3.674 \(\int \frac{1}{x \sqrt{d x^2} \left (a+b x^2\right )} \, dx\)

Optimal. Leaf size=50 \[ -\frac{\sqrt{b} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{d x^2}}-\frac{1}{a \sqrt{d x^2}} \]

[Out]

-(1/(a*Sqrt[d*x^2])) - (Sqrt[b]*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[d*x
^2])

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Rubi [A]  time = 0.0418573, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{\sqrt{b} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{a^{3/2} \sqrt{d x^2}}-\frac{1}{a \sqrt{d x^2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x*Sqrt[d*x^2]*(a + b*x^2)),x]

[Out]

-(1/(a*Sqrt[d*x^2])) - (Sqrt[b]*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(a^(3/2)*Sqrt[d*x
^2])

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Rubi in Sympy [A]  time = 17.7112, size = 53, normalized size = 1.06 \[ - \frac{1}{a \sqrt{d x^{2}}} - \frac{\sqrt{b} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d x^{2}}}{\sqrt{a} \sqrt{d}} \right )}}{a^{\frac{3}{2}} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(a*sqrt(d*x**2)) - sqrt(b)*atan(sqrt(b)*sqrt(d*x**2)/(sqrt(a)*sqrt(d)))/(a**(
3/2)*sqrt(d))

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Mathematica [A]  time = 0.0273057, size = 46, normalized size = 0.92 \[ -\frac{d x^2 \left (\sqrt{b} x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )+\sqrt{a}\right )}{a^{3/2} \left (d x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

-((d*x^2*(Sqrt[a] + Sqrt[b]*x*ArcTan[(Sqrt[b]*x)/Sqrt[a]]))/(a^(3/2)*(d*x^2)^(3/
2)))

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Maple [A]  time = 0.008, size = 36, normalized size = 0.7 \[ -{\frac{1}{a} \left ( b\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ) x+\sqrt{ab} \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-(b*arctan(x*b/(a*b)^(1/2))*x+(a*b)^(1/2))/(d*x^2)^(1/2)/a/(a*b)^(1/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24417, size = 1, normalized size = 0.02 \[ \left [\frac{d x^{2} \sqrt{-\frac{b}{a d}} \log \left (-\frac{2 \, a d x^{2} \sqrt{-\frac{b}{a d}} -{\left (b x^{2} - a\right )} \sqrt{d x^{2}}}{b x^{3} + a x}\right ) - 2 \, \sqrt{d x^{2}}}{2 \, a d x^{2}}, -\frac{d x^{2} \sqrt{\frac{b}{a d}} \arctan \left (\frac{\sqrt{d x^{2}} b}{a d \sqrt{\frac{b}{a d}}}\right ) + \sqrt{d x^{2}}}{a d x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(d*x^2*sqrt(-b/(a*d))*log(-(2*a*d*x^2*sqrt(-b/(a*d)) - (b*x^2 - a)*sqrt(d*x
^2))/(b*x^3 + a*x)) - 2*sqrt(d*x^2))/(a*d*x^2), -(d*x^2*sqrt(b/(a*d))*arctan(sqr
t(d*x^2)*b/(a*d*sqrt(b/(a*d)))) + sqrt(d*x^2))/(a*d*x^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x \sqrt{d x^{2}} \left (a + b x^{2}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x*sqrt(d*x**2)*(a + b*x**2)), x)

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GIAC/XCAS [A]  time = 0.230399, size = 65, normalized size = 1.3 \[ -d{\left (\frac{b \arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d} a d} + \frac{1}{\sqrt{d x^{2}} a d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-d*(b*arctan(sqrt(d*x^2)*b/sqrt(a*b*d))/(sqrt(a*b*d)*a*d) + 1/(sqrt(d*x^2)*a*d))