3.142 \(\int \frac{1}{x^7 \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=63 \[ \frac{b^3 \log \left (a+b x^2\right )}{2 a^4}-\frac{b^3 \log (x)}{a^4}-\frac{b^2}{2 a^3 x^2}+\frac{b}{4 a^2 x^4}-\frac{1}{6 a x^6} \]

[Out]

-1/(6*a*x^6) + b/(4*a^2*x^4) - b^2/(2*a^3*x^2) - (b^3*Log[x])/a^4 + (b^3*Log[a +
b*x^2])/(2*a^4)

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Rubi [A]  time = 0.0885751, antiderivative size = 63, 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 \[ \frac{b^3 \log \left (a+b x^2\right )}{2 a^4}-\frac{b^3 \log (x)}{a^4}-\frac{b^2}{2 a^3 x^2}+\frac{b}{4 a^2 x^4}-\frac{1}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^7*(a + b*x^2)),x]

[Out]

-1/(6*a*x^6) + b/(4*a^2*x^4) - b^2/(2*a^3*x^2) - (b^3*Log[x])/a^4 + (b^3*Log[a +
b*x^2])/(2*a^4)

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Rubi in Sympy [A]  time = 13.9522, size = 60, normalized size = 0.95 \[ - \frac{1}{6 a x^{6}} + \frac{b}{4 a^{2} x^{4}} - \frac{b^{2}}{2 a^{3} x^{2}} - \frac{b^{3} \log{\left (x^{2} \right )}}{2 a^{4}} + \frac{b^{3} \log{\left (a + b x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**7/(b*x**2+a),x)

[Out]

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

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Mathematica [A]  time = 0.0127059, size = 63, normalized size = 1. \[ \frac{b^3 \log \left (a+b x^2\right )}{2 a^4}-\frac{b^3 \log (x)}{a^4}-\frac{b^2}{2 a^3 x^2}+\frac{b}{4 a^2 x^4}-\frac{1}{6 a x^6} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^7*(a + b*x^2)),x]

[Out]

-1/(6*a*x^6) + b/(4*a^2*x^4) - b^2/(2*a^3*x^2) - (b^3*Log[x])/a^4 + (b^3*Log[a +
b*x^2])/(2*a^4)

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Maple [A]  time = 0.01, size = 56, normalized size = 0.9 \[ -{\frac{1}{6\,a{x}^{6}}}+{\frac{b}{4\,{a}^{2}{x}^{4}}}-{\frac{{b}^{2}}{2\,{a}^{3}{x}^{2}}}-{\frac{{b}^{3}\ln \left ( x \right ) }{{a}^{4}}}+{\frac{{b}^{3}\ln \left ( b{x}^{2}+a \right ) }{2\,{a}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^7/(b*x^2+a),x)

[Out]

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

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Maxima [A]  time = 1.35327, size = 78, normalized size = 1.24 \[ \frac{b^{3} \log \left (b x^{2} + a\right )}{2 \, a^{4}} - \frac{b^{3} \log \left (x^{2}\right )}{2 \, a^{4}} - \frac{6 \, b^{2} x^{4} - 3 \, a b x^{2} + 2 \, a^{2}}{12 \, a^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*x^7),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.20758, size = 78, normalized size = 1.24 \[ \frac{6 \, b^{3} x^{6} \log \left (b x^{2} + a\right ) - 12 \, b^{3} x^{6} \log \left (x\right ) - 6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, a^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*x^7),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.18698, size = 56, normalized size = 0.89 \[ - \frac{2 a^{2} - 3 a b x^{2} + 6 b^{2} x^{4}}{12 a^{3} x^{6}} - \frac{b^{3} \log{\left (x \right )}}{a^{4}} + \frac{b^{3} \log{\left (\frac{a}{b} + x^{2} \right )}}{2 a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**7/(b*x**2+a),x)

[Out]

-(2*a**2 - 3*a*b*x**2 + 6*b**2*x**4)/(12*a**3*x**6) - b**3*log(x)/a**4 + b**3*lo
g(a/b + x**2)/(2*a**4)

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GIAC/XCAS [A]  time = 0.209489, size = 95, normalized size = 1.51 \[ -\frac{b^{3}{\rm ln}\left (x^{2}\right )}{2 \, a^{4}} + \frac{b^{3}{\rm ln}\left ({\left | b x^{2} + a \right |}\right )}{2 \, a^{4}} + \frac{11 \, b^{3} x^{6} - 6 \, a b^{2} x^{4} + 3 \, a^{2} b x^{2} - 2 \, a^{3}}{12 \, a^{4} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*x^2 + a)*x^7),x, algorithm="giac")

[Out]

-1/2*b^3*ln(x^2)/a^4 + 1/2*b^3*ln(abs(b*x^2 + a))/a^4 + 1/12*(11*b^3*x^6 - 6*a*b
^2*x^4 + 3*a^2*b*x^2 - 2*a^3)/(a^4*x^6)