3.772 \(\int \frac{\sqrt{a+c x^4}}{x^3} \, dx\)

Optimal. Leaf size=49 \[ \frac{1}{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-\frac{\sqrt{a+c x^4}}{2 x^2} \]

[Out]

-Sqrt[a + c*x^4]/(2*x^2) + (Sqrt[c]*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/2

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Rubi [A]  time = 0.0257567, antiderivative size = 49, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267, Rules used = {275, 277, 217, 206} \[ \frac{1}{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )-\frac{\sqrt{a+c x^4}}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + c*x^4]/x^3,x]

[Out]

-Sqrt[a + c*x^4]/(2*x^2) + (Sqrt[c]*ArcTanh[(Sqrt[c]*x^2)/Sqrt[a + c*x^4]])/2

Rule 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

Rule 277

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

Rule 217

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\sqrt{a+c x^4}}{x^3} \, dx &=\frac{1}{2} \operatorname{Subst}\left (\int \frac{\sqrt{a+c x^2}}{x^2} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{a+c x^4}}{2 x^2}+\frac{1}{2} c \operatorname{Subst}\left (\int \frac{1}{\sqrt{a+c x^2}} \, dx,x,x^2\right )\\ &=-\frac{\sqrt{a+c x^4}}{2 x^2}+\frac{1}{2} c \operatorname{Subst}\left (\int \frac{1}{1-c x^2} \, dx,x,\frac{x^2}{\sqrt{a+c x^4}}\right )\\ &=-\frac{\sqrt{a+c x^4}}{2 x^2}+\frac{1}{2} \sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )\\ \end{align*}

Mathematica [A]  time = 0.075015, size = 69, normalized size = 1.41 \[ -\frac{-\sqrt{a} \sqrt{c} x^2 \sqrt{\frac{c x^4}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )+a+c x^4}{2 x^2 \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + c*x^4]/x^3,x]

[Out]

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

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Maple [A]  time = 0.01, size = 60, normalized size = 1.2 \begin{align*} -{\frac{1}{2\,a{x}^{2}} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}}+{\frac{c{x}^{2}}{2\,a}\sqrt{c{x}^{4}+a}}+{\frac{1}{2}\sqrt{c}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^(1/2)/x^3,x)

[Out]

-1/2/a/x^2*(c*x^4+a)^(3/2)+1/2*c/a*x^2*(c*x^4+a)^(1/2)+1/2*c^(1/2)*ln(x^2*c^(1/2)+(c*x^4+a)^(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((c*x^4+a)^(1/2)/x^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.76932, size = 239, normalized size = 4.88 \begin{align*} \left [\frac{\sqrt{c} x^{2} \log \left (-2 \, c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{c} x^{2} - a\right ) - 2 \, \sqrt{c x^{4} + a}}{4 \, x^{2}}, -\frac{\sqrt{-c} x^{2} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + a}}\right ) + \sqrt{c x^{4} + a}}{2 \, x^{2}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^3,x, algorithm="fricas")

[Out]

[1/4*(sqrt(c)*x^2*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(c)*x^2 - a) - 2*sqrt(c*x^4 + a))/x^2, -1/2*(sqrt(-c)*x
^2*arctan(sqrt(-c)*x^2/sqrt(c*x^4 + a)) + sqrt(c*x^4 + a))/x^2]

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Sympy [A]  time = 1.91903, size = 66, normalized size = 1.35 \begin{align*} - \frac{\sqrt{a}}{2 x^{2} \sqrt{1 + \frac{c x^{4}}{a}}} + \frac{\sqrt{c} \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2} - \frac{c x^{2}}{2 \sqrt{a} \sqrt{1 + \frac{c x^{4}}{a}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**(1/2)/x**3,x)

[Out]

-sqrt(a)/(2*x**2*sqrt(1 + c*x**4/a)) + sqrt(c)*asinh(sqrt(c)*x**2/sqrt(a))/2 - c*x**2/(2*sqrt(a)*sqrt(1 + c*x*
*4/a))

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Giac [A]  time = 1.12833, size = 49, normalized size = 1. \begin{align*} -\frac{c \arctan \left (\frac{\sqrt{c + \frac{a}{x^{4}}}}{\sqrt{-c}}\right )}{2 \, \sqrt{-c}} - \frac{1}{2} \, \sqrt{c + \frac{a}{x^{4}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^(1/2)/x^3,x, algorithm="giac")

[Out]

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