3.1154 \(\int \frac{x^4}{\left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=74 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{5/4}}-\frac{x}{b \sqrt [4]{a+b x^4}} \]

[Out]

-(x/(b*(a + b*x^4)^(1/4))) + ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b^(5/4)) +
ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b^(5/4))

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Rubi [A]  time = 0.0598711, antiderivative size = 74, 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 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{5/4}}-\frac{x}{b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^4/(a + b*x^4)^(5/4),x]

[Out]

-(x/(b*(a + b*x^4)^(1/4))) + ArcTan[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b^(5/4)) +
ArcTanh[(b^(1/4)*x)/(a + b*x^4)^(1/4)]/(2*b^(5/4))

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Rubi in Sympy [A]  time = 7.61321, size = 63, normalized size = 0.85 \[ - \frac{x}{b \sqrt [4]{a + b x^{4}}} + \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 b^{\frac{5}{4}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 b^{\frac{5}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4/(b*x**4+a)**(5/4),x)

[Out]

-x/(b*(a + b*x**4)**(1/4)) + atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(2*b**(5/4)) +
atanh(b**(1/4)*x/(a + b*x**4)**(1/4))/(2*b**(5/4))

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Mathematica [A]  time = 0.0992203, size = 94, normalized size = 1.27 \[ \frac{-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{4 b^{5/4}}-\frac{x}{b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^4/(a + b*x^4)^(5/4),x]

[Out]

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

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Maple [F]  time = 0.043, size = 0, normalized size = 0. \[ \int{{x}^{4} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4/(b*x^4+a)^(5/4),x)

[Out]

int(x^4/(b*x^4+a)^(5/4),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)^(5/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259574, size = 252, normalized size = 3.41 \[ \frac{4 \,{\left (b^{2} x^{4} + a b\right )} \frac{1}{b^{5}}^{\frac{1}{4}} \arctan \left (\frac{b^{4} \frac{1}{b^{5}}^{\frac{3}{4}} x}{x \sqrt{\frac{b^{3} \sqrt{\frac{1}{b^{5}}} x^{2} + \sqrt{b x^{4} + a}}{x^{2}}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\right ) +{\left (b^{2} x^{4} + a b\right )} \frac{1}{b^{5}}^{\frac{1}{4}} \log \left (\frac{b^{4} \frac{1}{b^{5}}^{\frac{3}{4}} x +{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right ) -{\left (b^{2} x^{4} + a b\right )} \frac{1}{b^{5}}^{\frac{1}{4}} \log \left (-\frac{b^{4} \frac{1}{b^{5}}^{\frac{3}{4}} x -{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right ) - 4 \,{\left (b x^{4} + a\right )}^{\frac{3}{4}} x}{4 \,{\left (b^{2} x^{4} + a b\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^4 + a)^(5/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.01745, size = 37, normalized size = 0.5 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} \Gamma \left (\frac{9}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4/(b*x**4+a)**(5/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^4/(b*x^4 + a)^(5/4),x, algorithm="giac")

[Out]

integrate(x^4/(b*x^4 + a)^(5/4), x)