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

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

[Out]

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

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Rubi [A]  time = 0.0705767, antiderivative size = 68, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{5/2} \sqrt{c}}+\frac{3 x^2}{16 a^2 \left (a+c x^4\right )}+\frac{x^2}{8 a \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x/(a + c*x^4)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 8.34, size = 60, normalized size = 0.88 \[ \frac{x^{2}}{8 a \left (a + c x^{4}\right )^{2}} + \frac{3 x^{2}}{16 a^{2} \left (a + c x^{4}\right )} + \frac{3 \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{5}{2}} \sqrt{c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0785993, size = 58, normalized size = 0.85 \[ \frac{1}{16} \left (\frac{3 \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{a^{5/2} \sqrt{c}}+\frac{5 a x^2+3 c x^6}{a^2 \left (a+c x^4\right )^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x/(a + c*x^4)^3,x]

[Out]

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

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Maple [A]  time = 0.009, size = 57, normalized size = 0.8 \[{\frac{{x}^{2}}{8\,a \left ( c{x}^{4}+a \right ) ^{2}}}+{\frac{3\,{x}^{2}}{16\,{a}^{2} \left ( c{x}^{4}+a \right ) }}+{\frac{3}{16\,{a}^{2}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x/(c*x^4+a)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23838, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \log \left (\frac{2 \, a c x^{2} +{\left (c x^{4} - a\right )} \sqrt{-a c}}{c x^{4} + a}\right ) + 2 \,{\left (3 \, c x^{6} + 5 \, a x^{2}\right )} \sqrt{-a c}}{32 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \sqrt{-a c}}, -\frac{3 \,{\left (c^{2} x^{8} + 2 \, a c x^{4} + a^{2}\right )} \arctan \left (\frac{a}{\sqrt{a c} x^{2}}\right ) -{\left (3 \, c x^{6} + 5 \, a x^{2}\right )} \sqrt{a c}}{16 \,{\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )} \sqrt{a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

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Sympy [A]  time = 4.94295, size = 110, normalized size = 1.62 \[ - \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (- a^{3} \sqrt{- \frac{1}{a^{5} c}} + x^{2} \right )}}{32} + \frac{3 \sqrt{- \frac{1}{a^{5} c}} \log{\left (a^{3} \sqrt{- \frac{1}{a^{5} c}} + x^{2} \right )}}{32} + \frac{5 a x^{2} + 3 c x^{6}}{16 a^{4} + 32 a^{3} c x^{4} + 16 a^{2} c^{2} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x/(c*x**4+a)**3,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.221665, size = 66, normalized size = 0.97 \[ \frac{3 \, \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{2}} + \frac{3 \, c x^{6} + 5 \, a x^{2}}{16 \,{\left (c x^{4} + a\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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