3.1857 \(\int \frac{(d+e x)^2}{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=38 \[ \frac{\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}+\frac{e x}{c d} \]

[Out]

(e*x)/(c*d) + ((c*d^2 - a*e^2)*Log[a*e + c*d*x])/(c^2*d^2)

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Rubi [A]  time = 0.0804165, antiderivative size = 38, 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 \[ \frac{\left (c d^2-a e^2\right ) \log (a e+c d x)}{c^2 d^2}+\frac{e x}{c d} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(e*x)/(c*d) + ((c*d^2 - a*e^2)*Log[a*e + c*d*x])/(c^2*d^2)

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{\int e\, dx}{c d} - \frac{\left (a e^{2} - c d^{2}\right ) \log{\left (a e + c d x \right )}}{c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

Integral(e, x)/(c*d) - (a*e**2 - c*d**2)*log(a*e + c*d*x)/(c**2*d**2)

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Mathematica [A]  time = 0.0180704, size = 35, normalized size = 0.92 \[ \frac{\left (c d^2-a e^2\right ) \log (a e+c d x)+c d e x}{c^2 d^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2),x]

[Out]

(c*d*e*x + (c*d^2 - a*e^2)*Log[a*e + c*d*x])/(c^2*d^2)

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Maple [A]  time = 0.003, size = 45, normalized size = 1.2 \[{\frac{ex}{cd}}-{\frac{\ln \left ( cdx+ae \right ) a{e}^{2}}{{c}^{2}{d}^{2}}}+{\frac{\ln \left ( cdx+ae \right ) }{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^2/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

e*x/c/d-1/c^2/d^2*ln(c*d*x+a*e)*a*e^2+1/c*ln(c*d*x+a*e)

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Maxima [A]  time = 0.718649, size = 51, normalized size = 1.34 \[ \frac{e x}{c d} + \frac{{\left (c d^{2} - a e^{2}\right )} \log \left (c d x + a e\right )}{c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")

[Out]

e*x/(c*d) + (c*d^2 - a*e^2)*log(c*d*x + a*e)/(c^2*d^2)

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Fricas [A]  time = 0.212928, size = 47, normalized size = 1.24 \[ \frac{c d e x +{\left (c d^{2} - a e^{2}\right )} \log \left (c d x + a e\right )}{c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="fricas")

[Out]

(c*d*e*x + (c*d^2 - a*e^2)*log(c*d*x + a*e))/(c^2*d^2)

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Sympy [A]  time = 1.3361, size = 32, normalized size = 0.84 \[ \frac{e x}{c d} - \frac{\left (a e^{2} - c d^{2}\right ) \log{\left (a e + c d x \right )}}{c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x+d)**2/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

e*x/(c*d) - (a*e**2 - c*d**2)*log(a*e + c*d*x)/(c**2*d**2)

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GIAC/XCAS [A]  time = 0.219023, size = 215, normalized size = 5.66 \[ \frac{x e}{c d} + \frac{{\left (c d^{2} - a e^{2}\right )}{\rm ln}\left (c d x^{2} e + c d^{2} x + a x e^{2} + a d e\right )}{2 \, c^{2} d^{2}} + \frac{{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac{2 \, c d x e + c d^{2} + a e^{2}}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}}}\right )}{\sqrt{-c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4}} c^{2} d^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^2/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")

[Out]

x*e/(c*d) + 1/2*(c*d^2 - a*e^2)*ln(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)/(c^2*d
^2) + (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*arctan((2*c*d*x*e + c*d^2 + a*e^2)/sqr
t(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4))/(sqrt(-c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4)
*c^2*d^2)