3.768 \(\int x^5 \sqrt{a+c x^4} \, dx\)

Optimal. Leaf size=74 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{16 c^{3/2}}+\frac{1}{8} x^6 \sqrt{a+c x^4}+\frac{a x^2 \sqrt{a+c x^4}}{16 c} \]

[Out]

(a*x^2*Sqrt[a + c*x^4])/(16*c) + (x^6*Sqrt[a + c*x^4])/8 - (a^2*ArcTanh[(Sqrt[c]
*x^2)/Sqrt[a + c*x^4]])/(16*c^(3/2))

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Rubi [A]  time = 0.108596, antiderivative size = 74, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{a^2 \tanh ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a+c x^4}}\right )}{16 c^{3/2}}+\frac{1}{8} x^6 \sqrt{a+c x^4}+\frac{a x^2 \sqrt{a+c x^4}}{16 c} \]

Antiderivative was successfully verified.

[In]  Int[x^5*Sqrt[a + c*x^4],x]

[Out]

(a*x^2*Sqrt[a + c*x^4])/(16*c) + (x^6*Sqrt[a + c*x^4])/8 - (a^2*ArcTanh[(Sqrt[c]
*x^2)/Sqrt[a + c*x^4]])/(16*c^(3/2))

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Rubi in Sympy [A]  time = 11.5531, size = 63, normalized size = 0.85 \[ - \frac{a^{2} \operatorname{atanh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a + c x^{4}}} \right )}}{16 c^{\frac{3}{2}}} + \frac{a x^{2} \sqrt{a + c x^{4}}}{16 c} + \frac{x^{6} \sqrt{a + c x^{4}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(c*x**4+a)**(1/2),x)

[Out]

-a**2*atanh(sqrt(c)*x**2/sqrt(a + c*x**4))/(16*c**(3/2)) + a*x**2*sqrt(a + c*x**
4)/(16*c) + x**6*sqrt(a + c*x**4)/8

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Mathematica [A]  time = 0.052737, size = 67, normalized size = 0.91 \[ \frac{\sqrt{c} x^2 \sqrt{a+c x^4} \left (a+2 c x^4\right )-a^2 \log \left (\sqrt{c} \sqrt{a+c x^4}+c x^2\right )}{16 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5*Sqrt[a + c*x^4],x]

[Out]

(Sqrt[c]*x^2*Sqrt[a + c*x^4]*(a + 2*c*x^4) - a^2*Log[c*x^2 + Sqrt[c]*Sqrt[a + c*
x^4]])/(16*c^(3/2))

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Maple [A]  time = 0.019, size = 63, normalized size = 0.9 \[{\frac{{x}^{2}}{8\,c} \left ( c{x}^{4}+a \right ) ^{{\frac{3}{2}}}}-{\frac{a{x}^{2}}{16\,c}\sqrt{c{x}^{4}+a}}-{\frac{{a}^{2}}{16}\ln \left ({x}^{2}\sqrt{c}+\sqrt{c{x}^{4}+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(c*x^4+a)^(1/2),x)

[Out]

1/8*x^2*(c*x^4+a)^(3/2)/c-1/16*a*x^2*(c*x^4+a)^(1/2)/c-1/16*a^2/c^(3/2)*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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.253611, size = 1, normalized size = 0.01 \[ \left [\frac{a^{2} \log \left (2 \, \sqrt{c x^{4} + a} c x^{2} -{\left (2 \, c x^{4} + a\right )} \sqrt{c}\right ) + 2 \,{\left (2 \, c x^{6} + a x^{2}\right )} \sqrt{c x^{4} + a} \sqrt{c}}{32 \, c^{\frac{3}{2}}}, -\frac{a^{2} \arctan \left (\frac{\sqrt{-c} x^{2}}{\sqrt{c x^{4} + a}}\right ) -{\left (2 \, c x^{6} + a x^{2}\right )} \sqrt{c x^{4} + a} \sqrt{-c}}{16 \, \sqrt{-c} c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)*x^5,x, algorithm="fricas")

[Out]

[1/32*(a^2*log(2*sqrt(c*x^4 + a)*c*x^2 - (2*c*x^4 + a)*sqrt(c)) + 2*(2*c*x^6 + a
*x^2)*sqrt(c*x^4 + a)*sqrt(c))/c^(3/2), -1/16*(a^2*arctan(sqrt(-c)*x^2/sqrt(c*x^
4 + a)) - (2*c*x^6 + a*x^2)*sqrt(c*x^4 + a)*sqrt(-c))/(sqrt(-c)*c)]

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Sympy [A]  time = 11.9362, size = 95, normalized size = 1.28 \[ \frac{a^{\frac{3}{2}} x^{2}}{16 c \sqrt{1 + \frac{c x^{4}}{a}}} + \frac{3 \sqrt{a} x^{6}}{16 \sqrt{1 + \frac{c x^{4}}{a}}} - \frac{a^{2} \operatorname{asinh}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{16 c^{\frac{3}{2}}} + \frac{c x^{10}}{8 \sqrt{a} \sqrt{1 + \frac{c x^{4}}{a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(c*x**4+a)**(1/2),x)

[Out]

a**(3/2)*x**2/(16*c*sqrt(1 + c*x**4/a)) + 3*sqrt(a)*x**6/(16*sqrt(1 + c*x**4/a))
- a**2*asinh(sqrt(c)*x**2/sqrt(a))/(16*c**(3/2)) + c*x**10/(8*sqrt(a)*sqrt(1 +
c*x**4/a))

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GIAC/XCAS [A]  time = 0.230994, size = 73, normalized size = 0.99 \[ \frac{1}{16} \, \sqrt{c x^{4} + a}{\left (2 \, x^{4} + \frac{a}{c}\right )} x^{2} + \frac{a^{2}{\rm ln}\left ({\left | -\sqrt{c} x^{2} + \sqrt{c x^{4} + a} \right |}\right )}{16 \, c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^4 + a)*x^5,x, algorithm="giac")

[Out]

1/16*sqrt(c*x^4 + a)*(2*x^4 + a/c)*x^2 + 1/16*a^2*ln(abs(-sqrt(c)*x^2 + sqrt(c*x
^4 + a)))/c^(3/2)