3.642 \(\int \frac{x^9}{a+c x^4} \, dx\)

Optimal. Leaf size=51 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{5/2}}-\frac{a x^2}{2 c^2}+\frac{x^6}{6 c} \]

[Out]

-(a*x^2)/(2*c^2) + x^6/(6*c) + (a^(3/2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*c^(5/2
))

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Rubi [A]  time = 0.0767684, antiderivative size = 51, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{a^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{2 c^{5/2}}-\frac{a x^2}{2 c^2}+\frac{x^6}{6 c} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + c*x^4),x]

[Out]

-(a*x^2)/(2*c^2) + x^6/(6*c) + (a^(3/2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(2*c^(5/2
))

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \frac{a^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{2 c^{\frac{5}{2}}} + \frac{x^{6}}{6 c} - \frac{\int ^{x^{2}} a\, dx}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(c*x**4+a),x)

[Out]

a**(3/2)*atan(sqrt(c)*x**2/sqrt(a))/(2*c**(5/2)) + x**6/(6*c) - Integral(a, (x,
x**2))/(2*c**2)

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Mathematica [A]  time = 0.0559989, size = 48, normalized size = 0.94 \[ \frac{1}{6} \left (\frac{3 a^{3/2} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{c^{5/2}}+\frac{c x^6-3 a x^2}{c^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + c*x^4),x]

[Out]

((-3*a*x^2 + c*x^6)/c^2 + (3*a^(3/2)*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/c^(5/2))/6

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Maple [A]  time = 0.008, size = 43, normalized size = 0.8 \[{\frac{{x}^{6}}{6\,c}}-{\frac{a{x}^{2}}{2\,{c}^{2}}}+{\frac{{a}^{2}}{2\,{c}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(c*x^4+a),x)

[Out]

1/6*x^6/c-1/2*a*x^2/c^2+1/2*a^2/c^2/(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^9/(c*x^4 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237071, size = 1, normalized size = 0.02 \[ \left [\frac{2 \, c x^{6} - 6 \, a x^{2} + 3 \, a \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} + 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right )}{12 \, c^{2}}, \frac{c x^{6} - 3 \, a x^{2} + 3 \, a \sqrt{\frac{a}{c}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{c}}}\right )}{6 \, c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + a),x, algorithm="fricas")

[Out]

[1/12*(2*c*x^6 - 6*a*x^2 + 3*a*sqrt(-a/c)*log((c*x^4 + 2*c*x^2*sqrt(-a/c) - a)/(
c*x^4 + a)))/c^2, 1/6*(c*x^6 - 3*a*x^2 + 3*a*sqrt(a/c)*arctan(x^2/sqrt(a/c)))/c^
2]

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Sympy [A]  time = 1.39009, size = 87, normalized size = 1.71 \[ - \frac{a x^{2}}{2 c^{2}} - \frac{\sqrt{- \frac{a^{3}}{c^{5}}} \log{\left (x^{2} - \frac{c^{2} \sqrt{- \frac{a^{3}}{c^{5}}}}{a} \right )}}{4} + \frac{\sqrt{- \frac{a^{3}}{c^{5}}} \log{\left (x^{2} + \frac{c^{2} \sqrt{- \frac{a^{3}}{c^{5}}}}{a} \right )}}{4} + \frac{x^{6}}{6 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(c*x**4+a),x)

[Out]

-a*x**2/(2*c**2) - sqrt(-a**3/c**5)*log(x**2 - c**2*sqrt(-a**3/c**5)/a)/4 + sqrt
(-a**3/c**5)*log(x**2 + c**2*sqrt(-a**3/c**5)/a)/4 + x**6/(6*c)

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GIAC/XCAS [A]  time = 0.223756, size = 61, normalized size = 1.2 \[ \frac{a^{2} \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} c^{2}} + \frac{c^{2} x^{6} - 3 \, a c x^{2}}{6 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + a),x, algorithm="giac")

[Out]

1/2*a^2*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*c^2) + 1/6*(c^2*x^6 - 3*a*c*x^2)/c^3