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

Optimal. Leaf size=44 \[ \frac{8 a^3}{b (a-b x)}+\frac{12 a^2 \log (a-b x)}{b}+5 a x+\frac{b x^2}{2} \]

[Out]

5*a*x + (b*x^2)/2 + (8*a^3)/(b*(a - b*x)) + (12*a^2*Log[a - b*x])/b

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Rubi [A]  time = 0.0881752, antiderivative size = 44, 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 \[ \frac{8 a^3}{b (a-b x)}+\frac{12 a^2 \log (a-b x)}{b}+5 a x+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

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

[Out]

5*a*x + (b*x^2)/2 + (8*a^3)/(b*(a - b*x)) + (12*a^2*Log[a - b*x])/b

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{8 a^{3}}{b \left (a - b x\right )} + \frac{12 a^{2} \log{\left (a - b x \right )}}{b} + 5 a x + b \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

8*a**3/(b*(a - b*x)) + 12*a**2*log(a - b*x)/b + 5*a*x + b*Integral(x, x)

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Mathematica [A]  time = 0.0400747, size = 45, normalized size = 1.02 \[ -\frac{8 a^3}{b (b x-a)}+\frac{12 a^2 \log (a-b x)}{b}+5 a x+\frac{b x^2}{2} \]

Antiderivative was successfully verified.

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

[Out]

5*a*x + (b*x^2)/2 - (8*a^3)/(b*(-a + b*x)) + (12*a^2*Log[a - b*x])/b

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Maple [A]  time = 0.009, size = 45, normalized size = 1. \[{\frac{b{x}^{2}}{2}}+5\,ax+12\,{\frac{{a}^{2}\ln \left ( bx-a \right ) }{b}}-8\,{\frac{{a}^{3}}{b \left ( bx-a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b*x^2+5*a*x+12/b*a^2*ln(b*x-a)-8/b*a^3/(b*x-a)

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Maxima [A]  time = 0.680898, size = 59, normalized size = 1.34 \[ \frac{1}{2} \, b x^{2} - \frac{8 \, a^{3}}{b^{2} x - a b} + 5 \, a x + \frac{12 \, a^{2} \log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*b*x^2 - 8*a^3/(b^2*x - a*b) + 5*a*x + 12*a^2*log(b*x - a)/b

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Fricas [A]  time = 0.211432, size = 88, normalized size = 2. \[ \frac{b^{3} x^{3} + 9 \, a b^{2} x^{2} - 10 \, a^{2} b x - 16 \, a^{3} + 24 \,{\left (a^{2} b x - a^{3}\right )} \log \left (b x - a\right )}{2 \,{\left (b^{2} x - a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(b^3*x^3 + 9*a*b^2*x^2 - 10*a^2*b*x - 16*a^3 + 24*(a^2*b*x - a^3)*log(b*x -
a))/(b^2*x - a*b)

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Sympy [A]  time = 1.49665, size = 37, normalized size = 0.84 \[ - \frac{8 a^{3}}{- a b + b^{2} x} + \frac{12 a^{2} \log{\left (- a + b x \right )}}{b} + 5 a x + \frac{b x^{2}}{2} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-8*a**3/(-a*b + b**2*x) + 12*a**2*log(-a + b*x)/b + 5*a*x + b*x**2/2

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GIAC/XCAS [A]  time = 0.216457, size = 74, normalized size = 1.68 \[ \frac{12 \, a^{2}{\rm ln}\left ({\left | b x - a \right |}\right )}{b} - \frac{8 \, a^{3}}{{\left (b x - a\right )} b} + \frac{b^{5} x^{2} + 10 \, a b^{4} x}{2 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

12*a^2*ln(abs(b*x - a))/b - 8*a^3/((b*x - a)*b) + 1/2*(b^5*x^2 + 10*a*b^4*x)/b^4