3.2899 \(\int \frac{1}{(c e+d e x) \left (a+b (c+d x)^3\right )^3} \, dx\)

Optimal. Leaf size=94 \[ -\frac{\log \left (a+b (c+d x)^3\right )}{3 a^3 d e}+\frac{\log (c+d x)}{a^3 d e}+\frac{1}{3 a^2 d e \left (a+b (c+d x)^3\right )}+\frac{1}{6 a d e \left (a+b (c+d x)^3\right )^2} \]

[Out]

1/(6*a*d*e*(a + b*(c + d*x)^3)^2) + 1/(3*a^2*d*e*(a + b*(c + d*x)^3)) + Log[c +
d*x]/(a^3*d*e) - Log[a + b*(c + d*x)^3]/(3*a^3*d*e)

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Rubi [A]  time = 0.21203, antiderivative size = 94, 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 \[ -\frac{\log \left (a+b (c+d x)^3\right )}{3 a^3 d e}+\frac{\log (c+d x)}{a^3 d e}+\frac{1}{3 a^2 d e \left (a+b (c+d x)^3\right )}+\frac{1}{6 a d e \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

1/(6*a*d*e*(a + b*(c + d*x)^3)^2) + 1/(3*a^2*d*e*(a + b*(c + d*x)^3)) + Log[c +
d*x]/(a^3*d*e) - Log[a + b*(c + d*x)^3]/(3*a^3*d*e)

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Rubi in Sympy [A]  time = 19.3734, size = 76, normalized size = 0.81 \[ \frac{1}{6 a d e \left (a + b \left (c + d x\right )^{3}\right )^{2}} + \frac{1}{3 a^{2} d e \left (a + b \left (c + d x\right )^{3}\right )} - \frac{\log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 a^{3} d e} + \frac{\log{\left (\left (c + d x\right )^{3} \right )}}{3 a^{3} d e} \]

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)**3,x)

[Out]

1/(6*a*d*e*(a + b*(c + d*x)**3)**2) + 1/(3*a**2*d*e*(a + b*(c + d*x)**3)) - log(
a + b*(c + d*x)**3)/(3*a**3*d*e) + log((c + d*x)**3)/(3*a**3*d*e)

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Mathematica [A]  time = 0.087772, size = 66, normalized size = 0.7 \[ \frac{\frac{a \left (2 \left (a+b (c+d x)^3\right )+a\right )}{\left (a+b (c+d x)^3\right )^2}-2 \log \left (a+b (c+d x)^3\right )+6 \log (c+d x)}{6 a^3 d e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 304, normalized size = 3.2 \[{\frac{b{d}^{2}{x}^{3}}{3\,e{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}+{\frac{bcd{x}^{2}}{e{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}+{\frac{b{c}^{2}x}{e{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}+{\frac{b{c}^{3}}{3\,e{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}+{\frac{1}{2\,ae \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,e{a}^{3}d}}+{\frac{\ln \left ( dx+c \right ) }{e{a}^{3}d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.44509, size = 348, normalized size = 3.7 \[ \frac{2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + 3 \, a}{6 \,{\left (a^{2} b^{2} d^{7} e x^{6} + 6 \, a^{2} b^{2} c d^{6} e x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} e x^{4} + 2 \,{\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} e x^{3} + 3 \,{\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} e x^{2} + 6 \,{\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} e x +{\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d e\right )}} - \frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d e} + \frac{\log \left (d x + c\right )}{a^{3} d e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(2*b*d^3*x^3 + 6*b*c*d^2*x^2 + 6*b*c^2*d*x + 2*b*c^3 + 3*a)/(a^2*b^2*d^7*e*x
^6 + 6*a^2*b^2*c*d^6*e*x^5 + 15*a^2*b^2*c^2*d^5*e*x^4 + 2*(10*a^2*b^2*c^3 + a^3*
b)*d^4*e*x^3 + 3*(5*a^2*b^2*c^4 + 2*a^3*b*c)*d^3*e*x^2 + 6*(a^2*b^2*c^5 + a^3*b*
c^2)*d^2*e*x + (a^2*b^2*c^6 + 2*a^3*b*c^3 + a^4)*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^3*d*e) + log(d*x + c)/(a^3*d*e)

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Fricas [A]  time = 0.262632, size = 640, normalized size = 6.81 \[ \frac{2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2} - 2 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \,{\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \,{\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \,{\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \,{\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (d x + c\right )}{6 \,{\left (a^{3} b^{2} d^{7} e x^{6} + 6 \, a^{3} b^{2} c d^{6} e x^{5} + 15 \, a^{3} b^{2} c^{2} d^{5} e x^{4} + 2 \,{\left (10 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{4} e x^{3} + 3 \,{\left (5 \, a^{3} b^{2} c^{4} + 2 \, a^{4} b c\right )} d^{3} e x^{2} + 6 \,{\left (a^{3} b^{2} c^{5} + a^{4} b c^{2}\right )} d^{2} e x +{\left (a^{3} b^{2} c^{6} + 2 \, a^{4} b c^{3} + a^{5}\right )} d e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6*(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 + 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 + 2*(10*b^2*c^3 + a*b
)*d^3*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*x^2 + 6*(b^2*c^5 + a*b*c^2)*
d*x + a^2)*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 + 2*(10*b^2*c^3 + a*b)*d^3
*x^3 + 2*a*b*c^3 + 3*(5*b^2*c^4 + 2*a*b*c)*d^2*x^2 + 6*(b^2*c^5 + a*b*c^2)*d*x +
a^2)*log(d*x + c))/(a^3*b^2*d^7*e*x^6 + 6*a^3*b^2*c*d^6*e*x^5 + 15*a^3*b^2*c^2*
d^5*e*x^4 + 2*(10*a^3*b^2*c^3 + a^4*b)*d^4*e*x^3 + 3*(5*a^3*b^2*c^4 + 2*a^4*b*c)
*d^3*e*x^2 + 6*(a^3*b^2*c^5 + a^4*b*c^2)*d^2*e*x + (a^3*b^2*c^6 + 2*a^4*b*c^3 +
a^5)*d*e)

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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)/(a+b*(d*x+c)**3)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.220463, size = 203, normalized size = 2.16 \[ -\frac{e^{\left (-1\right )}{\rm ln}\left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{3} d} + \frac{e^{\left (-1\right )}{\rm ln}\left ({\left | d x + c \right |}\right )}{a^{3} d} + \frac{{\left (2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2}\right )} e^{\left (-1\right )}}{6 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)^3*(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^3*d)
+ e^(-1)*ln(abs(d*x + c))/(a^3*d) + 1/6*(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 + 3*a^2)*e^(-1)/((b*d^3*x^3 + 3*b*c*d^2*x^2 + 3*b*c^2*d*x
+ b*c^3 + a)^2*a^3*d)