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

Optimal. Leaf size=71 \[ \frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b}+\frac{1}{4 a^3 b (a-b x)}-\frac{1}{8 a^3 b (a+b x)}+\frac{1}{8 a^2 b (a-b x)^2} \]

[Out]

1/(8*a^2*b*(a - b*x)^2) + 1/(4*a^3*b*(a - b*x)) - 1/(8*a^3*b*(a + b*x)) + (3*Arc
Tanh[(b*x)/a])/(8*a^4*b)

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Rubi [A]  time = 0.108575, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{3 \tanh ^{-1}\left (\frac{b x}{a}\right )}{8 a^4 b}+\frac{1}{4 a^3 b (a-b x)}-\frac{1}{8 a^3 b (a+b x)}+\frac{1}{8 a^2 b (a-b x)^2} \]

Antiderivative was successfully verified.

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

[Out]

1/(8*a^2*b*(a - b*x)^2) + 1/(4*a^3*b*(a - b*x)) - 1/(8*a^3*b*(a + b*x)) + (3*Arc
Tanh[(b*x)/a])/(8*a^4*b)

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Rubi in Sympy [A]  time = 24.9187, size = 56, normalized size = 0.79 \[ \frac{1}{8 a^{2} b \left (a - b x\right )^{2}} - \frac{1}{8 a^{3} b \left (a + b x\right )} + \frac{1}{4 a^{3} b \left (a - b x\right )} + \frac{3 \operatorname{atanh}{\left (\frac{b x}{a} \right )}}{8 a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0598378, size = 65, normalized size = 0.92 \[ \frac{\frac{2 a \left (2 a^2+3 a b x-3 b^2 x^2\right )}{(a-b x)^2 (a+b x)}-3 \log (a-b x)+3 \log (a+b x)}{16 a^4 b} \]

Antiderivative was successfully verified.

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

[Out]

((2*a*(2*a^2 + 3*a*b*x - 3*b^2*x^2))/((a - b*x)^2*(a + b*x)) - 3*Log[a - b*x] +
3*Log[a + b*x])/(16*a^4*b)

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Maple [A]  time = 0.016, size = 81, normalized size = 1.1 \[ -{\frac{3\,\ln \left ( bx-a \right ) }{16\,{a}^{4}b}}-{\frac{1}{4\,{a}^{3}b \left ( bx-a \right ) }}+{\frac{1}{8\,{a}^{2}b \left ( bx-a \right ) ^{2}}}+{\frac{3\,\ln \left ( bx+a \right ) }{16\,{a}^{4}b}}-{\frac{1}{8\,{a}^{3}b \left ( bx+a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)/(-b^2*x^2+a^2)^3,x)

[Out]

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

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Maxima [A]  time = 0.682811, size = 122, normalized size = 1.72 \[ -\frac{3 \, b^{2} x^{2} - 3 \, a b x - 2 \, a^{2}}{8 \,{\left (a^{3} b^{4} x^{3} - a^{4} b^{3} x^{2} - a^{5} b^{2} x + a^{6} b\right )}} + \frac{3 \, \log \left (b x + a\right )}{16 \, a^{4} b} - \frac{3 \, \log \left (b x - a\right )}{16 \, a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/8*(3*b^2*x^2 - 3*a*b*x - 2*a^2)/(a^3*b^4*x^3 - a^4*b^3*x^2 - a^5*b^2*x + a^6*
b) + 3/16*log(b*x + a)/(a^4*b) - 3/16*log(b*x - a)/(a^4*b)

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Fricas [A]  time = 0.212063, size = 181, normalized size = 2.55 \[ -\frac{6 \, a b^{2} x^{2} - 6 \, a^{2} b x - 4 \, a^{3} - 3 \,{\left (b^{3} x^{3} - a b^{2} x^{2} - a^{2} b x + a^{3}\right )} \log \left (b x + a\right ) + 3 \,{\left (b^{3} x^{3} - a b^{2} x^{2} - a^{2} b x + a^{3}\right )} \log \left (b x - a\right )}{16 \,{\left (a^{4} b^{4} x^{3} - a^{5} b^{3} x^{2} - a^{6} b^{2} x + a^{7} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(6*a*b^2*x^2 - 6*a^2*b*x - 4*a^3 - 3*(b^3*x^3 - a*b^2*x^2 - a^2*b*x + a^3)
*log(b*x + a) + 3*(b^3*x^3 - a*b^2*x^2 - a^2*b*x + a^3)*log(b*x - a))/(a^4*b^4*x
^3 - a^5*b^3*x^2 - a^6*b^2*x + a^7*b)

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Sympy [A]  time = 2.22736, size = 87, normalized size = 1.23 \[ - \frac{- 2 a^{2} - 3 a b x + 3 b^{2} x^{2}}{8 a^{6} b - 8 a^{5} b^{2} x - 8 a^{4} b^{3} x^{2} + 8 a^{3} b^{4} x^{3}} - \frac{\frac{3 \log{\left (- \frac{a}{b} + x \right )}}{16} - \frac{3 \log{\left (\frac{a}{b} + x \right )}}{16}}{a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(-2*a**2 - 3*a*b*x + 3*b**2*x**2)/(8*a**6*b - 8*a**5*b**2*x - 8*a**4*b**3*x**2
+ 8*a**3*b**4*x**3) - (3*log(-a/b + x)/16 - 3*log(a/b + x)/16)/(a**4*b)

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GIAC/XCAS [A]  time = 0.218779, size = 107, normalized size = 1.51 \[ \frac{3 \,{\rm ln}\left ({\left | b x + a \right |}\right )}{16 \, a^{4} b} - \frac{3 \,{\rm ln}\left ({\left | b x - a \right |}\right )}{16 \, a^{4} b} - \frac{3 \, a b^{2} x^{2} - 3 \, a^{2} b x - 2 \, a^{3}}{8 \,{\left (b x + a\right )}{\left (b x - a\right )}^{2} a^{4} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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