3.2883 \(\int \frac{1}{(c e+d e x) \left (a+b (c+d x)^3\right )} \, dx\)

Optimal. Leaf size=42 \[ \frac{\log (c+d x)}{a d e}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d e} \]

[Out]

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

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Rubi [A]  time = 0.0976713, antiderivative size = 42, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{\log (c+d x)}{a d e}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d e} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*e + d*e*x)*(a + b*(c + d*x)^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.6847, size = 32, normalized size = 0.76 \[ - \frac{\log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 a d e} + \frac{\log{\left (\left (c + d x\right )^{3} \right )}}{3 a d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*e*x+c*e)/(a+b*(d*x+c)**3),x)

[Out]

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

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Mathematica [A]  time = 0.0156859, size = 42, normalized size = 1. \[ \frac{\log (c+d x)}{a d e}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d e} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*e + d*e*x)*(a + b*(c + d*x)^3)),x]

[Out]

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

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Maple [A]  time = 0.006, size = 63, normalized size = 1.5 \[ -{\frac{\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,aed}}+{\frac{\ln \left ( dx+c \right ) }{aed}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*e*x+c*e)/(a+b*(d*x+c)^3),x)

[Out]

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

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Maxima [A]  time = 1.41228, size = 84, normalized size = 2. \[ -\frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a d e} + \frac{\log \left (d x + c\right )}{a d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)*(d*e*x + c*e)),x, algorithm="maxima")

[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)/(a*d*e) + log(d*x
+ c)/(a*d*e)

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Fricas [A]  time = 0.203951, size = 73, normalized size = 1.74 \[ -\frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \, \log \left (d x + c\right )}{3 \, a d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)*(d*e*x + c*e)),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) - 3*log(d*x + c))
/(a*d*e)

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Sympy [A]  time = 1.78082, size = 53, normalized size = 1.26 \[ \frac{\log{\left (\frac{c}{d} + x \right )}}{a d e} - \frac{\log{\left (\frac{3 c^{2} x}{d^{2}} + \frac{3 c x^{2}}{d} + x^{3} + \frac{a + b c^{3}}{b d^{3}} \right )}}{3 a d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*e*x+c*e)/(a+b*(d*x+c)**3),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.220951, size = 84, normalized size = 2. \[ -\frac{e^{\left (-1\right )}{\rm ln}\left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a d} + \frac{e^{\left (-1\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)*(d*e*x + c*e)),x, algorithm="giac")

[Out]

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