3.2907 \(\int \frac{(c+d x)^3}{a+b (c+d x)^4} \, dx\)

Optimal. Leaf size=22 \[ \frac{\log \left (a+b (c+d x)^4\right )}{4 b d} \]

[Out]

Log[a + b*(c + d*x)^4]/(4*b*d)

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Rubi [A]  time = 0.0200594, 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 \[ \frac{\log \left (a+b (c+d x)^4\right )}{4 b d} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^3/(a + b*(c + d*x)^4),x]

[Out]

Log[a + b*(c + d*x)^4]/(4*b*d)

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Rubi in Sympy [A]  time = 3.96708, size = 15, normalized size = 0.68 \[ \frac{\log{\left (a + b \left (c + d x\right )^{4} \right )}}{4 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**3/(a+b*(d*x+c)**4),x)

[Out]

log(a + b*(c + d*x)**4)/(4*b*d)

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Mathematica [A]  time = 0.0149451, size = 22, normalized size = 1. \[ \frac{\log \left (a+b (c+d x)^4\right )}{4 b d} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^3/(a + b*(c + d*x)^4),x]

[Out]

Log[a + b*(c + d*x)^4]/(4*b*d)

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Maple [B]  time = 0.002, size = 55, normalized size = 2.5 \[{\frac{\ln \left ( b{d}^{4}{x}^{4}+4\,bc{d}^{3}{x}^{3}+6\,b{c}^{2}{d}^{2}{x}^{2}+4\,b{c}^{3}dx+b{c}^{4}+a \right ) }{4\,bd}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^3/(a+b*(d*x+c)^4),x)

[Out]

1/4/b/d*ln(b*d^4*x^4+4*b*c*d^3*x^3+6*b*c^2*d^2*x^2+4*b*c^3*d*x+b*c^4+a)

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Maxima [A]  time = 1.37055, size = 27, normalized size = 1.23 \[ \frac{\log \left ({\left (d x + c\right )}^{4} b + a\right )}{4 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/((d*x + c)^4*b + a),x, algorithm="maxima")

[Out]

1/4*log((d*x + c)^4*b + a)/(b*d)

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Fricas [A]  time = 0.200964, size = 73, normalized size = 3.32 \[ \frac{\log \left (b d^{4} x^{4} + 4 \, b c d^{3} x^{3} + 6 \, b c^{2} d^{2} x^{2} + 4 \, b c^{3} d x + b c^{4} + a\right )}{4 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/((d*x + c)^4*b + a),x, algorithm="fricas")

[Out]

1/4*log(b*d^4*x^4 + 4*b*c*d^3*x^3 + 6*b*c^2*d^2*x^2 + 4*b*c^3*d*x + b*c^4 + a)/(
b*d)

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Sympy [A]  time = 1.84045, size = 56, normalized size = 2.55 \[ \frac{\log{\left (a + b c^{4} + 4 b c^{3} d x + 6 b c^{2} d^{2} x^{2} + 4 b c d^{3} x^{3} + b d^{4} x^{4} \right )}}{4 b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**3/(a+b*(d*x+c)**4),x)

[Out]

log(a + b*c**4 + 4*b*c**3*d*x + 6*b*c**2*d**2*x**2 + 4*b*c*d**3*x**3 + b*d**4*x*
*4)/(4*b*d)

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GIAC/XCAS [A]  time = 0.217113, size = 28, normalized size = 1.27 \[ \frac{{\rm ln}\left ({\left |{\left (d x + c\right )}^{4} b + a \right |}\right )}{4 \, b d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^3/((d*x + c)^4*b + a),x, algorithm="giac")

[Out]

1/4*ln(abs((d*x + c)^4*b + a))/(b*d)