3.1451 \(\int \frac{x^5}{a+b x^8} \, dx\)

Optimal. Leaf size=193 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} b^{3/4}}+\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} \sqrt [4]{a} b^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} \sqrt [4]{a} b^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.348357, antiderivative size = 193, 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 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}\right )}{4 \sqrt{2} \sqrt [4]{a} b^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x^2}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} \sqrt [4]{a} b^{3/4}}+\frac{\log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} \sqrt [4]{a} b^{3/4}}-\frac{\log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^2+\sqrt{a}+\sqrt{b} x^4\right )}{8 \sqrt{2} \sqrt [4]{a} b^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x^5/(a + b*x^8),x]

[Out]

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

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Rubi in Sympy [A]  time = 55.0707, size = 178, normalized size = 0.92 \[ \frac{\sqrt{2} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{2} + \sqrt{a} + \sqrt{b} x^{4} \right )}}{16 \sqrt [4]{a} b^{\frac{3}{4}}} - \frac{\sqrt{2} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} x^{2} + \sqrt{a} + \sqrt{b} x^{4} \right )}}{16 \sqrt [4]{a} b^{\frac{3}{4}}} - \frac{\sqrt{2} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} x^{2}}{\sqrt [4]{a}} \right )}}{8 \sqrt [4]{a} b^{\frac{3}{4}}} + \frac{\sqrt{2} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} x^{2}}{\sqrt [4]{a}} \right )}}{8 \sqrt [4]{a} b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5/(b*x**8+a),x)

[Out]

sqrt(2)*log(-sqrt(2)*a**(1/4)*b**(1/4)*x**2 + sqrt(a) + sqrt(b)*x**4)/(16*a**(1/
4)*b**(3/4)) - sqrt(2)*log(sqrt(2)*a**(1/4)*b**(1/4)*x**2 + sqrt(a) + sqrt(b)*x*
*4)/(16*a**(1/4)*b**(3/4)) - sqrt(2)*atan(1 - sqrt(2)*b**(1/4)*x**2/a**(1/4))/(8
*a**(1/4)*b**(3/4)) + sqrt(2)*atan(1 + sqrt(2)*b**(1/4)*x**2/a**(1/4))/(8*a**(1/
4)*b**(3/4))

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Mathematica [A]  time = 0.421377, size = 279, normalized size = 1.45 \[ -\frac{\log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )+\log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \sin \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\log \left (-2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-\log \left (2 \sqrt [8]{a} \sqrt [8]{b} x \cos \left (\frac{\pi }{8}\right )+\sqrt [4]{a}+\sqrt [4]{b} x^2\right )-2 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}-\tan \left (\frac{\pi }{8}\right )\right )+2 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \sec \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\tan \left (\frac{\pi }{8}\right )\right )+2 \tan ^{-1}\left (\cot \left (\frac{\pi }{8}\right )-\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}\right )+2 \tan ^{-1}\left (\frac{\sqrt [8]{b} x \csc \left (\frac{\pi }{8}\right )}{\sqrt [8]{a}}+\cot \left (\frac{\pi }{8}\right )\right )}{8 \sqrt{2} \sqrt [4]{a} b^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^5/(a + b*x^8),x]

[Out]

-(2*ArcTan[Cot[Pi/8] - (b^(1/8)*x*Csc[Pi/8])/a^(1/8)] + 2*ArcTan[Cot[Pi/8] + (b^
(1/8)*x*Csc[Pi/8])/a^(1/8)] - 2*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) - Tan[Pi/8]
] + 2*ArcTan[(b^(1/8)*x*Sec[Pi/8])/a^(1/8) + Tan[Pi/8]] - Log[a^(1/4) + b^(1/4)*
x^2 - 2*a^(1/8)*b^(1/8)*x*Cos[Pi/8]] - Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(
1/8)*x*Cos[Pi/8]] + Log[a^(1/4) + b^(1/4)*x^2 - 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]] +
Log[a^(1/4) + b^(1/4)*x^2 + 2*a^(1/8)*b^(1/8)*x*Sin[Pi/8]])/(8*Sqrt[2]*a^(1/4)*
b^(3/4))

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Maple [A]  time = 0.003, size = 136, normalized size = 0.7 \[{\frac{\sqrt{2}}{16\,b}\ln \left ({1 \left ({x}^{4}-\sqrt [4]{{\frac{a}{b}}}{x}^{2}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ({x}^{4}+\sqrt [4]{{\frac{a}{b}}}{x}^{2}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}}{8\,b}\arctan \left ({{x}^{2}\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{\sqrt{2}}{8\,b}\arctan \left ({{x}^{2}\sqrt{2}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5/(b*x^8+a),x)

[Out]

1/16/b/(a/b)^(1/4)*2^(1/2)*ln((x^4-(a/b)^(1/4)*x^2*2^(1/2)+(a/b)^(1/2))/(x^4+(a/
b)^(1/4)*x^2*2^(1/2)+(a/b)^(1/2)))+1/8/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b
)^(1/4)*x^2+1)+1/8/b/(a/b)^(1/4)*2^(1/2)*arctan(2^(1/2)/(a/b)^(1/4)*x^2-1)

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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^5/(b*x^8 + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231538, size = 162, normalized size = 0.84 \[ \frac{1}{2} \, \left (-\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{a b^{2} \left (-\frac{1}{a b^{3}}\right )^{\frac{3}{4}}}{x^{2} + \sqrt{x^{4} - a b \sqrt{-\frac{1}{a b^{3}}}}}\right ) + \frac{1}{8} \, \left (-\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \log \left (a b^{2} \left (-\frac{1}{a b^{3}}\right )^{\frac{3}{4}} + x^{2}\right ) - \frac{1}{8} \, \left (-\frac{1}{a b^{3}}\right )^{\frac{1}{4}} \log \left (-a b^{2} \left (-\frac{1}{a b^{3}}\right )^{\frac{3}{4}} + x^{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^8 + a),x, algorithm="fricas")

[Out]

1/2*(-1/(a*b^3))^(1/4)*arctan(a*b^2*(-1/(a*b^3))^(3/4)/(x^2 + sqrt(x^4 - a*b*sqr
t(-1/(a*b^3))))) + 1/8*(-1/(a*b^3))^(1/4)*log(a*b^2*(-1/(a*b^3))^(3/4) + x^2) -
1/8*(-1/(a*b^3))^(1/4)*log(-a*b^2*(-1/(a*b^3))^(3/4) + x^2)

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Sympy [A]  time = 0.546369, size = 27, normalized size = 0.14 \[ \operatorname{RootSum}{\left (4096 t^{4} a b^{3} + 1, \left ( t \mapsto t \log{\left (512 t^{3} a b^{2} + x^{2} \right )} \right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5/(b*x**8+a),x)

[Out]

RootSum(4096*_t**4*a*b**3 + 1, Lambda(_t, _t*log(512*_t**3*a*b**2 + x**2)))

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GIAC/XCAS [A]  time = 0.231252, size = 269, normalized size = 1.39 \[ \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} x^{4} \arctan \left (\frac{\sqrt{2}{\left (2 \, x^{2} + \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a b} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} x^{4} \arctan \left (\frac{\sqrt{2}{\left (2 \, x^{2} - \sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, a b} + \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} x^{4}{\rm ln}\left (x^{4} + \sqrt{2} x^{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{16 \, a b} - \frac{\sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} x^{4}{\rm ln}\left (x^{4} - \sqrt{2} x^{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{\frac{a}{b}}\right )}{16 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^5/(b*x^8 + a),x, algorithm="giac")

[Out]

1/8*sqrt(2)*(a*b^3)^(1/4)*x^4*arctan(1/2*sqrt(2)*(2*x^2 + sqrt(2)*(a/b)^(1/4))/(
a/b)^(1/4))/(a*b) + 1/8*sqrt(2)*(a*b^3)^(1/4)*x^4*arctan(1/2*sqrt(2)*(2*x^2 - sq
rt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a*b) + 1/16*sqrt(2)*(a*b^3)^(1/4)*x^4*ln(x^4 +
sqrt(2)*x^2*(a/b)^(1/4) + sqrt(a/b))/(a*b) - 1/16*sqrt(2)*(a*b^3)^(1/4)*x^4*ln(x
^4 - sqrt(2)*x^2*(a/b)^(1/4) + sqrt(a/b))/(a*b)