3.679 \(\int \frac{x^8}{\left (a+c x^4\right )^3} \, dx\)

Optimal. Leaf size=221 \[ -\frac{5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{3/4} c^{9/4}}+\frac{5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{3/4} c^{9/4}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{3/4} c^{9/4}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{3/4} c^{9/4}}-\frac{5 x}{32 c^2 \left (a+c x^4\right )}-\frac{x^5}{8 c \left (a+c x^4\right )^2} \]

[Out]

-x^5/(8*c*(a + c*x^4)^2) - (5*x)/(32*c^2*(a + c*x^4)) - (5*ArcTan[1 - (Sqrt[2]*c
^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(3/4)*c^(9/4)) + (5*ArcTan[1 + (Sqrt[2]*c^(1/4
)*x)/a^(1/4)])/(64*Sqrt[2]*a^(3/4)*c^(9/4)) - (5*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c
^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(3/4)*c^(9/4)) + (5*Log[Sqrt[a] + Sqrt[2
]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(3/4)*c^(9/4))

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Rubi [A]  time = 0.287622, antiderivative size = 221, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 7, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.538 \[ -\frac{5 \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{3/4} c^{9/4}}+\frac{5 \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{128 \sqrt{2} a^{3/4} c^{9/4}}-\frac{5 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt{2} a^{3/4} c^{9/4}}+\frac{5 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{64 \sqrt{2} a^{3/4} c^{9/4}}-\frac{5 x}{32 c^2 \left (a+c x^4\right )}-\frac{x^5}{8 c \left (a+c x^4\right )^2} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + c*x^4)^3,x]

[Out]

-x^5/(8*c*(a + c*x^4)^2) - (5*x)/(32*c^2*(a + c*x^4)) - (5*ArcTan[1 - (Sqrt[2]*c
^(1/4)*x)/a^(1/4)])/(64*Sqrt[2]*a^(3/4)*c^(9/4)) + (5*ArcTan[1 + (Sqrt[2]*c^(1/4
)*x)/a^(1/4)])/(64*Sqrt[2]*a^(3/4)*c^(9/4)) - (5*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c
^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(3/4)*c^(9/4)) + (5*Log[Sqrt[a] + Sqrt[2
]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(128*Sqrt[2]*a^(3/4)*c^(9/4))

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Rubi in Sympy [A]  time = 59.6547, size = 209, normalized size = 0.95 \[ - \frac{x^{5}}{8 c \left (a + c x^{4}\right )^{2}} - \frac{5 x}{32 c^{2} \left (a + c x^{4}\right )} - \frac{5 \sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 a^{\frac{3}{4}} c^{\frac{9}{4}}} + \frac{5 \sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x + \sqrt{a} + \sqrt{c} x^{2} \right )}}{256 a^{\frac{3}{4}} c^{\frac{9}{4}}} - \frac{5 \sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{3}{4}} c^{\frac{9}{4}}} + \frac{5 \sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}} \right )}}{128 a^{\frac{3}{4}} c^{\frac{9}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(c*x**4+a)**3,x)

[Out]

-x**5/(8*c*(a + c*x**4)**2) - 5*x/(32*c**2*(a + c*x**4)) - 5*sqrt(2)*log(-sqrt(2
)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(256*a**(3/4)*c**(9/4)) + 5*sqrt
(2)*log(sqrt(2)*a**(1/4)*c**(1/4)*x + sqrt(a) + sqrt(c)*x**2)/(256*a**(3/4)*c**(
9/4)) - 5*sqrt(2)*atan(1 - sqrt(2)*c**(1/4)*x/a**(1/4))/(128*a**(3/4)*c**(9/4))
+ 5*sqrt(2)*atan(1 + sqrt(2)*c**(1/4)*x/a**(1/4))/(128*a**(3/4)*c**(9/4))

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Mathematica [A]  time = 0.183343, size = 201, normalized size = 0.91 \[ \frac{-\frac{5 \sqrt{2} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{3/4}}+\frac{5 \sqrt{2} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt{a}+\sqrt{c} x^2\right )}{a^{3/4}}-\frac{10 \sqrt{2} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{a^{3/4}}+\frac{10 \sqrt{2} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right )}{a^{3/4}}-\frac{72 \sqrt [4]{c} x}{a+c x^4}+\frac{32 a \sqrt [4]{c} x}{\left (a+c x^4\right )^2}}{256 c^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + c*x^4)^3,x]

[Out]

((32*a*c^(1/4)*x)/(a + c*x^4)^2 - (72*c^(1/4)*x)/(a + c*x^4) - (10*Sqrt[2]*ArcTa
n[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/a^(3/4) + (10*Sqrt[2]*ArcTan[1 + (Sqrt[2]*c^
(1/4)*x)/a^(1/4)])/a^(3/4) - (5*Sqrt[2]*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x
+ Sqrt[c]*x^2])/a^(3/4) + (5*Sqrt[2]*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + S
qrt[c]*x^2])/a^(3/4))/(256*c^(9/4))

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Maple [A]  time = 0.015, size = 163, normalized size = 0.7 \[{\frac{1}{ \left ( c{x}^{4}+a \right ) ^{2}} \left ( -{\frac{9\,{x}^{5}}{32\,c}}-{\frac{5\,ax}{32\,{c}^{2}}} \right ) }+{\frac{5\,\sqrt{2}}{256\,{c}^{2}a}\sqrt [4]{{\frac{a}{c}}}\ln \left ({1 \left ({x}^{2}+\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) \left ({x}^{2}-\sqrt [4]{{\frac{a}{c}}}x\sqrt{2}+\sqrt{{\frac{a}{c}}} \right ) ^{-1}} \right ) }+{\frac{5\,\sqrt{2}}{128\,{c}^{2}a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}+1 \right ) }+{\frac{5\,\sqrt{2}}{128\,{c}^{2}a}\sqrt [4]{{\frac{a}{c}}}\arctan \left ({x\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{c}}}}}}-1 \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(c*x^4+a)^3,x)

[Out]

(-9/32/c*x^5-5/32*a/c^2*x)/(c*x^4+a)^2+5/256/c^2*(a/c)^(1/4)/a*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)))+5/128
/c^2*(a/c)^(1/4)/a*2^(1/2)*arctan(2^(1/2)/(a/c)^(1/4)*x+1)+5/128/c^2*(a/c)^(1/4)
/a*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^8/(c*x^4 + a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.247375, size = 306, normalized size = 1.38 \[ -\frac{36 \, c x^{5} + 20 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{a c^{2} \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{1}{4}}}{x + \sqrt{a^{2} c^{4} \sqrt{-\frac{1}{a^{3} c^{9}}} + x^{2}}}\right ) - 5 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{1}{4}} \log \left (a c^{2} \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{1}{4}} + x\right ) + 5 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )} \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{1}{4}} \log \left (-a c^{2} \left (-\frac{1}{a^{3} c^{9}}\right )^{\frac{1}{4}} + x\right ) + 20 \, a x}{128 \,{\left (c^{4} x^{8} + 2 \, a c^{3} x^{4} + a^{2} c^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(c*x^4 + a)^3,x, algorithm="fricas")

[Out]

-1/128*(36*c*x^5 + 20*(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a^3*c^9))^(1/4)*arc
tan(a*c^2*(-1/(a^3*c^9))^(1/4)/(x + sqrt(a^2*c^4*sqrt(-1/(a^3*c^9)) + x^2))) - 5
*(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a^3*c^9))^(1/4)*log(a*c^2*(-1/(a^3*c^9))
^(1/4) + x) + 5*(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)*(-1/(a^3*c^9))^(1/4)*log(-a*c^
2*(-1/(a^3*c^9))^(1/4) + x) + 20*a*x)/(c^4*x^8 + 2*a*c^3*x^4 + a^2*c^2)

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Sympy [A]  time = 4.97007, size = 66, normalized size = 0.3 \[ - \frac{5 a x + 9 c x^{5}}{32 a^{2} c^{2} + 64 a c^{3} x^{4} + 32 c^{4} x^{8}} + \operatorname{RootSum}{\left (268435456 t^{4} a^{3} c^{9} + 625, \left ( t \mapsto t \log{\left (\frac{128 t a c^{2}}{5} + x \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(c*x**4+a)**3,x)

[Out]

-(5*a*x + 9*c*x**5)/(32*a**2*c**2 + 64*a*c**3*x**4 + 32*c**4*x**8) + RootSum(268
435456*_t**4*a**3*c**9 + 625, Lambda(_t, _t*log(128*_t*a*c**2/5 + x)))

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GIAC/XCAS [A]  time = 0.224414, size = 275, normalized size = 1.24 \[ \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x + \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a c^{3}} + \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (2 \, x - \sqrt{2} \left (\frac{a}{c}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{c}\right )^{\frac{1}{4}}}\right )}{128 \, a c^{3}} + \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} + \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a c^{3}} - \frac{5 \, \sqrt{2} \left (a c^{3}\right )^{\frac{1}{4}}{\rm ln}\left (x^{2} - \sqrt{2} x \left (\frac{a}{c}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{c}}\right )}{256 \, a c^{3}} - \frac{9 \, c x^{5} + 5 \, a x}{32 \,{\left (c x^{4} + a\right )}^{2} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(c*x^4 + a)^3,x, algorithm="giac")

[Out]

5/128*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*c^3) + 5/128*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*c^3) + 5/256*sqrt(2)*(a*c^3)^(1/4)*ln(x^2 + sqrt(2)
*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^3) - 5/256*sqrt(2)*(a*c^3)^(1/4)*ln(x^2 - sqrt(
2)*x*(a/c)^(1/4) + sqrt(a/c))/(a*c^3) - 1/32*(9*c*x^5 + 5*a*x)/((c*x^4 + a)^2*c^
2)