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

Optimal. Leaf size=222 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{7/4} c^{5/4}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{7/4} c^{5/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} c^{5/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{7/4} c^{5/4}}+\frac{x}{32 a c \left (a+c x^4\right )}-\frac{x}{8 c \left (a+c x^4\right )^2} \]

[Out]

-x/(8*c*(a + c*x^4)^2) + x/(32*a*c*(a + c*x^4)) - (3*ArcTan[1 - (Sqrt[2]*c^(1/4)
*x)/a^(1/4)])/(64*Sqrt[2]*a^(7/4)*c^(5/4)) + (3*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a
^(1/4)])/(64*Sqrt[2]*a^(7/4)*c^(5/4)) - (3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)
*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(7/4)*c^(5/4)) + (3*Log[Sqrt[a] + Sqrt[2]*a^(1
/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(7/4)*c^(5/4))

_______________________________________________________________________________________

Rubi [A]  time = 0.277425, antiderivative size = 222, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.615 \[ -\frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{7/4} c^{5/4}}+\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{7/4} c^{5/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{7/4} c^{5/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{7/4} c^{5/4}}+\frac{x}{32 a c \left (a+c x^4\right )}-\frac{x}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + c*x^4)^3,x]

[Out]

-x/(8*c*(a + c*x^4)^2) + x/(32*a*c*(a + c*x^4)) - (3*ArcTan[1 - (Sqrt[2]*c^(1/4)
*x)/a^(1/4)])/(64*Sqrt[2]*a^(7/4)*c^(5/4)) + (3*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a
^(1/4)])/(64*Sqrt[2]*a^(7/4)*c^(5/4)) - (3*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)
*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(7/4)*c^(5/4)) + (3*Log[Sqrt[a] + Sqrt[2]*a^(1
/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(7/4)*c^(5/4))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 58.266, size = 206, normalized size = 0.93 \[ - \frac{x}{8 c \left (a + c x^{4}\right )^{2}} + \frac{x}{32 a c \left (a + c x^{4}\right )} - \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 a^{\frac{7}{4}} c^{\frac{5}{4}}} + \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 a^{\frac{7}{4}} c^{\frac{5}{4}}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{7}{4}} c^{\frac{5}{4}}} + \frac{3 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{7}{4}} c^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-x/(8*c*(a + c*x**4)**2) + x/(32*a*c*(a + c*x**4)) - 3*sqrt(2)*log(-sqrt(2)*a**(
1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(256*a**(7/4)*c**(5/4)) + 3*sqrt(2)*lo
g(sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(256*a**(7/4)*c**(5/4))
- 3*sqrt(2)*atan(1 - sqrt(2)*c**(1/4)*x/a**(1/4))/(128*a**(7/4)*c**(5/4)) + 3*sq
rt(2)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))/(128*a**(7/4)*c**(5/4))

_______________________________________________________________________________________

Mathematica [A]  time = 0.187814, size = 203, normalized size = 0.91 \[ \frac{-\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{7/4}}+\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{7/4}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{7/4}}+\frac{8 \sqrt [4]{c} x}{a^2+a c x^4}-\frac{32 \sqrt [4]{c} x}{\left (a+c x^4\right )^2}}{256 c^{5/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + c*x^4)^3,x]

[Out]

((-32*c^(1/4)*x)/(a + c*x^4)^2 + (8*c^(1/4)*x)/(a^2 + a*c*x^4) - (6*Sqrt[2]*ArcT
an[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(7/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^
(1/4)*x)/a^(1/4)])/a^(7/4) - (3*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x
+ Sqrt[c]*x^2])/a^(7/4) + (3*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + S
qrt[c]*x^2])/a^(7/4))/(256*c^(5/4))

_______________________________________________________________________________________

Maple [A]  time = 0.017, size = 162, normalized size = 0.7 \[{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ({\frac{{x}^{5}}{32\,a}}-{\frac{3\,x}{32\,c}} \right ) }+{\frac{3\,\sqrt{2}}{256\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{3\,\sqrt{2}}{128\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{3\,\sqrt{2}}{128\,{a}^{2}c}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(c*x^4+a)^3,x)

[Out]

(1/32*x^5/a-3/32*x/c)/(c*x^4+a)^2+3/256/c/a^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)))+3/128/c/a^
2*(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+3/128/c/a^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^4/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.25148, size = 312, normalized size = 1.41 \[ \frac{4 \, c x^{5} - 12 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{2} c \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{1}{4}}}{x + \sqrt{a^{4} c^{2} \sqrt{-\frac{1}{a^{7} c^{5}}} + x^{2}}}\right ) + 3 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{1}{4}} \log \left (a^{2} c \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{1}{4}} + x\right ) - 3 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{1}{4}} \log \left (-a^{2} c \left (-\frac{1}{a^{7} c^{5}}\right )^{\frac{1}{4}} + x\right ) - 12 \, a x}{128 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/128*(4*c*x^5 - 12*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^7*c^5))^(1/4)*arc
tan(a^2*c*(-1/(a^7*c^5))^(1/4)/(x + sqrt(a^4*c^2*sqrt(-1/(a^7*c^5)) + x^2))) + 3
*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^7*c^5))^(1/4)*log(a^2*c*(-1/(a^7*c^5
))^(1/4) + x) - 3*(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)*(-1/(a^7*c^5))^(1/4)*log(-
a^2*c*(-1/(a^7*c^5))^(1/4) + x) - 12*a*x)/(a*c^3*x^8 + 2*a^2*c^2*x^4 + a^3*c)

_______________________________________________________________________________________

Sympy [A]  time = 4.87081, size = 66, normalized size = 0.3 \[ \frac{- 3 a x + c x^{5}}{32 a^{3} c + 64 a^{2} c^{2} x^{4} + 32 a c^{3} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} a^{7} c^{5} + 81, \left ( t \mapsto t \log{\left (\frac{128 t a^{2} c}{3} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(c*x**4+a)**3,x)

[Out]

(-3*a*x + c*x**5)/(32*a**3*c + 64*a**2*c**2*x**4 + 32*a*c**3*x**8) + RootSum(268
435456*_t**4*a**7*c**5 + 81, Lambda(_t, _t*log(128*_t*a**2*c/3 + x)))

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.234133, size = 278, normalized size = 1.25 \[ \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{2} c^{2}} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{2} c^{2}} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{2} c^{2}} - \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{2} c^{2}} + \frac{c x^{5} - 3 \, a x}{32 \,{\left (c x^{4} + a\right )}^{2} a c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/128*sqrt(2)*(a*c^3)^(1/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)
^(1/4))/(a^2*c^2) + 3/128*sqrt(2)*(a*c^3)^(1/4)*arctan(1/2*sqrt(2)*(2*x - sqrt(2
)*(a/c)^(1/4))/(a/c)^(1/4))/(a^2*c^2) + 3/256*sqrt(2)*(a*c^3)^(1/4)*ln(x^2 + sqr
t(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^2) - 3/256*sqrt(2)*(a*c^3)^(1/4)*ln(x^2 -
sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^2*c^2) + 1/32*(c*x^5 - 3*a*x)/((c*x^4 + a
)^2*a*c)