3.680 \(\int \frac{x^6}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=226 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{5/4} c^{7/4}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{5/4} c^{7/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{5/4} c^{7/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{5/4} c^{7/4}}+\frac{3 x^3}{32 a c \left (a+c x^4\right )}-\frac{x^3}{8 c \left (a+c x^4\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.289421, antiderivative size = 226, 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 \[ \frac{3 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{5/4} c^{7/4}}-\frac{3 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{5/4} c^{7/4}}-\frac{3 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{5/4} c^{7/4}}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{5/4} c^{7/4}}+\frac{3 x^3}{32 a c \left (a+c x^4\right )}-\frac{x^3}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^6/(a + c*x^4)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 60.376, size = 211, normalized size = 0.93 \[ - \frac{x^{3}}{8 c \left (a + c x^{4}\right )^{2}} + \frac{3 x^{3}}{32 a c \left (a + c x^{4}\right )} + \frac{3 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 a^{\frac{5}{4}} c^{\frac{7}{4}}} - \frac{3 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 a^{\frac{5}{4}} c^{\frac{7}{4}}} - \frac{3 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{5}{4}} c^{\frac{7}{4}}} + \frac{3 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{5}{4}} c^{\frac{7}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6/(c*x**4+a)**3,x)

[Out]

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

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Mathematica [A]  time = 0.2015, size = 207, normalized size = 0.92 \[ \frac{\frac{3 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{5/4}}-\frac{3 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{5/4}}-\frac{6 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{5/4}}+\frac{6 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{5/4}}+\frac{24 c^{3/4} x^3}{a^2+a c x^4}-\frac{32 c^{3/4} x^3}{\left (a+c x^4\right )^2}}{256 c^{7/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6/(a + c*x^4)^3,x]

[Out]

((-32*c^(3/4)*x^3)/(a + c*x^4)^2 + (24*c^(3/4)*x^3)/(a^2 + a*c*x^4) - (6*Sqrt[2]
*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(5/4) + (6*Sqrt[2]*ArcTan[1 + (Sqrt[
2]*c^(1/4)*x)/a^(1/4)])/a^(5/4) + (3*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/
4)*x + Sqrt[c]*x^2])/a^(5/4) - (3*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*
x + Sqrt[c]*x^2])/a^(5/4))/(256*c^(7/4))

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Maple [A]  time = 0.015, size = 164, normalized size = 0.7 \[{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ({\frac{3\,{x}^{7}}{32\,a}}-{\frac{{x}^{3}}{32\,c}} \right ) }+{\frac{3\,\sqrt{2}}{256\,{c}^{2}a}\ln \left ({1 \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,\sqrt{2}}{128\,{c}^{2}a}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+{\frac{3\,\sqrt{2}}{128\,{c}^{2}a}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6/(c*x^4+a)^3,x)

[Out]

(3/32*x^7/a-1/32/c*x^3)/(c*x^4+a)^2+3/256/c^2/a/(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^
2/a/(a/c)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+3/128/c^2/a/(a/c)^(1/4)*
2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x-1)

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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^6/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.246098, size = 324, normalized size = 1.43 \[ \frac{12 \, c x^{7} - 4 \, a x^{3} + 12 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac{1}{a^{5} c^{7}}\right )^{\frac{1}{4}} \arctan \left (\frac{a^{4} c^{5} \left (-\frac{1}{a^{5} c^{7}}\right )^{\frac{3}{4}}}{x + \sqrt{-a^{3} c^{3} \sqrt{-\frac{1}{a^{5} c^{7}}} + x^{2}}}\right ) + 3 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac{1}{a^{5} c^{7}}\right )^{\frac{1}{4}} \log \left (a^{4} c^{5} \left (-\frac{1}{a^{5} c^{7}}\right )^{\frac{3}{4}} + x\right ) - 3 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )} \left (-\frac{1}{a^{5} c^{7}}\right )^{\frac{1}{4}} \log \left (-a^{4} c^{5} \left (-\frac{1}{a^{5} c^{7}}\right )^{\frac{3}{4}} + x\right )}{128 \,{\left (a c^{3} x^{8} + 2 \, a^{2} c^{2} x^{4} + a^{3} c\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.0495, size = 71, normalized size = 0.31 \[ \frac{- a x^{3} + 3 c x^{7}}{32 a^{3} c + 64 a^{2} c^{2} x^{4} + 32 a c^{3} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} a^{5} c^{7} + 81, \left ( t \mapsto t \log{\left (\frac{2097152 t^{3} a^{4} c^{5}}{27} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6/(c*x**4+a)**3,x)

[Out]

(-a*x**3 + 3*c*x**7)/(32*a**3*c + 64*a**2*c**2*x**4 + 32*a*c**3*x**8) + RootSum(
268435456*_t**4*a**5*c**7 + 81, Lambda(_t, _t*log(2097152*_t**3*a**4*c**5/27 + x
)))

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GIAC/XCAS [A]  time = 0.224173, size = 282, normalized size = 1.25 \[ \frac{3 \, c x^{7} - a x^{3}}{32 \,{\left (c x^{4} + a\right )}^{2} a c} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{2} c^{4}} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a^{2} c^{4}} - \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{2} c^{4}} + \frac{3 \, \sqrt{2} \left (a c^{3}\right )^{\frac{3}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a^{2} c^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^6/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

1/32*(3*c*x^7 - a*x^3)/((c*x^4 + a)^2*a*c) + 3/128*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^2*c^4) + 3/128*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^2*c^
4) - 3/256*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 + sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^
2*c^4) + 3/256*sqrt(2)*(a*c^3)^(3/4)*ln(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt(a/c))
/(a^2*c^4)