3.173 \(\int \frac{x^5}{\left (a+b x^2\right )^3} \, dx\)

Optimal. Leaf size=49 \[ -\frac{a^2}{4 b^3 \left (a+b x^2\right )^2}+\frac{a}{b^3 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^3} \]

[Out]

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

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Rubi [A]  time = 0.0943527, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ -\frac{a^2}{4 b^3 \left (a+b x^2\right )^2}+\frac{a}{b^3 \left (a+b x^2\right )}+\frac{\log \left (a+b x^2\right )}{2 b^3} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^2)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 12.7735, size = 41, normalized size = 0.84 \[ - \frac{a^{2}}{4 b^{3} \left (a + b x^{2}\right )^{2}} + \frac{a}{b^{3} \left (a + b x^{2}\right )} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**2+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0283806, size = 39, normalized size = 0.8 \[ \frac{\frac{a \left (3 a+4 b x^2\right )}{\left (a+b x^2\right )^2}+2 \log \left (a+b x^2\right )}{4 b^3} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^2)^3,x]

[Out]

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

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Maple [A]  time = 0.013, size = 46, normalized size = 0.9 \[ -{\frac{{a}^{2}}{4\,{b}^{3} \left ( b{x}^{2}+a \right ) ^{2}}}+{\frac{a}{{b}^{3} \left ( b{x}^{2}+a \right ) }}+{\frac{\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^2+a)^3,x)

[Out]

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

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Maxima [A]  time = 1.35084, size = 74, normalized size = 1.51 \[ \frac{4 \, a b x^{2} + 3 \, a^{2}}{4 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} + \frac{\log \left (b x^{2} + a\right )}{2 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^2 + a)^3,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.22632, size = 93, normalized size = 1.9 \[ \frac{4 \, a b x^{2} + 3 \, a^{2} + 2 \,{\left (b^{2} x^{4} + 2 \, a b x^{2} + a^{2}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + a^{2} b^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^2 + a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.82636, size = 53, normalized size = 1.08 \[ \frac{3 a^{2} + 4 a b x^{2}}{4 a^{2} b^{3} + 8 a b^{4} x^{2} + 4 b^{5} x^{4}} + \frac{\log{\left (a + b x^{2} \right )}}{2 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**2+a)**3,x)

[Out]

(3*a**2 + 4*a*b*x**2)/(4*a**2*b**3 + 8*a*b**4*x**2 + 4*b**5*x**4) + log(a + b*x*
*2)/(2*b**3)

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GIAC/XCAS [A]  time = 0.221898, size = 57, normalized size = 1.16 \[ \frac{{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{3}} - \frac{3 \, b x^{4} + 2 \, a x^{2}}{4 \,{\left (b x^{2} + a\right )}^{2} b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^2 + a)^3,x, algorithm="giac")

[Out]

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