Optimal. Leaf size=62 \[ -\frac{c d x \left (c d^2-a e^2\right )}{e^2}+\frac{\left (c d^2-a e^2\right )^2 \log (d+e x)}{e^3}+\frac{(a e+c d x)^2}{2 e} \]
[Out]
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Rubi [A] time = 0.0888126, antiderivative size = 62, 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 \[ -\frac{c d x \left (c d^2-a e^2\right )}{e^2}+\frac{\left (c d^2-a e^2\right )^2 \log (d+e x)}{e^3}+\frac{(a e+c d x)^2}{2 e} \]
Antiderivative was successfully verified.
[In] Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^3,x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ \frac{d \left (a e^{2} - c d^{2}\right ) \int c\, dx}{e^{2}} + \frac{\left (a e + c d x\right )^{2}}{2 e} + \frac{\left (a e^{2} - c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d)**3,x)
[Out]
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Mathematica [A] time = 0.0410935, size = 52, normalized size = 0.84 \[ \frac{2 \left (c d^2-a e^2\right )^2 \log (d+e x)+c d e x \left (4 a e^2+c d (e x-2 d)\right )}{2 e^3} \]
Antiderivative was successfully verified.
[In] Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2/(d + e*x)^3,x]
[Out]
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Maple [A] time = 0.004, size = 77, normalized size = 1.2 \[{\frac{{x}^{2}{c}^{2}{d}^{2}}{2\,e}}+2\,cdax-{\frac{{c}^{2}{d}^{3}x}{{e}^{2}}}+e\ln \left ( ex+d \right ){a}^{2}-2\,{\frac{\ln \left ( ex+d \right ) ac{d}^{2}}{e}}+{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{4}}{{e}^{3}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2/(e*x+d)^3,x)
[Out]
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Maxima [A] time = 0.717049, size = 97, normalized size = 1.56 \[ \frac{c^{2} d^{2} e x^{2} - 2 \,{\left (c^{2} d^{3} - 2 \, a c d e^{2}\right )} x}{2 \, e^{2}} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (e x + d\right )}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2/(e*x + d)^3,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.234921, size = 97, normalized size = 1.56 \[ \frac{c^{2} d^{2} e^{2} x^{2} - 2 \,{\left (c^{2} d^{3} e - 2 \, a c d e^{3}\right )} x + 2 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (e x + d\right )}{2 \, e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2/(e*x + d)^3,x, algorithm="fricas")
[Out]
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Sympy [A] time = 1.66198, size = 56, normalized size = 0.9 \[ \frac{c^{2} d^{2} x^{2}}{2 e} + \frac{x \left (2 a c d e^{2} - c^{2} d^{3}\right )}{e^{2}} + \frac{\left (a e^{2} - c d^{2}\right )^{2} \log{\left (d + e x \right )}}{e^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2/(e*x+d)**3,x)
[Out]
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GIAC/XCAS [A] time = 0.210526, size = 96, normalized size = 1.55 \[{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{1}{2} \,{\left (c^{2} d^{2} x^{2} e^{5} - 2 \, c^{2} d^{3} x e^{4} + 4 \, a c d x e^{6}\right )} e^{\left (-6\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2/(e*x + d)^3,x, algorithm="giac")
[Out]