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

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])

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Rubi [A]  time = 0.0082262, 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, Rules used = {15, 205} \[ \frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{d x^2}} \]

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])

Rule 15

Int[(u_.)*((a_.)*(x_)^(n_))^(m_), x_Symbol] :> Dist[(a^IntPart[m]*(a*x^n)^FracPart[m])/x^(n*FracPart[m]), Int[
u*x^(m*n), x], x] /; FreeQ[{a, m, n}, x] &&  !IntegerQ[m]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x}{\sqrt{d x^2} \left (a+b x^2\right )} \, dx &=\frac{x \int \frac{1}{a+b x^2} \, dx}{\sqrt{d x^2}}\\ &=\frac{x \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )}{\sqrt{a} \sqrt{b} \sqrt{d x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0072771, 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])

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Maple [A]  time = 0.004, size = 24, normalized size = 0.7 \begin{align*}{x\arctan \left ({bx{\frac{1}{\sqrt{ab}}}} \right ){\frac{1}{\sqrt{d{x}^{2}}}}{\frac{1}{\sqrt{ab}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.35109, size = 205, normalized size = 6.03 \begin{align*} \left [-\frac{\sqrt{-a b d} \log \left (\frac{b d x^{2} - a d - 2 \, \sqrt{-a b d} \sqrt{d x^{2}}}{b x^{2} + a}\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 ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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)

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Giac [A]  time = 1.12092, size = 31, normalized size = 0.91 \begin{align*} \frac{\arctan \left (\frac{\sqrt{d x^{2}} b}{\sqrt{a b d}}\right )}{\sqrt{a b d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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