3.377 \(\int \frac{\sqrt{a x^4}}{\sqrt{1+x^2}} \, dx\)

Optimal. Leaf size=44 \[ \frac{\sqrt{x^2+1} \sqrt{a x^4}}{2 x}-\frac{\sqrt{a x^4} \sinh ^{-1}(x)}{2 x^2} \]

[Out]

(Sqrt[a*x^4]*Sqrt[1 + x^2])/(2*x) - (Sqrt[a*x^4]*ArcSinh[x])/(2*x^2)

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Rubi [A]  time = 0.0064502, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158, Rules used = {15, 321, 215} \[ \frac{\sqrt{x^2+1} \sqrt{a x^4}}{2 x}-\frac{\sqrt{a x^4} \sinh ^{-1}(x)}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a*x^4]/Sqrt[1 + x^2],x]

[Out]

(Sqrt[a*x^4]*Sqrt[1 + x^2])/(2*x) - (Sqrt[a*x^4]*ArcSinh[x])/(2*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 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 215

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSinh[(Rt[b, 2]*x)/Sqrt[a]]/Rt[b, 2], x] /; FreeQ[{a, b},
x] && GtQ[a, 0] && PosQ[b]

Rubi steps

\begin{align*} \int \frac{\sqrt{a x^4}}{\sqrt{1+x^2}} \, dx &=\frac{\sqrt{a x^4} \int \frac{x^2}{\sqrt{1+x^2}} \, dx}{x^2}\\ &=\frac{\sqrt{a x^4} \sqrt{1+x^2}}{2 x}-\frac{\sqrt{a x^4} \int \frac{1}{\sqrt{1+x^2}} \, dx}{2 x^2}\\ &=\frac{\sqrt{a x^4} \sqrt{1+x^2}}{2 x}-\frac{\sqrt{a x^4} \sinh ^{-1}(x)}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0080166, size = 32, normalized size = 0.73 \[ \frac{\sqrt{a x^4} \left (x \sqrt{x^2+1}-\sinh ^{-1}(x)\right )}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a*x^4]/Sqrt[1 + x^2],x]

[Out]

(Sqrt[a*x^4]*(x*Sqrt[1 + x^2] - ArcSinh[x]))/(2*x^2)

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Maple [A]  time = 0.005, size = 26, normalized size = 0.6 \begin{align*} -{\frac{1}{2\,{x}^{2}}\sqrt{a{x}^{4}} \left ( -x\sqrt{{x}^{2}+1}+{\it Arcsinh} \left ( x \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^4)^(1/2)/(x^2+1)^(1/2),x)

[Out]

-1/2*(a*x^4)^(1/2)*(-x*(x^2+1)^(1/2)+arcsinh(x))/x^2

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{4}}}{\sqrt{x^{2} + 1}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4)^(1/2)/(x^2+1)^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*x^4)/sqrt(x^2 + 1), x)

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Fricas [A]  time = 1.29491, size = 104, normalized size = 2.36 \begin{align*} \frac{\sqrt{a x^{4}} \sqrt{x^{2} + 1} x + \sqrt{a x^{4}} \log \left (-x + \sqrt{x^{2} + 1}\right )}{2 \, x^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4)^(1/2)/(x^2+1)^(1/2),x, algorithm="fricas")

[Out]

1/2*(sqrt(a*x^4)*sqrt(x^2 + 1)*x + sqrt(a*x^4)*log(-x + sqrt(x^2 + 1)))/x^2

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a x^{4}}}{\sqrt{x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x**4)**(1/2)/(x**2+1)**(1/2),x)

[Out]

Integral(sqrt(a*x**4)/sqrt(x**2 + 1), x)

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Giac [A]  time = 1.12858, size = 36, normalized size = 0.82 \begin{align*} \frac{1}{2} \,{\left (\sqrt{x^{2} + 1} x + \log \left (-x + \sqrt{x^{2} + 1}\right )\right )} \sqrt{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a*x^4)^(1/2)/(x^2+1)^(1/2),x, algorithm="giac")

[Out]

1/2*(sqrt(x^2 + 1)*x + log(-x + sqrt(x^2 + 1)))*sqrt(a)