3.1094 \(\int \frac{x^4}{\sqrt [4]{a+b x^4}} \, dx\)

Optimal. Leaf size=78 \[ -\frac{a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{5/4}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{5/4}}+\frac{x \left (a+b x^4\right )^{3/4}}{4 b} \]

[Out]

(x*(a + b*x^4)^(3/4))/(4*b) - (a*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b^(5/
4)) - (a*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b^(5/4))

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Rubi [A]  time = 0.0603056, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ -\frac{a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{5/4}}-\frac{a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{5/4}}+\frac{x \left (a+b x^4\right )^{3/4}}{4 b} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^4)^(1/4),x]

[Out]

(x*(a + b*x^4)^(3/4))/(4*b) - (a*ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b^(5/
4)) - (a*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b^(5/4))

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Rubi in Sympy [A]  time = 7.66322, size = 68, normalized size = 0.87 \[ - \frac{a \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 b^{\frac{5}{4}}} - \frac{a \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 b^{\frac{5}{4}}} + \frac{x \left (a + b x^{4}\right )^{\frac{3}{4}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0618962, size = 97, normalized size = 1.24 \[ \frac{x \left (a+b x^4\right )^{3/4}}{4 b}-\frac{a \left (-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right )}{16 b^{5/4}} \]

Antiderivative was successfully verified.

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

[Out]

(x*(a + b*x^4)^(3/4))/(4*b) - (a*(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)]))/(1
6*b^(5/4))

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Maple [F]  time = 0.036, size = 0, normalized size = 0. \[ \int{{x}^{4}{\frac{1}{\sqrt [4]{b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^4+a)^(1/4),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.282128, size = 261, normalized size = 3.35 \[ -\frac{4 \, b \left (\frac{a^{4}}{b^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{4} x \left (\frac{a^{4}}{b^{5}}\right )^{\frac{3}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} + x \sqrt{\frac{a^{4} b^{3} x^{2} \sqrt{\frac{a^{4}}{b^{5}}} + \sqrt{b x^{4} + a} a^{6}}{x^{2}}}}\right ) + b \left (\frac{a^{4}}{b^{5}}\right )^{\frac{1}{4}} \log \left (\frac{b^{4} x \left (\frac{a^{4}}{b^{5}}\right )^{\frac{3}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3}}{x}\right ) - b \left (\frac{a^{4}}{b^{5}}\right )^{\frac{1}{4}} \log \left (-\frac{b^{4} x \left (\frac{a^{4}}{b^{5}}\right )^{\frac{3}{4}} -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3}}{x}\right ) - 4 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} x}{16 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*(4*b*(a^4/b^5)^(1/4)*arctan(b^4*x*(a^4/b^5)^(3/4)/((b*x^4 + a)^(1/4)*a^3 +
x*sqrt((a^4*b^3*x^2*sqrt(a^4/b^5) + sqrt(b*x^4 + a)*a^6)/x^2))) + b*(a^4/b^5)^(
1/4)*log((b^4*x*(a^4/b^5)^(3/4) + (b*x^4 + a)^(1/4)*a^3)/x) - b*(a^4/b^5)^(1/4)*
log(-(b^4*x*(a^4/b^5)^(3/4) - (b*x^4 + a)^(1/4)*a^3)/x) - 4*(b*x^4 + a)^(3/4)*x)
/b

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Sympy [A]  time = 4.2417, size = 37, normalized size = 0.47 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**5*gamma(5/4)*hyper((1/4, 5/4), (9/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(1/4)*
gamma(9/4))

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(x^4/(b*x^4 + a)^(1/4), x)