3.477 \(\int \frac{x^7}{a^2+2 a b x^2+b^2 x^4} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.12208, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{a^3}{2 b^4 \left (a+b x^2\right )}+\frac{3 a^2 \log \left (a+b x^2\right )}{2 b^4}-\frac{a x^2}{b^3}+\frac{x^4}{4 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^7/(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**7/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

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

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Mathematica [A]  time = 0.0272952, size = 49, normalized size = 0.86 \[ \frac{\frac{2 a^3}{a+b x^2}+6 a^2 \log \left (a+b x^2\right )-4 a b x^2+b^2 x^4}{4 b^4} \]

Antiderivative was successfully verified.

[In]  Integrate[x^7/(a^2 + 2*a*b*x^2 + b^2*x^4),x]

[Out]

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

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Maple [A]  time = 0.01, size = 52, normalized size = 0.9 \[ -{\frac{a{x}^{2}}{{b}^{3}}}+{\frac{{x}^{4}}{4\,{b}^{2}}}+{\frac{{a}^{3}}{2\,{b}^{4} \left ( b{x}^{2}+a \right ) }}+{\frac{3\,{a}^{2}\ln \left ( b{x}^{2}+a \right ) }{2\,{b}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^7/(b^2*x^4+2*a*b*x^2+a^2),x)

[Out]

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

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Maxima [A]  time = 0.682896, size = 73, normalized size = 1.28 \[ \frac{a^{3}}{2 \,{\left (b^{5} x^{2} + a b^{4}\right )}} + \frac{3 \, a^{2} \log \left (b x^{2} + a\right )}{2 \, b^{4}} + \frac{b x^{4} - 4 \, a x^{2}}{4 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.254004, size = 95, normalized size = 1.67 \[ \frac{b^{3} x^{6} - 3 \, a b^{2} x^{4} - 4 \, a^{2} b x^{2} + 2 \, a^{3} + 6 \,{\left (a^{2} b x^{2} + a^{3}\right )} \log \left (b x^{2} + a\right )}{4 \,{\left (b^{5} x^{2} + a b^{4}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.47057, size = 53, normalized size = 0.93 \[ \frac{a^{3}}{2 a b^{4} + 2 b^{5} x^{2}} + \frac{3 a^{2} \log{\left (a + b x^{2} \right )}}{2 b^{4}} - \frac{a x^{2}}{b^{3}} + \frac{x^{4}}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**7/(b**2*x**4+2*a*b*x**2+a**2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.269771, size = 90, normalized size = 1.58 \[ \frac{3 \, a^{2}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, b^{4}} + \frac{b^{2} x^{4} - 4 \, a b x^{2}}{4 \, b^{4}} - \frac{3 \, a^{2} b x^{2} + 2 \, a^{3}}{2 \,{\left (b x^{2} + a\right )} b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^7/(b^2*x^4 + 2*a*b*x^2 + a^2),x, algorithm="giac")

[Out]

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