3.673 \(\int \frac{x^5}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=71 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{3/2} c^{3/2}}+\frac{x^2}{16 a c \left (a+c x^4\right )}-\frac{x^2}{8 c \left (a+c x^4\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.0833789, antiderivative size = 71, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{3/2} c^{3/2}}+\frac{x^2}{16 a c \left (a+c x^4\right )}-\frac{x^2}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + c*x^4)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 11.3689, size = 56, normalized size = 0.79 \[ - \frac{x^{2}}{8 c \left (a + c x^{4}\right )^{2}} + \frac{x^{2}}{16 a c \left (a + c x^{4}\right )} + \frac{\operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{3}{2}} c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(c*x**4+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.0559769, size = 62, normalized size = 0.87 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )+\frac{\sqrt{a} \sqrt{c} x^2 \left (c x^4-a\right )}{\left (a+c x^4\right )^2}}{16 a^{3/2} c^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + c*x^4)^3,x]

[Out]

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

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Maple [A]  time = 0.014, size = 54, normalized size = 0.8 \[{\frac{1}{2\, \left ( c{x}^{4}+a \right ) ^{2}} \left ({\frac{{x}^{6}}{8\,a}}-{\frac{{x}^{2}}{8\,c}} \right ) }+{\frac{1}{16\,ac}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(c*x^4+a)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234141, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (\frac{2 \, a c x^{2} +{\left (c x^{4} - a\right )} \sqrt{-a c}}{c x^{4} + a}\right ) + 2 \,{\left (c x^{6} - a x^{2}\right )} \sqrt{-a c}}{32 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \sqrt{-a c}}, -\frac{{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \arctan \left (\frac{a}{\sqrt{a c} x^{2}}\right ) -{\left (c x^{6} - a x^{2}\right )} \sqrt{a c}}{16 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.98255, size = 116, normalized size = 1.63 \[ - \frac{\sqrt{- \frac{1}{a^{3} c^{3}}} \log{\left (- a^{2} c \sqrt{- \frac{1}{a^{3} c^{3}}} + x^{2} \right )}}{32} + \frac{\sqrt{- \frac{1}{a^{3} c^{3}}} \log{\left (a^{2} c \sqrt{- \frac{1}{a^{3} c^{3}}} + x^{2} \right )}}{32} + \frac{- a x^{2} + c x^{6}}{16 a^{3} c + 32 a^{2} c^{2} x^{4} + 16 a c^{3} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(c*x**4+a)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.226203, size = 73, normalized size = 1.03 \[ \frac{\arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a c} + \frac{c x^{6} - a x^{2}}{16 \,{\left (c x^{4} + a\right )}^{2} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

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