3.657 \(\int \frac{x^9}{\left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=59 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{5/2}}-\frac{x^6}{4 c \left (a+c x^4\right )}+\frac{3 x^2}{4 c^2} \]

[Out]

(3*x^2)/(4*c^2) - x^6/(4*c*(a + c*x^4)) - (3*Sqrt[a]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a
]])/(4*c^(5/2))

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Rubi [A]  time = 0.0777882, antiderivative size = 59, 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 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{5/2}}-\frac{x^6}{4 c \left (a+c x^4\right )}+\frac{3 x^2}{4 c^2} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + c*x^4)^2,x]

[Out]

(3*x^2)/(4*c^2) - x^6/(4*c*(a + c*x^4)) - (3*Sqrt[a]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a
]])/(4*c^(5/2))

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Rubi in Sympy [A]  time = 13.1356, size = 51, normalized size = 0.86 \[ - \frac{3 \sqrt{a} \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{4 c^{\frac{5}{2}}} - \frac{x^{6}}{4 c \left (a + c x^{4}\right )} + \frac{3 x^{2}}{4 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(c*x**4+a)**2,x)

[Out]

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

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Mathematica [A]  time = 0.0822974, size = 60, normalized size = 1.02 \[ -\frac{3 \sqrt{a} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{4 c^{5/2}}+\frac{a x^2}{4 c^2 \left (a+c x^4\right )}+\frac{x^2}{2 c^2} \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + c*x^4)^2,x]

[Out]

x^2/(2*c^2) + (a*x^2)/(4*c^2*(a + c*x^4)) - (3*Sqrt[a]*ArcTan[(Sqrt[c]*x^2)/Sqrt
[a]])/(4*c^(5/2))

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Maple [A]  time = 0.014, size = 50, normalized size = 0.9 \[{\frac{{x}^{2}}{2\,{c}^{2}}}+{\frac{a{x}^{2}}{4\,{c}^{2} \left ( c{x}^{4}+a \right ) }}-{\frac{3\,a}{4\,{c}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(c*x^4+a)^2,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24373, size = 1, normalized size = 0.02 \[ \left [\frac{4 \, c x^{6} + 6 \, a x^{2} + 3 \,{\left (c x^{4} + a\right )} \sqrt{-\frac{a}{c}} \log \left (\frac{c x^{4} - 2 \, c x^{2} \sqrt{-\frac{a}{c}} - a}{c x^{4} + a}\right )}{8 \,{\left (c^{3} x^{4} + a c^{2}\right )}}, \frac{2 \, c x^{6} + 3 \, a x^{2} - 3 \,{\left (c x^{4} + a\right )} \sqrt{\frac{a}{c}} \arctan \left (\frac{x^{2}}{\sqrt{\frac{a}{c}}}\right )}{4 \,{\left (c^{3} x^{4} + a c^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + a)^2,x, algorithm="fricas")

[Out]

[1/8*(4*c*x^6 + 6*a*x^2 + 3*(c*x^4 + a)*sqrt(-a/c)*log((c*x^4 - 2*c*x^2*sqrt(-a/
c) - a)/(c*x^4 + a)))/(c^3*x^4 + a*c^2), 1/4*(2*c*x^6 + 3*a*x^2 - 3*(c*x^4 + a)*
sqrt(a/c)*arctan(x^2/sqrt(a/c)))/(c^3*x^4 + a*c^2)]

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Sympy [A]  time = 2.13898, size = 92, normalized size = 1.56 \[ \frac{a x^{2}}{4 a c^{2} + 4 c^{3} x^{4}} + \frac{3 \sqrt{- \frac{a}{c^{5}}} \log{\left (- c^{2} \sqrt{- \frac{a}{c^{5}}} + x^{2} \right )}}{8} - \frac{3 \sqrt{- \frac{a}{c^{5}}} \log{\left (c^{2} \sqrt{- \frac{a}{c^{5}}} + x^{2} \right )}}{8} + \frac{x^{2}}{2 c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(c*x**4+a)**2,x)

[Out]

a*x**2/(4*a*c**2 + 4*c**3*x**4) + 3*sqrt(-a/c**5)*log(-c**2*sqrt(-a/c**5) + x**2
)/8 - 3*sqrt(-a/c**5)*log(c**2*sqrt(-a/c**5) + x**2)/8 + x**2/(2*c**2)

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GIAC/XCAS [A]  time = 0.221287, size = 66, normalized size = 1.12 \[ \frac{a x^{2}}{4 \,{\left (c x^{4} + a\right )} c^{2}} + \frac{x^{2}}{2 \, c^{2}} - \frac{3 \, a \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{4 \, \sqrt{a c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + a)^2,x, algorithm="giac")

[Out]

1/4*a*x^2/((c*x^4 + a)*c^2) + 1/2*x^2/c^2 - 3/4*a*arctan(c*x^2/sqrt(a*c))/(sqrt(
a*c)*c^2)