Optimal. Leaf size=100 \[ \frac{\left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^4 d^4}+\frac{e x \left (c d^2-a e^2\right )^2}{c^3 d^3}+\frac{(d+e x)^2 \left (c d^2-a e^2\right )}{2 c^2 d^2}+\frac{(d+e x)^3}{3 c d} \]
[Out]
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Rubi [A] time = 0.124234, antiderivative size = 100, 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 \[ \frac{\left (c d^2-a e^2\right )^3 \log (a e+c d x)}{c^4 d^4}+\frac{e x \left (c d^2-a e^2\right )^2}{c^3 d^3}+\frac{(d+e x)^2 \left (c d^2-a e^2\right )}{2 c^2 d^2}+\frac{(d+e x)^3}{3 c d} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{\left (d + e x\right )^{3}}{3 c d} - \frac{\left (d + e x\right )^{2} \left (a e^{2} - c d^{2}\right )}{2 c^{2} d^{2}} + \frac{\left (a e^{2} - c d^{2}\right )^{2} \int e\, dx}{c^{3} d^{3}} - \frac{\left (a e^{2} - c d^{2}\right )^{3} \log{\left (a e + c d x \right )}}{c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
[Out]
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Mathematica [A] time = 0.0651658, size = 91, normalized size = 0.91 \[ \frac{c d e x \left (6 a^2 e^4-3 a c d e^2 (6 d+e x)+c^2 d^2 \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+6 \left (c d^2-a e^2\right )^3 \log (a e+c d x)}{6 c^4 d^4} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^4/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]
[Out]
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Maple [A] time = 0.005, size = 157, normalized size = 1.6 \[{\frac{{e}^{3}{x}^{3}}{3\,cd}}-{\frac{{e}^{4}{x}^{2}a}{2\,{c}^{2}{d}^{2}}}+{\frac{3\,{e}^{2}{x}^{2}}{2\,c}}+{\frac{{a}^{2}{e}^{5}x}{{c}^{3}{d}^{3}}}-3\,{\frac{a{e}^{3}x}{{c}^{2}d}}+3\,{\frac{dex}{c}}-{\frac{\ln \left ( cdx+ae \right ){a}^{3}{e}^{6}}{{c}^{4}{d}^{4}}}+3\,{\frac{\ln \left ( cdx+ae \right ){a}^{2}{e}^{4}}{{c}^{3}{d}^{2}}}-3\,{\frac{\ln \left ( cdx+ae \right ) a{e}^{2}}{{c}^{2}}}+{\frac{{d}^{2}\ln \left ( cdx+ae \right ) }{c}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^4/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)
[Out]
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Maxima [A] time = 0.719797, size = 182, normalized size = 1.82 \[ \frac{2 \, c^{2} d^{2} e^{3} x^{3} + 3 \,{\left (3 \, c^{2} d^{3} e^{2} - a c d e^{4}\right )} x^{2} + 6 \,{\left (3 \, c^{2} d^{4} e - 3 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x}{6 \, c^{3} d^{3}} + \frac{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (c d x + a e\right )}{c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.218017, size = 185, normalized size = 1.85 \[ \frac{2 \, c^{3} d^{3} e^{3} x^{3} + 3 \,{\left (3 \, c^{3} d^{4} e^{2} - a c^{2} d^{2} e^{4}\right )} x^{2} + 6 \,{\left (3 \, c^{3} d^{5} e - 3 \, a c^{2} d^{3} e^{3} + a^{2} c d e^{5}\right )} x + 6 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} \log \left (c d x + a e\right )}{6 \, c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="fricas")
[Out]
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Sympy [A] time = 2.00613, size = 104, normalized size = 1.04 \[ \frac{e^{3} x^{3}}{3 c d} - \frac{x^{2} \left (a e^{4} - 3 c d^{2} e^{2}\right )}{2 c^{2} d^{2}} + \frac{x \left (a^{2} e^{5} - 3 a c d^{2} e^{3} + 3 c^{2} d^{4} e\right )}{c^{3} d^{3}} - \frac{\left (a e^{2} - c d^{2}\right )^{3} \log{\left (a e + c d x \right )}}{c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**4/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)
[Out]
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GIAC/XCAS [A] time = 0.221594, size = 374, normalized size = 3.74 \[ \frac{{\left (2 \, c^{2} d^{2} x^{3} e^{6} + 9 \, c^{2} d^{3} x^{2} e^{5} + 18 \, c^{2} d^{4} x e^{4} - 3 \, a c d x^{2} e^{7} - 18 \, a c d^{2} x e^{6} + 6 \, a^{2} x e^{8}\right )} e^{\left (-3\right )}}{6 \, c^{3} d^{3}} + \frac{{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )}{\rm ln}\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{4} d^{4}} + \frac{{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c^{4} d^{4}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^4/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")
[Out]