3.1125 \(\int \frac{x^2}{\left (a+b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=57 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{3/4}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.0518494, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.267 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{3/4}}-\frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{2 b^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a + b*x^4)^(3/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 7.82031, size = 49, normalized size = 0.86 \[ - \frac{\operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 b^{\frac{3}{4}}} + \frac{\operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{2 b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(b*x**4+a)**(3/4),x)

[Out]

-atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(2*b**(3/4)) + atanh(b**(1/4)*x/(a + b*x**
4)**(1/4))/(2*b**(3/4))

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Mathematica [C]  time = 0.0274257, size = 52, normalized size = 0.91 \[ \frac{x^3 \left (\frac{a+b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};-\frac{b x^4}{a}\right )}{3 \left (a+b x^4\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/(a + b*x^4)^(3/4),x]

[Out]

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

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \[ \int{{x}^{2} \left ( b{x}^{4}+a \right ) ^{-{\frac{3}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(b*x^4+a)^(3/4),x)

[Out]

int(x^2/(b*x^4+a)^(3/4),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^2/(b*x^4 + a)^(3/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.266536, size = 165, normalized size = 2.89 \[ -\frac{1}{b^{3}}^{\frac{1}{4}} \arctan \left (\frac{b \frac{1}{b^{3}}^{\frac{1}{4}} x}{x \sqrt{\frac{b^{2} \sqrt{\frac{1}{b^{3}}} x^{2} + \sqrt{b x^{4} + a}}{x^{2}}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}}}\right ) + \frac{1}{4} \, \frac{1}{b^{3}}^{\frac{1}{4}} \log \left (\frac{b \frac{1}{b^{3}}^{\frac{1}{4}} x +{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right ) - \frac{1}{4} \, \frac{1}{b^{3}}^{\frac{1}{4}} \log \left (-\frac{b \frac{1}{b^{3}}^{\frac{1}{4}} x -{\left (b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x^4 + a)^(3/4),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 3.64719, size = 37, normalized size = 0.65 \[ \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{7}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(b*x**4+a)**(3/4),x)

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{{\left (b x^{4} + a\right )}^{\frac{3}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/(b*x^4 + a)^(3/4),x, algorithm="giac")

[Out]

integrate(x^2/(b*x^4 + a)^(3/4), x)