3.1093 \(\int \frac{x^8}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=104 \[ \frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}}+\frac{5 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}}-\frac{5 a x \left (a+b x^4\right )^{3/4}}{32 b^2}+\frac{x^5 \left (a+b x^4\right )^{3/4}}{8 b} \]

[Out]

(-5*a*x*(a + b*x^4)^(3/4))/(32*b^2) + (x^5*(a + b*x^4)^(3/4))/(8*b) + (5*a^2*Arc
Tan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(9/4)) + (5*a^2*ArcTanh[(b^(1/4)*x)/(a
+ b*x^4)^(1/4)])/(64*b^(9/4))

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Rubi [A]  time = 0.0930655, antiderivative size = 104, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}}+\frac{5 a^2 \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{64 b^{9/4}}-\frac{5 a x \left (a+b x^4\right )^{3/4}}{32 b^2}+\frac{x^5 \left (a+b x^4\right )^{3/4}}{8 b} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x^4)^(1/4),x]

[Out]

(-5*a*x*(a + b*x^4)^(3/4))/(32*b^2) + (x^5*(a + b*x^4)^(3/4))/(8*b) + (5*a^2*Arc
Tan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(64*b^(9/4)) + (5*a^2*ArcTanh[(b^(1/4)*x)/(a
+ b*x^4)^(1/4)])/(64*b^(9/4))

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Rubi in Sympy [A]  time = 11.2712, size = 97, normalized size = 0.93 \[ \frac{5 a^{2} \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 b^{\frac{9}{4}}} + \frac{5 a^{2} \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{64 b^{\frac{9}{4}}} - \frac{5 a x \left (a + b x^{4}\right )^{\frac{3}{4}}}{32 b^{2}} + \frac{x^{5} \left (a + b x^{4}\right )^{\frac{3}{4}}}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x**4+a)**(1/4),x)

[Out]

5*a**2*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(64*b**(9/4)) + 5*a**2*atanh(b**(1/4
)*x/(a + b*x**4)**(1/4))/(64*b**(9/4)) - 5*a*x*(a + b*x**4)**(3/4)/(32*b**2) + x
**5*(a + b*x**4)**(3/4)/(8*b)

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Mathematica [A]  time = 0.1428, size = 112, normalized size = 1.08 \[ \frac{5 a^2 \left (-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right )}{128 b^{9/4}}+\left (a+b x^4\right )^{3/4} \left (\frac{x^5}{8 b}-\frac{5 a x}{32 b^2}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + b*x^4)^(1/4),x]

[Out]

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

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Maple [F]  time = 0.037, size = 0, normalized size = 0. \[ \int{{x}^{8}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^4+a)^(1/4),x)

[Out]

int(x^8/(b*x^4+a)^(1/4),x)

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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/(b*x^4 + a)^(1/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276555, size = 285, normalized size = 2.74 \[ \frac{20 \, b^{2} \left (\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{7} x \left (\frac{a^{8}}{b^{9}}\right )^{\frac{3}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6} + x \sqrt{\frac{a^{8} b^{5} x^{2} \sqrt{\frac{a^{8}}{b^{9}}} + \sqrt{b x^{4} + a} a^{12}}{x^{2}}}}\right ) + 5 \, b^{2} \left (\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{125 \,{\left (b^{7} x \left (\frac{a^{8}}{b^{9}}\right )^{\frac{3}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6}\right )}}{x}\right ) - 5 \, b^{2} \left (\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \log \left (-\frac{125 \,{\left (b^{7} x \left (\frac{a^{8}}{b^{9}}\right )^{\frac{3}{4}} -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{6}\right )}}{x}\right ) + 4 \,{\left (4 \, b x^{5} - 5 \, a x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{128 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^4 + a)^(1/4),x, algorithm="fricas")

[Out]

1/128*(20*b^2*(a^8/b^9)^(1/4)*arctan(b^7*x*(a^8/b^9)^(3/4)/((b*x^4 + a)^(1/4)*a^
6 + x*sqrt((a^8*b^5*x^2*sqrt(a^8/b^9) + sqrt(b*x^4 + a)*a^12)/x^2))) + 5*b^2*(a^
8/b^9)^(1/4)*log(125*(b^7*x*(a^8/b^9)^(3/4) + (b*x^4 + a)^(1/4)*a^6)/x) - 5*b^2*
(a^8/b^9)^(1/4)*log(-125*(b^7*x*(a^8/b^9)^(3/4) - (b*x^4 + a)^(1/4)*a^6)/x) + 4*
(4*b*x^5 - 5*a*x)*(b*x^4 + a)^(3/4))/b^2

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Sympy [A]  time = 6.08321, size = 37, normalized size = 0.36 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**4+a)**(1/4),x)

[Out]

x**9*gamma(9/4)*hyper((1/4, 9/4), (13/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(1/4)
*gamma(13/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^4 + a)^(1/4),x, algorithm="giac")

[Out]

integrate(x^8/(b*x^4 + a)^(1/4), x)