3.215 \(\int \frac{x^4}{a x^2+b x^3} \, dx\)

Optimal. Leaf size=31 \[ \frac{a^2 \log (a+b x)}{b^3}-\frac{a x}{b^2}+\frac{x^2}{2 b} \]

[Out]

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

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Rubi [A]  time = 0.0426045, antiderivative size = 31, 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 \[ \frac{a^2 \log (a+b x)}{b^3}-\frac{a x}{b^2}+\frac{x^2}{2 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a*x^2 + b*x^3),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**3+a*x**2),x)

[Out]

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

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Mathematica [A]  time = 0.00529348, size = 31, normalized size = 1. \[ \frac{a^2 \log (a+b x)}{b^3}-\frac{a x}{b^2}+\frac{x^2}{2 b} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a*x^2 + b*x^3),x]

[Out]

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

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Maple [A]  time = 0.003, size = 30, normalized size = 1. \[ -{\frac{ax}{{b}^{2}}}+{\frac{{x}^{2}}{2\,b}}+{\frac{{a}^{2}\ln \left ( bx+a \right ) }{{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^3+a*x^2),x)

[Out]

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

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Maxima [A]  time = 1.37902, size = 39, normalized size = 1.26 \[ \frac{a^{2} \log \left (b x + a\right )}{b^{3}} + \frac{b x^{2} - 2 \, a x}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^3 + a*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.206531, size = 39, normalized size = 1.26 \[ \frac{b^{2} x^{2} - 2 \, a b x + 2 \, a^{2} \log \left (b x + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^3 + a*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.09562, size = 26, normalized size = 0.84 \[ \frac{a^{2} \log{\left (a + b x \right )}}{b^{3}} - \frac{a x}{b^{2}} + \frac{x^{2}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**3+a*x**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.216817, size = 41, normalized size = 1.32 \[ \frac{a^{2}{\rm ln}\left ({\left | b x + a \right |}\right )}{b^{3}} + \frac{b x^{2} - 2 \, a x}{2 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^3 + a*x^2),x, algorithm="giac")

[Out]

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