3.2881 \(\int \frac{(c e+d e x)^2}{a+b (c+d x)^3} \, dx\)

Optimal. Leaf size=25 \[ \frac{e^2 \log \left (a+b (c+d x)^3\right )}{3 b d} \]

[Out]

(e^2*Log[a + b*(c + d*x)^3])/(3*b*d)

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Rubi [A]  time = 0.0209557, antiderivative size = 25, 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 \[ \frac{e^2 \log \left (a+b (c+d x)^3\right )}{3 b d} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*Log[a + b*(c + d*x)^3])/(3*b*d)

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Rubi in Sympy [A]  time = 5.31493, size = 19, normalized size = 0.76 \[ \frac{e^{2} \log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 b d} \]

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),x)

[Out]

e**2*log(a + b*(c + d*x)**3)/(3*b*d)

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Mathematica [A]  time = 0.0138553, size = 25, normalized size = 1. \[ \frac{e^2 \log \left (a+b (c+d x)^3\right )}{3 b d} \]

Antiderivative was successfully verified.

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

[Out]

(e^2*Log[a + b*(c + d*x)^3])/(3*b*d)

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Maple [A]  time = 0.002, size = 46, normalized size = 1.8 \[{\frac{{e}^{2}\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,bd}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*e^2/b/d*ln(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 = 1.34596, size = 61, normalized size = 2.44 \[ \frac{e^{2} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.203188, size = 61, normalized size = 2.44 \[ \frac{e^{2} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.82105, size = 46, normalized size = 1.84 \[ \frac{e^{2} \log{\left (a + b c^{3} + 3 b c^{2} d x + 3 b c d^{2} x^{2} + b d^{3} x^{3} \right )}}{3 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.216519, size = 61, normalized size = 2.44 \[ \frac{e^{2}{\rm ln}\left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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