3.744 \(\int \frac{1}{(a+b x)^3 \left (a^2-b^2 x^2\right )} \, dx\)

Optimal. Leaf size=69 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b}-\frac{1}{8 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2}-\frac{1}{6 a b (a+b x)^3} \]

[Out]

-1/(6*a*b*(a + b*x)^3) - 1/(8*a^2*b*(a + b*x)^2) - 1/(8*a^3*b*(a + b*x)) + ArcTa
nh[(b*x)/a]/(8*a^4*b)

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Rubi [A]  time = 0.106382, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{\tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b}-\frac{1}{8 a^3 b (a+b x)}-\frac{1}{8 a^2 b (a+b x)^2}-\frac{1}{6 a b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

-1/(6*a*b*(a + b*x)^3) - 1/(8*a^2*b*(a + b*x)^2) - 1/(8*a^3*b*(a + b*x)) + ArcTa
nh[(b*x)/a]/(8*a^4*b)

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Rubi in Sympy [A]  time = 24.081, size = 54, normalized size = 0.78 \[ - \frac{1}{6 a b \left (a + b x\right )^{3}} - \frac{1}{8 a^{2} b \left (a + b x\right )^{2}} - \frac{1}{8 a^{3} b \left (a + b x\right )} + \frac{\operatorname{atanh}{\left (\frac{b x}{a} \right )}}{8 a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(6*a*b*(a + b*x)**3) - 1/(8*a**2*b*(a + b*x)**2) - 1/(8*a**3*b*(a + b*x)) + a
tanh(b*x/a)/(8*a**4*b)

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Mathematica [A]  time = 0.0396529, size = 71, normalized size = 1.03 \[ \frac{-2 a \left (10 a^2+9 a b x+3 b^2 x^2\right )-3 (a+b x)^3 \log (a-b x)+3 (a+b x)^3 \log (a+b x)}{48 a^4 b (a+b x)^3} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*x)^3*(a^2 - b^2*x^2)),x]

[Out]

(-2*a*(10*a^2 + 9*a*b*x + 3*b^2*x^2) - 3*(a + b*x)^3*Log[a - b*x] + 3*(a + b*x)^
3*Log[a + b*x])/(48*a^4*b*(a + b*x)^3)

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Maple [A]  time = 0.014, size = 77, normalized size = 1.1 \[ -{\frac{\ln \left ( bx-a \right ) }{16\,{a}^{4}b}}+{\frac{\ln \left ( bx+a \right ) }{16\,{a}^{4}b}}-{\frac{1}{8\,{a}^{3}b \left ( bx+a \right ) }}-{\frac{1}{8\,{a}^{2}b \left ( bx+a \right ) ^{2}}}-{\frac{1}{6\,ab \left ( bx+a \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16/a^4/b*ln(b*x-a)+1/16/a^4/b*ln(b*x+a)-1/8/a^3/b/(b*x+a)-1/8/a^2/b/(b*x+a)^2
-1/6/a/b/(b*x+a)^3

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Maxima [A]  time = 0.70203, size = 122, normalized size = 1.77 \[ -\frac{3 \, b^{2} x^{2} + 9 \, a b x + 10 \, a^{2}}{24 \,{\left (a^{3} b^{4} x^{3} + 3 \, a^{4} b^{3} x^{2} + 3 \, a^{5} b^{2} x + a^{6} b\right )}} + \frac{\log \left (b x + a\right )}{16 \, a^{4} b} - \frac{\log \left (b x - a\right )}{16 \, a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b^2*x^2 - a^2)*(b*x + a)^3),x, algorithm="maxima")

[Out]

-1/24*(3*b^2*x^2 + 9*a*b*x + 10*a^2)/(a^3*b^4*x^3 + 3*a^4*b^3*x^2 + 3*a^5*b^2*x
+ a^6*b) + 1/16*log(b*x + a)/(a^4*b) - 1/16*log(b*x - a)/(a^4*b)

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Fricas [A]  time = 0.216162, size = 181, normalized size = 2.62 \[ -\frac{6 \, a b^{2} x^{2} + 18 \, a^{2} b x + 20 \, a^{3} - 3 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x + a\right ) + 3 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \log \left (b x - a\right )}{48 \,{\left (a^{4} b^{4} x^{3} + 3 \, a^{5} b^{3} x^{2} + 3 \, a^{6} b^{2} x + a^{7} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-1/((b^2*x^2 - a^2)*(b*x + a)^3),x, algorithm="fricas")

[Out]

-1/48*(6*a*b^2*x^2 + 18*a^2*b*x + 20*a^3 - 3*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x
+ a^3)*log(b*x + a) + 3*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*log(b*x - a))/
(a^4*b^4*x^3 + 3*a^5*b^3*x^2 + 3*a^6*b^2*x + a^7*b)

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Sympy [A]  time = 2.29796, size = 83, normalized size = 1.2 \[ - \frac{10 a^{2} + 9 a b x + 3 b^{2} x^{2}}{24 a^{6} b + 72 a^{5} b^{2} x + 72 a^{4} b^{3} x^{2} + 24 a^{3} b^{4} x^{3}} - \frac{\frac{\log{\left (- \frac{a}{b} + x \right )}}{16} - \frac{\log{\left (\frac{a}{b} + x \right )}}{16}}{a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(b*x+a)**3/(-b**2*x**2+a**2),x)

[Out]

-(10*a**2 + 9*a*b*x + 3*b**2*x**2)/(24*a**6*b + 72*a**5*b**2*x + 72*a**4*b**3*x*
*2 + 24*a**3*b**4*x**3) - (log(-a/b + x)/16 - log(a/b + x)/16)/(a**4*b)

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GIAC/XCAS [A]  time = 0.217289, size = 95, normalized size = 1.38 \[ \frac{{\rm ln}\left ({\left | b x + a \right |}\right )}{16 \, a^{4} b} - \frac{{\rm ln}\left ({\left | b x - a \right |}\right )}{16 \, a^{4} b} - \frac{3 \, a b^{2} x^{2} + 9 \, a^{2} b x + 10 \, a^{3}}{24 \,{\left (b x + a\right )}^{3} a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/16*ln(abs(b*x + a))/(a^4*b) - 1/16*ln(abs(b*x - a))/(a^4*b) - 1/24*(3*a*b^2*x^
2 + 9*a^2*b*x + 10*a^3)/((b*x + a)^3*a^4*b)