3.664 \(\int \frac{x^6}{\left (a+c x^4\right )^2} \, dx\)

Optimal. Leaf size=204 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{x^3}{4 c \left (a+c x^4\right )} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.263406, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{16 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{8 \sqrt{2} \sqrt [4]{a} c^{7/4}}-\frac{x^3}{4 c \left (a+c x^4\right )} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + c*x^4)^2,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 53.8217, size = 192, normalized size = 0.94 \[ - \frac{x^{3}}{4 c \left (a + c x^{4}\right )} + \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 \sqrt [4]{a} c^{\frac{7}{4}}} - \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{32 \sqrt [4]{a} c^{\frac{7}{4}}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 \sqrt [4]{a} c^{\frac{7}{4}}} + \frac{3 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{16 \sqrt [4]{a} c^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(c*x**4+a)**2,x)

[Out]

-x**3/(4*c*(a + c*x**4)) + 3*sqrt(2)*log(-sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a)
+ sqrt(c)*x**2)/(32*a**(1/4)*c**(7/4)) - 3*sqrt(2)*log(sqrt(2)*a**(1/4)*c**(1/4)
*x + sqrt(a) + sqrt(c)*x**2)/(32*a**(1/4)*c**(7/4)) - 3*sqrt(2)*atan(1 - sqrt(2)
*c**(1/4)*x/a**(1/4))/(16*a**(1/4)*c**(7/4)) + 3*sqrt(2)*atan(1 + sqrt(2)*c**(1/
4)*x/a**(1/4))/(16*a**(1/4)*c**(7/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.300103, size = 185, normalized size = 0.91 \[ \frac{-\frac{8 c^{3/4} x^3}{a+c x^4}+\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{a}}-\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{\sqrt [4]{a}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{\sqrt [4]{a}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{\sqrt [4]{a}}}{32 c^{7/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + c*x^4)^2,x]

[Out]

((-8*c^(3/4)*x^3)/(a + c*x^4) - (6*Sqrt[2]*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4
)])/a^(1/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(1/4) + (3*S
qrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(1/4) - (3*Sqrt
[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/a^(1/4))/(32*c^(7/4)
)

_______________________________________________________________________________________

Maple [A]  time = 0.011, size = 145, normalized size = 0.7 \[ -{\frac{{x}^{3}}{4\,c \left ( c{x}^{4}+a \right ) }}+{\frac{3\,\sqrt{2}}{32\,{c}^{2}}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,\sqrt{2}}{16\,{c}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,\sqrt{2}}{16\,{c}^{2}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(c*x^4+a)^2,x)

[Out]

-1/4*x^3/c/(c*x^4+a)+3/32/c^2/(a/c)^(1/4)*2^(1/2)*ln((x^2-(a/c)^(1/4)*x*2^(1/2)+
(a/c)^(1/2))/(x^2+(a/c)^(1/4)*x*2^(1/2)+(a/c)^(1/2)))+3/16/c^2/(a/c)^(1/4)*2^(1/
2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+3/16/c^2/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(
a/c)^(1/4)*x-1)

_______________________________________________________________________________________

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^6/(c*x^4 + a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.247078, size = 228, normalized size = 1.12 \[ -\frac{4 \, x^{3} - 12 \,{\left (c^{2} x^{4} + a c\right )} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}}}{x + \sqrt{-a c^{3} \sqrt{-\frac{1}{a c^{7}}} + x^{2}}}\right ) - 3 \,{\left (c^{2} x^{4} + a c\right )} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{4}} \log \left (a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}} + x\right ) + 3 \,{\left (c^{2} x^{4} + a c\right )} \left (-\frac{1}{a c^{7}}\right )^{\frac{1}{4}} \log \left (-a c^{5} \left (-\frac{1}{a c^{7}}\right )^{\frac{3}{4}} + x\right )}{16 \,{\left (c^{2} x^{4} + a c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(c*x^4 + a)^2,x, algorithm="fricas")

[Out]

-1/16*(4*x^3 - 12*(c^2*x^4 + a*c)*(-1/(a*c^7))^(1/4)*arctan(a*c^5*(-1/(a*c^7))^(
3/4)/(x + sqrt(-a*c^3*sqrt(-1/(a*c^7)) + x^2))) - 3*(c^2*x^4 + a*c)*(-1/(a*c^7))
^(1/4)*log(a*c^5*(-1/(a*c^7))^(3/4) + x) + 3*(c^2*x^4 + a*c)*(-1/(a*c^7))^(1/4)*
log(-a*c^5*(-1/(a*c^7))^(3/4) + x))/(c^2*x^4 + a*c)

_______________________________________________________________________________________

Sympy [A]  time = 1.88773, size = 44, normalized size = 0.22 \[ - \frac{x^{3}}{4 a c + 4 c^{2} x^{4}} + \operatorname{RootSum}{\left (65536 t^{4} a c^{7} + 81, \left ( t \mapsto t \log{\left (\frac{4096 t^{3} a c^{5}}{27} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(c*x**4+a)**2,x)

[Out]

-x**3/(4*a*c + 4*c**2*x**4) + RootSum(65536*_t**4*a*c**7 + 81, Lambda(_t, _t*log
(4096*_t**3*a*c**5/27 + x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.22572, size = 265, normalized size = 1.3 \[ -\frac{x^{3}}{4 \,{\left (c x^{4} + a\right )} c} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a c^{4}} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{16 \, a c^{4}} - \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a c^{4}} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{32 \, a c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(c*x^4 + a)^2,x, algorithm="giac")

[Out]

-1/4*x^3/((c*x^4 + a)*c) + 3/16*sqrt(2)*(a*c^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x +
sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^4) + 3/16*sqrt(2)*(a*c^3)^(3/4)*arctan(1/
2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a*c^4) - 3/32*sqrt(2)*(a*c^3
)^(3/4)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^4) + 3/32*sqrt(2)*(a*c^
3)^(3/4)*ln(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^4)