Optimal. Leaf size=184 \[ \frac{e^5 (a e+c d x)^4}{4 c^6 d^6}-\frac{\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac{5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6}+\frac{5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac{5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac{10 e^2 x \left (c d^2-a e^2\right )^3}{c^5 d^5} \]
[Out]
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Rubi [A] time = 0.43705, antiderivative size = 184, 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 \[ \frac{e^5 (a e+c d x)^4}{4 c^6 d^6}-\frac{\left (c d^2-a e^2\right )^5}{c^6 d^6 (a e+c d x)}+\frac{5 e \left (c d^2-a e^2\right )^4 \log (a e+c d x)}{c^6 d^6}+\frac{5 e^4 \left (c d^2-a e^2\right ) (a e+c d x)^3}{3 c^6 d^6}+\frac{5 e^3 \left (c d^2-a e^2\right )^2 (a e+c d x)^2}{c^6 d^6}+\frac{10 e^2 x \left (c d^2-a e^2\right )^3}{c^5 d^5} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 77.3356, size = 173, normalized size = 0.94 \[ - \frac{10 e^{2} x \left (a e^{2} - c d^{2}\right )^{3}}{c^{5} d^{5}} + \frac{e^{5} \left (a e + c d x\right )^{4}}{4 c^{6} d^{6}} - \frac{5 e^{4} \left (a e + c d x\right )^{3} \left (a e^{2} - c d^{2}\right )}{3 c^{6} d^{6}} + \frac{5 e^{3} \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )^{2}}{c^{6} d^{6}} + \frac{5 e \left (a e^{2} - c d^{2}\right )^{4} \log{\left (a e + c d x \right )}}{c^{6} d^{6}} + \frac{\left (a e^{2} - c d^{2}\right )^{5}}{c^{6} d^{6} \left (a e + c d x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**7/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
[Out]
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Mathematica [A] time = 0.166755, size = 263, normalized size = 1.43 \[ \frac{12 a^5 e^{10}-12 a^4 c d e^8 (5 d+4 e x)+30 a^3 c^2 d^2 e^6 \left (4 d^2+6 d e x-e^2 x^2\right )-10 a^2 c^3 d^3 e^4 \left (12 d^3+24 d^2 e x-12 d e^2 x^2-e^3 x^3\right )+5 a c^4 d^4 e^2 \left (12 d^4+24 d^3 e x-36 d^2 e^2 x^2-8 d e^3 x^3-e^4 x^4\right )+60 e \left (c d^2-a e^2\right )^4 (a e+c d x) \log (a e+c d x)+c^5 d^5 \left (-12 d^5+120 d^3 e^2 x^2+60 d^2 e^3 x^3+20 d e^4 x^4+3 e^5 x^5\right )}{12 c^6 d^6 (a e+c d x)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^7/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
[Out]
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Maple [B] time = 0.014, size = 378, normalized size = 2.1 \[{\frac{{e}^{5}{x}^{4}}{4\,{c}^{2}{d}^{2}}}-{\frac{2\,{e}^{6}{x}^{3}a}{3\,{c}^{3}{d}^{3}}}+{\frac{5\,{e}^{4}{x}^{3}}{3\,{c}^{2}d}}+{\frac{3\,{e}^{7}{x}^{2}{a}^{2}}{2\,{c}^{4}{d}^{4}}}-5\,{\frac{{e}^{5}{x}^{2}a}{{c}^{3}{d}^{2}}}+5\,{\frac{{e}^{3}{x}^{2}}{{c}^{2}}}-4\,{\frac{{a}^{3}{e}^{8}x}{{c}^{5}{d}^{5}}}+15\,{\frac{{a}^{2}{e}^{6}x}{{c}^{4}{d}^{3}}}-20\,{\frac{a{e}^{4}x}{{c}^{3}d}}+10\,{\frac{d{e}^{2}x}{{c}^{2}}}+{\frac{{a}^{5}{e}^{10}}{{c}^{6}{d}^{6} \left ( cdx+ae \right ) }}-5\,{\frac{{a}^{4}{e}^{8}}{{c}^{5}{d}^{4} \left ( cdx+ae \right ) }}+10\,{\frac{{a}^{3}{e}^{6}}{{c}^{4}{d}^{2} \left ( cdx+ae \right ) }}-10\,{\frac{{a}^{2}{e}^{4}}{{c}^{3} \left ( cdx+ae \right ) }}+5\,{\frac{a{d}^{2}{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) }}-{\frac{{d}^{4}}{c \left ( cdx+ae \right ) }}+5\,{\frac{{e}^{9}\ln \left ( cdx+ae \right ){a}^{4}}{{c}^{6}{d}^{6}}}-20\,{\frac{{e}^{7}\ln \left ( cdx+ae \right ){a}^{3}}{{c}^{5}{d}^{4}}}+30\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{4}{d}^{2}}}-20\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) a}{{c}^{3}}}+5\,{\frac{{d}^{2}e\ln \left ( cdx+ae \right ) }{{c}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^7/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)
[Out]
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Maxima [A] time = 0.734513, size = 405, normalized size = 2.2 \[ -\frac{c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}}{c^{7} d^{7} x + a c^{6} d^{6} e} + \frac{3 \, c^{3} d^{3} e^{5} x^{4} + 4 \,{\left (5 \, c^{3} d^{4} e^{4} - 2 \, a c^{2} d^{2} e^{6}\right )} x^{3} + 6 \,{\left (10 \, c^{3} d^{5} e^{3} - 10 \, a c^{2} d^{3} e^{5} + 3 \, a^{2} c d e^{7}\right )} x^{2} + 12 \,{\left (10 \, c^{3} d^{6} e^{2} - 20 \, a c^{2} d^{4} e^{4} + 15 \, a^{2} c d^{2} e^{6} - 4 \, a^{3} e^{8}\right )} x}{12 \, c^{5} d^{5}} + \frac{5 \,{\left (c^{4} d^{8} e - 4 \, a c^{3} d^{6} e^{3} + 6 \, a^{2} c^{2} d^{4} e^{5} - 4 \, a^{3} c d^{2} e^{7} + a^{4} e^{9}\right )} \log \left (c d x + a e\right )}{c^{6} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^7/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.205255, size = 563, normalized size = 3.06 \[ \frac{3 \, c^{5} d^{5} e^{5} x^{5} - 12 \, c^{5} d^{10} + 60 \, a c^{4} d^{8} e^{2} - 120 \, a^{2} c^{3} d^{6} e^{4} + 120 \, a^{3} c^{2} d^{4} e^{6} - 60 \, a^{4} c d^{2} e^{8} + 12 \, a^{5} e^{10} + 5 \,{\left (4 \, c^{5} d^{6} e^{4} - a c^{4} d^{4} e^{6}\right )} x^{4} + 10 \,{\left (6 \, c^{5} d^{7} e^{3} - 4 \, a c^{4} d^{5} e^{5} + a^{2} c^{3} d^{3} e^{7}\right )} x^{3} + 30 \,{\left (4 \, c^{5} d^{8} e^{2} - 6 \, a c^{4} d^{6} e^{4} + 4 \, a^{2} c^{3} d^{4} e^{6} - a^{3} c^{2} d^{2} e^{8}\right )} x^{2} + 12 \,{\left (10 \, a c^{4} d^{7} e^{3} - 20 \, a^{2} c^{3} d^{5} e^{5} + 15 \, a^{3} c^{2} d^{3} e^{7} - 4 \, a^{4} c d e^{9}\right )} x + 60 \,{\left (a c^{4} d^{8} e^{2} - 4 \, a^{2} c^{3} d^{6} e^{4} + 6 \, a^{3} c^{2} d^{4} e^{6} - 4 \, a^{4} c d^{2} e^{8} + a^{5} e^{10} +{\left (c^{5} d^{9} e - 4 \, a c^{4} d^{7} e^{3} + 6 \, a^{2} c^{3} d^{5} e^{5} - 4 \, a^{3} c^{2} d^{3} e^{7} + a^{4} c d e^{9}\right )} x\right )} \log \left (c d x + a e\right )}{12 \,{\left (c^{7} d^{7} x + a c^{6} d^{6} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^7/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="fricas")
[Out]
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Sympy [A] time = 5.33305, size = 265, normalized size = 1.44 \[ \frac{a^{5} e^{10} - 5 a^{4} c d^{2} e^{8} + 10 a^{3} c^{2} d^{4} e^{6} - 10 a^{2} c^{3} d^{6} e^{4} + 5 a c^{4} d^{8} e^{2} - c^{5} d^{10}}{a c^{6} d^{6} e + c^{7} d^{7} x} + \frac{e^{5} x^{4}}{4 c^{2} d^{2}} - \frac{x^{3} \left (2 a e^{6} - 5 c d^{2} e^{4}\right )}{3 c^{3} d^{3}} + \frac{x^{2} \left (3 a^{2} e^{7} - 10 a c d^{2} e^{5} + 10 c^{2} d^{4} e^{3}\right )}{2 c^{4} d^{4}} - \frac{x \left (4 a^{3} e^{8} - 15 a^{2} c d^{2} e^{6} + 20 a c^{2} d^{4} e^{4} - 10 c^{3} d^{6} e^{2}\right )}{c^{5} d^{5}} + \frac{5 e \left (a e^{2} - c d^{2}\right )^{4} \log{\left (a e + c d x \right )}}{c^{6} d^{6}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**7/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.331119, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^7/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")
[Out]