3.2889 \(\int \frac{(c e+d e x)^2}{\left (a+b (c+d x)^3\right )^2} \, dx\)

Optimal. Leaf size=26 \[ -\frac{e^2}{3 b d \left (a+b (c+d x)^3\right )} \]

[Out]

-e^2/(3*b*d*(a + b*(c + d*x)^3))

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Rubi [A]  time = 0.0210926, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042 \[ -\frac{e^2}{3 b d \left (a+b (c+d x)^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-e^2/(3*b*d*(a + b*(c + d*x)^3))

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Rubi in Sympy [A]  time = 5.40842, size = 19, normalized size = 0.73 \[ - \frac{e^{2}}{3 b d \left (a + b \left (c + d x\right )^{3}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**2/(3*b*d*(a + b*(c + d*x)**3))

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Mathematica [A]  time = 0.029672, size = 26, normalized size = 1. \[ -\frac{e^2}{3 b d \left (a+b (c+d x)^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

-e^2/(3*b*d*(a + b*(c + d*x)^3))

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Maple [A]  time = 0., size = 47, normalized size = 1.8 \[ -{\frac{{e}^{2}}{3\,bd \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34842, size = 74, normalized size = 2.85 \[ -\frac{e^{2}}{3 \,{\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x +{\left (b^{2} c^{3} + a b\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^2/((d*x + c)^3*b + a)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.203604, size = 74, normalized size = 2.85 \[ -\frac{e^{2}}{3 \,{\left (b^{2} d^{4} x^{3} + 3 \, b^{2} c d^{3} x^{2} + 3 \, b^{2} c^{2} d^{2} x +{\left (b^{2} c^{3} + a b\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^2/((d*x + c)^3*b + a)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.28998, size = 60, normalized size = 2.31 \[ - \frac{e^{2}}{3 a b d + 3 b^{2} c^{3} d + 9 b^{2} c^{2} d^{2} x + 9 b^{2} c d^{3} x^{2} + 3 b^{2} d^{4} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e**2/(3*a*b*d + 3*b**2*c**3*d + 9*b**2*c**2*d**2*x + 9*b**2*c*d**3*x**2 + 3*b**
2*d**4*x**3)

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GIAC/XCAS [A]  time = 0.215955, size = 61, normalized size = 2.35 \[ -\frac{e^{2}}{3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*e*x + c*e)^2/((d*x + c)^3*b + a)^2,x, algorithm="giac")

[Out]

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