Optimal. Leaf size=145 \[ \frac{e^2 x \left (3 a^2 e^4-8 a c d^2 e^2+6 c^2 d^4\right )}{c^4 d^4}-\frac{\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac{4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5}+\frac{e^3 x^2 \left (2 c d^2-a e^2\right )}{c^3 d^3}+\frac{e^4 x^3}{3 c^2 d^2} \]
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Rubi [A] time = 0.336256, antiderivative size = 145, 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 \[ \frac{e^2 x \left (3 a^2 e^4-8 a c d^2 e^2+6 c^2 d^4\right )}{c^4 d^4}-\frac{\left (c d^2-a e^2\right )^4}{c^5 d^5 (a e+c d x)}+\frac{4 e \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^5 d^5}+\frac{e^3 x^2 \left (2 c d^2-a e^2\right )}{c^3 d^3}+\frac{e^4 x^3}{3 c^2 d^2} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{e^{4} x^{3}}{3 c^{2} d^{2}} - \frac{2 e^{3} \left (a e^{2} - 2 c d^{2}\right ) \int x\, dx}{c^{3} d^{3}} + \frac{6 e^{2} x \left (\frac{a e^{2} \left (3 a e^{2} - 8 c d^{2}\right )}{6} + c^{2} d^{4}\right )}{c^{4} d^{4}} - \frac{4 e \left (a e^{2} - c d^{2}\right )^{3} \log{\left (a e + c d x \right )}}{c^{5} d^{5}} - \frac{\left (a e^{2} - c d^{2}\right )^{4}}{c^{5} d^{5} \left (a e + c d x\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
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Mathematica [A] time = 0.142815, size = 196, normalized size = 1.35 \[ \frac{-3 a^4 e^8+3 a^3 c d e^6 (4 d+3 e x)-6 a^2 c^2 d^2 e^4 \left (3 d^2+4 d e x-e^2 x^2\right )+2 a c^3 d^3 e^2 \left (6 d^3+9 d^2 e x-9 d e^2 x^2-e^3 x^3\right )-12 e \left (a e^2-c d^2\right )^3 (a e+c d x) \log (a e+c d x)+c^4 d^4 \left (-3 d^4+18 d^2 e^2 x^2+6 d e^3 x^3+e^4 x^4\right )}{3 c^5 d^5 (a e+c d x)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^6/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]
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Maple [A] time = 0.014, size = 275, normalized size = 1.9 \[{\frac{{e}^{4}{x}^{3}}{3\,{c}^{2}{d}^{2}}}-{\frac{{e}^{5}{x}^{2}a}{{c}^{3}{d}^{3}}}+2\,{\frac{{e}^{3}{x}^{2}}{{c}^{2}d}}+3\,{\frac{{a}^{2}{e}^{6}x}{{c}^{4}{d}^{4}}}-8\,{\frac{a{e}^{4}x}{{c}^{3}{d}^{2}}}+6\,{\frac{{e}^{2}x}{{c}^{2}}}-{\frac{{a}^{4}{e}^{8}}{{c}^{5}{d}^{5} \left ( cdx+ae \right ) }}+4\,{\frac{{a}^{3}{e}^{6}}{{c}^{4}{d}^{3} \left ( cdx+ae \right ) }}-6\,{\frac{{a}^{2}{e}^{4}}{{c}^{3}d \left ( cdx+ae \right ) }}+4\,{\frac{ad{e}^{2}}{{c}^{2} \left ( cdx+ae \right ) }}-{\frac{{d}^{3}}{c \left ( cdx+ae \right ) }}-4\,{\frac{{e}^{7}\ln \left ( cdx+ae \right ){a}^{3}}{{c}^{5}{d}^{5}}}+12\,{\frac{{e}^{5}\ln \left ( cdx+ae \right ){a}^{2}}{{c}^{4}{d}^{3}}}-12\,{\frac{{e}^{3}\ln \left ( cdx+ae \right ) a}{{c}^{3}d}}+4\,{\frac{de\ln \left ( cdx+ae \right ) }{{c}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^6/(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 = 0.722421, size = 289, normalized size = 1.99 \[ -\frac{c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{c^{6} d^{6} x + a c^{5} d^{5} e} + \frac{c^{2} d^{2} e^{4} x^{3} + 3 \,{\left (2 \, c^{2} d^{3} e^{3} - a c d e^{5}\right )} x^{2} + 3 \,{\left (6 \, c^{2} d^{4} e^{2} - 8 \, a c d^{2} e^{4} + 3 \, a^{2} e^{6}\right )} x}{3 \, c^{4} d^{4}} + \frac{4 \,{\left (c^{3} d^{6} e - 3 \, a c^{2} d^{4} e^{3} + 3 \, a^{2} c d^{2} e^{5} - a^{3} e^{7}\right )} \log \left (c d x + a e\right )}{c^{5} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^6/(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 = 0.208212, size = 412, normalized size = 2.84 \[ \frac{c^{4} d^{4} e^{4} x^{4} - 3 \, c^{4} d^{8} + 12 \, a c^{3} d^{6} e^{2} - 18 \, a^{2} c^{2} d^{4} e^{4} + 12 \, a^{3} c d^{2} e^{6} - 3 \, a^{4} e^{8} + 2 \,{\left (3 \, c^{4} d^{5} e^{3} - a c^{3} d^{3} e^{5}\right )} x^{3} + 6 \,{\left (3 \, c^{4} d^{6} e^{2} - 3 \, a c^{3} d^{4} e^{4} + a^{2} c^{2} d^{2} e^{6}\right )} x^{2} + 3 \,{\left (6 \, a c^{3} d^{5} e^{3} - 8 \, a^{2} c^{2} d^{3} e^{5} + 3 \, a^{3} c d e^{7}\right )} x + 12 \,{\left (a c^{3} d^{6} e^{2} - 3 \, a^{2} c^{2} d^{4} e^{4} + 3 \, a^{3} c d^{2} e^{6} - a^{4} e^{8} +{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x\right )} \log \left (c d x + a e\right )}{3 \,{\left (c^{6} d^{6} x + a c^{5} d^{5} e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="fricas")
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Sympy [A] time = 4.2649, size = 187, normalized size = 1.29 \[ - \frac{a^{4} e^{8} - 4 a^{3} c d^{2} e^{6} + 6 a^{2} c^{2} d^{4} e^{4} - 4 a c^{3} d^{6} e^{2} + c^{4} d^{8}}{a c^{5} d^{5} e + c^{6} d^{6} x} + \frac{e^{4} x^{3}}{3 c^{2} d^{2}} - \frac{x^{2} \left (a e^{5} - 2 c d^{2} e^{3}\right )}{c^{3} d^{3}} + \frac{x \left (3 a^{2} e^{6} - 8 a c d^{2} e^{4} + 6 c^{2} d^{4} e^{2}\right )}{c^{4} d^{4}} - \frac{4 e \left (a e^{2} - c d^{2}\right )^{3} \log{\left (a e + c d x \right )}}{c^{5} d^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**6/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)
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GIAC/XCAS [A] time = 0.362545, size = 1, normalized size = 0.01 \[ \mathit{Done} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^6/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")
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