3.1824 \(\int \frac{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}{(d+e x)^3} \, dx\)

Optimal. Leaf size=33 \[ \frac{c d \log (d+e x)}{e^2}-\frac{a-\frac{c d^2}{e^2}}{d+e x} \]

[Out]

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

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Rubi [A]  time = 0.072832, antiderivative size = 33, 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 \[ \frac{c d \log (d+e x)}{e^2}-\frac{a-\frac{c d^2}{e^2}}{d+e x} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 15.7369, size = 31, normalized size = 0.94 \[ \frac{c d \log{\left (d + e x \right )}}{e^{2}} - \frac{a e^{2} - c d^{2}}{e^{2} \left (d + e x\right )} \]

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0244201, size = 36, normalized size = 1.09 \[ \frac{c d^2-a e^2}{e^2 (d+e x)}+\frac{c d \log (d+e x)}{e^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)/(d + e*x)^3,x]

[Out]

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

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Maple [A]  time = 0.008, size = 39, normalized size = 1.2 \[{\frac{cd\ln \left ( ex+d \right ) }{{e}^{2}}}-{\frac{a}{ex+d}}+{\frac{c{d}^{2}}{{e}^{2} \left ( ex+d \right ) }} \]

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)^3,x)

[Out]

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

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Maxima [A]  time = 0.725454, size = 53, normalized size = 1.61 \[ \frac{c d \log \left (e x + d\right )}{e^{2}} + \frac{c d^{2} - a e^{2}}{e^{3} x + d e^{2}} \]

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)^3,x, algorithm="maxima")

[Out]

c*d*log(e*x + d)/e^2 + (c*d^2 - a*e^2)/(e^3*x + d*e^2)

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Fricas [A]  time = 0.206861, size = 59, normalized size = 1.79 \[ \frac{c d^{2} - a e^{2} +{\left (c d e x + c d^{2}\right )} \log \left (e x + d\right )}{e^{3} x + d e^{2}} \]

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)^3,x, algorithm="fricas")

[Out]

(c*d^2 - a*e^2 + (c*d*e*x + c*d^2)*log(e*x + d))/(e^3*x + d*e^2)

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Sympy [A]  time = 1.51148, size = 32, normalized size = 0.97 \[ \frac{c d \log{\left (d + e x \right )}}{e^{2}} - \frac{a e^{2} - c d^{2}}{d e^{2} + e^{3} x} \]

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)**3,x)

[Out]

c*d*log(d + e*x)/e**2 - (a*e**2 - c*d**2)/(d*e**2 + e**3*x)

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GIAC/XCAS [A]  time = 0.211926, size = 70, normalized size = 2.12 \[ c d e^{\left (-2\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) + \frac{{\left (c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} \]

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)^3,x, algorithm="giac")

[Out]

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