3.1249 \(\int \frac{x^2}{\left (a-b x^4\right )^{3/4}} \, dx\)

Optimal. Leaf size=209 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{2 \sqrt{2} b^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{2 \sqrt{2} b^{3/4}}+\frac{\log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2} b^{3/4}}-\frac{\log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2} b^{3/4}} \]

[Out]

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

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Rubi [A]  time = 0.200375, antiderivative size = 209, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.438 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{2 \sqrt{2} b^{3/4}}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{2 \sqrt{2} b^{3/4}}+\frac{\log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2} b^{3/4}}-\frac{\log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{4 \sqrt{2} b^{3/4}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/(a - b*x^4)^(3/4),x]

[Out]

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

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Rubi in Sympy [A]  time = 32.2555, size = 185, normalized size = 0.89 \[ \frac{\sqrt{2} \log{\left (- \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{8 b^{\frac{3}{4}}} - \frac{\sqrt{2} \log{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{8 b^{\frac{3}{4}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} - 1 \right )}}{4 b^{\frac{3}{4}}} + \frac{\sqrt{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + 1 \right )}}{4 b^{\frac{3}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(2)*log(-sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) + sqrt(b)*x**2/sqrt(a - b*x*
*4) + 1)/(8*b**(3/4)) - sqrt(2)*log(sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) + sqr
t(b)*x**2/sqrt(a - b*x**4) + 1)/(8*b**(3/4)) + sqrt(2)*atan(sqrt(2)*b**(1/4)*x/(
a - b*x**4)**(1/4) - 1)/(4*b**(3/4)) + sqrt(2)*atan(sqrt(2)*b**(1/4)*x/(a - b*x*
*4)**(1/4) + 1)/(4*b**(3/4))

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Mathematica [C]  time = 0.0306848, size = 53, normalized size = 0.25 \[ \frac{x^3 \left (\frac{a-b x^4}{a}\right )^{3/4} \, _2F_1\left (\frac{3}{4},\frac{3}{4};\frac{7}{4};\frac{b x^4}{a}\right )}{3 \left (a-b x^4\right )^{3/4}} \]

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 = 0.035, 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 = 0.240085, size = 188, normalized size = 0.9 \[ \left (-\frac{1}{b^{3}}\right )^{\frac{1}{4}} \arctan \left (\frac{b x \left (-\frac{1}{b^{3}}\right )^{\frac{1}{4}}}{x \sqrt{\frac{b^{2} x^{2} \sqrt{-\frac{1}{b^{3}}} + \sqrt{-b x^{4} + a}}{x^{2}}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}\right ) - \frac{1}{4} \, \left (-\frac{1}{b^{3}}\right )^{\frac{1}{4}} \log \left (\frac{b x \left (-\frac{1}{b^{3}}\right )^{\frac{1}{4}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right ) + \frac{1}{4} \, \left (-\frac{1}{b^{3}}\right )^{\frac{1}{4}} \log \left (-\frac{b x \left (-\frac{1}{b^{3}}\right )^{\frac{1}{4}} -{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}{x}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 3.91033, size = 39, normalized size = 0.19 \[ \frac{x^{3} \Gamma \left (\frac{3}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{4}, \frac{3}{4} \\ \frac{7}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 a^{\frac{3}{4}} \Gamma \left (\frac{7}{4}\right )} \]

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(2*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)