3.644 \(\int \frac{x^5}{a+c x^4} \, dx\)

Optimal. Leaf size=40 \[ \frac{x^2}{2 c}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2}} \]

[Out]

x^2/(2*c) - (Sqrt[a]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*c^(3/2))

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Rubi [A]  time = 0.0548419, antiderivative size = 40, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{x^2}{2 c}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + c*x^4),x]

[Out]

x^2/(2*c) - (Sqrt[a]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*c^(3/2))

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Rubi in Sympy [A]  time = 9.08327, size = 32, normalized size = 0.8 \[ - \frac{\sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 c^{\frac{3}{2}}} + \frac{x^{2}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(a)*atan(sqrt(c)*x**2/sqrt(a))/(2*c**(3/2)) + x**2/(2*c)

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Mathematica [A]  time = 0.0216222, size = 40, normalized size = 1. \[ \frac{x^2}{2 c}-\frac{\sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + c*x^4),x]

[Out]

x^2/(2*c) - (Sqrt[a]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*c^(3/2))

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Maple [A]  time = 0.004, size = 32, normalized size = 0.8 \[{\frac{{x}^{2}}{2\,c}}-{\frac{a}{2\,c}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(c*x^4+a),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231979, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, x^{2} + \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} - 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right )}{4 \, c}, \frac{x^{2} - \sqrt{\frac{a}{c}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{c}}}\right )}{2 \, c}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 1.35723, size = 63, normalized size = 1.58 \[ \frac{\sqrt{- \frac{a}{c^{3}}} \log{\left (- c \sqrt{- \frac{a}{c^{3}}} + x^{2} \right )}}{4} - \frac{\sqrt{- \frac{a}{c^{3}}} \log{\left (c \sqrt{- \frac{a}{c^{3}}} + x^{2} \right )}}{4} + \frac{x^{2}}{2 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(c*x**4+a),x)

[Out]

sqrt(-a/c**3)*log(-c*sqrt(-a/c**3) + x**2)/4 - sqrt(-a/c**3)*log(c*sqrt(-a/c**3)
+ x**2)/4 + x**2/(2*c)

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GIAC/XCAS [A]  time = 0.221138, size = 42, normalized size = 1.05 \[ \frac{x^{2}}{2 \, c} - \frac{a \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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