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

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 = 0.0561052, 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 \[ \frac{1}{4} x^2 \sqrt{a+c x^4}+\frac{a \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{4 \sqrt{c}} \]

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])

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Rubi in Sympy [A]  time = 5.11774, size = 42, normalized size = 0.84 \[ \frac{a \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{4 \sqrt{c}} + \frac{x^{2} \sqrt{a + c x^{4}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x*(c*x**4+a)**(1/2),x)

[Out]

a*atanh(sqrt(c)*x**2/sqrt(a + c*x**4))/(4*sqrt(c)) + x**2*sqrt(a + c*x**4)/4

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Mathematica [A]  time = 0.0293181, size = 53, normalized size = 1.06 \[ \frac{1}{4} x^2 \sqrt{a+c x^4}+\frac{a \log \left (\sqrt{c} \sqrt{a+c x^4}+c x^2\right )}{4 \sqrt{c}} \]

Antiderivative was successfully verified.

[In]  Integrate[x*Sqrt[a + c*x^4],x]

[Out]

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

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Maple [A]  time = 0.009, size = 40, normalized size = 0.8 \[{\frac{{x}^{2}}{4}\sqrt{c{x}^{4}+a}}+{\frac{a}{4}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ){\frac{1}{\sqrt{c}}}} \]

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]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29485, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, \sqrt{c x^{4} + a} \sqrt{c} x^{2} + a \log \left (-2 \, \sqrt{c x^{4} + a} c x^{2} -{\left (2 \, c x^{4} + a\right )} \sqrt{c}\right )}{8 \, \sqrt{c}}, \frac{\sqrt{c x^{4} + a} \sqrt{-c} x^{2} + a \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + a}}\right )}{4 \, \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)*x,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 6.24094, size = 44, normalized size = 0.88 \[ \frac{\sqrt{a} x^{2} \sqrt{1 + \frac{c x^{4}}{a}}}{4} + \frac{a \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 \sqrt{c}} \]

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/XCAS [A]  time = 0.217435, size = 55, normalized size = 1.1 \[ \frac{1}{4} \, \sqrt{c x^{4} + a} x^{2} - \frac{a{\rm ln}\left ({\left | -\sqrt{c} x^{2} + \sqrt{c x^{4} + a} \right |}\right )}{4 \, \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)*x,x, algorithm="giac")

[Out]

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