3.2894 \(\int \frac{1}{(c e+d e x)^4 \left (a+b (c+d x)^3\right )^2} \, dx\)

Optimal. Leaf size=92 \[ -\frac{2 b \log (c+d x)}{a^3 d e^4}+\frac{2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d e^4}-\frac{b}{3 a^2 d e^4 \left (a+b (c+d x)^3\right )}-\frac{1}{3 a^2 d e^4 (c+d x)^3} \]

[Out]

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

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Rubi [A]  time = 0.1896, antiderivative size = 92, 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 \[ -\frac{2 b \log (c+d x)}{a^3 d e^4}+\frac{2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d e^4}-\frac{b}{3 a^2 d e^4 \left (a+b (c+d x)^3\right )}-\frac{1}{3 a^2 d e^4 (c+d x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 19.9334, size = 87, normalized size = 0.95 \[ - \frac{b}{3 a^{2} d e^{4} \left (a + b \left (c + d x\right )^{3}\right )} - \frac{1}{3 a^{2} d e^{4} \left (c + d x\right )^{3}} + \frac{2 b \log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 a^{3} d e^{4}} - \frac{2 b \log{\left (\left (c + d x\right )^{3} \right )}}{3 a^{3} d e^{4}} \]

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)**2,x)

[Out]

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

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Mathematica [A]  time = 0.12536, size = 63, normalized size = 0.68 \[ -\frac{a \left (\frac{b}{a+b (c+d x)^3}+\frac{1}{(c+d x)^3}\right )-2 b \log \left (a+b (c+d x)^3\right )+6 b \log (c+d x)}{3 a^3 d e^4} \]

Antiderivative was successfully verified.

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

[Out]

-(a*((c + d*x)^(-3) + b/(a + b*(c + d*x)^3)) + 6*b*Log[c + d*x] - 2*b*Log[a + b*
(c + d*x)^3])/(3*a^3*d*e^4)

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Maple [A]  time = 0.015, size = 131, normalized size = 1.4 \[ -{\frac{b}{3\,{e}^{4}{a}^{2}d \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }}+{\frac{2\,b\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,{e}^{4}{a}^{3}d}}-{\frac{1}{3\,{e}^{4}{a}^{2}d \left ( dx+c \right ) ^{3}}}-2\,{\frac{b\ln \left ( dx+c \right ) }{{e}^{4}{a}^{3}d}} \]

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)^2,x)

[Out]

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

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Maxima [A]  time = 1.41964, size = 336, normalized size = 3.65 \[ -\frac{2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + a}{3 \,{\left (a^{2} b d^{7} e^{4} x^{6} + 6 \, a^{2} b c d^{6} e^{4} x^{5} + 15 \, a^{2} b c^{2} d^{5} e^{4} x^{4} +{\left (20 \, a^{2} b c^{3} + a^{3}\right )} d^{4} e^{4} x^{3} + 3 \,{\left (5 \, a^{2} b c^{4} + a^{3} c\right )} d^{3} e^{4} x^{2} + 3 \,{\left (2 \, a^{2} b c^{5} + a^{3} c^{2}\right )} d^{2} e^{4} x +{\left (a^{2} b c^{6} + a^{3} c^{3}\right )} d e^{4}\right )}} + \frac{2 \, b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d e^{4}} - \frac{2 \, b \log \left (d x + c\right )}{a^{3} d e^{4}} \]

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)^4),x, algorithm="maxima")

[Out]

-1/3*(2*b*d^3*x^3 + 6*b*c*d^2*x^2 + 6*b*c^2*d*x + 2*b*c^3 + a)/(a^2*b*d^7*e^4*x^
6 + 6*a^2*b*c*d^6*e^4*x^5 + 15*a^2*b*c^2*d^5*e^4*x^4 + (20*a^2*b*c^3 + a^3)*d^4*
e^4*x^3 + 3*(5*a^2*b*c^4 + a^3*c)*d^3*e^4*x^2 + 3*(2*a^2*b*c^5 + a^3*c^2)*d^2*e^
4*x + (a^2*b*c^6 + a^3*c^3)*d*e^4) + 2/3*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^3*d*e^4) - 2*b*log(d*x + c)/(a^3*d*e^4)

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Fricas [A]  time = 0.250942, size = 610, normalized size = 6.63 \[ -\frac{2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + a^{2} - 2 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} +{\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \,{\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} +{\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \,{\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (d x + c\right )}{3 \,{\left (a^{3} b d^{7} e^{4} x^{6} + 6 \, a^{3} b c d^{6} e^{4} x^{5} + 15 \, a^{3} b c^{2} d^{5} e^{4} x^{4} +{\left (20 \, a^{3} b c^{3} + a^{4}\right )} d^{4} e^{4} x^{3} + 3 \,{\left (5 \, a^{3} b c^{4} + a^{4} c\right )} d^{3} e^{4} x^{2} + 3 \,{\left (2 \, a^{3} b c^{5} + a^{4} c^{2}\right )} d^{2} e^{4} x +{\left (a^{3} b c^{6} + a^{4} c^{3}\right )} d e^{4}\right )}} \]

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)^4),x, algorithm="fricas")

[Out]

-1/3*(2*a*b*d^3*x^3 + 6*a*b*c*d^2*x^2 + 6*a*b*c^2*d*x + 2*a*b*c^3 + a^2 - 2*(b^2
*d^6*x^6 + 6*b^2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + b^2*c^6 + (20*b^2*c^3 + a*b)*d
^3*x^3 + a*b*c^3 + 3*(5*b^2*c^4 + a*b*c)*d^2*x^2 + 3*(2*b^2*c^5 + a*b*c^2)*d*x)*
log(b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x + b*c^3 + a) + 6*(b^2*d^6*x^6 + 6*b^
2*c*d^5*x^5 + 15*b^2*c^2*d^4*x^4 + b^2*c^6 + (20*b^2*c^3 + a*b)*d^3*x^3 + a*b*c^
3 + 3*(5*b^2*c^4 + a*b*c)*d^2*x^2 + 3*(2*b^2*c^5 + a*b*c^2)*d*x)*log(d*x + c))/(
a^3*b*d^7*e^4*x^6 + 6*a^3*b*c*d^6*e^4*x^5 + 15*a^3*b*c^2*d^5*e^4*x^4 + (20*a^3*b
*c^3 + a^4)*d^4*e^4*x^3 + 3*(5*a^3*b*c^4 + a^4*c)*d^3*e^4*x^2 + 3*(2*a^3*b*c^5 +
a^4*c^2)*d^2*e^4*x + (a^3*b*c^6 + a^4*c^3)*d*e^4)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219227, size = 213, normalized size = 2.32 \[ \frac{2 \, b e^{\left (-4\right )}{\rm ln}\left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{3} d} - \frac{2 \, b e^{\left (-4\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{a^{3} d} - \frac{{\left (2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + a^{2}\right )} e^{\left (-4\right )}}{3 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{\left (d x + c\right )}^{3} a^{3} d} \]

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)^4),x, algorithm="giac")

[Out]

2/3*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^3*d
) - 2*b*e^(-4)*ln(abs(d*x + c))/(a^3*d) - 1/3*(2*a*b*d^3*x^3 + 6*a*b*c*d^2*x^2 +
6*a*b*c^2*d*x + 2*a*b*c^3 + a^2)*e^(-4)/((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d
*x + b*c^3 + a)*(d*x + c)^3*a^3*d)