3.2860 \(\int \frac{1}{(c+d x)^4 \left (a+b (c+d x)^3\right )} \, dx\)

Optimal. Leaf size=56 \[ -\frac{b \log (c+d x)}{a^2 d}+\frac{b \log \left (a+b (c+d x)^3\right )}{3 a^2 d}-\frac{1}{3 a d (c+d x)^3} \]

[Out]

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

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Rubi [A]  time = 0.114263, antiderivative size = 56, 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 \[ -\frac{b \log (c+d x)}{a^2 d}+\frac{b \log \left (a+b (c+d x)^3\right )}{3 a^2 d}-\frac{1}{3 a d (c+d x)^3} \]

Antiderivative was successfully verified.

[In]  Int[1/((c + d*x)^4*(a + b*(c + d*x)^3)),x]

[Out]

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

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Rubi in Sympy [A]  time = 14.28, size = 49, normalized size = 0.88 \[ - \frac{1}{3 a d \left (c + d x\right )^{3}} + \frac{b \log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 a^{2} d} - \frac{b \log{\left (\left (c + d x\right )^{3} \right )}}{3 a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(d*x+c)**4/(a+b*(d*x+c)**3),x)

[Out]

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

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Mathematica [A]  time = 0.0372345, size = 44, normalized size = 0.79 \[ \frac{b \log \left (a+b (c+d x)^3\right )-\frac{a}{(c+d x)^3}-3 b \log (c+d x)}{3 a^2 d} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c + d*x)^4*(a + b*(c + d*x)^3)),x]

[Out]

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

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Maple [A]  time = 0.012, size = 75, normalized size = 1.3 \[{\frac{b\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,{a}^{2}d}}-{\frac{1}{3\,ad \left ( dx+c \right ) ^{3}}}-{\frac{b\ln \left ( dx+c \right ) }{{a}^{2}d}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(d*x+c)^4/(a+b*(d*x+c)^3),x)

[Out]

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

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Maxima [A]  time = 1.35626, size = 132, normalized size = 2.36 \[ -\frac{1}{3 \,{\left (a d^{4} x^{3} + 3 \, a c d^{3} x^{2} + 3 \, a c^{2} d^{2} x + a c^{3} d\right )}} + \frac{b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{2} d} - \frac{b \log \left (d x + c\right )}{a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)*(d*x + c)^4),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.212311, size = 213, normalized size = 3.8 \[ \frac{{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (d x + c\right ) - a}{3 \,{\left (a^{2} d^{4} x^{3} + 3 \, a^{2} c d^{3} x^{2} + 3 \, a^{2} c^{2} d^{2} x + a^{2} c^{3} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)*(d*x + c)^4),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)*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)*log(d*x + c) - a)/(a^2*d^4*x^3 + 3*a^2*c*d^3*x^2 + 3*a^2*c^2*d^2*x + a^2*c
^3*d)

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Sympy [A]  time = 7.53402, size = 100, normalized size = 1.79 \[ - \frac{1}{3 a c^{3} d + 9 a c^{2} d^{2} x + 9 a c d^{3} x^{2} + 3 a d^{4} x^{3}} - \frac{b \log{\left (\frac{c}{d} + x \right )}}{a^{2} d} + \frac{b \log{\left (\frac{3 c^{2} x}{d^{2}} + \frac{3 c x^{2}}{d} + x^{3} + \frac{a + b c^{3}}{b d^{3}} \right )}}{3 a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(d*x+c)**4/(a+b*(d*x+c)**3),x)

[Out]

-1/(3*a*c**3*d + 9*a*c**2*d**2*x + 9*a*c*d**3*x**2 + 3*a*d**4*x**3) - b*log(c/d
+ x)/(a**2*d) + b*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 = 0.221809, size = 55, normalized size = 0.98 \[ \frac{b{\rm ln}\left ({\left | -b - \frac{a}{{\left (d x + c\right )}^{3}} \right |}\right )}{3 \, a^{2} d} - \frac{1}{3 \,{\left (d x + c\right )}^{3} a d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)*(d*x + c)^4),x, algorithm="giac")

[Out]

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