3.756 \(\int \frac{(a+b x)^8}{\left (a^2-b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=71 \[ \frac{16 a^5}{b (a-b x)^2}-\frac{80 a^4}{b (a-b x)}-\frac{80 a^3 \log (a-b x)}{b}-31 a^2 x-4 a b x^2-\frac{b^2 x^3}{3} \]

[Out]

-31*a^2*x - 4*a*b*x^2 - (b^2*x^3)/3 + (16*a^5)/(b*(a - b*x)^2) - (80*a^4)/(b*(a
- b*x)) - (80*a^3*Log[a - b*x])/b

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Rubi [A]  time = 0.130351, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{16 a^5}{b (a-b x)^2}-\frac{80 a^4}{b (a-b x)}-\frac{80 a^3 \log (a-b x)}{b}-31 a^2 x-4 a b x^2-\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^8/(a^2 - b^2*x^2)^3,x]

[Out]

-31*a^2*x - 4*a*b*x^2 - (b^2*x^3)/3 + (16*a^5)/(b*(a - b*x)^2) - (80*a^4)/(b*(a
- b*x)) - (80*a^3*Log[a - b*x])/b

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{16 a^{5}}{b \left (a - b x\right )^{2}} - \frac{80 a^{4}}{b \left (a - b x\right )} - \frac{80 a^{3} \log{\left (a - b x \right )}}{b} - 31 a^{2} x - 8 a b \int x\, dx - \frac{b^{2} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**8/(-b**2*x**2+a**2)**3,x)

[Out]

16*a**5/(b*(a - b*x)**2) - 80*a**4/(b*(a - b*x)) - 80*a**3*log(a - b*x)/b - 31*a
**2*x - 8*a*b*Integral(x, x) - b**2*x**3/3

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Mathematica [A]  time = 0.0677913, size = 73, normalized size = 1.03 \[ \frac{16 a^5}{b (b x-a)^2}+\frac{80 a^4}{b (b x-a)}-\frac{80 a^3 \log (a-b x)}{b}-31 a^2 x-4 a b x^2-\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^8/(a^2 - b^2*x^2)^3,x]

[Out]

-31*a^2*x - 4*a*b*x^2 - (b^2*x^3)/3 + (16*a^5)/(b*(-a + b*x)^2) + (80*a^4)/(b*(-
a + b*x)) - (80*a^3*Log[a - b*x])/b

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Maple [A]  time = 0.01, size = 73, normalized size = 1. \[ -{\frac{{b}^{2}{x}^{3}}{3}}-4\,ab{x}^{2}-31\,{a}^{2}x-80\,{\frac{{a}^{3}\ln \left ( bx-a \right ) }{b}}+80\,{\frac{{a}^{4}}{b \left ( bx-a \right ) }}+16\,{\frac{{a}^{5}}{b \left ( bx-a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^8/(-b^2*x^2+a^2)^3,x)

[Out]

-1/3*b^2*x^3-4*a*b*x^2-31*a^2*x-80/b*a^3*ln(b*x-a)+80*a^4/b/(b*x-a)+16*a^5/b/(b*
x-a)^2

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Maxima [A]  time = 0.68521, size = 101, normalized size = 1.42 \[ -\frac{1}{3} \, b^{2} x^{3} - 4 \, a b x^{2} - 31 \, a^{2} x - \frac{80 \, a^{3} \log \left (b x - a\right )}{b} + \frac{16 \,{\left (5 \, a^{4} b x - 4 \, a^{5}\right )}}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^8/(b^2*x^2 - a^2)^3,x, algorithm="maxima")

[Out]

-1/3*b^2*x^3 - 4*a*b*x^2 - 31*a^2*x - 80*a^3*log(b*x - a)/b + 16*(5*a^4*b*x - 4*
a^5)/(b^3*x^2 - 2*a*b^2*x + a^2*b)

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Fricas [A]  time = 0.209457, size = 143, normalized size = 2.01 \[ -\frac{b^{5} x^{5} + 10 \, a b^{4} x^{4} + 70 \, a^{2} b^{3} x^{3} - 174 \, a^{3} b^{2} x^{2} - 147 \, a^{4} b x + 192 \, a^{5} + 240 \,{\left (a^{3} b^{2} x^{2} - 2 \, a^{4} b x + a^{5}\right )} \log \left (b x - a\right )}{3 \,{\left (b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^8/(b^2*x^2 - a^2)^3,x, algorithm="fricas")

[Out]

-1/3*(b^5*x^5 + 10*a*b^4*x^4 + 70*a^2*b^3*x^3 - 174*a^3*b^2*x^2 - 147*a^4*b*x +
192*a^5 + 240*(a^3*b^2*x^2 - 2*a^4*b*x + a^5)*log(b*x - a))/(b^3*x^2 - 2*a*b^2*x
+ a^2*b)

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Sympy [A]  time = 1.98818, size = 70, normalized size = 0.99 \[ - \frac{80 a^{3} \log{\left (- a + b x \right )}}{b} - 31 a^{2} x - 4 a b x^{2} - \frac{b^{2} x^{3}}{3} + \frac{- 64 a^{5} + 80 a^{4} b x}{a^{2} b - 2 a b^{2} x + b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**8/(-b**2*x**2+a**2)**3,x)

[Out]

-80*a**3*log(-a + b*x)/b - 31*a**2*x - 4*a*b*x**2 - b**2*x**3/3 + (-64*a**5 + 80
*a**4*b*x)/(a**2*b - 2*a*b**2*x + b**3*x**2)

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GIAC/XCAS [A]  time = 0.218417, size = 103, normalized size = 1.45 \[ -\frac{80 \, a^{3}{\rm ln}\left ({\left | b x - a \right |}\right )}{b} + \frac{16 \,{\left (5 \, a^{4} b x - 4 \, a^{5}\right )}}{{\left (b x - a\right )}^{2} b} - \frac{b^{11} x^{3} + 12 \, a b^{10} x^{2} + 93 \, a^{2} b^{9} x}{3 \, b^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^8/(b^2*x^2 - a^2)^3,x, algorithm="giac")

[Out]

-80*a^3*ln(abs(b*x - a))/b + 16*(5*a^4*b*x - 4*a^5)/((b*x - a)^2*b) - 1/3*(b^11*
x^3 + 12*a*b^10*x^2 + 93*a^2*b^9*x)/b^9