3.2875 \(\int \frac{1}{(c+d x) \left (a+b (c+d x)^3\right )^3} \, dx\)

Optimal. Leaf size=82 \[ -\frac{\log \left (a+b (c+d x)^3\right )}{3 a^3 d}+\frac{\log (c+d x)}{a^3 d}+\frac{1}{3 a^2 d \left (a+b (c+d x)^3\right )}+\frac{1}{6 a d \left (a+b (c+d x)^3\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.187008, antiderivative size = 82, 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 \[ -\frac{\log \left (a+b (c+d x)^3\right )}{3 a^3 d}+\frac{\log (c+d x)}{a^3 d}+\frac{1}{3 a^2 d \left (a+b (c+d x)^3\right )}+\frac{1}{6 a d \left (a+b (c+d x)^3\right )^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.5114, size = 70, normalized size = 0.85 \[ \frac{1}{6 a d \left (a + b \left (c + d x\right )^{3}\right )^{2}} + \frac{1}{3 a^{2} d \left (a + b \left (c + d x\right )^{3}\right )} - \frac{\log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 a^{3} d} + \frac{\log{\left (\left (c + d x\right )^{3} \right )}}{3 a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0842358, size = 63, normalized size = 0.77 \[ \frac{\frac{a \left (2 \left (a+b (c+d x)^3\right )+a\right )}{\left (a+b (c+d x)^3\right )^2}-2 \log \left (a+b (c+d x)^3\right )+6 \log (c+d x)}{6 a^3 d} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c + d*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)

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Maple [B]  time = 0.036, size = 283, normalized size = 3.5 \[{\frac{b{d}^{2}{x}^{3}}{3\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}+{\frac{bcd{x}^{2}}{{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}+{\frac{b{c}^{2}x}{{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}}}+{\frac{b{c}^{3}}{3\,{a}^{2} \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}+{\frac{1}{2\,a \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) ^{2}d}}-{\frac{\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,{a}^{3}d}}+{\frac{\ln \left ( dx+c \right ) }{{a}^{3}d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*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+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+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*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/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/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

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Maxima [A]  time = 1.38505, size = 331, normalized size = 4.04 \[ \frac{2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + 3 \, a}{6 \,{\left (a^{2} b^{2} d^{7} x^{6} + 6 \, a^{2} b^{2} c d^{6} x^{5} + 15 \, a^{2} b^{2} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, a^{2} b^{2} c^{3} + a^{3} b\right )} d^{4} x^{3} + 3 \,{\left (5 \, a^{2} b^{2} c^{4} + 2 \, a^{3} b c\right )} d^{3} x^{2} + 6 \,{\left (a^{2} b^{2} c^{5} + a^{3} b c^{2}\right )} d^{2} x +{\left (a^{2} b^{2} c^{6} + 2 \, a^{3} b c^{3} + a^{4}\right )} d\right )}} - \frac{\log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d} + \frac{\log \left (d x + c\right )}{a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)^3*(d*x + c)),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*x^6
+ 6*a^2*b^2*c*d^6*x^5 + 15*a^2*b^2*c^2*d^5*x^4 + 2*(10*a^2*b^2*c^3 + a^3*b)*d^4
*x^3 + 3*(5*a^2*b^2*c^4 + 2*a^3*b*c)*d^3*x^2 + 6*(a^2*b^2*c^5 + a^3*b*c^2)*d^2*x
+ (a^2*b^2*c^6 + 2*a^3*b*c^3 + a^4)*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^3*d) + log(d*x + c)/(a^3*d)

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Fricas [A]  time = 0.26285, size = 630, normalized size = 7.68 \[ \frac{2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2} - 2 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \,{\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \,{\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} + 2 \,{\left (10 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + 2 \, a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + 2 \, a b c\right )} d^{2} x^{2} + 6 \,{\left (b^{2} c^{5} + a b c^{2}\right )} d x + a^{2}\right )} \log \left (d x + c\right )}{6 \,{\left (a^{3} b^{2} d^{7} x^{6} + 6 \, a^{3} b^{2} c d^{6} x^{5} + 15 \, a^{3} b^{2} c^{2} d^{5} x^{4} + 2 \,{\left (10 \, a^{3} b^{2} c^{3} + a^{4} b\right )} d^{4} x^{3} + 3 \,{\left (5 \, a^{3} b^{2} c^{4} + 2 \, a^{4} b c\right )} d^{3} x^{2} + 6 \,{\left (a^{3} b^{2} c^{5} + a^{4} b c^{2}\right )} d^{2} x +{\left (a^{3} b^{2} c^{6} + 2 \, a^{4} b c^{3} + a^{5}\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)^3*(d*x + c)),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*x^6 + 6*a^3*b^2*c*d^6*x^5 + 15*a^3*b^2*c^2*d^5*
x^4 + 2*(10*a^3*b^2*c^3 + a^4*b)*d^4*x^3 + 3*(5*a^3*b^2*c^4 + 2*a^4*b*c)*d^3*x^2
+ 6*(a^3*b^2*c^5 + a^4*b*c^2)*d^2*x + (a^3*b^2*c^6 + 2*a^4*b*c^3 + a^5)*d)

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.219365, size = 194, normalized size = 2.37 \[ -\frac{{\rm ln}\left ({\left | b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a \right |}\right )}{3 \, a^{3} d} + \frac{{\rm ln}\left ({\left | d x + c \right |}\right )}{a^{3} d} + \frac{2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + 3 \, a^{2}}{6 \,{\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}^{2} a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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