3.1218 \(\int \frac{x^4}{\sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=233 \[ -\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{5/4}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{8 \sqrt{2} b^{5/4}}-\frac{a \log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{16 \sqrt{2} b^{5/4}}+\frac{a \log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{16 \sqrt{2} b^{5/4}}-\frac{x \left (a-b x^4\right )^{3/4}}{4 b} \]

[Out]

-(x*(a - b*x^4)^(3/4))/(4*b) - (a*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/
4)])/(8*Sqrt[2]*b^(5/4)) + (a*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])
/(8*Sqrt[2]*b^(5/4)) - (a*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/
4)*x)/(a - b*x^4)^(1/4)])/(16*Sqrt[2]*b^(5/4)) + (a*Log[1 + (Sqrt[b]*x^2)/Sqrt[a
- b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(16*Sqrt[2]*b^(5/4))

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Rubi [A]  time = 0.240694, antiderivative size = 233, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{5/4}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{8 \sqrt{2} b^{5/4}}-\frac{a \log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{16 \sqrt{2} b^{5/4}}+\frac{a \log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{16 \sqrt{2} b^{5/4}}-\frac{x \left (a-b x^4\right )^{3/4}}{4 b} \]

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[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/
4)])/(8*Sqrt[2]*b^(5/4)) + (a*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])
/(8*Sqrt[2]*b^(5/4)) - (a*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/
4)*x)/(a - b*x^4)^(1/4)])/(16*Sqrt[2]*b^(5/4)) + (a*Log[1 + (Sqrt[b]*x^2)/Sqrt[a
- b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(16*Sqrt[2]*b^(5/4))

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Rubi in Sympy [A]  time = 34.9547, size = 207, normalized size = 0.89 \[ - \frac{\sqrt{2} a \log{\left (- \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{32 b^{\frac{5}{4}}} + \frac{\sqrt{2} a \log{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{32 b^{\frac{5}{4}}} + \frac{\sqrt{2} a \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} - 1 \right )}}{16 b^{\frac{5}{4}}} + \frac{\sqrt{2} a \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + 1 \right )}}{16 b^{\frac{5}{4}}} - \frac{x \left (a - b x^{4}\right )^{\frac{3}{4}}}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*a*log(-sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) + sqrt(b)*x**2/sqrt(a - b
*x**4) + 1)/(32*b**(5/4)) + sqrt(2)*a*log(sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4)
+ sqrt(b)*x**2/sqrt(a - b*x**4) + 1)/(32*b**(5/4)) + sqrt(2)*a*atan(sqrt(2)*b**
(1/4)*x/(a - b*x**4)**(1/4) - 1)/(16*b**(5/4)) + sqrt(2)*a*atan(sqrt(2)*b**(1/4)
*x/(a - b*x**4)**(1/4) + 1)/(16*b**(5/4)) - x*(a - b*x**4)**(3/4)/(4*b)

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Mathematica [A]  time = 0.125308, size = 195, normalized size = 0.84 \[ \frac{a \left (-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )-\log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )+\log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )\right )}{16 \sqrt{2} b^{5/4}}-\frac{x \left (a-b x^4\right )^{3/4}}{4 b} \]

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[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)
^(1/4)] + 2*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)] - Log[1 + (Sqrt[b]
*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)] + Log[1 + (Sqrt[b
]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)]))/(16*Sqrt[2]*b^
(5/4))

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Maple [F]  time = 0.037, 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 = 0.248336, size = 279, normalized size = 1.2 \[ -\frac{4 \, b \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{4} x \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{3}{4}}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{3} + x \sqrt{-\frac{a^{4} b^{3} x^{2} \sqrt{-\frac{a^{4}}{b^{5}}} - \sqrt{-b x^{4} + a} a^{6}}{x^{2}}}}\right ) + b \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{1}{4}} \log \left (\frac{b^{4} x \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{3}{4}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{3}}{x}\right ) - b \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{1}{4}} \log \left (-\frac{b^{4} x \left (-\frac{a^{4}}{b^{5}}\right )^{\frac{3}{4}} -{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{3}}{x}\right ) + 4 \,{\left (-b x^{4} + a\right )}^{\frac{3}{4}} x}{16 \, b} \]

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 = 4.43741, size = 39, normalized size = 0.17 \[ \frac{x^{5} \Gamma \left (\frac{5}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{5}{4} \\ \frac{9}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{9}{4}\right )} \]

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(2*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)