3.677 \(\int \frac{1}{x^3 \left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=78 \[ -\frac{15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{7/2}}-\frac{15}{16 a^3 x^2}+\frac{5}{16 a^2 x^2 \left (a+c x^4\right )}+\frac{1}{8 a x^2 \left (a+c x^4\right )^2} \]

[Out]

-15/(16*a^3*x^2) + 1/(8*a*x^2*(a + c*x^4)^2) + 5/(16*a^2*x^2*(a + c*x^4)) - (15*
Sqrt[c]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(16*a^(7/2))

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Rubi [A]  time = 0.102196, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308 \[ -\frac{15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x^2}{\sqrt{a}}\right )}{16 a^{7/2}}-\frac{15}{16 a^3 x^2}+\frac{5}{16 a^2 x^2 \left (a+c x^4\right )}+\frac{1}{8 a x^2 \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^3*(a + c*x^4)^3),x]

[Out]

-15/(16*a^3*x^2) + 1/(8*a*x^2*(a + c*x^4)^2) + 5/(16*a^2*x^2*(a + c*x^4)) - (15*
Sqrt[c]*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(16*a^(7/2))

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Rubi in Sympy [A]  time = 16.3619, size = 71, normalized size = 0.91 \[ \frac{1}{8 a x^{2} \left (a + c x^{4}\right )^{2}} + \frac{5}{16 a^{2} x^{2} \left (a + c x^{4}\right )} - \frac{15}{16 a^{3} x^{2}} - \frac{15 \sqrt{c} \operatorname{atan}{\left (\frac{\sqrt{c} x^{2}}{\sqrt{a}} \right )}}{16 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(c*x**4+a)**3,x)

[Out]

1/(8*a*x**2*(a + c*x**4)**2) + 5/(16*a**2*x**2*(a + c*x**4)) - 15/(16*a**3*x**2)
- 15*sqrt(c)*atan(sqrt(c)*x**2/sqrt(a))/(16*a**(7/2))

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Mathematica [A]  time = 0.177981, size = 105, normalized size = 1.35 \[ \frac{-\frac{\sqrt{a} \left (8 a^2+25 a c x^4+15 c^2 x^8\right )}{x^2 \left (a+c x^4\right )^2}+15 \sqrt{c} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )+15 \sqrt{c} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{16 a^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^3*(a + c*x^4)^3),x]

[Out]

(-((Sqrt[a]*(8*a^2 + 25*a*c*x^4 + 15*c^2*x^8))/(x^2*(a + c*x^4)^2)) + 15*Sqrt[c]
*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)] + 15*Sqrt[c]*ArcTan[1 + (Sqrt[2]*c^(1/4
)*x)/a^(1/4)])/(16*a^(7/2))

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Maple [A]  time = 0.02, size = 70, normalized size = 0.9 \[ -{\frac{1}{2\,{x}^{2}{a}^{3}}}-{\frac{7\,{c}^{2}{x}^{6}}{16\,{a}^{3} \left ( c{x}^{4}+a \right ) ^{2}}}-{\frac{9\,c{x}^{2}}{16\,{a}^{2} \left ( c{x}^{4}+a \right ) ^{2}}}-{\frac{15\,c}{16\,{a}^{3}}\arctan \left ({c{x}^{2}{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(c*x^4+a)^3,x)

[Out]

-1/2/x^2/a^3-7/16*c^2/a^3/(c*x^4+a)^2*x^6-9/16*c/a^2/(c*x^4+a)^2*x^2-15/16*c/a^3
/(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(1/((c*x^4 + a)^3*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24088, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, c^{2} x^{8} + 50 \, a c x^{4} - 15 \,{\left (c^{2} x^{10} + 2 \, a c x^{6} + a^{2} x^{2}\right )} \sqrt{-\frac{c}{a}} \log \left (\frac{c x^{4} - 2 \, a x^{2} \sqrt{-\frac{c}{a}} - a}{c x^{4} + a}\right ) + 16 \, a^{2}}{32 \,{\left (a^{3} c^{2} x^{10} + 2 \, a^{4} c x^{6} + a^{5} x^{2}\right )}}, -\frac{15 \, c^{2} x^{8} + 25 \, a c x^{4} - 15 \,{\left (c^{2} x^{10} + 2 \, a c x^{6} + a^{2} x^{2}\right )} \sqrt{\frac{c}{a}} \arctan \left (\frac{a \sqrt{\frac{c}{a}}}{c x^{2}}\right ) + 8 \, a^{2}}{16 \,{\left (a^{3} c^{2} x^{10} + 2 \, a^{4} c x^{6} + a^{5} x^{2}\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^3*x^3),x, algorithm="fricas")

[Out]

[-1/32*(30*c^2*x^8 + 50*a*c*x^4 - 15*(c^2*x^10 + 2*a*c*x^6 + a^2*x^2)*sqrt(-c/a)
*log((c*x^4 - 2*a*x^2*sqrt(-c/a) - a)/(c*x^4 + a)) + 16*a^2)/(a^3*c^2*x^10 + 2*a
^4*c*x^6 + a^5*x^2), -1/16*(15*c^2*x^8 + 25*a*c*x^4 - 15*(c^2*x^10 + 2*a*c*x^6 +
a^2*x^2)*sqrt(c/a)*arctan(a*sqrt(c/a)/(c*x^2)) + 8*a^2)/(a^3*c^2*x^10 + 2*a^4*c
*x^6 + a^5*x^2)]

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Sympy [A]  time = 34.2568, size = 119, normalized size = 1.53 \[ \frac{15 \sqrt{- \frac{c}{a^{7}}} \log{\left (- \frac{a^{4} \sqrt{- \frac{c}{a^{7}}}}{c} + x^{2} \right )}}{32} - \frac{15 \sqrt{- \frac{c}{a^{7}}} \log{\left (\frac{a^{4} \sqrt{- \frac{c}{a^{7}}}}{c} + x^{2} \right )}}{32} - \frac{8 a^{2} + 25 a c x^{4} + 15 c^{2} x^{8}}{16 a^{5} x^{2} + 32 a^{4} c x^{6} + 16 a^{3} c^{2} x^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/x**3/(c*x**4+a)**3,x)

[Out]

15*sqrt(-c/a**7)*log(-a**4*sqrt(-c/a**7)/c + x**2)/32 - 15*sqrt(-c/a**7)*log(a**
4*sqrt(-c/a**7)/c + x**2)/32 - (8*a**2 + 25*a*c*x**4 + 15*c**2*x**8)/(16*a**5*x*
*2 + 32*a**4*c*x**6 + 16*a**3*c**2*x**10)

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GIAC/XCAS [A]  time = 0.21984, size = 82, normalized size = 1.05 \[ -\frac{15 \, c \arctan \left (\frac{c x^{2}}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{3}} - \frac{7 \, c^{2} x^{6} + 9 \, a c x^{2}}{16 \,{\left (c x^{4} + a\right )}^{2} a^{3}} - \frac{1}{2 \, a^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*x^4 + a)^3*x^3),x, algorithm="giac")

[Out]

-15/16*c*arctan(c*x^2/sqrt(a*c))/(sqrt(a*c)*a^3) - 1/16*(7*c^2*x^6 + 9*a*c*x^2)/
((c*x^4 + a)^2*a^3) - 1/2/(a^3*x^2)