3.2886 \(\int \frac{1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )} \, dx\)

Optimal. Leaf size=65 \[ -\frac{b \log (c+d x)}{a^2 d e^4}+\frac{b \log \left (a+b (c+d x)^3\right )}{3 a^2 d e^4}-\frac{1}{3 a d e^4 (c+d x)^3} \]

[Out]

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

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Rubi [A]  time = 0.125712, antiderivative size = 65, 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 \[ -\frac{b \log (c+d x)}{a^2 d e^4}+\frac{b \log \left (a+b (c+d x)^3\right )}{3 a^2 d e^4}-\frac{1}{3 a d e^4 (c+d x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 15.4078, size = 60, normalized size = 0.92 \[ - \frac{1}{3 a d e^{4} \left (c + d x\right )^{3}} + \frac{b \log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 a^{2} d e^{4}} - \frac{b \log{\left (\left (c + d x\right )^{3} \right )}}{3 a^{2} d e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0405102, size = 47, normalized size = 0.72 \[ \frac{b \log \left (a+b (c+d x)^3\right )-\frac{a}{(c+d x)^3}-3 b \log (c+d x)}{3 a^2 d e^4} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c*e + d*e*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*e^4)

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Maple [A]  time = 0.006, size = 84, normalized size = 1.3 \[{\frac{b\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,{e}^{4}{a}^{2}d}}-{\frac{1}{3\,ad{e}^{4} \left ( dx+c \right ) ^{3}}}-{\frac{b\ln \left ( dx+c \right ) }{{e}^{4}{a}^{2}d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.35391, size = 157, normalized size = 2.42 \[ -\frac{1}{3 \,{\left (a d^{4} e^{4} x^{3} + 3 \, a c d^{3} e^{4} x^{2} + 3 \, a c^{2} d^{2} e^{4} x + a c^{3} d e^{4}\right )}} + \frac{b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{2} d e^{4}} - \frac{b \log \left (d x + c\right )}{a^{2} d e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.217469, size = 230, normalized size = 3.54 \[ \frac{{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) - 3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \log \left (d x + c\right ) - a}{3 \,{\left (a^{2} d^{4} e^{4} x^{3} + 3 \, a^{2} c d^{3} e^{4} x^{2} + 3 \, a^{2} c^{2} d^{2} e^{4} x + a^{2} c^{3} d e^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 8.96032, size = 121, normalized size = 1.86 \[ - \frac{1}{3 a c^{3} d e^{4} + 9 a c^{2} d^{2} e^{4} x + 9 a c d^{3} e^{4} x^{2} + 3 a d^{4} e^{4} x^{3}} - \frac{b \log{\left (\frac{c}{d} + x \right )}}{a^{2} d e^{4}} + \frac{b \log{\left (\frac{3 c^{2} x}{d^{2}} + \frac{3 c x^{2}}{d} + x^{3} + \frac{a + b c^{3}}{b d^{3}} \right )}}{3 a^{2} d e^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.219776, size = 111, normalized size = 1.71 \[ \frac{b e^{\left (-4\right )}{\rm ln}\left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{2} d} - \frac{b e^{\left (-4\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{a^{2} d} - \frac{e^{\left (-4\right )}}{3 \,{\left (d x + c\right )}^{3} a d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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