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]

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

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Rubi [A]  time = 0.00, antiderivative size = 22, normalized size of antiderivative = 1.00, 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*}

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Mathematica [A]  time = 0.00, size = 22, normalized size = 1.00 \[ \frac {\sqrt {x^2+1} \sqrt {a x^2}}{x} \]

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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fricas [A]  time = 0.42, size = 18, normalized size = 0.82 \[ \frac {\sqrt {a x^{2}} \sqrt {x^{2} + 1}}{x} \]

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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giac [A]  time = 0.16, size = 19, normalized size = 0.86 \[ {\left (\sqrt {x^{2} + 1} \mathrm {sgn}\relax (x) - \mathrm {sgn}\relax (x)\right )} \sqrt {a} \]

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)

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maple [A]  time = 0.00, size = 19, normalized size = 0.86 \[ \frac {\sqrt {a \,x^{2}}\, \sqrt {x^{2}+1}}{x} \]

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 = 1.90, size = 19, normalized size = 0.86 \[ \frac {\sqrt {a} x^{2} + \sqrt {a}}{\sqrt {x^{2} + 1}} \]

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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mupad [B]  time = 2.67, size = 19, normalized size = 0.86 \[ \frac {\sqrt {a}\,\sqrt {x^2+1}\,\sqrt {x^2}}{x} \]

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^(1/2)*(x^2 + 1)^(1/2)*(x^2)^(1/2))/x

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sympy [A]  time = 0.48, size = 20, normalized size = 0.91 \[ \frac {\sqrt {a} \sqrt {x^{2} + 1} \sqrt {x^{2}}}{x} \]

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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