3.1217 \(\int \frac{x^8}{\sqrt [4]{a-b x^4}} \, dx\)

Optimal. Leaf size=264 \[ -\frac{5 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}+\frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt{2} b^{9/4}}-\frac{5 a^2 \log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{128 \sqrt{2} b^{9/4}}+\frac{5 a^2 \log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{128 \sqrt{2} b^{9/4}}-\frac{5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b} \]

[Out]

(-5*a*x*(a - b*x^4)^(3/4))/(32*b^2) - (x^5*(a - b*x^4)^(3/4))/(8*b) - (5*a^2*Arc
Tan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(64*Sqrt[2]*b^(9/4)) + (5*a^2*Ar
cTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(64*Sqrt[2]*b^(9/4)) - (5*a^2*L
og[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(
128*Sqrt[2]*b^(9/4)) + (5*a^2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b
^(1/4)*x)/(a - b*x^4)^(1/4)])/(128*Sqrt[2]*b^(9/4))

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Rubi [A]  time = 0.319607, antiderivative size = 264, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 8, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.5 \[ -\frac{5 a^2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )}{64 \sqrt{2} b^{9/4}}+\frac{5 a^2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )}{64 \sqrt{2} b^{9/4}}-\frac{5 a^2 \log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{128 \sqrt{2} b^{9/4}}+\frac{5 a^2 \log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )}{128 \sqrt{2} b^{9/4}}-\frac{5 a x \left (a-b x^4\right )^{3/4}}{32 b^2}-\frac{x^5 \left (a-b x^4\right )^{3/4}}{8 b} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a - b*x^4)^(1/4),x]

[Out]

(-5*a*x*(a - b*x^4)^(3/4))/(32*b^2) - (x^5*(a - b*x^4)^(3/4))/(8*b) - (5*a^2*Arc
Tan[1 - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(64*Sqrt[2]*b^(9/4)) + (5*a^2*Ar
cTan[1 + (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(64*Sqrt[2]*b^(9/4)) - (5*a^2*L
og[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] - (Sqrt[2]*b^(1/4)*x)/(a - b*x^4)^(1/4)])/(
128*Sqrt[2]*b^(9/4)) + (5*a^2*Log[1 + (Sqrt[b]*x^2)/Sqrt[a - b*x^4] + (Sqrt[2]*b
^(1/4)*x)/(a - b*x^4)^(1/4)])/(128*Sqrt[2]*b^(9/4))

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Rubi in Sympy [A]  time = 41.0717, size = 243, normalized size = 0.92 \[ - \frac{5 \sqrt{2} a^{2} \log{\left (- \frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{256 b^{\frac{9}{4}}} + \frac{5 \sqrt{2} a^{2} \log{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + \frac{\sqrt{b} x^{2}}{\sqrt{a - b x^{4}}} + 1 \right )}}{256 b^{\frac{9}{4}}} + \frac{5 \sqrt{2} a^{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} - 1 \right )}}{128 b^{\frac{9}{4}}} + \frac{5 \sqrt{2} a^{2} \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a - b x^{4}}} + 1 \right )}}{128 b^{\frac{9}{4}}} - \frac{5 a x \left (a - b x^{4}\right )^{\frac{3}{4}}}{32 b^{2}} - \frac{x^{5} \left (a - b x^{4}\right )^{\frac{3}{4}}}{8 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(-b*x**4+a)**(1/4),x)

[Out]

-5*sqrt(2)*a**2*log(-sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) + sqrt(b)*x**2/sqrt(
a - b*x**4) + 1)/(256*b**(9/4)) + 5*sqrt(2)*a**2*log(sqrt(2)*b**(1/4)*x/(a - b*x
**4)**(1/4) + sqrt(b)*x**2/sqrt(a - b*x**4) + 1)/(256*b**(9/4)) + 5*sqrt(2)*a**2
*atan(sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) - 1)/(128*b**(9/4)) + 5*sqrt(2)*a**
2*atan(sqrt(2)*b**(1/4)*x/(a - b*x**4)**(1/4) + 1)/(128*b**(9/4)) - 5*a*x*(a - b
*x**4)**(3/4)/(32*b**2) - x**5*(a - b*x**4)**(3/4)/(8*b)

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Mathematica [A]  time = 0.324832, size = 210, normalized size = 0.8 \[ \frac{5 a^2 \left (-2 \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}\right )+2 \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+1\right )-\log \left (-\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )+\log \left (\frac{\sqrt{2} \sqrt [4]{b} x}{\sqrt [4]{a-b x^4}}+\frac{\sqrt{b} x^2}{\sqrt{a-b x^4}}+1\right )\right )}{128 \sqrt{2} b^{9/4}}+\left (a-b x^4\right )^{3/4} \left (-\frac{5 a x}{32 b^2}-\frac{x^5}{8 b}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a - b*x^4)^(1/4),x]

[Out]

(a - b*x^4)^(3/4)*((-5*a*x)/(32*b^2) - x^5/(8*b)) + (5*a^2*(-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)]))/(128*Sqrt[2]*b^(9/4))

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Maple [F]  time = 0.042, size = 0, normalized size = 0. \[ \int{{x}^{8}{\frac{1}{\sqrt [4]{-b{x}^{4}+a}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(-b*x^4+a)^(1/4),x)

[Out]

int(x^8/(-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^8/(-b*x^4 + a)^(1/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.248765, size = 304, normalized size = 1.15 \[ -\frac{20 \, b^{2} \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{7} x \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{3}{4}}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{6} + x \sqrt{-\frac{a^{8} b^{5} x^{2} \sqrt{-\frac{a^{8}}{b^{9}}} - \sqrt{-b x^{4} + a} a^{12}}{x^{2}}}}\right ) + 5 \, b^{2} \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{125 \,{\left (b^{7} x \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{3}{4}} +{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{6}\right )}}{x}\right ) - 5 \, b^{2} \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{1}{4}} \log \left (-\frac{125 \,{\left (b^{7} x \left (-\frac{a^{8}}{b^{9}}\right )^{\frac{3}{4}} -{\left (-b x^{4} + a\right )}^{\frac{1}{4}} a^{6}\right )}}{x}\right ) + 4 \,{\left (4 \, b x^{5} + 5 \, a x\right )}{\left (-b x^{4} + a\right )}^{\frac{3}{4}}}{128 \, b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(-b*x^4 + a)^(1/4),x, algorithm="fricas")

[Out]

-1/128*(20*b^2*(-a^8/b^9)^(1/4)*arctan(b^7*x*(-a^8/b^9)^(3/4)/((-b*x^4 + a)^(1/4
)*a^6 + x*sqrt(-(a^8*b^5*x^2*sqrt(-a^8/b^9) - sqrt(-b*x^4 + a)*a^12)/x^2))) + 5*
b^2*(-a^8/b^9)^(1/4)*log(125*(b^7*x*(-a^8/b^9)^(3/4) + (-b*x^4 + a)^(1/4)*a^6)/x
) - 5*b^2*(-a^8/b^9)^(1/4)*log(-125*(b^7*x*(-a^8/b^9)^(3/4) - (-b*x^4 + a)^(1/4)
*a^6)/x) + 4*(4*b*x^5 + 5*a*x)*(-b*x^4 + a)^(3/4))/b^2

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Sympy [A]  time = 6.35276, size = 39, normalized size = 0.15 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{1}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{2 i \pi }}{a}} \right )}}{4 \sqrt [4]{a} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(-b*x**4+a)**(1/4),x)

[Out]

x**9*gamma(9/4)*hyper((1/4, 9/4), (13/4,), b*x**4*exp_polar(2*I*pi)/a)/(4*a**(1/
4)*gamma(13/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (-b x^{4} + a\right )}^{\frac{1}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(-b*x^4 + a)^(1/4),x, algorithm="giac")

[Out]

integrate(x^8/(-b*x^4 + a)^(1/4), x)