3.2866 \(\int \frac{1}{(c+d x) \left (a+b (c+d x)^3\right )^2} \, dx\)

Optimal. Leaf size=59 \[ -\frac{\log \left (a+b (c+d x)^3\right )}{3 a^2 d}+\frac{\log (c+d x)}{a^2 d}+\frac{1}{3 a d \left (a+b (c+d x)^3\right )} \]

[Out]

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

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Rubi [A]  time = 0.137501, antiderivative size = 59, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ -\frac{\log \left (a+b (c+d x)^3\right )}{3 a^2 d}+\frac{\log (c+d x)}{a^2 d}+\frac{1}{3 a d \left (a+b (c+d x)^3\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 14.1872, size = 49, normalized size = 0.83 \[ \frac{1}{3 a d \left (a + b \left (c + d x\right )^{3}\right )} - \frac{\log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 a^{2} d} + \frac{\log{\left (\left (c + d x\right )^{3} \right )}}{3 a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0395537, size = 48, normalized size = 0.81 \[ \frac{\frac{a}{a+b (c+d x)^3}-\log \left (a+b (c+d x)^3\right )+3 \log (c+d x)}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c + d*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)

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Maple [A]  time = 0.023, size = 100, normalized size = 1.7 \[{\frac{1}{3\,ad \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }}-{\frac{\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,{a}^{2}d}}+{\frac{\ln \left ( dx+c \right ) }{{a}^{2}d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3/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/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

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Maxima [A]  time = 1.8593, size = 140, normalized size = 2.37 \[ \frac{1}{3 \,{\left (a b d^{4} x^{3} + 3 \, a b c d^{3} x^{2} + 3 \, a b c^{2} d^{2} x +{\left (a b c^{3} + a^{2}\right )} d\right )}} - \frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{2} d} + \frac{\log \left (d x + c\right )}{a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.213629, size = 228, normalized size = 3.86 \[ -\frac{{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} \log \left (d x + c\right ) - a}{3 \,{\left (a^{2} b d^{4} x^{3} + 3 \, a^{2} b c d^{3} x^{2} + 3 \, a^{2} b c^{2} d^{2} x +{\left (a^{2} b c^{3} + a^{3}\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 8.83108, size = 110, normalized size = 1.86 \[ \frac{1}{3 a^{2} d + 3 a b c^{3} d + 9 a b c^{2} d^{2} x + 9 a b c d^{3} x^{2} + 3 a b d^{4} x^{3}} + \frac{\log{\left (\frac{c}{d} + x \right )}}{a^{2} d} - \frac{\log{\left (\frac{3 c^{2} x}{d^{2}} + \frac{3 c x^{2}}{d} + x^{3} + \frac{a + b c^{3}}{b d^{3}} \right )}}{3 a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217147, size = 136, normalized size = 2.31 \[ -\frac{{\rm ln}\left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{2} d} + \frac{{\rm ln}\left ({\left | d x + c \right |}\right )}{a^{2} d} + \frac{1}{3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )} a d} \]

Verification of antiderivative is not currently implemented for this CAS.

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