3.2854 \(\int \frac{(c+d x)^2}{a+b (c+d x)^3} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log \left (a+b (c+d x)^3\right )}{3 b d} \]

[Out]

Log[a + b*(c + d*x)^3]/(3*b*d)

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Rubi [A]  time = 0.0162174, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048 \[ \frac{\log \left (a+b (c+d x)^3\right )}{3 b d} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*(c + d*x)^3]/(3*b*d)

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Rubi in Sympy [A]  time = 3.89799, size = 15, normalized size = 0.68 \[ \frac{\log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*(c + d*x)**3)/(3*b*d)

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Mathematica [A]  time = 0.0117895, size = 22, normalized size = 1. \[ \frac{\log \left (a+b (c+d x)^3\right )}{3 b d} \]

Antiderivative was successfully verified.

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

[Out]

Log[a + b*(c + d*x)^3]/(3*b*d)

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Maple [B]  time = 0.003, size = 43, normalized size = 2. \[{\frac{\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,bd}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3/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 = 1.40982, size = 27, normalized size = 1.23 \[ \frac{\log \left ({\left (d x + c\right )}^{3} b + a\right )}{3 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*log((d*x + c)^3*b + a)/(b*d)

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Fricas [A]  time = 0.206996, size = 57, normalized size = 2.59 \[ \frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*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 = 1.59537, size = 42, normalized size = 1.91 \[ \frac{\log{\left (a + b c^{3} + 3 b c^{2} d x + 3 b c d^{2} x^{2} + b d^{3} x^{3} \right )}}{3 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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 = 0.219181, size = 28, normalized size = 1.27 \[ \frac{{\rm ln}\left ({\left |{\left (d x + c\right )}^{3} b + a \right |}\right )}{3 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*ln(abs((d*x + c)^3*b + a))/(b*d)