3.1823 \(\int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^2} \, dx\)

Optimal. Leaf size=26 \[ \left (a-\frac{c d^2}{e^2}\right ) \log (d+e x)+\frac{c d x}{e} \]

[Out]

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

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Rubi [A]  time = 0.0737887, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.061 \[ \left (a-\frac{c d^2}{e^2}\right ) \log (d+e x)+\frac{c d x}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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

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)/(e*x+d)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0172544, size = 30, normalized size = 1.15 \[ \frac{\left (a e^2-c d^2\right ) \log (d+e x)}{e^2}+\frac{c d x}{e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.004, size = 32, normalized size = 1.2 \[{\frac{cdx}{e}}+\ln \left ( ex+d \right ) a-{\frac{\ln \left ( ex+d \right ) c{d}^{2}}{{e}^{2}}} \]

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)/(e*x+d)^2,x)

[Out]

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

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Maxima [A]  time = 0.720564, size = 42, normalized size = 1.62 \[ \frac{c d x}{e} - \frac{{\left (c d^{2} - a e^{2}\right )} \log \left (e x + d\right )}{e^{2}} \]

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)/(e*x + d)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.204831, size = 41, normalized size = 1.58 \[ \frac{c d e x -{\left (c d^{2} - a e^{2}\right )} \log \left (e x + d\right )}{e^{2}} \]

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)/(e*x + d)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.26032, size = 26, normalized size = 1. \[ \frac{c d x}{e} + \frac{\left (a e^{2} - c d^{2}\right ) \log{\left (d + e x \right )}}{e^{2}} \]

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)/(e*x+d)**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.212219, size = 158, normalized size = 6.08 \[{\left (2 \, d e^{\left (-3\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) +{\left (x e + d\right )} e^{\left (-3\right )} - \frac{d^{2} e^{\left (-3\right )}}{x e + d}\right )} c d e -{\left (c d^{2} + a e^{2}\right )}{\left (e^{\left (-1\right )}{\rm ln}\left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - \frac{d e^{\left (-1\right )}}{x e + d}\right )} e^{\left (-1\right )} - \frac{a d}{x e + d} \]

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)/(e*x + d)^2,x, algorithm="giac")

[Out]

(2*d*e^(-3)*ln(abs(x*e + d)*e^(-1)/(x*e + d)^2) + (x*e + d)*e^(-3) - d^2*e^(-3)/
(x*e + d))*c*d*e - (c*d^2 + a*e^2)*(e^(-1)*ln(abs(x*e + d)*e^(-1)/(x*e + d)^2) -
d*e^(-1)/(x*e + d))*e^(-1) - a*d/(x*e + d)