3.197 \(\int \frac{x^4}{(a+b x)^4} \, dx\)

Optimal. Leaf size=65 \[ -\frac{a^4}{3 b^5 (a+b x)^3}+\frac{2 a^3}{b^5 (a+b x)^2}-\frac{6 a^2}{b^5 (a+b x)}-\frac{4 a \log (a+b x)}{b^5}+\frac{x}{b^4} \]

[Out]

x/b^4 - a^4/(3*b^5*(a + b*x)^3) + (2*a^3)/(b^5*(a + b*x)^2) - (6*a^2)/(b^5*(a +
b*x)) - (4*a*Log[a + b*x])/b^5

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Rubi [A]  time = 0.080038, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ -\frac{a^4}{3 b^5 (a+b x)^3}+\frac{2 a^3}{b^5 (a+b x)^2}-\frac{6 a^2}{b^5 (a+b x)}-\frac{4 a \log (a+b x)}{b^5}+\frac{x}{b^4} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x)^4,x]

[Out]

x/b^4 - a^4/(3*b^5*(a + b*x)^3) + (2*a^3)/(b^5*(a + b*x)^2) - (6*a^2)/(b^5*(a +
b*x)) - (4*a*Log[a + b*x])/b^5

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{a^{4}}{3 b^{5} \left (a + b x\right )^{3}} + \frac{2 a^{3}}{b^{5} \left (a + b x\right )^{2}} - \frac{6 a^{2}}{b^{5} \left (a + b x\right )} - \frac{4 a \log{\left (a + b x \right )}}{b^{5}} + \int \frac{1}{b^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x+a)**4,x)

[Out]

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

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Mathematica [A]  time = 0.08023, size = 51, normalized size = 0.78 \[ -\frac{\frac{a^2 \left (13 a^2+30 a b x+18 b^2 x^2\right )}{(a+b x)^3}+12 a \log (a+b x)-3 b x}{3 b^5} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x)^4,x]

[Out]

-(-3*b*x + (a^2*(13*a^2 + 30*a*b*x + 18*b^2*x^2))/(a + b*x)^3 + 12*a*Log[a + b*x
])/(3*b^5)

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Maple [A]  time = 0.01, size = 64, normalized size = 1. \[{\frac{x}{{b}^{4}}}-{\frac{{a}^{4}}{3\,{b}^{5} \left ( bx+a \right ) ^{3}}}+2\,{\frac{{a}^{3}}{{b}^{5} \left ( bx+a \right ) ^{2}}}-6\,{\frac{{a}^{2}}{{b}^{5} \left ( bx+a \right ) }}-4\,{\frac{a\ln \left ( bx+a \right ) }{{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x+a)^4,x)

[Out]

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

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Maxima [A]  time = 1.33958, size = 107, normalized size = 1.65 \[ -\frac{18 \, a^{2} b^{2} x^{2} + 30 \, a^{3} b x + 13 \, a^{4}}{3 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} + \frac{x}{b^{4}} - \frac{4 \, a \log \left (b x + a\right )}{b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x + a)^4,x, algorithm="maxima")

[Out]

-1/3*(18*a^2*b^2*x^2 + 30*a^3*b*x + 13*a^4)/(b^8*x^3 + 3*a*b^7*x^2 + 3*a^2*b^6*x
+ a^3*b^5) + x/b^4 - 4*a*log(b*x + a)/b^5

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Fricas [A]  time = 0.216938, size = 157, normalized size = 2.42 \[ \frac{3 \, b^{4} x^{4} + 9 \, a b^{3} x^{3} - 9 \, a^{2} b^{2} x^{2} - 27 \, a^{3} b x - 13 \, a^{4} - 12 \,{\left (a b^{3} x^{3} + 3 \, a^{2} b^{2} x^{2} + 3 \, a^{3} b x + a^{4}\right )} \log \left (b x + a\right )}{3 \,{\left (b^{8} x^{3} + 3 \, a b^{7} x^{2} + 3 \, a^{2} b^{6} x + a^{3} b^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x + a)^4,x, algorithm="fricas")

[Out]

1/3*(3*b^4*x^4 + 9*a*b^3*x^3 - 9*a^2*b^2*x^2 - 27*a^3*b*x - 13*a^4 - 12*(a*b^3*x
^3 + 3*a^2*b^2*x^2 + 3*a^3*b*x + a^4)*log(b*x + a))/(b^8*x^3 + 3*a*b^7*x^2 + 3*a
^2*b^6*x + a^3*b^5)

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Sympy [A]  time = 2.00459, size = 80, normalized size = 1.23 \[ - \frac{4 a \log{\left (a + b x \right )}}{b^{5}} - \frac{13 a^{4} + 30 a^{3} b x + 18 a^{2} b^{2} x^{2}}{3 a^{3} b^{5} + 9 a^{2} b^{6} x + 9 a b^{7} x^{2} + 3 b^{8} x^{3}} + \frac{x}{b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x+a)**4,x)

[Out]

-4*a*log(a + b*x)/b**5 - (13*a**4 + 30*a**3*b*x + 18*a**2*b**2*x**2)/(3*a**3*b**
5 + 9*a**2*b**6*x + 9*a*b**7*x**2 + 3*b**8*x**3) + x/b**4

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GIAC/XCAS [A]  time = 0.206389, size = 74, normalized size = 1.14 \[ \frac{x}{b^{4}} - \frac{4 \, a{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5}} - \frac{18 \, a^{2} b^{2} x^{2} + 30 \, a^{3} b x + 13 \, a^{4}}{3 \,{\left (b x + a\right )}^{3} b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x + a)^4,x, algorithm="giac")

[Out]

x/b^4 - 4*a*ln(abs(b*x + a))/b^5 - 1/3*(18*a^2*b^2*x^2 + 30*a^3*b*x + 13*a^4)/((
b*x + a)^3*b^5)