac{22 a^6 e^{12}-4 a^5 c d e^{10} (27 d+4 e x)+2 a^4 c^2 d^2 e^8 \left (105 d^2+12 d e x-34 e^2 x^2\right )-4 a^3 c^3 d^3 e^6 \left (50 d^3-15 d^2 e x-63 d e^2 x^2+5 e^3 x^3\right )+5 a^2 c^4 d^4 e^4 \left (18 d^4-32 d^3 e x-66 d^2 e^2 x^2+16 d e^3 x^3+e^4 x^4\right )-2 a c^5 d^5 e^2 \left (6 d^5-60 d^4 e x-80 d^3 e^2 x^2+60 d^2 e^3 x^3+10 d e^4 x^4+e^5 x^5\right )+60 e^2 \left (c d^2-a e^2\right )^4 (a e+c d x)^2 \log (a e+c d x)+c^6 d^6 \left (-2 d^6-24 d^5 e x+80 d^3 e^3 x^3+30 d^2 e^4 x^4+8 d e^5 x^5+e^6 x^6\right )}{4 c^7 d^7 (a e+c d x)^2} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^9/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]
[Out]
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Maple [B] time = 0.017, size = 544, normalized size = 2.5 \[ -60\,{\frac{{a}^{2}{e}^{5}}{d{c}^{4} \left ( cdx+ae \right ) }}+30\,{\frac{ad{e}^{3}}{{c}^{3} \left ( cdx+ae \right ) }}-60\,{\frac{{e}^{8}\ln \left ( cdx+ae \right ){a}^{3}}{{c}^{6}{d}^{5}}}-{\frac{{e}^{7}{x}^{3}a}{{c}^{4}{d}^{4}}}+3\,{\frac{{e}^{8}{x}^{2}{a}^{2}}{{c}^{5}{d}^{5}}}+10\,{\frac{{a}^{3}{e}^{6}}{d{c}^{4} \left ( cdx+ae \right ) ^{2}}}-{\frac{15\,{a}^{2}d{e}^{4}}{2\,{c}^{3} \left ( cdx+ae \right ) ^{2}}}+3\,{\frac{{d}^{3}{e}^{2}a}{{c}^{2} \left ( cdx+ae \right ) ^{2}}}+90\,{\frac{{e}^{6}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{5}{d}^{3}}}+15\,{\frac{{e}^{10}\ln \left ( cdx+ae \right ){a}^{4}}{{c}^{7}{d}^{7}}}-30\,{\frac{{e}^{9}{a}^{4}}{{c}^{6}{d}^{5} \left ( cdx+ae \right ) }}+60\,{\frac{{e}^{7}{a}^{3}}{{c}^{5}{d}^{3} \left ( cdx+ae \right ) }}-{\frac{{a}^{6}{e}^{12}}{2\,{c}^{7}{d}^{7} \left ( cdx+ae \right ) ^{2}}}+3\,{\frac{{a}^{5}{e}^{10}}{{c}^{6}{d}^{5} \left ( cdx+ae \right ) ^{2}}}-{\frac{15\,{a}^{4}{e}^{8}}{2\,{c}^{5}{d}^{3} \left ( cdx+ae \right ) ^{2}}}-10\,{\frac{{a}^{3}{e}^{9}x}{{c}^{6}{d}^{6}}}+{\frac{{e}^{6}{x}^{4}}{4\,{c}^{3}{d}^{3}}}+2\,{\frac{{e}^{5}{x}^{3}}{{c}^{3}{d}^{2}}}+{\frac{15\,{e}^{4}{x}^{2}}{2\,{c}^{3}d}}-6\,{\frac{{d}^{3}e}{{c}^{2} \left ( cdx+ae \right ) }}+15\,{\frac{d{e}^{2}\ln \left ( cdx+ae \right ) }{{c}^{3}}}+36\,{\frac{{a}^{2}{e}^{7}x}{{c}^{5}{d}^{4}}}-45\,{\frac{a{e}^{5}x}{{c}^{4}{d}^{2}}}+6\,{\frac{{a}^{5}{e}^{11}}{{c}^{7}{d}^{7} \left ( cdx+ae \right ) }}+20\,{\frac{{e}^{3}x}{{c}^{3}}}-{\frac{{d}^{5}}{2\,c \left ( cdx+ae \right ) ^{2}}}-9\,{\frac{{e}^{6}{x}^{2}a}{{c}^{4}{d}^{3}}}-60\,{\frac{{e}^{4}\ln \left ( cdx+ae \right ) a}{d{c}^{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^9/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)
[Out]
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Maxima [A] time = 0.740405, size = 551, normalized size = 2.49 \[ -\frac{c^{6} d^{12} + 6 \, a c^{5} d^{10} e^{2} - 45 \, a^{2} c^{4} d^{8} e^{4} + 100 \, a^{3} c^{3} d^{6} e^{6} - 105 \, a^{4} c^{2} d^{4} e^{8} + 54 \, a^{5} c d^{2} e^{10} - 11 \, a^{6} e^{12} + 12 \,{\left (c^{6} d^{11} e - 5 \, a c^{5} d^{9} e^{3} + 10 \, a^{2} c^{4} d^{7} e^{5} - 10 \, a^{3} c^{3} d^{5} e^{7} + 5 \, a^{4} c^{2} d^{3} e^{9} - a^{5} c d e^{11}\right )} x}{2 \,{\left (c^{9} d^{9} x^{2} + 2 \, a c^{8} d^{8} e x + a^{2} c^{7} d^{7} e^{2}\right )}} + \frac{c^{3} d^{3} e^{6} x^{4} + 4 \,{\left (2 \, c^{3} d^{4} e^{5} - a c^{2} d^{2} e^{7}\right )} x^{3} + 6 \,{\left (5 \, c^{3} d^{5} e^{4} - 6 \, a c^{2} d^{3} e^{6} + 2 \, a^{2} c d e^{8}\right )} x^{2} + 4 \,{\left (20 \, c^{3} d^{6} e^{3} - 45 \, a c^{2} d^{4} e^{5} + 36 \, a^{2} c d^{2} e^{7} - 10 \, a^{3} e^{9}\right )} x}{4 \, c^{6} d^{6}} + \frac{15 \,{\left (c^{4} d^{8} e^{2} - 4 \, a c^{3} d^{6} e^{4} + 6 \, a^{2} c^{2} d^{4} e^{6} - 4 \, a^{3} c d^{2} e^{8} + a^{4} e^{10}\right )} \log \left (c d x + a e\right )}{c^{7} d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^9/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.208076, size = 818, normalized size = 3.7 \[ \frac{c^{6} d^{6} e^{6} x^{6} - 2 \, c^{6} d^{12} - 12 \, a c^{5} d^{10} e^{2} + 90 \, a^{2} c^{4} d^{8} e^{4} - 200 \, a^{3} c^{3} d^{6} e^{6} + 210 \, a^{4} c^{2} d^{4} e^{8} - 108 \, a^{5} c d^{2} e^{10} + 22 \, a^{6} e^{12} + 2 \,{\left (4 \, c^{6} d^{7} e^{5} - a c^{5} d^{5} e^{7}\right )} x^{5} + 5 \,{\left (6 \, c^{6} d^{8} e^{4} - 4 \, a c^{5} d^{6} e^{6} + a^{2} c^{4} d^{4} e^{8}\right )} x^{4} + 20 \,{\left (4 \, c^{6} d^{9} e^{3} - 6 \, a c^{5} d^{7} e^{5} + 4 \, a^{2} c^{4} d^{5} e^{7} - a^{3} c^{3} d^{3} e^{9}\right )} x^{3} + 2 \,{\left (80 \, a c^{5} d^{8} e^{4} - 165 \, a^{2} c^{4} d^{6} e^{6} + 126 \, a^{3} c^{3} d^{4} e^{8} - 34 \, a^{4} c^{2} d^{2} e^{10}\right )} x^{2} - 4 \,{\left (6 \, c^{6} d^{11} e - 30 \, a c^{5} d^{9} e^{3} + 40 \, a^{2} c^{4} d^{7} e^{5} - 15 \, a^{3} c^{3} d^{5} e^{7} - 6 \, a^{4} c^{2} d^{3} e^{9} + 4 \, a^{5} c d e^{11}\right )} x + 60 \,{\left (a^{2} c^{4} d^{8} e^{4} - 4 \, a^{3} c^{3} d^{6} e^{6} + 6 \, a^{4} c^{2} d^{4} e^{8} - 4 \, a^{5} c d^{2} e^{10} + a^{6} e^{12} +{\left (c^{6} d^{10} e^{2} - 4 \, a c^{5} d^{8} e^{4} + 6 \, a^{2} c^{4} d^{6} e^{6} - 4 \, a^{3} c^{3} d^{4} e^{8} + a^{4} c^{2} d^{2} e^{10}\right )} x^{2} + 2 \,{\left (a c^{5} d^{9} e^{3} - 4 \, a^{2} c^{4} d^{7} e^{5} + 6 \, a^{3} c^{3} d^{5} e^{7} - 4 \, a^{4} c^{2} d^{3} e^{9} + a^{5} c d e^{11}\right )} x\right )} \log \left (c d x + a e\right )}{4 \,{\left (c^{9} d^{9} x^{2} + 2 \, a c^{8} d^{8} e x + a^{2} c^{7} d^{7} e^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^9/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 39.2669, size = 386, normalized size = 1.75 \[ \frac{11 a^{6} e^{12} - 54 a^{5} c d^{2} e^{10} + 105 a^{4} c^{2} d^{4} e^{8} - 100 a^{3} c^{3} d^{6} e^{6} + 45 a^{2} c^{4} d^{8} e^{4} - 6 a c^{5} d^{10} e^{2} - c^{6} d^{12} + x \left (12 a^{5} c d e^{11} - 60 a^{4} c^{2} d^{3} e^{9} + 120 a^{3} c^{3} d^{5} e^{7} - 120 a^{2} c^{4} d^{7} e^{5} + 60 a c^{5} d^{9} e^{3} - 12 c^{6} d^{11} e\right )}{2 a^{2} c^{7} d^{7} e^{2} + 4 a c^{8} d^{8} e x + 2 c^{9} d^{9} x^{2}} + \frac{e^{6} x^{4}}{4 c^{3} d^{3}} - \frac{x^{3} \left (a e^{7} - 2 c d^{2} e^{5}\right )}{c^{4} d^{4}} + \frac{x^{2} \left (6 a^{2} e^{8} - 18 a c d^{2} e^{6} + 15 c^{2} d^{4} e^{4}\right )}{2 c^{5} d^{5}} - \frac{x \left (10 a^{3} e^{9} - 36 a^{2} c d^{2} e^{7} + 45 a c^{2} d^{4} e^{5} - 20 c^{3} d^{6} e^{3}\right )}{c^{6} d^{6}} + \frac{15 e^{2} \left (a e^{2} - c d^{2}\right )^{4} \log{\left (a e + c d x \right )}}{c^{7} d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**9/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)
[Out]
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GIAC/XCAS [A] time = 10.9392, size = 1, normalized size = 0. \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^9/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")
[Out]