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

Optimal. Leaf size=105 \[ \frac{5 \tanh ^{-1}\left (\frac{b x}{a}\right )}{16 a^6 b}+\frac{1}{8 a^5 b (a-b x)}-\frac{3}{16 a^5 b (a+b x)}+\frac{1}{32 a^4 b (a-b x)^2}-\frac{3}{32 a^4 b (a+b x)^2}-\frac{1}{24 a^3 b (a+b x)^3} \]

[Out]

1/(32*a^4*b*(a - b*x)^2) + 1/(8*a^5*b*(a - b*x)) - 1/(24*a^3*b*(a + b*x)^3) - 3/
(32*a^4*b*(a + b*x)^2) - 3/(16*a^5*b*(a + b*x)) + (5*ArcTanh[(b*x)/a])/(16*a^6*b
)

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Rubi [A]  time = 0.163289, antiderivative size = 105, 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 \[ \frac{5 \tanh ^{-1}\left (\frac{b x}{a}\right )}{16 a^6 b}+\frac{1}{8 a^5 b (a-b x)}-\frac{3}{16 a^5 b (a+b x)}+\frac{1}{32 a^4 b (a-b x)^2}-\frac{3}{32 a^4 b (a+b x)^2}-\frac{1}{24 a^3 b (a+b x)^3} \]

Antiderivative was successfully verified.

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

[Out]

1/(32*a^4*b*(a - b*x)^2) + 1/(8*a^5*b*(a - b*x)) - 1/(24*a^3*b*(a + b*x)^3) - 3/
(32*a^4*b*(a + b*x)^2) - 3/(16*a^5*b*(a + b*x)) + (5*ArcTanh[(b*x)/a])/(16*a^6*b
)

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Rubi in Sympy [A]  time = 32.7663, size = 87, normalized size = 0.83 \[ - \frac{1}{24 a^{3} b \left (a + b x\right )^{3}} - \frac{3}{32 a^{4} b \left (a + b x\right )^{2}} + \frac{1}{32 a^{4} b \left (a - b x\right )^{2}} - \frac{3}{16 a^{5} b \left (a + b x\right )} + \frac{1}{8 a^{5} b \left (a - b x\right )} + \frac{5 \operatorname{atanh}{\left (\frac{b x}{a} \right )}}{16 a^{6} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(24*a**3*b*(a + b*x)**3) - 3/(32*a**4*b*(a + b*x)**2) + 1/(32*a**4*b*(a - b*x
)**2) - 3/(16*a**5*b*(a + b*x)) + 1/(8*a**5*b*(a - b*x)) + 5*atanh(b*x/a)/(16*a*
*6*b)

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Mathematica [A]  time = 0.0802668, size = 87, normalized size = 0.83 \[ \frac{-\frac{2 a \left (8 a^4-25 a^3 b x-25 a^2 b^2 x^2+15 a b^3 x^3+15 b^4 x^4\right )}{(a-b x)^2 (a+b x)^3}-15 \log (a-b x)+15 \log (a+b x)}{96 a^6 b} \]

Antiderivative was successfully verified.

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

[Out]

((-2*a*(8*a^4 - 25*a^3*b*x - 25*a^2*b^2*x^2 + 15*a*b^3*x^3 + 15*b^4*x^4))/((a -
b*x)^2*(a + b*x)^3) - 15*Log[a - b*x] + 15*Log[a + b*x])/(96*a^6*b)

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Maple [A]  time = 0.017, size = 111, normalized size = 1.1 \[ -{\frac{5\,\ln \left ( bx-a \right ) }{32\,{a}^{6}b}}-{\frac{1}{8\,{a}^{5}b \left ( bx-a \right ) }}+{\frac{1}{32\,{a}^{4}b \left ( bx-a \right ) ^{2}}}+{\frac{5\,\ln \left ( bx+a \right ) }{32\,{a}^{6}b}}-{\frac{3}{16\,{a}^{5}b \left ( bx+a \right ) }}-{\frac{3}{32\,{a}^{4}b \left ( bx+a \right ) ^{2}}}-{\frac{1}{24\,{a}^{3}b \left ( bx+a \right ) ^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-5/32/a^6/b*ln(b*x-a)-1/8/a^5/b/(b*x-a)+1/32/a^4/b/(b*x-a)^2+5/32/a^6/b*ln(b*x+a
)-3/16/a^5/b/(b*x+a)-3/32/a^4/b/(b*x+a)^2-1/24/a^3/b/(b*x+a)^3

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Maxima [A]  time = 0.699461, size = 178, normalized size = 1.7 \[ -\frac{15 \, b^{4} x^{4} + 15 \, a b^{3} x^{3} - 25 \, a^{2} b^{2} x^{2} - 25 \, a^{3} b x + 8 \, a^{4}}{48 \,{\left (a^{5} b^{6} x^{5} + a^{6} b^{5} x^{4} - 2 \, a^{7} b^{4} x^{3} - 2 \, a^{8} b^{3} x^{2} + a^{9} b^{2} x + a^{10} b\right )}} + \frac{5 \, \log \left (b x + a\right )}{32 \, a^{6} b} - \frac{5 \, \log \left (b x - a\right )}{32 \, a^{6} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/48*(15*b^4*x^4 + 15*a*b^3*x^3 - 25*a^2*b^2*x^2 - 25*a^3*b*x + 8*a^4)/(a^5*b^6
*x^5 + a^6*b^5*x^4 - 2*a^7*b^4*x^3 - 2*a^8*b^3*x^2 + a^9*b^2*x + a^10*b) + 5/32*
log(b*x + a)/(a^6*b) - 5/32*log(b*x - a)/(a^6*b)

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Fricas [A]  time = 0.214059, size = 292, normalized size = 2.78 \[ -\frac{30 \, a b^{4} x^{4} + 30 \, a^{2} b^{3} x^{3} - 50 \, a^{3} b^{2} x^{2} - 50 \, a^{4} b x + 16 \, a^{5} - 15 \,{\left (b^{5} x^{5} + a b^{4} x^{4} - 2 \, a^{2} b^{3} x^{3} - 2 \, a^{3} b^{2} x^{2} + a^{4} b x + a^{5}\right )} \log \left (b x + a\right ) + 15 \,{\left (b^{5} x^{5} + a b^{4} x^{4} - 2 \, a^{2} b^{3} x^{3} - 2 \, a^{3} b^{2} x^{2} + a^{4} b x + a^{5}\right )} \log \left (b x - a\right )}{96 \,{\left (a^{6} b^{6} x^{5} + a^{7} b^{5} x^{4} - 2 \, a^{8} b^{4} x^{3} - 2 \, a^{9} b^{3} x^{2} + a^{10} b^{2} x + a^{11} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/96*(30*a*b^4*x^4 + 30*a^2*b^3*x^3 - 50*a^3*b^2*x^2 - 50*a^4*b*x + 16*a^5 - 15
*(b^5*x^5 + a*b^4*x^4 - 2*a^2*b^3*x^3 - 2*a^3*b^2*x^2 + a^4*b*x + a^5)*log(b*x +
a) + 15*(b^5*x^5 + a*b^4*x^4 - 2*a^2*b^3*x^3 - 2*a^3*b^2*x^2 + a^4*b*x + a^5)*l
og(b*x - a))/(a^6*b^6*x^5 + a^7*b^5*x^4 - 2*a^8*b^4*x^3 - 2*a^9*b^3*x^2 + a^10*b
^2*x + a^11*b)

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Sympy [A]  time = 3.22386, size = 134, normalized size = 1.28 \[ - \frac{8 a^{4} - 25 a^{3} b x - 25 a^{2} b^{2} x^{2} + 15 a b^{3} x^{3} + 15 b^{4} x^{4}}{48 a^{10} b + 48 a^{9} b^{2} x - 96 a^{8} b^{3} x^{2} - 96 a^{7} b^{4} x^{3} + 48 a^{6} b^{5} x^{4} + 48 a^{5} b^{6} x^{5}} - \frac{\frac{5 \log{\left (- \frac{a}{b} + x \right )}}{32} - \frac{5 \log{\left (\frac{a}{b} + x \right )}}{32}}{a^{6} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(8*a**4 - 25*a**3*b*x - 25*a**2*b**2*x**2 + 15*a*b**3*x**3 + 15*b**4*x**4)/(48*
a**10*b + 48*a**9*b**2*x - 96*a**8*b**3*x**2 - 96*a**7*b**4*x**3 + 48*a**6*b**5*
x**4 + 48*a**5*b**6*x**5) - (5*log(-a/b + x)/32 - 5*log(a/b + x)/32)/(a**6*b)

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GIAC/XCAS [A]  time = 0.218135, size = 136, normalized size = 1.3 \[ \frac{5 \,{\rm ln}\left ({\left | b x + a \right |}\right )}{32 \, a^{6} b} - \frac{5 \,{\rm ln}\left ({\left | b x - a \right |}\right )}{32 \, a^{6} b} - \frac{15 \, a b^{4} x^{4} + 15 \, a^{2} b^{3} x^{3} - 25 \, a^{3} b^{2} x^{2} - 25 \, a^{4} b x + 8 \, a^{5}}{48 \,{\left (b x + a\right )}^{3}{\left (b x - a\right )}^{2} a^{6} b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/32*ln(abs(b*x + a))/(a^6*b) - 5/32*ln(abs(b*x - a))/(a^6*b) - 1/48*(15*a*b^4*x
^4 + 15*a^2*b^3*x^3 - 25*a^3*b^2*x^2 - 25*a^4*b*x + 8*a^5)/((b*x + a)^3*(b*x - a
)^2*a^6*b)