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

Optimal. Leaf size=34 \[ \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{d x^2}} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.0232445, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 12.2338, size = 39, normalized size = 1.15 \[ \frac{\operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{d x^{2}}}{\sqrt{a} \sqrt{d}} \right )}}{\sqrt{a} \sqrt{b} \sqrt{d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

atan(sqrt(b)*sqrt(d*x**2)/(sqrt(a)*sqrt(d)))/(sqrt(a)*sqrt(b)*sqrt(d))

_______________________________________________________________________________________

Mathematica [A]  time = 0.012135, size = 34, normalized size = 1. \[ \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{d x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.005, size = 24, normalized size = 0.7 \[{x\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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(x/((b*x^2 + a)*sqrt(d*x^2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.225072, size = 1, normalized size = 0.03 \[ \left [-\frac{\sqrt{-a b d} \log \left (\frac{2 \, a b d x^{2} + \sqrt{-a b d}{\left (b x^{2} - a\right )} \sqrt{d x^{2}}}{b x^{3} + a x}\right )}{2 \, a b d}, \frac{\sqrt{a b d} \arctan \left (\frac{\sqrt{a b d} \sqrt{d x^{2}}}{a d}\right )}{a b d}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.23842, size = 31, normalized size = 0.91 \[ \frac{\arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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