3.771 \(\int x \sqrt{a+c x^4} \, dx\)

Optimal. Leaf size=50 \[ \frac{1}{4} x^2 \sqrt{a+c x^4}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{4 \sqrt{c}} \]

[Out]

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

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Rubi [A]  time = 0.0248935, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used = {275, 195, 217, 206} \[ \frac{1}{4} x^2 \sqrt{a+c x^4}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{4 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]

Int[x*Sqrt[a + c*x^4],x]

[Out]

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

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 195

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(x*(a + b*x^n)^p)/(n*p + 1), x] + Dist[(a*n*p)/(n*p + 1),
Int[(a + b*x^n)^(p - 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && GtQ[p, 0] && (IntegerQ[2*p] || (EqQ[n, 2
] && IntegerQ[4*p]) || (EqQ[n, 2] && IntegerQ[3*p]) || LtQ[Denominator[p + 1/n], Denominator[p]])

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

Mathematica [A]  time = 0.052265, size = 72, normalized size = 1.44 \[ \frac{a^{3/2} \sqrt{\frac{c x^4}{a}+1} \sinh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )+\sqrt{c} x^2 \left (a+c x^4\right )}{4 \sqrt{c} \sqrt{a+c x^4}} \]

Antiderivative was successfully verified.

[In]

Integrate[x*Sqrt[a + c*x^4],x]

[Out]

(Sqrt[c]*x^2*(a + c*x^4) + a^(3/2)*Sqrt[1 + (c*x^4)/a]*ArcSinh[(Sqrt[c]*x^2)/Sqrt[a]])/(4*Sqrt[c]*Sqrt[a + c*x
^4])

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Maple [A]  time = 0.006, size = 40, normalized size = 0.8 \begin{align*}{\frac{{x}^{2}}{4}\sqrt{c{x}^{4}+a}}+{\frac{a}{4}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ){\frac{1}{\sqrt{c}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.56374, size = 243, normalized size = 4.86 \begin{align*} \left [\frac{2 \, \sqrt{c x^{4} + a} c x^{2} + a \sqrt{c} \log \left (-2 \, c x^{4} - 2 \, \sqrt{c x^{4} + a} \sqrt{c} x^{2} - a\right )}{8 \, c}, \frac{\sqrt{c x^{4} + a} c x^{2} - a \sqrt{-c} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + a}}\right )}{4 \, c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/8*(2*sqrt(c*x^4 + a)*c*x^2 + a*sqrt(c)*log(-2*c*x^4 - 2*sqrt(c*x^4 + a)*sqrt(c)*x^2 - a))/c, 1/4*(sqrt(c*x^
4 + a)*c*x^2 - a*sqrt(-c)*arctan(sqrt(-c)*x^2/sqrt(c*x^4 + a)))/c]

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Sympy [A]  time = 2.44994, size = 44, normalized size = 0.88 \begin{align*} \frac{\sqrt{a} x^{2} \sqrt{1 + \frac{c x^{4}}{a}}}{4} + \frac{a \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.10949, size = 55, normalized size = 1.1 \begin{align*} \frac{1}{4} \, \sqrt{c x^{4} + a} x^{2} - \frac{a \log \left ({\left | -\sqrt{c} x^{2} + \sqrt{c x^{4} + a} \right |}\right )}{4 \, \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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