3.671 \(\int \frac{x^9}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=68 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 \sqrt{a} c^{5/2}}-\frac{3 x^2}{16 c^2 \left (a+c x^4\right )}-\frac{x^6}{8 c \left (a+c x^4\right )^2} \]

[Out]

-x^6/(8*c*(a + c*x^4)^2) - (3*x^2)/(16*c^2*(a + c*x^4)) + (3*ArcTan[(Sqrt[c]*x^2
)/Sqrt[a]])/(16*Sqrt[a]*c^(5/2))

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Rubi [A]  time = 0.0882107, antiderivative size = 68, 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 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 \sqrt{a} c^{5/2}}-\frac{3 x^2}{16 c^2 \left (a+c x^4\right )}-\frac{x^6}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^9/(a + c*x^4)^3,x]

[Out]

-x^6/(8*c*(a + c*x^4)^2) - (3*x^2)/(16*c^2*(a + c*x^4)) + (3*ArcTan[(Sqrt[c]*x^2
)/Sqrt[a]])/(16*Sqrt[a]*c^(5/2))

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Rubi in Sympy [A]  time = 12.8702, size = 60, normalized size = 0.88 \[ - \frac{x^{6}}{8 c \left (a + c x^{4}\right )^{2}} - \frac{3 x^{2}}{16 c^{2} \left (a + c x^{4}\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{16 \sqrt{a} c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**9/(c*x**4+a)**3,x)

[Out]

-x**6/(8*c*(a + c*x**4)**2) - 3*x**2/(16*c**2*(a + c*x**4)) + 3*atan(sqrt(c)*x**
2/sqrt(a))/(16*sqrt(a)*c**(5/2))

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Mathematica [A]  time = 0.0923042, size = 58, normalized size = 0.85 \[ \frac{1}{16} \left (\frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{\sqrt{a} c^{5/2}}+\frac{-3 a x^2-5 c x^6}{c^2 \left (a+c x^4\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^9/(a + c*x^4)^3,x]

[Out]

((-3*a*x^2 - 5*c*x^6)/(c^2*(a + c*x^4)^2) + (3*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(S
qrt[a]*c^(5/2)))/16

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Maple [A]  time = 0.015, size = 52, normalized size = 0.8 \[{\frac{1}{2\, \left ( c{x}^{4}+a \right ) ^{2}} \left ( -{\frac{5\,{x}^{6}}{8\,c}}-{\frac{3\,a{x}^{2}}{8\,{c}^{2}}} \right ) }+{\frac{3}{16\,{c}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^9/(c*x^4+a)^3,x)

[Out]

1/2*(-5/8*x^6/c-3/8*a*x^2/c^2)/(c*x^4+a)^2+3/16/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)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.232998, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (\frac{2 \, a c x^{2} +{\left (c x^{4} - a\right )} \sqrt{-a c}}{c x^{4} + a}\right ) - 2 \,{\left (5 \, c x^{6} + 3 \, a x^{2}\right )} \sqrt{-a c}}{32 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \sqrt{-a c}}, -\frac{3 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \arctan \left (\frac{a}{\sqrt{a c} x^{2}}\right ) +{\left (5 \, c x^{6} + 3 \, a x^{2}\right )} \sqrt{a c}}{16 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

[1/32*(3*(c^2*x^8 + 2*a*c*x^4 + a^2)*log((2*a*c*x^2 + (c*x^4 - a)*sqrt(-a*c))/(c
*x^4 + a)) - 2*(5*c*x^6 + 3*a*x^2)*sqrt(-a*c))/((c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2
)*sqrt(-a*c)), -1/16*(3*(c^2*x^8 + 2*a*c*x^4 + a^2)*arctan(a/(sqrt(a*c)*x^2)) +
(5*c*x^6 + 3*a*x^2)*sqrt(a*c))/((c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*sqrt(a*c))]

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Sympy [A]  time = 5.10263, size = 114, normalized size = 1.68 \[ - \frac{3 \sqrt{- \frac{1}{a c^{5}}} \log{\left (- a c^{2} \sqrt{- \frac{1}{a c^{5}}} + x^{2} \right )}}{32} + \frac{3 \sqrt{- \frac{1}{a c^{5}}} \log{\left (a c^{2} \sqrt{- \frac{1}{a c^{5}}} + x^{2} \right )}}{32} - \frac{3 a x^{2} + 5 c x^{6}}{16 a^{2} c^{2} + 32 a c^{3} x^{4} + 16 c^{4} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**9/(c*x**4+a)**3,x)

[Out]

-3*sqrt(-1/(a*c**5))*log(-a*c**2*sqrt(-1/(a*c**5)) + x**2)/32 + 3*sqrt(-1/(a*c**
5))*log(a*c**2*sqrt(-1/(a*c**5)) + x**2)/32 - (3*a*x**2 + 5*c*x**6)/(16*a**2*c**
2 + 32*a*c**3*x**4 + 16*c**4*x**8)

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GIAC/XCAS [A]  time = 0.222273, size = 66, normalized size = 0.97 \[ \frac{3 \, \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} c^{2}} - \frac{5 \, c x^{6} + 3 \, a x^{2}}{16 \,{\left (c x^{4} + a\right )}^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^9/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

3/16*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*c^2) - 1/16*(5*c*x^6 + 3*a*x^2)/((c*x^4
+ a)^2*c^2)