3.1188 \(\int x^2 \sqrt [4]{a-b x^4} \, dx\)

Optimal. Leaf size=232 \[ -\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{3/4}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{8 \sqrt{2} b^{3/4}}+\frac{a \log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{16 \sqrt{2} b^{3/4}}-\frac{a \log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{16 \sqrt{2} b^{3/4}}+\frac{1}{4} x^3 \sqrt [4]{a-b x^4} \]

[Out]

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

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Rubi [A]  time = 0.259012, antiderivative size = 232, 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 \[ -\frac{a \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{8 \sqrt{2} b^{3/4}}+\frac{a \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{8 \sqrt{2} b^{3/4}}+\frac{a \log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{16 \sqrt{2} b^{3/4}}-\frac{a \log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{16 \sqrt{2} b^{3/4}}+\frac{1}{4} x^3 \sqrt [4]{a-b x^4} \]

Antiderivative was successfully verified.

[In]  Int[x^2*(a - b*x^4)^(1/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 35.1296, size = 207, normalized size = 0.89 \[ \frac{\sqrt{2} a \log{\left (- \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{32 b^{\frac{3}{4}}} - \frac{\sqrt{2} a \log{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{32 b^{\frac{3}{4}}} + \frac{\sqrt{2} a \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} - 1 \right )}}{16 b^{\frac{3}{4}}} + \frac{\sqrt{2} a \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + 1 \right )}}{16 b^{\frac{3}{4}}} + \frac{x^{3} \sqrt [4]{a - b x^{4}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2*(-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**(3/4)) - 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**(3/4)) + sqrt(2)*a*atan(sqrt(2)*b**(
1/4)*x/(a - b*x**4)**(1/4) - 1)/(16*b**(3/4)) + sqrt(2)*a*atan(sqrt(2)*b**(1/4)*
x/(a - b*x**4)**(1/4) + 1)/(16*b**(3/4)) + x**3*(a - b*x**4)**(1/4)/4

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Mathematica [C]  time = 0.0461211, size = 64, normalized size = 0.28 \[ \frac{x^3 \left (a \left (1-\frac{b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )+3 a-3 b x^4\right )}{12 \left (a-b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2*(a - b*x^4)^(1/4),x]

[Out]

(x^3*(3*a - 3*b*x^4 + a*(1 - (b*x^4)/a)^(3/4)*Hypergeometric2F1[3/4, 3/4, 7/4, (
b*x^4)/a]))/(12*(a - b*x^4)^(3/4))

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Maple [F]  time = 0.029, size = 0, normalized size = 0. \[ \int{x}^{2}\sqrt [4]{-b{x}^{4}+a}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(-b*x^4+a)^(1/4),x)

[Out]

int(x^2*(-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((-b*x^4 + a)^(1/4)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.260762, size = 250, normalized size = 1.08 \[ \frac{1}{4} \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{3} + \frac{1}{4} \, \left (-\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{\left (-\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} b x}{x \sqrt{\frac{\sqrt{-\frac{a^{4}}{b^{3}}} b^{2} x^{2} + \sqrt{-b x^{4} + a} a^{2}}{x^{2}}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a}\right ) - \frac{1}{16} \, \left (-\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} \log \left (\frac{\left (-\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} b x +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a}{x}\right ) + \frac{1}{16} \, \left (-\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} \log \left (-\frac{\left (-\frac{a^{4}}{b^{3}}\right )^{\frac{1}{4}} b x -{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.4794, size = 41, normalized size = 0.18 \[ \frac{\sqrt [4]{a} x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2*(-b*x**4+a)**(1/4),x)

[Out]

a**(1/4)*x**3*gamma(3/4)*hyper((-1/4, 3/4), (7/4,), b*x**4*exp_polar(2*I*pi)/a)/
(4*gamma(7/4))

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GIAC/XCAS [A]  time = 0.266707, size = 265, normalized size = 1.14 \[ \frac{1}{32} \,{\left (\frac{8 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}} x^{3}}{a} - \frac{2 \, \sqrt{2} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} + \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, b^{\frac{1}{4}}}\right )}{b^{\frac{3}{4}}} - \frac{2 \, \sqrt{2} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} b^{\frac{1}{4}} - \frac{2 \,{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right )}}{2 \, b^{\frac{1}{4}}}\right )}{b^{\frac{3}{4}}} - \frac{\sqrt{2}{\rm ln}\left (\sqrt{b} + \frac{\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{\frac{1}{4}}}{x} + \frac{\sqrt{-b x^{4} + a}}{x^{2}}\right )}{b^{\frac{3}{4}}} + \frac{\sqrt{2}{\rm ln}\left (\sqrt{b} - \frac{\sqrt{2}{\left (-b x^{4} + a\right )}^{\frac{1}{4}} b^{\frac{1}{4}}}{x} + \frac{\sqrt{-b x^{4} + a}}{x^{2}}\right )}{b^{\frac{3}{4}}}\right )} a \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x^4 + a)^(1/4)*x^2,x, algorithm="giac")

[Out]

1/32*(8*(-b*x^4 + a)^(1/4)*x^3/a - 2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*b^(1/4)
 + 2*(-b*x^4 + a)^(1/4)/x)/b^(1/4))/b^(3/4) - 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqr
t(2)*b^(1/4) - 2*(-b*x^4 + a)^(1/4)/x)/b^(1/4))/b^(3/4) - sqrt(2)*ln(sqrt(b) + s
qrt(2)*(-b*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^(3/4) + sqrt(2)*ln
(sqrt(b) - sqrt(2)*(-b*x^4 + a)^(1/4)*b^(1/4)/x + sqrt(-b*x^4 + a)/x^2)/b^(3/4))
*a