3.379 \(\int \frac{\sqrt{a x^2}}{\sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=22 \[ \frac{\sqrt{x^2+1} \sqrt{a x^2}}{x} \]

[Out]

(Sqrt[a*x^2]*Sqrt[1 + x^2])/x

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Rubi [A]  time = 0.0031639, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {15, 261} \[ \frac{\sqrt{x^2+1} \sqrt{a x^2}}{x} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^2]/Sqrt[1 + x^2],x]

[Out]

(Sqrt[a*x^2]*Sqrt[1 + x^2])/x

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 261

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a + b*x^n)^(p + 1)/(b*n*(p + 1)), x] /; FreeQ
[{a, b, m, n, p}, x] && EqQ[m, n - 1] && NeQ[p, -1]

Rubi steps

\begin{align*} \int \frac{\sqrt{a x^2}}{\sqrt{1+x^2}} \, dx &=\frac{\sqrt{a x^2} \int \frac{x}{\sqrt{1+x^2}} \, dx}{x}\\ &=\frac{\sqrt{a x^2} \sqrt{1+x^2}}{x}\\ \end{align*}

Mathematica [A]  time = 0.0046102, size = 22, normalized size = 1. \[ \frac{\sqrt{x^2+1} \sqrt{a x^2}}{x} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x^2]/Sqrt[1 + x^2],x]

[Out]

(Sqrt[a*x^2]*Sqrt[1 + x^2])/x

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Maple [A]  time = 0.001, size = 19, normalized size = 0.9 \begin{align*}{\frac{1}{x}\sqrt{a{x}^{2}}\sqrt{{x}^{2}+1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^2)^(1/2)/(x^2+1)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.56989, size = 26, normalized size = 1.18 \begin{align*} \frac{\sqrt{a} x^{2} + \sqrt{a}}{\sqrt{x^{2} + 1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(a)*x^2 + sqrt(a))/sqrt(x^2 + 1)

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Fricas [A]  time = 1.27875, size = 39, normalized size = 1.77 \begin{align*} \frac{\sqrt{a x^{2}} \sqrt{x^{2} + 1}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(a*x^2)*sqrt(x^2 + 1)/x

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Sympy [A]  time = 0.401803, size = 20, normalized size = 0.91 \begin{align*} \frac{\sqrt{a} \sqrt{x^{2} + 1} \sqrt{x^{2}}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**2)**(1/2)/(x**2+1)**(1/2),x)

[Out]

sqrt(a)*sqrt(x**2 + 1)*sqrt(x**2)/x

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Giac [A]  time = 1.14934, size = 26, normalized size = 1.18 \begin{align*}{\left (\sqrt{x^{2} + 1} \mathrm{sgn}\left (x\right ) - \mathrm{sgn}\left (x\right )\right )} \sqrt{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(sqrt(x^2 + 1)*sgn(x) - sgn(x))*sqrt(a)