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

Optimal. Leaf size=18 \[ \frac{\log \left (\sqrt{a+b x^2}+1\right )}{b} \]

[Out]

Log[1 + Sqrt[a + b*x^2]]/b

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Rubi [A]  time = 0.115315, antiderivative size = 18, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{\log \left (\sqrt{a+b x^2}+1\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

Log[1 + Sqrt[a + b*x^2]]/b

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Rubi in Sympy [A]  time = 4.47737, size = 14, normalized size = 0.78 \[ \frac{\log{\left (\sqrt{a + b x^{2}} + 1 \right )}}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sqrt(a + b*x**2) + 1)/b

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Mathematica [A]  time = 0.0170817, size = 18, normalized size = 1. \[ \frac{\log \left (\sqrt{a+b x^2}+1\right )}{b} \]

Antiderivative was successfully verified.

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

[Out]

Log[1 + Sqrt[a + b*x^2]]/b

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Maple [B]  time = 0.057, size = 1059, normalized size = 58.8 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 0.728139, size = 22, normalized size = 1.22 \[ \frac{\log \left (\sqrt{b x^{2} + a} + 1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

log(sqrt(b*x^2 + a) + 1)/b

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Fricas [A]  time = 0.276914, size = 90, normalized size = 5. \[ \frac{2 \, \log \left (b x^{2} + a - 1\right ) + \log \left (\frac{b x^{2} + a + 2 \, \sqrt{b x^{2} + a} + 1}{x^{2}}\right ) - \log \left (\frac{b x^{2} + a - 2 \, \sqrt{b x^{2} + a} + 1}{x^{2}}\right )}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x}{a + b x^{2} + \sqrt{a + b x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x/(a + b*x**2 + sqrt(a + b*x**2)), x)

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GIAC/XCAS [A]  time = 0.261379, size = 22, normalized size = 1.22 \[ \frac{{\rm ln}\left (\sqrt{b x^{2} + a} + 1\right )}{b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

ln(sqrt(b*x^2 + a) + 1)/b