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

Optimal. Leaf size=44 \[ \frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

[Out]

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

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Rubi [A]  time = 0.0786467, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b \sqrt{a^2+2 a b x^2+b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

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Rubi in Sympy [A]  time = 3.98391, size = 29, normalized size = 0.66 \[ \frac{\sqrt{\left (a + b x^{2}\right )^{2}} \log{\left (a + b x^{2} \right )}}{2 b \left (a + b x^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt((a + b*x**2)**2)*log(a + b*x**2)/(2*b*(a + b*x**2))

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Mathematica [A]  time = 0.0118819, size = 35, normalized size = 0.8 \[ \frac{\left (a+b x^2\right ) \log \left (a+b x^2\right )}{2 b \sqrt{\left (a+b x^2\right )^2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x/Sqrt[a^2 + 2*a*b*x^2 + b^2*x^4],x]

[Out]

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

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Maple [A]  time = 0.007, size = 32, normalized size = 0.7 \[{\frac{ \left ( b{x}^{2}+a \right ) \ln \left ( b{x}^{2}+a \right ) }{2\,b}{\frac{1}{\sqrt{ \left ( b{x}^{2}+a \right ) ^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/((b*x^2+a)^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 0.698751, size = 23, normalized size = 0.52 \[ \frac{1}{2} \, \sqrt{\frac{1}{b^{2}}} \log \left (x^{2} + \frac{a}{b}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(b^(-2))*log(x^2 + a/b)

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Fricas [A]  time = 0.258657, size = 18, normalized size = 0.41 \[ \frac{\log \left (b x^{2} + a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*log(b*x^2 + a)/b

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Sympy [A]  time = 0.292081, size = 10, normalized size = 0.23 \[ \frac{\log{\left (a + b x^{2} \right )}}{2 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(a + b*x**2)/(2*b)

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GIAC/XCAS [A]  time = 0.270073, size = 30, normalized size = 0.68 \[ \frac{{\rm ln}\left ({\left | b x^{2} + a \right |}\right ){\rm sign}\left (b x^{2} + a\right )}{2 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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