3.1055 \(\int \frac{1}{(a+b x) (a c-b c x)^3} \, dx\)

Optimal. Leaf size=63 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^3 b c^3}+\frac{1}{4 a^2 b c^3 (a-b x)}+\frac{1}{4 a b c^3 (a-b x)^2} \]

[Out]

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

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Rubi [A]  time = 0.0850115, antiderivative size = 63, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{4 a^3 b c^3}+\frac{1}{4 a^2 b c^3 (a-b x)}+\frac{1}{4 a b c^3 (a-b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 25.5496, size = 49, normalized size = 0.78 \[ \frac{1}{4 a b c^{3} \left (a - b x\right )^{2}} + \frac{1}{4 a^{2} b c^{3} \left (a - b x\right )} + \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{4 a^{3} b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/(4*a*b*c**3*(a - b*x)**2) + 1/(4*a**2*b*c**3*(a - b*x)) + atanh(b*x/a)/(4*a**3
*b*c**3)

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Mathematica [A]  time = 0.030103, size = 65, normalized size = 1.03 \[ \frac{2 a (2 a-b x)+(a-b x)^2 (-\log (a-b x))+(a-b x)^2 \log (a+b x)}{8 a^3 b c^3 (a-b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*a*(2*a - b*x) - (a - b*x)^2*Log[a - b*x] + (a - b*x)^2*Log[a + b*x])/(8*a^3*b
*c^3*(a - b*x)^2)

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Maple [A]  time = 0.013, size = 78, normalized size = 1.2 \[{\frac{\ln \left ( bx+a \right ) }{8\,{c}^{3}{a}^{3}b}}-{\frac{\ln \left ( bx-a \right ) }{8\,{c}^{3}{a}^{3}b}}-{\frac{1}{4\,{c}^{3}{a}^{2}b \left ( bx-a \right ) }}+{\frac{1}{4\,{c}^{3}ba \left ( bx-a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(b*x+a)/(-b*c*x+a*c)^3,x)

[Out]

1/8/c^3/a^3/b*ln(b*x+a)-1/8/c^3/a^3/b*ln(b*x-a)-1/4/c^3/a^2/b/(b*x-a)+1/4/c^3/b/
a/(b*x-a)^2

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Maxima [A]  time = 1.35308, size = 111, normalized size = 1.76 \[ -\frac{b x - 2 \, a}{4 \,{\left (a^{2} b^{3} c^{3} x^{2} - 2 \, a^{3} b^{2} c^{3} x + a^{4} b c^{3}\right )}} + \frac{\log \left (b x + a\right )}{8 \, a^{3} b c^{3}} - \frac{\log \left (b x - a\right )}{8 \, a^{3} b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*(b*x - 2*a)/(a^2*b^3*c^3*x^2 - 2*a^3*b^2*c^3*x + a^4*b*c^3) + 1/8*log(b*x +
a)/(a^3*b*c^3) - 1/8*log(b*x - a)/(a^3*b*c^3)

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Fricas [A]  time = 0.209453, size = 132, normalized size = 2.1 \[ -\frac{2 \, a b x - 4 \, a^{2} -{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x + a\right ) +{\left (b^{2} x^{2} - 2 \, a b x + a^{2}\right )} \log \left (b x - a\right )}{8 \,{\left (a^{3} b^{3} c^{3} x^{2} - 2 \, a^{4} b^{2} c^{3} x + a^{5} b c^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(2*a*b*x - 4*a^2 - (b^2*x^2 - 2*a*b*x + a^2)*log(b*x + a) + (b^2*x^2 - 2*a*
b*x + a^2)*log(b*x - a))/(a^3*b^3*c^3*x^2 - 2*a^4*b^2*c^3*x + a^5*b*c^3)

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Sympy [A]  time = 2.00199, size = 71, normalized size = 1.13 \[ - \frac{- 2 a + b x}{4 a^{4} b c^{3} - 8 a^{3} b^{2} c^{3} x + 4 a^{2} b^{3} c^{3} x^{2}} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{8} - \frac{\log{\left (\frac{a}{b} + x \right )}}{8}}{a^{3} b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*a + b*x)/(4*a**4*b*c**3 - 8*a**3*b**2*c**3*x + 4*a**2*b**3*c**3*x**2) - (lo
g(-a/b + x)/8 - log(a/b + x)/8)/(a**3*b*c**3)

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GIAC/XCAS [A]  time = 0.208682, size = 93, normalized size = 1.48 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{8 \, a^{3} b c^{3}} - \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{8 \, a^{3} b c^{3}} - \frac{a b x - 2 \, a^{2}}{4 \,{\left (b x - a\right )}^{2} a^{3} b c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*ln(abs(b*x + a))/(a^3*b*c^3) - 1/8*ln(abs(b*x - a))/(a^3*b*c^3) - 1/4*(a*b*x
- 2*a^2)/((b*x - a)^2*a^3*b*c^3)