3.1834 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2}{(d+e x)^4} \, dx\)

Optimal. Leaf size=63 \[ -\frac{\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac{2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3}+\frac{c^2 d^2 x}{e^2} \]

[Out]

(c^2*d^2*x)/e^2 - (c*d^2 - a*e^2)^2/(e^3*(d + e*x)) - (2*c*d*(c*d^2 - a*e^2)*Log
[d + e*x])/e^3

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Rubi [A]  time = 0.123023, antiderivative size = 63, 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 \[ -\frac{\left (c d^2-a e^2\right )^2}{e^3 (d+e x)}-\frac{2 c d \left (c d^2-a e^2\right ) \log (d+e x)}{e^3}+\frac{c^2 d^2 x}{e^2} \]

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)^4,x]

[Out]

(c^2*d^2*x)/e^2 - (c*d^2 - a*e^2)^2/(e^3*(d + e*x)) - (2*c*d*(c*d^2 - a*e^2)*Log
[d + e*x])/e^3

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{2 c d \left (a e^{2} - c d^{2}\right ) \log{\left (d + e x \right )}}{e^{3}} + \frac{d^{2} \int c^{2}\, dx}{e^{2}} - \frac{\left (a e^{2} - c d^{2}\right )^{2}}{e^{3} \left (d + e x\right )} \]

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)**2/(e*x+d)**4,x)

[Out]

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

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Mathematica [A]  time = 0.0797276, size = 59, normalized size = 0.94 \[ \frac{-\frac{\left (c d^2-a e^2\right )^2}{d+e x}+2 c d \left (a e^2-c d^2\right ) \log (d+e x)+c^2 d^2 e x}{e^3} \]

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)^4,x]

[Out]

(c^2*d^2*e*x - (c*d^2 - a*e^2)^2/(d + e*x) + 2*c*d*(-(c*d^2) + a*e^2)*Log[d + e*
x])/e^3

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Maple [A]  time = 0.01, size = 92, normalized size = 1.5 \[{\frac{{c}^{2}{d}^{2}x}{{e}^{2}}}+2\,{\frac{cd\ln \left ( ex+d \right ) a}{e}}-2\,{\frac{{c}^{2}{d}^{3}\ln \left ( ex+d \right ) }{{e}^{3}}}-{\frac{e{a}^{2}}{ex+d}}+2\,{\frac{ac{d}^{2}}{e \left ( ex+d \right ) }}-{\frac{{c}^{2}{d}^{4}}{{e}^{3} \left ( ex+d \right ) }} \]

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)^2/(e*x+d)^4,x)

[Out]

c^2*d^2*x/e^2+2*d/e*c*ln(e*x+d)*a-2*d^3/e^3*c^2*ln(e*x+d)-e/(e*x+d)*a^2+2/e/(e*x
+d)*a*c*d^2-1/e^3/(e*x+d)*c^2*d^4

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Maxima [A]  time = 0.719576, size = 107, normalized size = 1.7 \[ \frac{c^{2} d^{2} x}{e^{2}} - \frac{c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{e^{4} x + d e^{3}} - \frac{2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \log \left (e x + d\right )}{e^{3}} \]

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)^4,x, algorithm="maxima")

[Out]

c^2*d^2*x/e^2 - (c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)/(e^4*x + d*e^3) - 2*(c^2*d^3
- a*c*d*e^2)*log(e*x + d)/e^3

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Fricas [A]  time = 0.25359, size = 146, normalized size = 2.32 \[ \frac{c^{2} d^{2} e^{2} x^{2} + c^{2} d^{3} e x - c^{2} d^{4} + 2 \, a c d^{2} e^{2} - a^{2} e^{4} - 2 \,{\left (c^{2} d^{4} - a c d^{2} e^{2} +{\left (c^{2} d^{3} e - a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{e^{4} x + d e^{3}} \]

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)^4,x, algorithm="fricas")

[Out]

(c^2*d^2*e^2*x^2 + c^2*d^3*e*x - c^2*d^4 + 2*a*c*d^2*e^2 - a^2*e^4 - 2*(c^2*d^4
- a*c*d^2*e^2 + (c^2*d^3*e - a*c*d*e^3)*x)*log(e*x + d))/(e^4*x + d*e^3)

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Sympy [A]  time = 2.29685, size = 71, normalized size = 1.13 \[ \frac{c^{2} d^{2} x}{e^{2}} + \frac{2 c d \left (a e^{2} - c d^{2}\right ) \log{\left (d + e x \right )}}{e^{3}} - \frac{a^{2} e^{4} - 2 a c d^{2} e^{2} + c^{2} d^{4}}{d e^{3} + e^{4} x} \]

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)**2/(e*x+d)**4,x)

[Out]

c**2*d**2*x/e**2 + 2*c*d*(a*e**2 - c*d**2)*log(d + e*x)/e**3 - (a**2*e**4 - 2*a*
c*d**2*e**2 + c**2*d**4)/(d*e**3 + e**4*x)

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GIAC/XCAS [A]  time = 0.212514, size = 181, normalized size = 2.87 \[ c^{2} d^{2} x e^{\left (-2\right )} - 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} e^{\left (-3\right )}{\rm ln}\left ({\left | x e + d \right |}\right ) - \frac{{\left (c^{2} d^{6} - 2 \, a c d^{4} e^{2} + a^{2} d^{2} e^{4} +{\left (c^{2} d^{4} e^{2} - 2 \, a c d^{2} e^{4} + a^{2} e^{6}\right )} x^{2} + 2 \,{\left (c^{2} d^{5} e - 2 \, a c d^{3} e^{3} + a^{2} d e^{5}\right )} x\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} \]

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)^4,x, algorithm="giac")

[Out]

c^2*d^2*x*e^(-2) - 2*(c^2*d^3 - a*c*d*e^2)*e^(-3)*ln(abs(x*e + d)) - (c^2*d^6 -
2*a*c*d^4*e^2 + a^2*d^2*e^4 + (c^2*d^4*e^2 - 2*a*c*d^2*e^4 + a^2*e^6)*x^2 + 2*(c
^2*d^5*e - 2*a*c*d^3*e^3 + a^2*d*e^5)*x)*e^(-3)/(x*e + d)^3