3.746 \(\int \frac{(a+b x)^7}{\left (a^2-b^2 x^2\right )^2} \, dx\)

Optimal. Leaf size=70 \[ \frac{32 a^5}{b (a-b x)}+\frac{80 a^4 \log (a-b x)}{b}+49 a^3 x+\frac{23}{2} a^2 b x^2+\frac{7}{3} a b^2 x^3+\frac{b^3 x^4}{4} \]

[Out]

49*a^3*x + (23*a^2*b*x^2)/2 + (7*a*b^2*x^3)/3 + (b^3*x^4)/4 + (32*a^5)/(b*(a - b
*x)) + (80*a^4*Log[a - b*x])/b

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Rubi [A]  time = 0.126808, antiderivative size = 70, 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 \[ \frac{32 a^5}{b (a-b x)}+\frac{80 a^4 \log (a-b x)}{b}+49 a^3 x+\frac{23}{2} a^2 b x^2+\frac{7}{3} a b^2 x^3+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x)^7/(a^2 - b^2*x^2)^2,x]

[Out]

49*a^3*x + (23*a^2*b*x^2)/2 + (7*a*b^2*x^3)/3 + (b^3*x^4)/4 + (32*a^5)/(b*(a - b
*x)) + (80*a^4*Log[a - b*x])/b

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{32 a^{5}}{b \left (a - b x\right )} + \frac{80 a^{4} \log{\left (a - b x \right )}}{b} + 49 a^{3} x + 23 a^{2} b \int x\, dx + \frac{7 a b^{2} x^{3}}{3} + \frac{b^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**7/(-b**2*x**2+a**2)**2,x)

[Out]

32*a**5/(b*(a - b*x)) + 80*a**4*log(a - b*x)/b + 49*a**3*x + 23*a**2*b*Integral(
x, x) + 7*a*b**2*x**3/3 + b**3*x**4/4

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Mathematica [A]  time = 0.0505599, size = 71, normalized size = 1.01 \[ -\frac{32 a^5}{b (b x-a)}+\frac{80 a^4 \log (a-b x)}{b}+49 a^3 x+\frac{23}{2} a^2 b x^2+\frac{7}{3} a b^2 x^3+\frac{b^3 x^4}{4} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x)^7/(a^2 - b^2*x^2)^2,x]

[Out]

49*a^3*x + (23*a^2*b*x^2)/2 + (7*a*b^2*x^3)/3 + (b^3*x^4)/4 - (32*a^5)/(b*(-a +
b*x)) + (80*a^4*Log[a - b*x])/b

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Maple [A]  time = 0.01, size = 67, normalized size = 1. \[{\frac{{b}^{3}{x}^{4}}{4}}+{\frac{7\,a{b}^{2}{x}^{3}}{3}}+{\frac{23\,{a}^{2}b{x}^{2}}{2}}+49\,{a}^{3}x+80\,{\frac{{a}^{4}\ln \left ( bx-a \right ) }{b}}-32\,{\frac{{a}^{5}}{b \left ( bx-a \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^7/(-b^2*x^2+a^2)^2,x)

[Out]

1/4*b^3*x^4+7/3*a*b^2*x^3+23/2*a^2*b*x^2+49*a^3*x+80*a^4/b*ln(b*x-a)-32*a^5/b/(b
*x-a)

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Maxima [A]  time = 0.684259, size = 89, normalized size = 1.27 \[ \frac{1}{4} \, b^{3} x^{4} + \frac{7}{3} \, a b^{2} x^{3} + \frac{23}{2} \, a^{2} b x^{2} - \frac{32 \, a^{5}}{b^{2} x - a b} + 49 \, a^{3} x + \frac{80 \, a^{4} \log \left (b x - a\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/(b^2*x^2 - a^2)^2,x, algorithm="maxima")

[Out]

1/4*b^3*x^4 + 7/3*a*b^2*x^3 + 23/2*a^2*b*x^2 - 32*a^5/(b^2*x - a*b) + 49*a^3*x +
80*a^4*log(b*x - a)/b

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Fricas [A]  time = 0.223523, size = 119, normalized size = 1.7 \[ \frac{3 \, b^{5} x^{5} + 25 \, a b^{4} x^{4} + 110 \, a^{2} b^{3} x^{3} + 450 \, a^{3} b^{2} x^{2} - 588 \, a^{4} b x - 384 \, a^{5} + 960 \,{\left (a^{4} b x - a^{5}\right )} \log \left (b x - a\right )}{12 \,{\left (b^{2} x - a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/(b^2*x^2 - a^2)^2,x, algorithm="fricas")

[Out]

1/12*(3*b^5*x^5 + 25*a*b^4*x^4 + 110*a^2*b^3*x^3 + 450*a^3*b^2*x^2 - 588*a^4*b*x
- 384*a^5 + 960*(a^4*b*x - a^5)*log(b*x - a))/(b^2*x - a*b)

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Sympy [A]  time = 1.66252, size = 65, normalized size = 0.93 \[ - \frac{32 a^{5}}{- a b + b^{2} x} + \frac{80 a^{4} \log{\left (- a + b x \right )}}{b} + 49 a^{3} x + \frac{23 a^{2} b x^{2}}{2} + \frac{7 a b^{2} x^{3}}{3} + \frac{b^{3} x^{4}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**7/(-b**2*x**2+a**2)**2,x)

[Out]

-32*a**5/(-a*b + b**2*x) + 80*a**4*log(-a + b*x)/b + 49*a**3*x + 23*a**2*b*x**2/
2 + 7*a*b**2*x**3/3 + b**3*x**4/4

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GIAC/XCAS [A]  time = 0.216877, size = 105, normalized size = 1.5 \[ \frac{80 \, a^{4}{\rm ln}\left ({\left | b x - a \right |}\right )}{b} - \frac{32 \, a^{5}}{{\left (b x - a\right )} b} + \frac{3 \, b^{11} x^{4} + 28 \, a b^{10} x^{3} + 138 \, a^{2} b^{9} x^{2} + 588 \, a^{3} b^{8} x}{12 \, b^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)^7/(b^2*x^2 - a^2)^2,x, algorithm="giac")

[Out]

80*a^4*ln(abs(b*x - a))/b - 32*a^5/((b*x - a)*b) + 1/12*(3*b^11*x^4 + 28*a*b^10*
x^3 + 138*a^2*b^9*x^2 + 588*a^3*b^8*x)/b^8