3.737 \(\int \frac{(a+b x)^5}{a^2-b^2 x^2} \, dx\)

Optimal. Leaf size=66 \[ -\frac{16 a^4 \log (a-b x)}{b}-8 a^3 x-\frac{2 a^2 (a+b x)^2}{b}-\frac{2 a (a+b x)^3}{3 b}-\frac{(a+b x)^4}{4 b} \]

[Out]

-8*a^3*x - (2*a^2*(a + b*x)^2)/b - (2*a*(a + b*x)^3)/(3*b) - (a + b*x)^4/(4*b) -
(16*a^4*Log[a - b*x])/b

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Rubi [A]  time = 0.0671753, antiderivative size = 66, 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 \[ -\frac{16 a^4 \log (a-b x)}{b}-8 a^3 x-\frac{2 a^2 (a+b x)^2}{b}-\frac{2 a (a+b x)^3}{3 b}-\frac{(a+b x)^4}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^5/(a^2 - b^2*x^2),x]

[Out]

-8*a^3*x - (2*a^2*(a + b*x)^2)/b - (2*a*(a + b*x)^3)/(3*b) - (a + b*x)^4/(4*b) -
(16*a^4*Log[a - b*x])/b

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Rubi in Sympy [A]  time = 14.7718, size = 58, normalized size = 0.88 \[ - \frac{16 a^{4} \log{\left (a - b x \right )}}{b} - 8 a^{3} x - \frac{2 a^{2} \left (a + b x\right )^{2}}{b} - \frac{2 a \left (a + b x\right )^{3}}{3 b} - \frac{\left (a + b x\right )^{4}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**5/(-b**2*x**2+a**2),x)

[Out]

-16*a**4*log(a - b*x)/b - 8*a**3*x - 2*a**2*(a + b*x)**2/b - 2*a*(a + b*x)**3/(3
*b) - (a + b*x)**4/(4*b)

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Mathematica [A]  time = 0.0112669, size = 54, normalized size = 0.82 \[ -\frac{16 a^4 \log (a-b x)}{b}-15 a^3 x-\frac{11}{2} a^2 b x^2-\frac{5}{3} a b^2 x^3-\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^5/(a^2 - b^2*x^2),x]

[Out]

-15*a^3*x - (11*a^2*b*x^2)/2 - (5*a*b^2*x^3)/3 - (b^3*x^4)/4 - (16*a^4*Log[a - b
*x])/b

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Maple [A]  time = 0.005, size = 50, normalized size = 0.8 \[ -{\frac{{b}^{3}{x}^{4}}{4}}-{\frac{5\,a{b}^{2}{x}^{3}}{3}}-{\frac{11\,{a}^{2}b{x}^{2}}{2}}-15\,{a}^{3}x-16\,{\frac{{a}^{4}\ln \left ( bx-a \right ) }{b}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^5/(-b^2*x^2+a^2),x)

[Out]

-1/4*b^3*x^4-5/3*a*b^2*x^3-11/2*a^2*b*x^2-15*a^3*x-16*a^4/b*ln(b*x-a)

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Maxima [A]  time = 0.690689, size = 66, normalized size = 1. \[ -\frac{1}{4} \, b^{3} x^{4} - \frac{5}{3} \, a b^{2} x^{3} - \frac{11}{2} \, a^{2} b x^{2} - 15 \, a^{3} x - \frac{16 \, a^{4} \log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^5/(b^2*x^2 - a^2),x, algorithm="maxima")

[Out]

-1/4*b^3*x^4 - 5/3*a*b^2*x^3 - 11/2*a^2*b*x^2 - 15*a^3*x - 16*a^4*log(b*x - a)/b

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Fricas [A]  time = 0.211825, size = 73, normalized size = 1.11 \[ -\frac{3 \, b^{4} x^{4} + 20 \, a b^{3} x^{3} + 66 \, a^{2} b^{2} x^{2} + 180 \, a^{3} b x + 192 \, a^{4} \log \left (b x - a\right )}{12 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^5/(b^2*x^2 - a^2),x, algorithm="fricas")

[Out]

-1/12*(3*b^4*x^4 + 20*a*b^3*x^3 + 66*a^2*b^2*x^2 + 180*a^3*b*x + 192*a^4*log(b*x
- a))/b

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Sympy [A]  time = 1.3761, size = 53, normalized size = 0.8 \[ - \frac{16 a^{4} \log{\left (- a + b x \right )}}{b} - 15 a^{3} x - \frac{11 a^{2} b x^{2}}{2} - \frac{5 a b^{2} x^{3}}{3} - \frac{b^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**5/(-b**2*x**2+a**2),x)

[Out]

-16*a**4*log(-a + b*x)/b - 15*a**3*x - 11*a**2*b*x**2/2 - 5*a*b**2*x**3/3 - b**3
*x**4/4

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GIAC/XCAS [A]  time = 0.215304, size = 82, normalized size = 1.24 \[ -\frac{16 \, a^{4}{\rm ln}\left ({\left | b x - a \right |}\right )}{b} - \frac{3 \, b^{7} x^{4} + 20 \, a b^{6} x^{3} + 66 \, a^{2} b^{5} x^{2} + 180 \, a^{3} b^{4} x}{12 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(-(b*x + a)^5/(b^2*x^2 - a^2),x, algorithm="giac")

[Out]

-16*a^4*ln(abs(b*x - a))/b - 1/12*(3*b^7*x^4 + 20*a*b^6*x^3 + 66*a^2*b^5*x^2 + 1
80*a^3*b^4*x)/b^4