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

Optimal. Leaf size=83 \[ -\frac{32 a^5 \log (a-b x)}{b}-16 a^4 x-\frac{4 a^3 (a+b x)^2}{b}-\frac{4 a^2 (a+b x)^3}{3 b}-\frac{a (a+b x)^4}{2 b}-\frac{(a+b x)^5}{5 b} \]

[Out]

-16*a^4*x - (4*a^3*(a + b*x)^2)/b - (4*a^2*(a + b*x)^3)/(3*b) - (a*(a + b*x)^4)/
(2*b) - (a + b*x)^5/(5*b) - (32*a^5*Log[a - b*x])/b

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Rubi [A]  time = 0.0815265, antiderivative size = 83, 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 \[ -\frac{32 a^5 \log (a-b x)}{b}-16 a^4 x-\frac{4 a^3 (a+b x)^2}{b}-\frac{4 a^2 (a+b x)^3}{3 b}-\frac{a (a+b x)^4}{2 b}-\frac{(a+b x)^5}{5 b} \]

Antiderivative was successfully verified.

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

[Out]

-16*a^4*x - (4*a^3*(a + b*x)^2)/b - (4*a^2*(a + b*x)^3)/(3*b) - (a*(a + b*x)^4)/
(2*b) - (a + b*x)^5/(5*b) - (32*a^5*Log[a - b*x])/b

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Rubi in Sympy [A]  time = 17.1518, size = 71, normalized size = 0.86 \[ - \frac{32 a^{5} \log{\left (a - b x \right )}}{b} - 16 a^{4} x - \frac{4 a^{3} \left (a + b x\right )^{2}}{b} - \frac{4 a^{2} \left (a + b x\right )^{3}}{3 b} - \frac{a \left (a + b x\right )^{4}}{2 b} - \frac{\left (a + b x\right )^{5}}{5 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32*a**5*log(a - b*x)/b - 16*a**4*x - 4*a**3*(a + b*x)**2/b - 4*a**2*(a + b*x)**
3/(3*b) - a*(a + b*x)**4/(2*b) - (a + b*x)**5/(5*b)

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Mathematica [A]  time = 0.0133411, size = 65, normalized size = 0.78 \[ -\frac{32 a^5 \log (a-b x)}{b}-31 a^4 x-13 a^3 b x^2-\frac{16}{3} a^2 b^2 x^3-\frac{3}{2} a b^3 x^4-\frac{b^4 x^5}{5} \]

Antiderivative was successfully verified.

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

[Out]

-31*a^4*x - 13*a^3*b*x^2 - (16*a^2*b^2*x^3)/3 - (3*a*b^3*x^4)/2 - (b^4*x^5)/5 -
(32*a^5*Log[a - b*x])/b

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Maple [A]  time = 0.005, size = 61, normalized size = 0.7 \[ -{\frac{{x}^{5}{b}^{4}}{5}}-{\frac{3\,{x}^{4}a{b}^{3}}{2}}-{\frac{16\,{x}^{3}{a}^{2}{b}^{2}}{3}}-13\,{a}^{3}b{x}^{2}-31\,{a}^{4}x-32\,{\frac{{a}^{5}\ln \left ( bx-a \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^6/(-b^2*x^2+a^2),x)

[Out]

-1/5*x^5*b^4-3/2*x^4*a*b^3-16/3*x^3*a^2*b^2-13*a^3*b*x^2-31*a^4*x-32*a^5/b*ln(b*
x-a)

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Maxima [A]  time = 0.679213, size = 81, normalized size = 0.98 \[ -\frac{1}{5} \, b^{4} x^{5} - \frac{3}{2} \, a b^{3} x^{4} - \frac{16}{3} \, a^{2} b^{2} x^{3} - 13 \, a^{3} b x^{2} - 31 \, a^{4} x - \frac{32 \, a^{5} \log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/5*b^4*x^5 - 3/2*a*b^3*x^4 - 16/3*a^2*b^2*x^3 - 13*a^3*b*x^2 - 31*a^4*x - 32*a
^5*log(b*x - a)/b

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Fricas [A]  time = 0.213309, size = 88, normalized size = 1.06 \[ -\frac{6 \, b^{5} x^{5} + 45 \, a b^{4} x^{4} + 160 \, a^{2} b^{3} x^{3} + 390 \, a^{3} b^{2} x^{2} + 930 \, a^{4} b x + 960 \, a^{5} \log \left (b x - a\right )}{30 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(6*b^5*x^5 + 45*a*b^4*x^4 + 160*a^2*b^3*x^3 + 390*a^3*b^2*x^2 + 930*a^4*b*
x + 960*a^5*log(b*x - a))/b

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Sympy [A]  time = 1.39712, size = 65, normalized size = 0.78 \[ - \frac{32 a^{5} \log{\left (- a + b x \right )}}{b} - 31 a^{4} x - 13 a^{3} b x^{2} - \frac{16 a^{2} b^{2} x^{3}}{3} - \frac{3 a b^{3} x^{4}}{2} - \frac{b^{4} x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32*a**5*log(-a + b*x)/b - 31*a**4*x - 13*a**3*b*x**2 - 16*a**2*b**2*x**3/3 - 3*
a*b**3*x**4/2 - b**4*x**5/5

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GIAC/XCAS [A]  time = 0.215259, size = 97, normalized size = 1.17 \[ -\frac{32 \, a^{5}{\rm ln}\left ({\left | b x - a \right |}\right )}{b} - \frac{6 \, b^{9} x^{5} + 45 \, a b^{8} x^{4} + 160 \, a^{2} b^{7} x^{3} + 390 \, a^{3} b^{6} x^{2} + 930 \, a^{4} b^{5} x}{30 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-32*a^5*ln(abs(b*x - a))/b - 1/30*(6*b^9*x^5 + 45*a*b^8*x^4 + 160*a^2*b^7*x^3 +
390*a^3*b^6*x^2 + 930*a^4*b^5*x)/b^5