3.1153 \(\int \frac{x^8}{\left (a+b x^4\right )^{5/4}} \, dx\)

Optimal. Leaf size=97 \[ -\frac{5 a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{9/4}}-\frac{5 a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{9/4}}+\frac{5 x \left (a+b x^4\right )^{3/4}}{4 b^2}-\frac{x^5}{b \sqrt [4]{a+b x^4}} \]

[Out]

-(x^5/(b*(a + b*x^4)^(1/4))) + (5*x*(a + b*x^4)^(3/4))/(4*b^2) - (5*a*ArcTan[(b^
(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b^(9/4)) - (5*a*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^
(1/4)])/(8*b^(9/4))

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Rubi [A]  time = 0.0911436, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4 \[ -\frac{5 a \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{9/4}}-\frac{5 a \tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )}{8 b^{9/4}}+\frac{5 x \left (a+b x^4\right )^{3/4}}{4 b^2}-\frac{x^5}{b \sqrt [4]{a+b x^4}} \]

Antiderivative was successfully verified.

[In]  Int[x^8/(a + b*x^4)^(5/4),x]

[Out]

-(x^5/(b*(a + b*x^4)^(1/4))) + (5*x*(a + b*x^4)^(3/4))/(4*b^2) - (5*a*ArcTan[(b^
(1/4)*x)/(a + b*x^4)^(1/4)])/(8*b^(9/4)) - (5*a*ArcTanh[(b^(1/4)*x)/(a + b*x^4)^
(1/4)])/(8*b^(9/4))

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Rubi in Sympy [A]  time = 11.3282, size = 90, normalized size = 0.93 \[ - \frac{5 a \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 b^{\frac{9}{4}}} - \frac{5 a \operatorname{atanh}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a + b x^{4}}} \right )}}{8 b^{\frac{9}{4}}} - \frac{x^{5}}{b \sqrt [4]{a + b x^{4}}} + \frac{5 x \left (a + b x^{4}\right )^{\frac{3}{4}}}{4 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**8/(b*x**4+a)**(5/4),x)

[Out]

-5*a*atan(b**(1/4)*x/(a + b*x**4)**(1/4))/(8*b**(9/4)) - 5*a*atanh(b**(1/4)*x/(a
+ b*x**4)**(1/4))/(8*b**(9/4)) - x**5/(b*(a + b*x**4)**(1/4)) + 5*x*(a + b*x**4
)**(3/4)/(4*b**2)

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Mathematica [A]  time = 0.200891, size = 106, normalized size = 1.09 \[ \frac{x \left (5 a+b x^4\right )}{4 b^2 \sqrt [4]{a+b x^4}}-\frac{5 a \left (-\log \left (1-\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )+\log \left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}+1\right )+2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a+b x^4}}\right )\right )}{16 b^{9/4}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^8/(a + b*x^4)^(5/4),x]

[Out]

(x*(5*a + b*x^4))/(4*b^2*(a + b*x^4)^(1/4)) - (5*a*(2*ArcTan[(b^(1/4)*x)/(a + b*
x^4)^(1/4)] - Log[1 - (b^(1/4)*x)/(a + b*x^4)^(1/4)] + Log[1 + (b^(1/4)*x)/(a +
b*x^4)^(1/4)]))/(16*b^(9/4))

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Maple [F]  time = 0.062, size = 0, normalized size = 0. \[ \int{{x}^{8} \left ( b{x}^{4}+a \right ) ^{-{\frac{5}{4}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^8/(b*x^4+a)^(5/4),x)

[Out]

int(x^8/(b*x^4+a)^(5/4),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)^(5/4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.265023, size = 340, normalized size = 3.51 \[ -\frac{20 \,{\left (b^{3} x^{4} + a b^{2}\right )} \left (\frac{a^{4}}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{7} x \left (\frac{a^{4}}{b^{9}}\right )^{\frac{3}{4}}}{{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3} + x \sqrt{\frac{a^{4} b^{5} x^{2} \sqrt{\frac{a^{4}}{b^{9}}} + \sqrt{b x^{4} + a} a^{6}}{x^{2}}}}\right ) + 5 \,{\left (b^{3} x^{4} + a b^{2}\right )} \left (\frac{a^{4}}{b^{9}}\right )^{\frac{1}{4}} \log \left (\frac{125 \,{\left (b^{7} x \left (\frac{a^{4}}{b^{9}}\right )^{\frac{3}{4}} +{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3}\right )}}{x}\right ) - 5 \,{\left (b^{3} x^{4} + a b^{2}\right )} \left (\frac{a^{4}}{b^{9}}\right )^{\frac{1}{4}} \log \left (-\frac{125 \,{\left (b^{7} x \left (\frac{a^{4}}{b^{9}}\right )^{\frac{3}{4}} -{\left (b x^{4} + a\right )}^{\frac{1}{4}} a^{3}\right )}}{x}\right ) - 4 \,{\left (b x^{5} + 5 \, a x\right )}{\left (b x^{4} + a\right )}^{\frac{3}{4}}}{16 \,{\left (b^{3} x^{4} + a b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^4 + a)^(5/4),x, algorithm="fricas")

[Out]

-1/16*(20*(b^3*x^4 + a*b^2)*(a^4/b^9)^(1/4)*arctan(b^7*x*(a^4/b^9)^(3/4)/((b*x^4
+ a)^(1/4)*a^3 + x*sqrt((a^4*b^5*x^2*sqrt(a^4/b^9) + sqrt(b*x^4 + a)*a^6)/x^2))
) + 5*(b^3*x^4 + a*b^2)*(a^4/b^9)^(1/4)*log(125*(b^7*x*(a^4/b^9)^(3/4) + (b*x^4
+ a)^(1/4)*a^3)/x) - 5*(b^3*x^4 + a*b^2)*(a^4/b^9)^(1/4)*log(-125*(b^7*x*(a^4/b^
9)^(3/4) - (b*x^4 + a)^(1/4)*a^3)/x) - 4*(b*x^5 + 5*a*x)*(b*x^4 + a)^(3/4))/(b^3
*x^4 + a*b^2)

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Sympy [A]  time = 5.41894, size = 37, normalized size = 0.38 \[ \frac{x^{9} \Gamma \left (\frac{9}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{5}{4}, \frac{9}{4} \\ \frac{13}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{5}{4}} \Gamma \left (\frac{13}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**8/(b*x**4+a)**(5/4),x)

[Out]

x**9*gamma(9/4)*hyper((5/4, 9/4), (13/4,), b*x**4*exp_polar(I*pi)/a)/(4*a**(5/4)
*gamma(13/4))

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{8}}{{\left (b x^{4} + a\right )}^{\frac{5}{4}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^8/(b*x^4 + a)^(5/4),x, algorithm="giac")

[Out]

integrate(x^8/(b*x^4 + a)^(5/4), x)