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

Optimal. Leaf size=57 \[ \frac{a^4 \log (a+b x)}{b^5}-\frac{a^3 x}{b^4}+\frac{a^2 x^2}{2 b^3}-\frac{a x^3}{3 b^2}+\frac{x^4}{4 b} \]

[Out]

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

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Rubi [A]  time = 0.0569483, antiderivative size = 57, 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 \[ \frac{a^4 \log (a+b x)}{b^5}-\frac{a^3 x}{b^4}+\frac{a^2 x^2}{2 b^3}-\frac{a x^3}{3 b^2}+\frac{x^4}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.00613247, size = 57, normalized size = 1. \[ \frac{a^4 \log (a+b x)}{b^5}-\frac{a^3 x}{b^4}+\frac{a^2 x^2}{2 b^3}-\frac{a x^3}{3 b^2}+\frac{x^4}{4 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x),x]

[Out]

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

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Maple [A]  time = 0.004, size = 52, normalized size = 0.9 \[ -{\frac{{a}^{3}x}{{b}^{4}}}+{\frac{{a}^{2}{x}^{2}}{2\,{b}^{3}}}-{\frac{a{x}^{3}}{3\,{b}^{2}}}+{\frac{{x}^{4}}{4\,b}}+{\frac{{a}^{4}\ln \left ( bx+a \right ) }{{b}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x+a),x)

[Out]

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

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Maxima [A]  time = 1.33445, size = 70, normalized size = 1.23 \[ \frac{a^{4} \log \left (b x + a\right )}{b^{5}} + \frac{3 \, b^{3} x^{4} - 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} - 12 \, a^{3} x}{12 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.189942, size = 70, normalized size = 1.23 \[ \frac{3 \, b^{4} x^{4} - 4 \, a b^{3} x^{3} + 6 \, a^{2} b^{2} x^{2} - 12 \, a^{3} b x + 12 \, a^{4} \log \left (b x + a\right )}{12 \, b^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.13154, size = 49, normalized size = 0.86 \[ \frac{a^{4} \log{\left (a + b x \right )}}{b^{5}} - \frac{a^{3} x}{b^{4}} + \frac{a^{2} x^{2}}{2 b^{3}} - \frac{a x^{3}}{3 b^{2}} + \frac{x^{4}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x+a),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.22616, size = 72, normalized size = 1.26 \[ \frac{a^{4}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{5}} + \frac{3 \, b^{3} x^{4} - 4 \, a b^{2} x^{3} + 6 \, a^{2} b x^{2} - 12 \, a^{3} x}{12 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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