3.4.77 \(\int \frac {\sqrt {a x^4}}{\sqrt {1+x^2}} \, dx\) [377]

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 44, normalized size of antiderivative = 1.00, 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, 327, 221} \begin {gather*} \frac {\sqrt {x^2+1} \sqrt {a x^4}}{2 x}-\frac {\sqrt {a x^4} \sinh ^{-1}(x)}{2 x^2} \end {gather*}

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 221

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]

Rule 327

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]

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*}

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Mathematica [A]
time = 0.03, size = 42, normalized size = 0.95 \begin {gather*} \frac {\sqrt {a x^4} \left (x \sqrt {1+x^2}-\tanh ^{-1}\left (\frac {x}{\sqrt {1+x^2}}\right )\right )}{2 x^2} \end {gather*}

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] - ArcTanh[x/Sqrt[1 + x^2]]))/(2*x^2)

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Maple [A]
time = 0.20, size = 27, normalized size = 0.61

method	result	size
default	\(\frac {\sqrt {a \,x^{4}}\, \left (x \sqrt {x^{2}+1}-\arcsinh \left (x \right )\right )}{2 x^{2}}\)	\(27\)
meijerg	\(\frac {\sqrt {a \,x^{4}}\, \left (\sqrt {\pi }\, x \sqrt {x^{2}+1}-\sqrt {\pi }\, \arcsinh \left (x \right )\right )}{2 x^{2} \sqrt {\pi }}\)	\(36\)
risch	\(\frac {\sqrt {a \,x^{4}}\, \sqrt {x^{2}+1}}{2 x}-\frac {\ln \left (x \sqrt {a}+\sqrt {a \,x^{2}+a}\right ) \sqrt {a \,x^{4}}\, \sqrt {\left (x^{2}+1\right ) a}}{2 \sqrt {a}\, x^{2} \sqrt {x^{2}+1}}\)	\(68\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.33, size = 42, normalized size = 0.95 \begin {gather*} \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 {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a x^{4}}}{\sqrt {x^{2} + 1}}\, dx \end {gather*}

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 = 3.62, size = 27, normalized size = 0.61 \begin {gather*} \frac {1}{2} \, {\left (\sqrt {x^{2} + 1} x + \log \left (-x + \sqrt {x^{2} + 1}\right )\right )} \sqrt {a} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {\sqrt {a\,x^4}}{\sqrt {x^2+1}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x^4)^(1/2)/(x^2 + 1)^(1/2),x)

[Out]

int((a*x^4)^(1/2)/(x^2 + 1)^(1/2), x)

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