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

Optimal. Leaf size=55 \[ \frac{16 a^4}{b (a-b x)}+\frac{32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3} \]

[Out]

17*a^2*x + 3*a*b*x^2 + (b^2*x^3)/3 + (16*a^4)/(b*(a - b*x)) + (32*a^3*Log[a - b*
x])/b

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Rubi [A]  time = 0.097854, antiderivative size = 55, 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 \[ \frac{16 a^4}{b (a-b x)}+\frac{32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

17*a^2*x + 3*a*b*x^2 + (b^2*x^3)/3 + (16*a^4)/(b*(a - b*x)) + (32*a^3*Log[a - b*
x])/b

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{16 a^{4}}{b \left (a - b x\right )} + \frac{32 a^{3} \log{\left (a - b x \right )}}{b} + 17 a^{2} x + 6 a b \int x\, dx + \frac{b^{2} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

16*a**4/(b*(a - b*x)) + 32*a**3*log(a - b*x)/b + 17*a**2*x + 6*a*b*Integral(x, x
) + b**2*x**3/3

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Mathematica [A]  time = 0.0569253, size = 56, normalized size = 1.02 \[ -\frac{16 a^4}{b (b x-a)}+\frac{32 a^3 \log (a-b x)}{b}+17 a^2 x+3 a b x^2+\frac{b^2 x^3}{3} \]

Antiderivative was successfully verified.

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

[Out]

17*a^2*x + 3*a*b*x^2 + (b^2*x^3)/3 - (16*a^4)/(b*(-a + b*x)) + (32*a^3*Log[a - b
*x])/b

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Maple [A]  time = 0.01, size = 56, normalized size = 1. \[{\frac{{b}^{2}{x}^{3}}{3}}+3\,ab{x}^{2}+17\,{a}^{2}x+32\,{\frac{{a}^{3}\ln \left ( bx-a \right ) }{b}}-16\,{\frac{{a}^{4}}{b \left ( bx-a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b^2*x^3+3*a*b*x^2+17*a^2*x+32/b*a^3*ln(b*x-a)-16*a^4/b/(b*x-a)

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Maxima [A]  time = 0.680885, size = 74, normalized size = 1.35 \[ \frac{1}{3} \, b^{2} x^{3} + 3 \, a b x^{2} - \frac{16 \, a^{4}}{b^{2} x - a b} + 17 \, a^{2} x + \frac{32 \, a^{3} \log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*b^2*x^3 + 3*a*b*x^2 - 16*a^4/(b^2*x - a*b) + 17*a^2*x + 32*a^3*log(b*x - a)/
b

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Fricas [A]  time = 0.215655, size = 103, normalized size = 1.87 \[ \frac{b^{4} x^{4} + 8 \, a b^{3} x^{3} + 42 \, a^{2} b^{2} x^{2} - 51 \, a^{3} b x - 48 \, a^{4} + 96 \,{\left (a^{3} b x - a^{4}\right )} \log \left (b x - a\right )}{3 \,{\left (b^{2} x - a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*(b^4*x^4 + 8*a*b^3*x^3 + 42*a^2*b^2*x^2 - 51*a^3*b*x - 48*a^4 + 96*(a^3*b*x
- a^4)*log(b*x - a))/(b^2*x - a*b)

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Sympy [A]  time = 1.56669, size = 49, normalized size = 0.89 \[ - \frac{16 a^{4}}{- a b + b^{2} x} + \frac{32 a^{3} \log{\left (- a + b x \right )}}{b} + 17 a^{2} x + 3 a b x^{2} + \frac{b^{2} x^{3}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-16*a**4/(-a*b + b**2*x) + 32*a**3*log(-a + b*x)/b + 17*a**2*x + 3*a*b*x**2 + b*
*2*x**3/3

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GIAC/XCAS [A]  time = 0.216618, size = 89, normalized size = 1.62 \[ \frac{32 \, a^{3}{\rm ln}\left ({\left | b x - a \right |}\right )}{b} - \frac{16 \, a^{4}}{{\left (b x - a\right )} b} + \frac{b^{8} x^{3} + 9 \, a b^{7} x^{2} + 51 \, a^{2} b^{6} x}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

32*a^3*ln(abs(b*x - a))/b - 16*a^4/((b*x - a)*b) + 1/3*(b^8*x^3 + 9*a*b^7*x^2 +
51*a^2*b^6*x)/b^6