3.2869 \(\int \frac{1}{(c+d x)^4 \left (a+b (c+d x)^3\right )^2} \, dx\)

Optimal. Leaf size=80 \[ -\frac{2 b \log (c+d x)}{a^3 d}+\frac{2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d}-\frac{b}{3 a^2 d \left (a+b (c+d x)^3\right )}-\frac{1}{3 a^2 d (c+d x)^3} \]

[Out]

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

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Rubi [A]  time = 0.168354, antiderivative size = 80, 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 \[ -\frac{2 b \log (c+d x)}{a^3 d}+\frac{2 b \log \left (a+b (c+d x)^3\right )}{3 a^3 d}-\frac{b}{3 a^2 d \left (a+b (c+d x)^3\right )}-\frac{1}{3 a^2 d (c+d x)^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.4529, size = 73, normalized size = 0.91 \[ - \frac{b}{3 a^{2} d \left (a + b \left (c + d x\right )^{3}\right )} - \frac{1}{3 a^{2} d \left (c + d x\right )^{3}} + \frac{2 b \log{\left (a + b \left (c + d x\right )^{3} \right )}}{3 a^{3} d} - \frac{2 b \log{\left (\left (c + d x\right )^{3} \right )}}{3 a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.116521, size = 60, normalized size = 0.75 \[ -\frac{a \left (\frac{b}{a+b (c+d x)^3}+\frac{1}{(c+d x)^3}\right )-2 b \log \left (a+b (c+d x)^3\right )+6 b \log (c+d x)}{3 a^3 d} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((c + d*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)

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Maple [A]  time = 0.026, size = 119, normalized size = 1.5 \[ -{\frac{b}{3\,{a}^{2}d \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }}+{\frac{2\,b\ln \left ( b{d}^{3}{x}^{3}+3\,bc{d}^{2}{x}^{2}+3\,b{c}^{2}dx+b{c}^{3}+a \right ) }{3\,{a}^{3}d}}-{\frac{1}{3\,{a}^{2}d \left ( dx+c \right ) ^{3}}}-2\,{\frac{b\ln \left ( dx+c \right ) }{{a}^{3}d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.47557, size = 300, normalized size = 3.75 \[ -\frac{2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 6 \, b c^{2} d x + 2 \, b c^{3} + a}{3 \,{\left (a^{2} b d^{7} x^{6} + 6 \, a^{2} b c d^{6} x^{5} + 15 \, a^{2} b c^{2} d^{5} x^{4} +{\left (20 \, a^{2} b c^{3} + a^{3}\right )} d^{4} x^{3} + 3 \,{\left (5 \, a^{2} b c^{4} + a^{3} c\right )} d^{3} x^{2} + 3 \,{\left (2 \, a^{2} b c^{5} + a^{3} c^{2}\right )} d^{2} x +{\left (a^{2} b c^{6} + a^{3} c^{3}\right )} d\right )}} + \frac{2 \, b \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right )}{3 \, a^{3} d} - \frac{2 \, b \log \left (d x + c\right )}{a^{3} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)^2*(d*x + c)^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*x^6 +
6*a^2*b*c*d^6*x^5 + 15*a^2*b*c^2*d^5*x^4 + (20*a^2*b*c^3 + a^3)*d^4*x^3 + 3*(5*a
^2*b*c^4 + a^3*c)*d^3*x^2 + 3*(2*a^2*b*c^5 + a^3*c^2)*d^2*x + (a^2*b*c^6 + a^3*c
^3)*d) + 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)
- 2*b*log(d*x + c)/(a^3*d)

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Fricas [A]  time = 0.248337, size = 582, normalized size = 7.28 \[ -\frac{2 \, a b d^{3} x^{3} + 6 \, a b c d^{2} x^{2} + 6 \, a b c^{2} d x + 2 \, a b c^{3} + a^{2} - 2 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} +{\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \,{\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3} + a\right ) + 6 \,{\left (b^{2} d^{6} x^{6} + 6 \, b^{2} c d^{5} x^{5} + 15 \, b^{2} c^{2} d^{4} x^{4} + b^{2} c^{6} +{\left (20 \, b^{2} c^{3} + a b\right )} d^{3} x^{3} + a b c^{3} + 3 \,{\left (5 \, b^{2} c^{4} + a b c\right )} d^{2} x^{2} + 3 \,{\left (2 \, b^{2} c^{5} + a b c^{2}\right )} d x\right )} \log \left (d x + c\right )}{3 \,{\left (a^{3} b d^{7} x^{6} + 6 \, a^{3} b c d^{6} x^{5} + 15 \, a^{3} b c^{2} d^{5} x^{4} +{\left (20 \, a^{3} b c^{3} + a^{4}\right )} d^{4} x^{3} + 3 \,{\left (5 \, a^{3} b c^{4} + a^{4} c\right )} d^{3} x^{2} + 3 \,{\left (2 \, a^{3} b c^{5} + a^{4} c^{2}\right )} d^{2} x +{\left (a^{3} b c^{6} + a^{4} c^{3}\right )} d\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)^2*(d*x + c)^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*x^6 + 6*a^3*b*c*d^6*x^5 + 15*a^3*b*c^2*d^5*x^4 + (20*a^3*b*c^3 + a^4)*
d^4*x^3 + 3*(5*a^3*b*c^4 + a^4*c)*d^3*x^2 + 3*(2*a^3*b*c^5 + a^4*c^2)*d^2*x + (a
^3*b*c^6 + a^4*c^3)*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)**4/(a+b*(d*x+c)**3)**2,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224616, size = 88, normalized size = 1.1 \[ \frac{2 \, b{\rm ln}\left ({\left | -b - \frac{a}{{\left (d x + c\right )}^{3}} \right |}\right )}{3 \, a^{3} d} + \frac{b^{2}}{3 \, a^{3}{\left (b + \frac{a}{{\left (d x + c\right )}^{3}}\right )} d} - \frac{1}{3 \,{\left (d x + c\right )}^{3} a^{2} d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(((d*x + c)^3*b + a)^2*(d*x + c)^4),x, algorithm="giac")

[Out]

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