3.758 \(\int \frac{(a+b x)^6}{\left (a^2-b^2 x^2\right )^3} \, dx\)

Optimal. Leaf size=49 \[ \frac{4 a^3}{b (a-b x)^2}-\frac{12 a^2}{b (a-b x)}-\frac{6 a \log (a-b x)}{b}-x \]

[Out]

-x + (4*a^3)/(b*(a - b*x)^2) - (12*a^2)/(b*(a - b*x)) - (6*a*Log[a - b*x])/b

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Rubi [A]  time = 0.0764532, antiderivative size = 49, 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 \[ \frac{4 a^3}{b (a-b x)^2}-\frac{12 a^2}{b (a-b x)}-\frac{6 a \log (a-b x)}{b}-x \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^6/(a^2 - b^2*x^2)^3,x]

[Out]

-x + (4*a^3)/(b*(a - b*x)^2) - (12*a^2)/(b*(a - b*x)) - (6*a*Log[a - b*x])/b

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Rubi in Sympy [A]  time = 15.6476, size = 37, normalized size = 0.76 \[ \frac{4 a^{3}}{b \left (a - b x\right )^{2}} - \frac{12 a^{2}}{b \left (a - b x\right )} - \frac{6 a \log{\left (a - b x \right )}}{b} - x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**6/(-b**2*x**2+a**2)**3,x)

[Out]

4*a**3/(b*(a - b*x)**2) - 12*a**2/(b*(a - b*x)) - 6*a*log(a - b*x)/b - x

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Mathematica [A]  time = 0.0845142, size = 41, normalized size = 0.84 \[ \frac{4 a^2 (3 b x-2 a)}{b (a-b x)^2}-\frac{6 a \log (a-b x)}{b}-x \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^6/(a^2 - b^2*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 53, normalized size = 1.1 \[ -x-6\,{\frac{a\ln \left ( bx-a \right ) }{b}}+12\,{\frac{{a}^{2}}{b \left ( bx-a \right ) }}+4\,{\frac{{a}^{3}}{b \left ( bx-a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6/(-b^2*x^2+a^2)^3,x)

[Out]

-x-6*a/b*ln(b*x-a)+12/b*a^2/(b*x-a)+4/b*a^3/(b*x-a)^2

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Maxima [A]  time = 0.685486, size = 74, normalized size = 1.51 \[ -x - \frac{6 \, a \log \left (b x - a\right )}{b} + \frac{4 \,{\left (3 \, a^{2} b x - 2 \, a^{3}\right )}}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^6/(b^2*x^2 - a^2)^3,x, algorithm="maxima")

[Out]

-x - 6*a*log(b*x - a)/b + 4*(3*a^2*b*x - 2*a^3)/(b^3*x^2 - 2*a*b^2*x + a^2*b)

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Fricas [A]  time = 0.21379, size = 111, normalized size = 2.27 \[ -\frac{b^{3} x^{3} - 2 \, a b^{2} x^{2} - 11 \, a^{2} b x + 8 \, a^{3} + 6 \,{\left (a b^{2} x^{2} - 2 \, a^{2} b x + a^{3}\right )} \log \left (b x - a\right )}{b^{3} x^{2} - 2 \, a b^{2} x + a^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^6/(b^2*x^2 - a^2)^3,x, algorithm="fricas")

[Out]

-(b^3*x^3 - 2*a*b^2*x^2 - 11*a^2*b*x + 8*a^3 + 6*(a*b^2*x^2 - 2*a^2*b*x + a^3)*l
og(b*x - a))/(b^3*x^2 - 2*a*b^2*x + a^2*b)

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Sympy [A]  time = 1.66382, size = 46, normalized size = 0.94 \[ - \frac{6 a \log{\left (- a + b x \right )}}{b} - x + \frac{- 8 a^{3} + 12 a^{2} b x}{a^{2} b - 2 a b^{2} x + b^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**6/(-b**2*x**2+a**2)**3,x)

[Out]

-6*a*log(-a + b*x)/b - x + (-8*a**3 + 12*a**2*b*x)/(a**2*b - 2*a*b**2*x + b**3*x
**2)

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GIAC/XCAS [A]  time = 0.219657, size = 62, normalized size = 1.27 \[ -x - \frac{6 \, a{\rm ln}\left ({\left | b x - a \right |}\right )}{b} + \frac{4 \,{\left (3 \, a^{2} b x - 2 \, a^{3}\right )}}{{\left (b x - a\right )}^{2} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^6/(b^2*x^2 - a^2)^3,x, algorithm="giac")

[Out]

-x - 6*a*ln(abs(b*x - a))/b + 4*(3*a^2*b*x - 2*a^3)/((b*x - a)^2*b)