3.2891 \(\int \frac{1}{(c e+d e x) \left (a+b (c+d x)^3\right )^2} \, dx\)

Optimal. Leaf size=68 \[ -\frac{\log \left (a+b (c+d x)^3\right )}{3 a^2 d e}+\frac{\log (c+d x)}{a^2 d e}+\frac{1}{3 a d e \left (a+b (c+d x)^3\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.156223, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\log \left (a+b (c+d x)^3\right )}{3 a^2 d e}+\frac{\log (c+d x)}{a^2 d e}+\frac{1}{3 a d e \left (a+b (c+d x)^3\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((c*e + d*e*x)*(a + b*(c + d*x)^3)^2),x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 15.7981, size = 54, normalized size = 0.79 \[ \frac{1}{3 a d e \left (a + b \left (c + d x\right )^{3}\right )} - \frac{\log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 a^{2} d e} + \frac{\log{\left (\left (c + d x\right )^{3} \right )}}{3 a^{2} d e} \]

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)**2,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.0418403, size = 51, normalized size = 0.75 \[ \frac{\frac{a}{a+b (c+d x)^3}-\log \left (a+b (c+d x)^3\right )+3 \log (c+d x)}{3 a^2 d e} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*e + d*e*x)*(a + b*(c + d*x)^3)^2),x]

[Out]

(a/(a + b*(c + d*x)^3) + 3*Log[c + d*x] - Log[a + b*(c + d*x)^3])/(3*a^2*d*e)

_______________________________________________________________________________________

Maple [A]  time = 0.013, size = 109, normalized size = 1.6 \[{\frac{1}{3\,aed \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }}-{\frac{\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,e{a}^{2}d}}+{\frac{\ln \left ( dx+c \right ) }{e{a}^{2}d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*e*x+c*e)/(a+b*(d*x+c)^3)^2,x)

[Out]

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

_______________________________________________________________________________________

Maxima [A]  time = 1.38429, size = 154, normalized size = 2.26 \[ \frac{1}{3 \,{\left (a b d^{4} e x^{3} + 3 \, a b c d^{3} e x^{2} + 3 \, a b c^{2} d^{2} e x +{\left (a b c^{3} + a^{2}\right )} d e\right )}} - \frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{2} d e} + \frac{\log \left (d x + c\right )}{a^{2} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)^2*(d*e*x + c*e)),x, algorithm="maxima")

[Out]

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

_______________________________________________________________________________________

Fricas [A]  time = 0.215677, size = 234, normalized size = 3.44 \[ -\frac{{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (d x + c\right ) - a}{3 \,{\left (a^{2} b d^{4} e x^{3} + 3 \, a^{2} b c d^{3} e x^{2} + 3 \, a^{2} b c^{2} d^{2} e x +{\left (a^{2} b c^{3} + a^{3}\right )} d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)^2*(d*e*x + c*e)),x, algorithm="fricas")

[Out]

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

_______________________________________________________________________________________

Sympy [A]  time = 10.1886, size = 122, normalized size = 1.79 \[ \frac{1}{3 a^{2} d e + 3 a b c^{3} d e + 9 a b c^{2} d^{2} e x + 9 a b c d^{3} e x^{2} + 3 a b d^{4} e x^{3}} + \frac{\log{\left (\frac{c}{d} + x \right )}}{a^{2} d e} - \frac{\log{\left (\frac{3 c^{2} x}{d^{2}} + \frac{3 c x^{2}}{d} + x^{3} + \frac{a + b c^{3}}{b d^{3}} \right )}}{3 a^{2} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*e*x+c*e)/(a+b*(d*x+c)**3)**2,x)

[Out]

1/(3*a**2*d*e + 3*a*b*c**3*d*e + 9*a*b*c**2*d**2*e*x + 9*a*b*c*d**3*e*x**2 + 3*a
*b*d**4*e*x**3) + log(c/d + x)/(a**2*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**2*d*e)

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.220223, size = 144, normalized size = 2.12 \[ -\frac{e^{\left (-1\right )}{\rm ln}\left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{2} d} + \frac{e^{\left (-1\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{a^{2} d} + \frac{e^{\left (-1\right )}}{3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} a d} \]

Verification of antiderivative is not currently implemented for this CAS.

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