3.2857 \(\int \frac{1}{(c+d x) \left (a+b (c+d x)^3\right )} \, dx\)

Optimal. Leaf size=36 \[ \frac{\log (c+d x)}{a d}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.083706, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238 \[ \frac{\log (c+d x)}{a d}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 10.4335, size = 29, normalized size = 0.81 \[ - \frac{\log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 a d} + \frac{\log{\left (\left (c + d x\right )^{3} \right )}}{3 a d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0140258, size = 36, normalized size = 1. \[ \frac{\log (c+d x)}{a d}-\frac{\log \left (a+b (c+d x)^3\right )}{3 a d} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.007, size = 57, normalized size = 1.6 \[ -{\frac{\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,ad}}+{\frac{\ln \left ( dx+c \right ) }{ad}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3/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/d

_______________________________________________________________________________________

Maxima [A]  time = 1.36648, size = 76, normalized size = 2.11 \[ -\frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a d} + \frac{\log \left (d x + c\right )}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

Fricas [A]  time = 0.205578, size = 69, normalized size = 1.92 \[ -\frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \, \log \left (d x + c\right )}{3 \, a d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

_______________________________________________________________________________________

Sympy [A]  time = 1.39657, size = 49, normalized size = 1.36 \[ \frac{\log{\left (\frac{c}{d} + x \right )}}{a d} - \frac{\log{\left (\frac{3 c^{2} x}{d^{2}} + \frac{3 c x^{2}}{d} + x^{3} + \frac{a + b c^{3}}{b d^{3}} \right )}}{3 a d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220093, size = 78, normalized size = 2.17 \[ -\frac{{\rm ln}\left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a d} + \frac{{\rm ln}\left ({\left | d x + c \right |}\right )}{a d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*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) + ln(abs
(d*x + c))/(a*d)