3.222 \(\int \frac{x^8}{\left (a x^2+b x^3\right )^2} \, dx\)

Optimal. Leaf size=58 \[ -\frac{a^4}{b^5 (a+b x)}-\frac{4 a^3 \log (a+b x)}{b^5}+\frac{3 a^2 x}{b^4}-\frac{a x^2}{b^3}+\frac{x^3}{3 b^2} \]

[Out]

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

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Rubi [A]  time = 0.0824263, antiderivative size = 58, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ -\frac{a^4}{b^5 (a+b x)}-\frac{4 a^3 \log (a+b x)}{b^5}+\frac{3 a^2 x}{b^4}-\frac{a x^2}{b^3}+\frac{x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a*x^2 + b*x^3)^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x**3+a*x**2)**2,x)

[Out]

-a**4/(b**5*(a + b*x)) - 4*a**3*log(a + b*x)/b**5 + 3*a**2*x/b**4 - 2*a*Integral
(x, x)/b**3 + x**3/(3*b**2)

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Mathematica [A]  time = 0.0327915, size = 54, normalized size = 0.93 \[ \frac{-\frac{3 a^4}{a+b x}-12 a^3 \log (a+b x)+9 a^2 b x-3 a b^2 x^2+b^3 x^3}{3 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a*x^2 + b*x^3)^2,x]

[Out]

(9*a^2*b*x - 3*a*b^2*x^2 + b^3*x^3 - (3*a^4)/(a + b*x) - 12*a^3*Log[a + b*x])/(3
*b^5)

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Maple [A]  time = 0.01, size = 57, normalized size = 1. \[ 3\,{\frac{{a}^{2}x}{{b}^{4}}}-{\frac{a{x}^{2}}{{b}^{3}}}+{\frac{{x}^{3}}{3\,{b}^{2}}}-{\frac{{a}^{4}}{{b}^{5} \left ( bx+a \right ) }}-4\,{\frac{{a}^{3}\ln \left ( bx+a \right ) }{{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^3+a*x^2)^2,x)

[Out]

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

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Maxima [A]  time = 1.37522, size = 80, normalized size = 1.38 \[ -\frac{a^{4}}{b^{6} x + a b^{5}} - \frac{4 \, a^{3} \log \left (b x + a\right )}{b^{5}} + \frac{b^{2} x^{3} - 3 \, a b x^{2} + 9 \, a^{2} x}{3 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^3 + a*x^2)^2,x, algorithm="maxima")

[Out]

-a^4/(b^6*x + a*b^5) - 4*a^3*log(b*x + a)/b^5 + 1/3*(b^2*x^3 - 3*a*b*x^2 + 9*a^2
*x)/b^4

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Fricas [A]  time = 0.209152, size = 99, normalized size = 1.71 \[ \frac{b^{4} x^{4} - 2 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} + 9 \, a^{3} b x - 3 \, a^{4} - 12 \,{\left (a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{3 \,{\left (b^{6} x + a b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^3 + a*x^2)^2,x, algorithm="fricas")

[Out]

1/3*(b^4*x^4 - 2*a*b^3*x^3 + 6*a^2*b^2*x^2 + 9*a^3*b*x - 3*a^4 - 12*(a^3*b*x + a
^4)*log(b*x + a))/(b^6*x + a*b^5)

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Sympy [A]  time = 1.41359, size = 54, normalized size = 0.93 \[ - \frac{a^{4}}{a b^{5} + b^{6} x} - \frac{4 a^{3} \log{\left (a + b x \right )}}{b^{5}} + \frac{3 a^{2} x}{b^{4}} - \frac{a x^{2}}{b^{3}} + \frac{x^{3}}{3 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**3+a*x**2)**2,x)

[Out]

-a**4/(a*b**5 + b**6*x) - 4*a**3*log(a + b*x)/b**5 + 3*a**2*x/b**4 - a*x**2/b**3
+ x**3/(3*b**2)

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GIAC/XCAS [A]  time = 0.218715, size = 84, normalized size = 1.45 \[ -\frac{4 \, a^{3}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5}} - \frac{a^{4}}{{\left (b x + a\right )} b^{5}} + \frac{b^{4} x^{3} - 3 \, a b^{3} x^{2} + 9 \, a^{2} b^{2} x}{3 \, b^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^3 + a*x^2)^2,x, algorithm="giac")

[Out]

-4*a^3*ln(abs(b*x + a))/b^5 - a^4/((b*x + a)*b^5) + 1/3*(b^4*x^3 - 3*a*b^3*x^2 +
9*a^2*b^2*x)/b^6